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inst/doc/documentation_rocbc.R
In rocbc: Statistical Inference for Box-Cox Based Receiver Operating Characteristic Curves
## ----eval=TRUE, error=FALSE, warning=FALSE, cache=FALSE, comment=FALSE, echo=FALSE, message=FALSE, results=FALSE----
# Hello, world!
#
# This is an example function named 'hello'
# which prints 'Hello, world!'.
#
# You can learn more about package authoring with RStudio at:
#
# http://r-pkgs.had.co.nz/
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# Some useful keyboard shortcuts for package authoring:
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# Install Package: 'Ctrl + Shift + B'
# Check Package: 'Ctrl + Shift + E'
# Test Package: 'Ctrl + Shift + T'
library(pracma)
library(clinfun)
library(splancs)
library(mvtnorm)
library(matlab)
library(formattable)
library(pROC)
library(MRMCaov)
## ----echo=FALSE, eval=TRUE----------------------------------------------------
rocboxcox<-function(marker, D, alpha, plots, printProgress = FALSE){
if (plots!="on"){plots="off"}
if (length(marker) != length(D)) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (sum(is.na(marker)) > 0 | sum(is.na(D)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if (alpha <= 0 | alpha >= 1) {
stop("ERROR: The level of significance, alpha, should be set between 0 and 1. A common choice is 0.05.")
} else if (min(marker<= 0)) {
stop("ERROR: All marker scores need to be positive")
} else if (sum(marker < 0) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
x=marker[D==0]
y=marker[D==1]
if (plots!="on"){plots="off"}
n1=length(x)
n2=length(y)
xor=x;yor=y;
Za=qnorm(1-alpha/2)
likbox<-function(x,y,h){
n=length(x);
m=length(y);
out=c();
for (i in 1:length(h)){
#print(i)
if (h[i]==0){
xh=log(x);
yh=log(y);
} else {
xh=((x^h[i])-1)/h[i];
yh=((y^h[i])-1)/h[i];
}
out[i]<-c( -n/2*log(sum((xh-sum(xh)/n)^2)/n) -m/2*log(sum((yh-sum(yh)/m)^2)/m) +(h[i]-1)*(sum(log(x))+sum(log(y))))
}
return(out)
}
boxcoxleo<-function(x,y){
init=1;
logL<-function(h){
-likbox(x,y,h)
}
#lam=fminsearch(logL,init)
#lam=c(lam$optbase$xopt)
lam<- optim(1,logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
transx=((x^lam)-1)/lam
transy=((y^lam)-1)/lam
#test1=print(shapiro.test(transx))
#test2=print(shapiro.test(transy))
return(list(transformation.parameter=lam,transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam ))
}
boxcoxleo2<-function(x,y){
init=1;
logL<-function(h){
-likbox(x,y,h)
}
lam<- optim(1,logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
#lam=fminsearch(logL,init)
#lam=c(lam$optbase$xopt)
transx=((x^lam)-1)/lam
transy=((y^lam)-1)/lam
return(list(transformation.parameter=lam,transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam ))
}
COVlam <-function (x,y){
#%Accepts X<Y Yields as an output the Variance Covariance matrix
#%of the mh, sh, md, sd, and lam, (with this order) where lam is the estimated boxcox
#%parameter Thus, the output will be a 5x5 matrix
#%---FOR TWO POPULATIONS:
n=length(x); #original scores
m=length(y);
# %mh-->g(1)
# %sh-->g(2)
# %md-->g(3)
# %sd-->g(4)
# %lam-->g(5)
# % logL=@(g) -n*log(sh)-1/(2*sh^2)*sum((xlam-mh)^2)+(lam-1)*sum(log(x)) -
# % m*log(sd)-1/(2*sd^2)*sum((ylam-md)^2)+(lam-1)*sum(log(y))
outt=boxcoxleo(x,y); #the original scores
transx=outt$transx
transy=outt$transy
lam=outt$transformation.parameter
mh=mean(transx); xlam=transx;
md=mean(transy); ylam=transy;
sh=sqrt(1/(length(transx))*sum((transx-mean(transx))^2));
sd=sqrt(1/(length(transy))*sum((transy-mean(transy))^2));
I=zeros(5,5);
I[1,1]=n/sh^2;
I[2,2]=-(n/sh^2-3/sh^4*sum((xlam-mh)^2));
I[3,3]=m/sd^2;
I[4,4]=-(m/sd^2-3/sd^4*sum((ylam-md)^2));
kk= sum(((mh - (x^lam - 1)/lam)*((2*(x^lam - 1))/lam^3 - (2*x^lam*log(x))/lam^2 + (x^lam*log(x)^2)/lam))/sh^2) + - sum(((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)^2/sd^2) + - sum(((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)^2/sh^2) + + sum(((md - (y^lam - 1)/lam)*((2*(y^lam - 1))/lam^3 - (2*y^lam*log(y))/lam^2 + (y^lam*log(y)^2)/lam))/sd^2);
I[5,5]=-kk ;
I[1,5]=sum(((2*(x^lam - 1))/lam^2 - (2*x^lam*log(x))/lam)/(2*sh^2));
I[2,5]=-sum((2*((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)*(mh - (x^lam - 1)/lam))/sh^3);
I[3,5]=-sum(-((2*(y^lam - 1))/lam^2 - (2*y^lam*log(y))/lam)/(2*sd^2));
I[4,5]=-sum((2*((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)*(md - (y^lam - 1)/lam))/sd^3);
I[5,1]=I[1,5];
I[5,2]=I[2,5];
I[5,3]=I[3,5];
I[5,4]=I[4,5];
S= inv(I); #%that's better
return(list(out=S))
}
cc=boxcoxleo(x,y)
transx=cc$transx
transy=cc$transy
#x11()
#qqnorm(x)
#title(main=" for X")
#x11()
#qqnorm(y)
#title(main=" for Y")
roc<-function(t,trans_x,trans_y){
na = length(trans_x)
nb = length(trans_y)
stransx=sqrt(var(trans_x)*(na-1)/na)
stransy=sqrt(var(trans_y)*(nb-1)/nb)
1-pnorm(qnorm(1-t,
mean=mean(trans_x),
sd=stransx),
mean=mean(trans_y),
sd=stransy)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
if (plots=="on"){
txt <- paste("Box-Cox Based ROC, alpha =", formattable(alpha, digits = 3))
plot(linspace(0,1,1000),roc(linspace(0,1,1000),transx,transy),main=" ",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="red")
lines(linspace(0,1,10),linspace(0,1,10),type="l", lty=2)
title(main=txt)
}
rocfun=function(t){1-pnorm(qnorm(1-t,mean=mean(transx),sd=sqrt(var(transx)*(length(transx)-1)/length(transx))),mean=mean(transy),sd=sqrt(var(transy)*(length(transy)-1)/length(transy)))}
m1hat=mean(transx)
m2hat=mean(transy)
s1hat=sqrt(var(transx)*(length(transx)-1)/length(transx))
s2hat=sqrt(var(transy)*(length(transy)-1)/length(transy))
n1=length(x)
n2=length(y)
#NOTE THAT: m1hat->m1
# m2hat->m2
# s1hat->s1hat
# s2hat->s2hat
#########################################
derde =c( (s2hat^2 + (s1hat*s2hat*(2*m1hat - 2*m2hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s2hat*(s1hat^2 - s2hat^2)), ((s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2) - 2*m2hat*s1hat + (s1hat*s2hat*(2*s1hat*log(s1hat^2/s2hat^2) + (2*(s1hat^2 - s2hat^2))/s1hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2) - (2*s1hat*(m1hat*s2hat^2 - m2hat*s1hat^2 + s1hat*s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2)^2)/s2hat, -((s1hat^2 + (s1hat*s2hat*(2*m1hat - 2*m2hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2) - 1)/s2hat, ((2*m1hat*s2hat + s1hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2) - (s1hat*s2hat*(2*s2hat*log(s1hat^2/s2hat^2) + (2*(s1hat^2 - s2hat^2))/s2hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2) + (2*s2hat*(m1hat*s2hat^2 - m2hat*s1hat^2 + s1hat*s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2)^2)/s2hat - (m2hat + (m1hat*s2hat^2 - m2hat*s1hat^2 + s1hat*s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2))/(s1hat^2 - s2hat^2))/s2hat^2);
derdp =c( -((s2hat^2 + (s1hat*s2hat*(2*m1hat - 2*m2hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2) + 1)/s1hat, (m1hat + (m1hat*s2hat^2 - m2hat*s1hat^2 + s1hat*s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2))/(s1hat^2 - s2hat^2))/s1hat^2 - ((s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2) - 2*m2hat*s1hat + (s1hat*s2hat*(2*s1hat*log(s1hat^2/s2hat^2) + (2*(s1hat^2 - s2hat^2))/s1hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2) - (2*s1hat*(m1hat*s2hat^2 - m2hat*s1hat^2 + s1hat*s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2)^2)/s1hat, (s1hat^2 + (s1hat*s2hat*(2*m1hat - 2*m2hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat*(s1hat^2 - s2hat^2)), -((2*m1hat*s2hat + s1hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2) - (s1hat*s2hat*(2*s2hat*log(s1hat^2/s2hat^2) + (2*(s1hat^2 - s2hat^2))/s2hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2) + (2*s2hat*(m1hat*s2hat^2 - m2hat*s1hat^2 + s1hat*s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2)^2)/s1hat);
outt=COVlam(xor,yor);
S=outt$out;
S=S[1:4,1:4];
# if (sum(is.infinite(S)) > 0 |
# sum(is.nan(S)) > 0 |
# sum(is.na(S)) > 0 |
# sum(diag(S) < 0) > 0) {
# stop("ERROR: The information matrix cannot be inverted - this might be due to the scale of the marker. Try rescaling the marker measurements by subtracting a constant or by dividing with a constant all marker measurements (of both groups) before you try again.")
# }
varde=t(derde)%*%S%*%t(t(derde));
vardp=t(derdp)%*%S%*%t(t(derdp));
covdedp=t(derde)%*%S%*%t(t(derdp));
##########################################
vardeltase=varde;
vardeltasp=vardp;
#########################
cutoff= (s1hat^2*m2hat-s2hat^2*m1hat-s1hat*s2hat*sqrt((m1hat-m2hat)^2+(s1hat^2-s2hat^2)*log(s1hat^2/s2hat^2)))/(s1hat^2-s2hat^2)
deltasp= (((s1hat^2*m2hat-s2hat^2*m1hat-s1hat*s2hat*sqrt((m1hat-m2hat)^2+(s1hat^2-s2hat^2)*log(s1hat^2./s2hat^2)))/(s1hat^2-s2hat^2))-m1hat)/(s1hat)
deltase= (m2hat-((s1hat^2*m2hat-s2hat^2*m1hat-s1hat*s2hat*sqrt((m1hat-m2hat)^2+(s1hat^2-s2hat^2)*log(s1hat^2/s2hat^2)))/(s1hat^2-s2hat^2)))/s2hat
Se=pnorm(deltase)
Sp=pnorm(deltasp)
Se
Sp
if (plots=="on"){
points(1-Sp,Se, col = "red")
}
covdeltas = matrix( c(vardeltasp, covdedp, covdedp, vardeltase), nrow=2, ncol=2, byrow = TRUE) # fill m
# if (sum(is.infinite(covdeltas)) > 0 |
# sum(is.nan(covdeltas)) > 0) {
# stop("ERROR: The information matrix cannot be inverted - this might be due to the scale of the marker. Try rescaling the marker measurements by subtracting a constant or by dividing with a constant all marker measurements (of both groups) before you try again.")
# }
svdA=svd(inv(covdeltas))
a=sqrt(qchisq(1-alpha,2))/sqrt(svdA$d[1])
b=sqrt(qchisq(1-alpha,2))/sqrt(svdA$d[2])
theta=seq(from=0,to=2*pi+1/40,by=1/40)
state1=a*cos(theta)
state2=b*sin(theta)
states=rbind(state1,state2)
X=svdA$v %*%states
x1=X[1,]+deltasp
x2=X[2,]+deltase
px1=pnorm(x1);
px2=pnorm(x2);
yegg=px2
xegg=1-px1
if (plots=="on"){
lines(xegg,yegg,col="green")
}
#----Area egg---------------
bxegg=c(xegg,xegg[1])
byegg=c(yegg,yegg[1])
ell <- cbind(bxegg, byegg)
areaegg=areapl(ell)
#----End Area egg---------------
margcisp=c(pnorm(deltasp-qnorm(1-alpha/2)*sqrt(vardeltasp)), pnorm(deltasp+qnorm(1-alpha/2)*sqrt(vardeltasp)))
margcise=c(pnorm(deltase-qnorm(1-alpha/2)*sqrt(vardeltase)), pnorm(deltase+qnorm(1-alpha/2)*sqrt(vardeltase)))
#-----Now deal with the rectangle-----
cisp=c(deltasp-qnorm(1-alpha/4)*sqrt(vardeltasp), deltasp+qnorm(1-alpha/4)*sqrt(vardeltasp))
cise=c(deltase-qnorm(1-alpha/4)*sqrt(vardeltase), deltase+qnorm(1-alpha/4)*sqrt(vardeltase))
bsenslb=pnorm(cise[1])#exp(cibootsse[1])/(1+exp(cibootsse[1])) #NOW trans back, going to construct a 95% rectangle for boots
bsensub=pnorm(cise[2])#exp(cibootsse[2])/(1+exp(cibootsse[2])) #NOW trans back, going to construct a 95% rectangle for boots
bspeclb=pnorm(cisp[1])#exp(cibootssp[1])/(1+exp(cibootssp[1])) #NOW trans back, going to construct a 95% rectangle for boots
bspecub=pnorm(cisp[2])#exp(cibootssp[2])/(1+exp(cibootssp[2])) #NOW trans back, going to construct a 95% rectangle for boots
#-----End of the rectangle------------
if (plots=="on"){
#-----Start--plot---Rectangle--------------------
lines(c(1-bspeclb,1-bspecub),c(bsenslb,bsenslb),col="black")
lines(c(1-bspecub,1-bspecub),c(bsenslb,bsensub),col="black")
lines(c(1-bspecub,1-bspeclb),c(bsensub,bsensub),col="black")
lines(c(1-bspeclb,1-bspeclb),c(bsenslb,bsensub),col="black")
#-----End--plot---Rectangle--------------------
}
brectx=c(1-bspeclb,1-bspecub, 1-bspecub,1-bspecub,1-bspecub,1-bspeclb,1-bspeclb,1-bspeclb)
brecty=c(bsenslb,bsenslb ,bsenslb,bsensub, bsensub,bsensub ,bsenslb,bsensub)
brectx=c(brectx, brectx[1]);
brecty=c(brecty, brecty[1]);
re <- cbind(brectx, brecty)
arearect=areapl(re)
#line for the Youden:
if (plots=="on"){
lines(c(1-Sp,1-Sp),c(rocuseless(1-Sp),Se),col="blue")
}
#================AUC================
invAUC= ((m2hat-m1hat)/s2hat)/(sqrt(1+(s1hat/s2hat)^2));
auc=pnorm(invAUC)
#====================AUC1==============================
dm1 =-1/(s2hat*sqrt(1+(s1hat/s2hat)^2));
ds1 = (s1hat*(m1hat-m2hat))/(s2hat^3*(1+(s1hat/s2hat)^2)^(3/2))
dm2 =1/(s2hat*sqrt(1+(s1hat/s2hat)^2));
ds2 =((m1hat-m2hat))/(s2hat^2*(1+(s1hat/s2hat)^2)^(3/2))
I=zeros(5,5);
sh=sqrt(1/(length(transx))*sum((transx-mean(transx))^2))
sd=sqrt(1/(length(transy))*sum((transy-mean(transy))^2))
mh=m1hat;
md=m2hat;
n=n1;
m=n2;
xlam=transx;
ylam=transy;
lam=cc$transformation.parameter
I[1,1]=n/sh^2;
I[2,2]=-(n/sh^2-3/sh^4*sum((xlam-mh)^2));
I[3,3]=m/sd^2;
I[4,4]=-(m/sd^2-3/sd^4*sum((ylam-md)^2));
kk= sum(((mh - (x^lam - 1)/lam)*((2*(x^lam - 1))/lam^3 - (2*x^lam*log(x))/lam^2 + (x^lam*log(x)^2)/lam))/sh^2) + - sum(((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)^2/sd^2) + - sum(((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)^2/sh^2) + + sum(((md - (y^lam - 1)/lam)*((2*(y^lam - 1))/lam^3 - (2*y^lam*log(y))/lam^2 + (y^lam*log(y)^2)/lam))/sd^2);
I[5,5]=-kk ;
I[1,5]=sum(((2*(x^lam - 1))/lam^2 - (2*x^lam*log(x))/lam)/(2*sh^2));
I[2,5]=-sum((2*((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)*(mh - (x^lam - 1)/lam))/sh^3);
I[3,5]=-sum(-((2*(y^lam - 1))/lam^2 - (2*y^lam*log(y))/lam)/(2*sd^2));
I[4,5]=-sum((2*((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)*(md - (y^lam - 1)/lam))/sd^3);
I[5,1]=I[1,5]
I[5,2]=I[2,5]
I[5,3]=I[3,5]
I[5,4]=I[4,5]
S=inv(I)
S=S[1:4,1:4]
## EVERYTHING ELSE ================================================================================
varauc=t(c(dm1,ds1,dm2,ds2))%*% S %*% t(t(c(dm1,ds1,dm2,ds2)))
CIauc=c(invAUC-Za*sqrt(varauc), invAUC+Za*sqrt(varauc))
CIauc=pnorm(CIauc)
CIauc
Zauc=invAUC/sqrt(varauc)
pvalauc=2*pnorm(-abs(Zauc))
#================= DeltaBClamJT================
mx=m1hat;
my=m2hat;
sx=s1hat;
sy=s2hat;
c=cutoff;
b=sy/sx;
a=my-mx;
Jhat=pnorm((m2hat-cutoff)/s2hat)+pnorm((cutoff-m1hat)/s1hat)-1;
radd=a^2+(b^2-1)*sx^2*log(b^2);
zy=(my-c)/sy;
zx=(c-mx)/sx;
dcdmx= (b^2 + a*b*radd^(-1/2)*(-1))/(b^2-1);
dcdsx= (-2*a*b^2)/((b^2-1)^2*sx) + (((b*(b^2+1))*(radd)^(1/2))/((b^2-1)^2*sx) - (sx*b*radd^(-1/2))/(b^2-1)*(log(b^2)+b^(2)-1) );
dcdmy= (-1 + a*b*radd^(-1/2))/(b^2-1);
dcdsy= (2*a*b)/((b^2-1)^2*sx) + (((-b^2-1)*(radd)^(1/2))/((b^2-1)^2*sx) + (sy*b*radd^(-1/2))/(b^2-1)*(log(b^2)+1-b^(-2)) );
d1=-sx^(-1)*dnorm(zx)+dcdmx*(sx^(-1)*dnorm(zx)-(sy^(-1)*dnorm(zy) ));
d2=-zx*sx^(-1)*dnorm(zx)+dcdsx*(sx^(-1)*dnorm(zx)-(sy^(-1)*dnorm(zy) )) ;
d3=sy^(-1)*dnorm(zy)+dcdmy*(sx^(-1)*dnorm(zx)-(sy^(-1)*dnorm(zy) )) ;
d4=-zy*sy^(-1)*dnorm(zy)+dcdsy*(sx^(-1)*dnorm(zx)-(sy^(-1)*dnorm(zy) ));
#varJ=t(c(d1, d2, d3, d4))%*%S%*%[d1 d2 d3 d4]';
varJ=t(c(d1,d2,d3,d4))%*% S %*% t(t(c(d1,d2,d3,d4)))
# Jstar=(Jhat+1)/2;
# Jstar=log(Jstar/(1-Jstar));
# VarJstar=(4/(Jhat^2 - 1)^2)*varJ
Jstar=qnorm(Jhat);
varJstar=(1/(dnorm(qnorm(Jhat))))^2*varJ
ZJstar=Jstar/sqrt(varJstar)
pvalJ=2*pnorm(-abs(ZJstar))
CIstar=c(Jstar-Za*sqrt(varJstar), Jstar+Za*sqrt(varJstar));
CI=c(0,0)
#CI[1]=exp(CIstar[1])/(1+exp(CIstar[1]))*2-1
#CI[2]=exp(CIstar[2])/(1+exp(CIstar[2]))*2-1
CI[1]=pnorm(CIstar[1])
CI[2]=pnorm(CIstar[2])
CIJ=CI;
Jhat
CIwidth=max(CI)-min(CI)
boots=0;c1hat_boots=0;c1hat_boots_or=0; c1hat=cutoff;
#===========CIs for the cutoff==================
for (boots in 1:1000){
#if (boots>=2){print((m1hat_boots*(b^2-1)-a+b*sqrt(a^2+(b^2-1)*s1hat_boots^2*log(b^2)))/(b^2-1))}
#tryCatch({
atx = sample(1:n1,n1,replace=T)
aty = sample(1:n2,n2,replace=T)
if (printProgress) {
if (boots==100){print(boots);print("bootstap samples out of 1000...")}
if (boots==200){print(boots);print("bootstap samples out of 1000...")}
if (boots==300){print(boots);print("bootstap samples out of 1000...")}
if (boots==400){print(boots);print("bootstap samples out of 1000...")}
if (boots==500){print(boots);print("bootstap samples out of 1000...")}
if (boots==600){print(boots);print("bootstap samples out of 1000...")}
if (boots==700){print(boots);print("bootstap samples out of 1000...")}
if (boots==800){print(boots);print("bootstap samples out of 1000...")}
if (boots==900){print(boots);print("bootstap samples out of 1000...")}
if (boots==1000){print(boots);print("Bootstrapping complete. You can request all results through the summary() function.")}
}
xboots=xor[atx]
yboots=yor[aty]
ccb=boxcoxleo2(xboots,yboots)
lam_boots=ccb$transformation.parameter
transx1_boots=ccb$transx;
transx2_boots=ccb$transy;
m1hat_boots=mean(transx1_boots)
m2hat_boots=mean(transx2_boots)
s1hat_boots=sqrt(var(transx1_boots)*(length(transx1_boots)-1)/length(transx1_boots))
s2hat_boots=sqrt(var(transx2_boots)*(length(transx2_boots)-1)/length(transx2_boots))
b=s2hat_boots/s1hat_boots
a=m2hat_boots-m1hat_boots
#print(b)
#print(a)
c1hat_boots[boots]=(m1hat_boots*(b^2-1)-a+b*sqrt(a^2+(b^2-1)*s1hat_boots^2*log(b^2)))/(b^2-1)
kr=(m1hat_boots*(b^2-1)-a+b*sqrt(a^2+(b^2-1)*s1hat_boots^2*log(b^2)))/(b^2-1)
c1hat_boots_or[boots] = ((kr*lam_boots+1))^(1/lam_boots)
#}, error=function(e){})
}
varboots=var(na.omit(c1hat_boots_or))
ccc=((c1hat*lam+1))^(1/lam)
CI_BCAN=c(ccc-Za*sqrt(var(na.omit(c1hat_boots_or))), ccc+Za*sqrt(var(na.omit(c1hat_boots_or))))
CIwidth=max(CI_BCAN)-min(CI_BCAN)
CIcutoff=CI_BCAN;
cutoff=ccc;
allc1=c1hat_boots_or
#CI_BCPB=c(quantile(na.omit(c1hat_boots_or),0.025,"type"=2), quantile(na.omit(c1hat_boots_or),0.975,"type"=2))
#CIwidth=max(CI_BCPB)-min(CI_BCPB)
#CI_BCPB
#legend("bottomright", legend=c("ROC estimate", "max of Youden index (J)", "rectangular 95% confidence region of the optimal operating point", "rectangular 95% confidence region of the optimal operating point"),
# col=c("red", "blue", "black", "green"), lty=c(1,1,1,1), cex=0.8)
if (plots=="on"){
legend("bottomright", legend=c(paste("ROC estimate with AUC = ",
formattable(auc, digits = 4, format = "f"),
", CI: (",
formattable(CIauc[1], digits = 4, format = "f"),
", ",
formattable(CIauc[2], digits = 4, format = "f") ,")", sep = ""),
paste("Maximized Youden index = ", formattable(Jhat, digits = 4, format = "f"),
", CI: (",
formattable((CIJ[1]), digits = 4, format = "f"),
", ",
formattable((CIJ[2]), digits = 4, format = "f"), ")", sep = ""),
paste("Optimal pair of (FPR,TPR): (",
formattable(1-Sp, digits = 4, format = "f"),
", ",
formattable(Se, digits = 4, format = "f"), ")", sep = ""),
paste("Area of the rect. conf. region =", formattable(arearect, digits = 4, format = "f")),
paste("Area of the egg conf. region =", formattable(areaegg, digits = 4, format = "f")),
paste("Youden based cutoff: ", formattable(cutoff, digits = 4, format = "f"),", CI: (",
formattable(CIcutoff[1], digits = 4, format = "f"),
", ",
formattable(CIcutoff[2], digits = 4, format = "f"), ")", sep = ""),
paste("Sp Marginal CI: (",
c(formattable(margcisp[1], digits = 4, format = "f")), ", ", formattable(margcisp[2], digits = 4, format = "f"),
")", sep = ""),
paste("Se Marginal CI: (",
c(formattable(margcise[1], digits = 4, format = "f")), ", ", formattable(margcise[2], digits = 4, format = "f"), ")",
sep = "")),
col=c("red", "blue", "red", "black", "green", "white", "white", "white"),
lty=c(1,1,1,1,1,1,NA, NA),
pch = c(NA, NA, 19, NA, NA, NA, NA, NA), cex=0.6)
}
#return(list(transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam , transformation.parameter=lam, AUC=auc, AUCCI=CIauc, J=Jhat, JCI=CIJ, Sens=Se, CImarginalSens=margcise, Spec=Sp, CImarginalSpec=margcisp, cutoff=cutoff, CIcutoff=CIcutoff, areaegg=areaegg, arearect=arearect, mxlam=mean(transx), sxlam=std(transx), mylam=mean(transy), sylam=std(transy)))
res <- matrix(c(auc,CIauc[1],CIauc[2],
Jhat,CIJ[1],CIJ[2],
cutoff,CIcutoff[1],CIcutoff[2],
Sp,margcisp[1],margcisp[2],
Se,margcise[1],margcise[2],
lam,NA,NA),ncol=3,byrow=TRUE)
colnames(res) <- c("Estimate","Confidence","Interval")
rownames(res) <- c("AUC","Jhat","cutoff","Sp","Se", "Lambda")
res <- data.frame(res)
res <- formattable(as.matrix(res), digits = 4, format = "f")
res
if (auc < 0.5) {
print("NOTE: AUC < 0.5; the ordering of the two groups may need to be reversed.")
}
return(list(transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam , transformation.parameter=lam, AUC=auc, AUCCI=CIauc, pvalueAUC=pvalauc, J=Jhat, JCI=CIJ, pvalueJ=pvalJ, Sens=Se, CImarginalSens=margcise, Spec=Sp, CImarginalSpec=margcisp, cutoff=cutoff, CIcutoff=CIcutoff, areaegg=areaegg, arearect=arearect, mxlam=mean(transx), sxlam=sqrt(var(transx)*(length(transx)-1)/length(transx)), mylam=mean(transy), sylam=sqrt(var(transy)*(length(transy)-1)/length(transy)), results=res , rocfun=rocfun))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
rocboxcoxCI<-function(marker, D, givenSP, givenSE, alpha, plots){
if (plots!="on"){plots="off"}
if (length(marker) != length(D)) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (sum(is.na(marker)) > 0 | sum(is.na(D)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if (alpha <= 0 | alpha >= 1) {
stop("ERROR: The level of significance, alpha, should be set between 0 and 1. A common choice is 0.05.")
} else if (!((sum(is.na(givenSP)) == 0 & sum(is.na(givenSE) > 0)) |
(sum(is.na(givenSE)) == 0 & sum(is.na(givenSP) > 0)))) {
stop("ERROR: Exactly one of 'givenSP' and 'givenSE' must be set to NA.")
} else if (sum(marker < 0) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
#graphics.off()
if (plots!="on"){plots="off"}
xor=marker[D==0]
yor=marker[D==1]
x = xor
y = yor
likbox<-function(x,y,h){
n=length(x);
m=length(y);
out=c();
for (i in 1:length(h)){
#print(i)
if (h[i]==0){
xh=log(x);
yh=log(y);
} else {
xh=((x^h[i])-1)/h[i];
yh=((y^h[i])-1)/h[i];
}
out[i]<-c( -n/2*log(sum((xh-sum(xh)/n)^2)/n) -m/2*log(sum((yh-sum(yh)/m)^2)/m) +(h[i]-1)*(sum(log(x))+sum(log(y))))
}
return(out)
}
boxcoxleo<-function(x,y){
init=1;
logL<-function(h){
-likbox(x,y,h)
}
#lam=fminsearch(logL,init)
#lam=c(lam$optbase$xopt)
lam<- optim(1,logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
transx=((x^lam)-1)/lam
transy=((y^lam)-1)/lam
#test1=print(shapiro.test(transx))
#test2=print(shapiro.test(transy))
return(list(transformation.parameter=lam,transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam ))
}
boxcoxleo2<-function(x,y){
init=1;
logL<-function(h){
-likbox(x,y,h)
}
lam<- optim(1,logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
#lam=fminsearch(logL,init)
#lam=c(lam$optbase$xopt)
transx=((x^lam)-1)/lam
transy=((y^lam)-1)/lam
return(list(transformation.parameter=lam,transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam ))
}
cc=boxcoxleo2(x,y)
transx=cc$transx
transy=cc$transy
#x11()
#qqnorm(x)
#title(main=" for X")
#x11()
#qqnorm(y)
#title(main=" for Y")
roc<-function(t,trans_x,trans_y){
na = length(trans_x)
nb = length(trans_y)
stransx=sqrt(var(trans_x)*(na-1)/na)
stransy=sqrt(var(trans_y)*(nb-1)/nb)
1-pnorm(qnorm(1-t,
mean=mean(trans_x),
sd=stransx),
mean=mean(trans_y),
sd=stransy)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
# x11()
#plot.new()
if (plots == "on") {
txt <- paste("Box-Cox Based ROC, alpha =", round(alpha,3) )
plot(linspace(0,1,1000),roc(linspace(0,1,1000),transx,transy),main=" ",xlab="FPR",ylab="TPR",type="l",col="red")
lines(linspace(0,1,10),linspace(0,1,10),type="l", lty=2)
title(main=txt)
}
m1hat=mean(transx)
m2hat=mean(transy)
# s1hat=std(transx)
# s2hat=std(transy)
n1=length(x)
n2=length(y)
I=zeros(5,5);
sh=sqrt(1/(length(transx))*sum((transx-mean(transx))^2))
sd=sqrt(1/(length(transy))*sum((transy-mean(transy))^2))
s1hat = sh
s2hat = sd
mh=m1hat;
md=m2hat;
n=n1;
m=n2;
xlam=transx;
ylam=transy;
lam=cc$transformation.parameter
I[1,1]=n/sh^2;
I[2,2]=-(n/sh^2-3/sh^4*sum((xlam-mh)^2));
I[3,3]=m/sd^2;
I[4,4]=-(m/sd^2-3/sd^4*sum((ylam-md)^2));
kk= sum(((mh - (x^lam - 1)/lam)*((2*(x^lam - 1))/lam^3 - (2*x^lam*log(x))/lam^2 + (x^lam*log(x)^2)/lam))/sh^2) + - sum(((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)^2/sd^2) + - sum(((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)^2/sh^2) + + sum(((md - (y^lam - 1)/lam)*((2*(y^lam - 1))/lam^3 - (2*y^lam*log(y))/lam^2 + (y^lam*log(y)^2)/lam))/sd^2);
I[5,5]=-kk ;
I[1,5]=sum(((2*(x^lam - 1))/lam^2 - (2*x^lam*log(x))/lam)/(2*sh^2));
I[2,5]=-sum((2*((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)*(mh - (x^lam - 1)/lam))/sh^3);
I[3,5]=-sum(-((2*(y^lam - 1))/lam^2 - (2*y^lam*log(y))/lam)/(2*sd^2));
I[4,5]=-sum((2*((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)*(md - (y^lam - 1)/lam))/sd^3);
I[5,1]=I[1,5]
I[5,2]=I[2,5]
I[5,3]=I[3,5]
I[5,4]=I[4,5]
S=inv(I)
S=S[1:4,1:4]
#=============================================
#GRID LINES:
kk=1;CIROC=c(1,1);CIROCvec1=0;CIROCvec2=0;Sehat=0;
tt=c(linspace(0,1,2000))
for (t in tt){
norminvROC= ((m2hat-m1hat)/s2hat+s1hat/s2hat*qnorm(t))
#====================ROC1==============================
dm1 =-1/s2hat;
ds1 =-(2^(1/2)*erfcinv(2*t))/s2hat;
dm2 =1/s2hat;
ds2 =(m1hat - m2hat)/s2hat^2 + (2^(1/2)*s1hat*erfcinv(2*t))/s2hat^2
vROC=t(c(dm1,ds1,dm2,ds2))%*% S %*% t(t(c(dm1,ds1,dm2,ds2)))
CIinvROC=c(norminvROC-qnorm(1-alpha/2)*sqrt(vROC), norminvROC+qnorm(1-alpha/2)*sqrt(vROC))
CIROC=c( pnorm(CIinvROC[1]), pnorm(CIinvROC[2]))
CIROCvec1[kk]=CIROC[1]
CIROCvec2[kk]=CIROC[2]
Sehat[kk]=pnorm(norminvROC)
#points(t,Sehat[length(Sehat)], col = "red")
#lines(c(t,t),c(CIROC[1],CIROC[1]),col="black",lty=2)
#lines(c(t,t),c(CIROC[2],CIROC[2]),col="black")
kk=kk+1
}
if (plots=="on"){
lines(c(tt),c(CIROCvec1),col="black",lty=2)
lines(c(tt),c(CIROCvec2),col="black",lty=2)
#====END GRID LINES=========================
#=============================================
# x11()
# plot.new()
txt <- paste("Box-Cox Based ROC, alpha =", round(alpha,3) )
plot(linspace(0,1,1000),roc(linspace(0,1,1000),transx,transy),main=" ",xlab="FPR",ylab="TPR",type="l",col="red")
lines(linspace(0,1,10),linspace(0,1,10),type="l", lty=2)
title(main=txt)
}
kk=1;CIROC=c(1,1);CIROCvec1=0;CIROCvec2=0;Sehat=0;
tt=c(givenSP);tt=1-givenSP;
#tt=c(0.05,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9)
for (t in tt){
norminvROC= ((m2hat-m1hat)/s2hat+s1hat/s2hat*qnorm(t))
#====================ROC1==============================
dm1 =-1/s2hat;
ds1 =-(2^(1/2)*erfcinv(2*t))/s2hat;
dm2 =1/s2hat;
ds2 =(m1hat - m2hat)/s2hat^2 + (2^(1/2)*s1hat*erfcinv(2*t))/s2hat^2
vROC=t(c(dm1,ds1,dm2,ds2))%*% S %*% t(t(c(dm1,ds1,dm2,ds2)))
CIinvROC=c(norminvROC-qnorm(1-alpha/2)*sqrt(vROC), norminvROC+qnorm(1-alpha/2)*sqrt(vROC))
CIROC=c( pnorm(CIinvROC[1]), pnorm(CIinvROC[2]))
CIROCvec1[kk]=CIROC[1]
CIROCvec2[kk]=CIROC[2]
Sehat[kk]=pnorm(norminvROC)
if (plots=="on"){
points(t,Sehat[length(Sehat)], col = "green")
lines(c(t,t),c(CIROC[1],CIROC[2]),col="green")
}
kk=kk+1
}
Spvalues=1-tt;FPRvalues=tt;
SEandCIs = matrix( c(Spvalues, Sehat, CIROCvec1,CIROCvec2), nrow=length(CIROCvec1), ncol=4)
colnames(SEandCIs) <- c("Given Sp", "Sehat","LL of 95%CI","UL of 95%CI")
SEandCIs
CIlowSe=CIROCvec1
CIuppSe=CIROCvec2
CIse=t(rbind(CIlowSe,CIuppSe))
#==================ROCinv1 (CIs for Sp)===================
mm1=m1hat;
mm2=m2hat;
ss1=s1hat;
ss2=s2hat;
m1hat=mm2;
m2hat=mm1;
s1hat=ss2;
s2hat=ss1;
kk=1;CIROC=c(1,1);CIROCvec1=0;CIROCvec2=0;Sphat=0;
mh=m1hat;
md=m2hat;
n=n1;
m=n2;
xlam=transx;
ylam=transy;
lam=cc$transformation.parameter
I[1,1]=n/sh^2;
I[2,2]=-(n/sh^2-3/sh^4*sum((xlam-mh)^2));
I[3,3]=m/sd^2;
I[4,4]=-(m/sd^2-3/sd^4*sum((ylam-md)^2));
kk= sum(((mh - (x^lam - 1)/lam)*((2*(x^lam - 1))/lam^3 - (2*x^lam*log(x))/lam^2 + (x^lam*log(x)^2)/lam))/sh^2) + - sum(((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)^2/sd^2) + - sum(((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)^2/sh^2) + + sum(((md - (y^lam - 1)/lam)*((2*(y^lam - 1))/lam^3 - (2*y^lam*log(y))/lam^2 + (y^lam*log(y)^2)/lam))/sd^2);
I[5,5]=-kk ;
I[1,5]=sum(((2*(x^lam - 1))/lam^2 - (2*x^lam*log(x))/lam)/(2*sh^2));
I[2,5]=-sum((2*((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)*(mh - (x^lam - 1)/lam))/sh^3);
I[3,5]=-sum(-((2*(y^lam - 1))/lam^2 - (2*y^lam*log(y))/lam)/(2*sd^2));
I[4,5]=-sum((2*((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)*(md - (y^lam - 1)/lam))/sd^3);
I[5,1]=I[1,5]
I[5,2]=I[2,5]
I[5,3]=I[3,5]
I[5,4]=I[4,5]
S=inv(I)
S=S[1:4,1:4]
pp=c(givenSE);
#pp=c(0.05,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9)
kk=1;CIROC=c(1,1);CIROCvec1=0;CIROCvec2=0;Sphat=0;
for (t in pp){
norminvROC= ((m2hat-m1hat)/s2hat+s1hat/s2hat*qnorm(t))
#====================ROC1==============================
dm1 =-1/s2hat;
ds1 =-(2^(1/2)*erfcinv(2*t))/s2hat;
dm2 =1/s2hat;
ds2 =(m1hat - m2hat)/s2hat^2 + (2^(1/2)*s1hat*erfcinv(2*t))/s2hat^2
vROC=t(c(dm1,ds1,dm2,ds2))%*% S %*% t(t(c(dm1,ds1,dm2,ds2)))
CIinvROC=c(norminvROC-qnorm(1-alpha/2)*sqrt(vROC), norminvROC+qnorm(1-alpha/2)*sqrt(vROC))
CIROC=c( pnorm(CIinvROC[1]), pnorm(CIinvROC[2]))
CIROCvec1[kk]=CIROC[1]
CIROCvec2[kk]=CIROC[2]
Sphat[kk]=pnorm(norminvROC)
if (plots=="on"){
points(Sphat[length(Sphat)],t, col = "purple")
lines(c(CIROC[1],CIROC[2]),c(t,t),col="purple")
}
kk=kk+1
}
#SPandCIs = matrix( c(Sphat, CIROCvec1,CIROCvec2), nrow=length(CIROCvec1), ncol=3)
#colnames(SPandCIs) <- c("Sphat","LL of 95%CI","UL of 95%CI")
#SPandCIs
Sevalues=givenSE;
SPandCIs = matrix( c(Sevalues, Sphat, CIROCvec1,CIROCvec2), nrow=length(CIROCvec1), ncol=4)
colnames(SPandCIs) <- c("Given Se", "Sphat","LL of 95%CI","UL of 95%CI")
SPandCIs
CIlowSp=CIROCvec1
CIuppSp=CIROCvec2
CIsp=t(rbind(CIlowSp,CIuppSp))
return(list(SPandCIs=formattable(as.matrix(SPandCIs), digits = 4, format = "f"),
SEandCIs=formattable(as.matrix(SEandCIs), digits = 4, format = "f"),
Sevalues=Sevalues,
Sphat=Sphat,
CIsp=formattable(as.matrix(CIsp), digits = 4, format = "f"),
Spvalues=Spvalues,
Sehat=Sehat,
CIse=formattable(as.matrix(CIse), digits = 4, format = "f")))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
checkboxcox<-function(marker, D, plots, printShapiro = FALSE){
oldpar <- par(no.readonly = TRUE)
on.exit(par(oldpar))
if (plots!="on"){plots="off"}
if (length(marker) != length(D)) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (sum(is.na(marker)) > 0 | sum(is.na(D)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if (sum(marker < 0) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
statusD=D;
xor=marker[D==0]
yor=marker[D==1]
x = xor
y = yor
scores = marker
# scores = c(x, y)
#====THIS IS THE LIKELIHOOD OF THE BOXCOX TRANSFORMATION=======================================
#INPUT ARGUMENTS: TWO GROUPS (x,y) and plots="on" or "off"====================================
likbox<-function(x,y,h){
n=length(x);
m=length(y);
out=c();
for (i in 1:length(h)){
#print(i)
if (h[i]==0){
xh=log(x);
yh=log(y);
} else {
xh=((x^h[i])-1)/h[i];
yh=((y^h[i])-1)/h[i];
}
out[i]<-c( -n/2*log(sum((xh-sum(xh)/n)^2)/n) -m/2*log(sum((yh-sum(yh)/m)^2)/m) +(h[i]-1)*(sum(log(x))+sum(log(y))))
}
return(out)
}
#====THIS THE FUNCTION NEEDED TO APPLY THE BOXCOX TRANSFORMATION INPUT ARGUMENTS:
#====TWO GROUPS (x,y) and plots="on" or "off"====================================
boxcoxleo<-function(x,y,plots){
if (plots!="on"){plots="off"}
init=1;
logL<-function(h){
-likbox(x,y,h)
}
#lam=fminsearch(logL,init)
# lam <- optimize(logL, c(-100, 100), tol = 0.0001)
# lam=c(lam$minimum) #c(lam$optbase$xopt)
lam<- optim(1,logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
transx=((x^lam)-1)/lam
transy=((y^lam)-1)/lam
if (!printShapiro) {
test1=shapiro.test(x)
test2=shapiro.test(y)
test3=shapiro.test(transx)
test4=shapiro.test(transy)
}
if (printShapiro) {
test1=print(shapiro.test(x))
test2=print(shapiro.test(y))
test3=print(shapiro.test(transx))
test4=print(shapiro.test(transy))
}
pval1=test1$p.value
pval2=test2$p.value
pval3=test3$p.value
pval4=test4$p.value
pvalues=c(pval1,pval2,pval3,pval4)
return(list(transformation.parameter=lam,transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam,pvalues=pvalues))
}
#====APPLY THE BOXCOX TRANSFORMATION WITH OR WITHOUT THE PLOTS================
cc=boxcoxleo(x,y,plots)
lam=cc$transformation.parameter
pvalues=cc$pvalues
transx=cc$transx #transformed scores for healthy
transy=cc$transy #transformed scores for diseased
if (plots!="on"){plots="off"}
#====IF PLOTS ARE REQUESTED PROVIDE THE HISTOGRAMS AND QQ-PLOTS FOR EACH GROUP
#====BEFORE AND AFTER THE TRANSFORMATION=======================================
if (plots=="on") {
# x11()
par(mfrow=c(2,2))
par(mar = c(4.1, 3.1, 3.1, 1.1))
hist(x, main = "Histogram of X", xlab = "X")
qqnorm(x, main = "Normal Q-Q Plot of X")
qqline(x, col = 2,lwd=2,lty=2)
hist(transx, main = "Histogram of Transformed X", xlab = "Transformed X")
qqnorm(transx, main = "Normal Q-Q Plot of Transformed X")
qqline(transx, col = 2,lwd=2,lty=2)
par(mfrow=c(2,2))
hist(y, main = "Histogram of Y", xlab = "Y")
qqnorm(y, main = "Normal Q-Q Plot of Y")
qqline(y, col = 2,lwd=2,lty=2)
hist(transy, main = "Histogram of Transformed Y", xlab = "Transformed Y")
qqnorm(transy, main = "Normal Q-Q Plot of Transformed Y")
qqline(transy, col = 2,lwd=2,lty=2)
par(mar = c(5.1, 4.1, 4.1, 2.1))
par(mfrow=c(1,1))
}
#====TWO ROC FUNCTIONS: ONE IS THE BOXCOX AND THE OTHER THE REFERENCE LINE================
roc<-function(t){
na = length(transx)
nb = length(transy)
stransx=sqrt( var(transx)*(na-1)/na )
stransy=sqrt( var(transy)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(transx),
sd=stransx),
mean=mean(transy),
sd=stransy)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
#================IF PLOTS ARE REQUESTED PLOT THE ROCS==
if (plots=="on") {
# x11()
par(mfrow = c(1, 1))
plot(linspace(0,1,1000),roc(linspace(0,1,1000)),main="Empirical and Box-Cox Based ROC",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="red")
lines(linspace(0,1,1000),linspace(0,1,1000),type="l", lty=2)
lines(roc.curve(scores, statusD), col=1)
#================LEGEND OF THE ROC PLOT================
legend("bottomright", legend=c(paste("Box-Cox ROC estimate "),
paste("Empirical ROC estimate ")),
col=c("red", "black"), lty=c(1, 1), pch = c(NA, NA), cex=0.8)
}
#================OUTPUT ARGUMENTS======================
return(list(transformation.parameter=lam,
transx=((x^lam)-1)/lam,
transy=((y^lam)-1)/lam,
pval_x=pvalues[1],
pval_y=pvalues[2],
pval_transx=pvalues[3],
pval_transy=pvalues[4],
rocfun=roc ))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
threerocs=function(marker, D, plots)
{
x=marker[D==0];
y=marker[D==1];
# Plot the empirical ROC curve
roc_obj <- roc(D, marker, quiet = TRUE)
auc_value <- auc(roc_obj)
tpr=roc_obj$sensitivities
fpr=1-roc_obj$specificities
if (plots == "on") {
plot(fpr, tpr, type = "l", col = "blue",
xlim = c(0, 1), ylim = c(0, 1), xlab = "FPR = 1 - Specificity",
ylab = "TPR = Sensitivity", main = "Three Estimates of the ROC Curve")
}
t=linspace(0,1,10000) #set a grid for the FPR
############## BOX-COX #############################
roctbc=checkboxcox(marker,D, plots="off", printShapiro = FALSE);
myroc=rocboxcox(marker,D,0.05,plots="off",FALSE)
rocboxc=myroc$rocbc
AUCbc=myroc$AUC
if (plots == "on") {lines(t,roctbc$roc(t),col="red")}
############## Metz #############################
curve2= roc_curves(D,marker,method="binormal")
rocmrm=points(curve2, metric="specificity", values=1-c(t))
rocmrm$TPR
AUCmrm=trapz(t,rocmrm$TPR)
if (plots == "on") {lines(t,rocmrm$TPR,col="forestgreen", lty = "dashed")}
#legend(x = "right", legend=c("one","two","three"))
if (plots == "on") {
legend("bottomright", legend = c(paste("Empirical (AUC = ", formattable(auc_value, digits = 4, format = "f"), ")", sep = ""),
paste("Box-Cox (AUC = ", formattable(AUCbc, digits = 4, format = "f"), ")", sep = ""),
paste("Metz (AUC = ", formattable(AUCmrm, digits = 4, format = "f"), ")", sep = "")),
col = c("blue", "red", "forestgreen"),
lty = c("solid", "dotted", "dashed"),
cex = 0.8)
}
return(list(AUC_Empirical = as.numeric(auc_value),
AUC_Metz = AUCmrm,
AUC_BoxCox = AUCbc))
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
checkboxcox2 <-function (marker1, marker2, D, plots, printShapiro = FALSE){
if ((length(marker1) != length(D)) | (length(marker2) != length(D))) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if ((sum(marker1 < 0) > 0) | (sum(marker2 < 2)) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
erf <- function (x) 2 * pnorm(x * sqrt(2)) - 1
erfc <- function (x) 2 * pnorm(x * sqrt(2), lower.tail = FALSE)
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower.tail = FALSE)/sqrt(2)
if (plots!="on"){plots="off"}
W1a=marker1[D==0]
W1b=marker1[D==1]
W2a=marker2[D==0]
W2b=marker2[D==1]
Wa=cbind(W1a,W2a);
Wb=cbind(W1b,W2b);
na=length(W1a)
nb=length(W1b)
likbox2D<-function(W1a,W1b,W2a,W2b,lam){
na=length(W1a);
nb=length(W1b);
out=c();
#for (i in 1:length(h)){
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
cova= cov(Walam)*(na-1)/na;
covb= cov(Wblam)*(nb-1)/nb;
mualam=c(mean(W1alam), mean(W2alam))
mublam=c(mean(W1blam), mean(W2blam))
out= -sum(log(dmvnorm(Walam,mualam,cova)))-sum(log(dmvnorm(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b));
return(out)
}
lam=c(2,2)
likbox2D(W1a,W1b,W2a,W2b,lam)
logL<-function(h){
likbox2D(W1a,W1b,W2a,W2b,h)
}
# init=c(0.8,0.8)
# lam=fminsearch(logL,init)
# lam=c(lam$optbase$xopt)
lam<- optim(c(1,1),logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
lam
out <- optim(c(1,1), logL, method = "Nelder-Mead")
lam = out$par
###################################
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
m1ahat=mean(W1alam)
m1bhat=mean(W1blam)
m2ahat=mean(W2alam)
m2bhat=mean(W2blam)
s1ahat=sqrt( var(W1alam)*(na-1)/na )
s1bhat=sqrt( var(W1blam)*(nb-1)/nb )
s2ahat=sqrt( var(W2alam)*(na-1)/na )
s2bhat=sqrt( var(W2blam)*(nb-1)/nb )
cova= cov(Walam)*(na-1)/na;cova=cova[1,2];
covb= cov(Wblam)*(nb-1)/nb;covb=covb[1,2];
lam1=lam[1]
lam2=lam[2]
# ==== PERFORM SHAPIRO-WILK TESTS =============================================================
x1 = W1a; y1 = W1b; transx1 = W1alam; transy1 = W1blam
x2 = W2a; y2 = W2b; transx2 = W2alam; transy2 = W2blam
if (!printShapiro) {
test_x1=shapiro.test(x1)
test_y1=shapiro.test(y1)
test_t_x1=shapiro.test(transx1)
test_t_y1=shapiro.test(transy1)
test_x2=shapiro.test(x2)
test_y2=shapiro.test(y2)
test_t_x2=shapiro.test(transx2)
test_t_y2=shapiro.test(transy2)
}
if (printShapiro) {
test_x1=print(shapiro.test(x1))
test_y1=print(shapiro.test(y1))
test_t_x1=print(shapiro.test(transx1))
test_t_y1=print(shapiro.test(transy1))
test_x2=print(shapiro.test(x2))
test_y2=print(shapiro.test(y2))
test_t_x2=print(shapiro.test(transx2))
test_t_y2=print(shapiro.test(transy2))
}
pval_x1=test_x1$p.value
pval_y1=test_y1$p.value
pval_t_x1=test_t_x1$p.value
pval_t_y1=test_t_y1$p.value
pval_x2=test_x2$p.value
pval_y2=test_y2$p.value
pval_t_x2=test_t_x2$p.value
pval_t_y2=test_t_y2$p.value
pvalues=c(pval_x1, pval_y1, pval_t_x1, pval_t_y1,
pval_x2, pval_y2, pval_t_x2, pval_t_y2)
res_shapiro = list(test_x1 = test_x1,
test_y1 = test_y1,
test_t_x1 = test_t_x1,
test_t_y1 = test_t_y1,
test_x2 = test_x2,
test_y2 = test_y2,
test_t_x2 = test_t_x2,
test_t_y2 = test_t_y2)
#====TWO ROC FUNCTIONS: ONE IS THE BOXCOX AND THE OTHER THE REFERENCE LINE================
roc1<-function(t){
na = length(W1alam)
nb = length(W1blam)
s1alam=sqrt( var(W1alam)*(na-1)/na )
s1blam=sqrt( var(W1blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W1alam),
sd=s1alam),
mean=mean(W1blam),
sd=s1blam)
}
roc2<-function(t){
na = length(W2alam)
nb = length(W2blam)
s2alam=sqrt( var(W2alam)*(na-1)/na )
s2blam=sqrt( var(W2blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W2alam),
sd=s2alam),
mean=mean(W2blam),
sd=s2blam)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
#================IF PLOTS ARE REQUESTED PLOT ALL OF THE THINGS====
if (plots=="on") {
par(mfrow=c(2,2))
par(mar = c(4.1, 3.1, 3.1, 1.1))
# HISTOGRAMS & QQ, X1
hist(W1a, main = "Histogram of X1", xlab = "X1")
qqnorm(W1a, main = "Normal Q-Q Plot of X1")
qqline(W1a, col = 2,lwd=2,lty=2)
hist(W1alam, main = "Histogram of Transformed X1", xlab = "Transformed X1")
qqnorm(W1alam, main = "Normal Q-Q Plot of Transformed X1")
qqline(W1alam, col = 2,lwd=2,lty=2)
# HISTOGRAMS & QQ, Y1
hist(W1b, main = "Histogram of Y1", xlab = "Y1")
qqnorm(W1b, main = "Normal Q-Q Plot of Y1")
qqline(W1b, col = 2,lwd=2,lty=2)
hist(W1blam, main = "Histogram of Transformed Y1", xlab = "Transformed Y1")
qqnorm(W1blam, main = "Normal Q-Q Plot of Transformed Y1")
qqline(W1blam, col = 2,lwd=2,lty=2)
# HISTOGRAMS & QQ, X2
hist(W2a, main = "Histogram of X2", xlab = "X2")
qqnorm(W2a, main = "Normal Q-Q Plot of X2")
qqline(W2a, col = 2,lwd=2,lty=2)
hist(W2alam, main = "Histogram of Transformed X2", xlab = "Transformed X2")
qqnorm(W2alam, main = "Normal Q-Q Plot of Transformed X2")
qqline(W2alam, col = 2,lwd=2,lty=2)
# HISTOGRAMS & QQ, Y2
hist(W2b, main = "Histogram of Y2", xlab = "Y2")
qqnorm(W2b, main = "Normal Q-Q Plot of Y2")
qqline(W2b, col = 2,lwd=2,lty=2)
hist(W2blam, main = "Histogram of Transformed Y2", xlab = "Transformed Y2")
qqnorm(W2blam, main = "Normal Q-Q Plot of Transformed Y2")
qqline(W2blam, col = 2,lwd=2,lty=2)
par(mar = c(5.1, 4.1, 4.1, 2.1))
# ROC PLOT 1
par(mfrow = c(1, 2))
plot(linspace(0,1,1000),
roc1(linspace(0,1,1000)),
main="Marker1",
xlab="FPR = 1 - Specificity",
ylab="TPR = Sensitivity",
type="l",
col="red")
lines(linspace(0,1,1000),
linspace(0,1,1000),
type="l",
lty=2)
lines(roc.curve(marker1, D), col="red", lty = 3, lwd = 1.5)
# LEGEND
legend("bottomright", legend=c(paste("Box-Cox ROC Estimate"),
paste("Empirical ROC Estimate")),
col=c("red", "red"), lty=c(1, 3), lwd = c(1, 1.5), pch = c(NA, NA), cex=0.8)
# ROC PLOT 2
plot(linspace(0,1,1000),
roc2(linspace(0,1,1000)),
main="Marker2",
xlab="FPR = 1 - Specificity",
ylab="TPR = Sensitivity",
type="l",
col="black")
lines(linspace(0,1,1000),
linspace(0,1,1000),
type="l",
lty=2)
lines(roc.curve(marker2, D), col="black", lty = 3, lwd = 1.5)
# LEGEND
legend("bottomright", legend=c(paste("Box-Cox ROC estimate"),
paste("Empirical ROC estimate")),
col=c("black", "black"), lty=c(1, 3), lwd = c(1, 1.5), pch = c(NA, NA), cex=0.8)
par(mfrow = c(1, 1))
}
return(list(res_shapiro = res_shapiro,
transformation.parameter.1 = lam[1],
transx1 = W1alam,
transy1 = W1blam,
transformation.parameter.2 = lam[2],
transx2 = W2alam,
transy2 = W2blam,
pval_x1 = pvalues[1],
pval_y1 = pvalues[2],
pval_transx1 = pvalues[3],
pval_transy1 = pvalues[4],
pval_x2 = pvalues[5],
pval_y2 = pvalues[6],
pval_transx2 = pvalues[7],
pval_transy2 = pvalues[8],
roc1 = roc1,
roc2 = roc2))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
comparebcAUC <-function (marker1, marker2, D, alpha, plots){
if ((length(marker1) != length(D)) | (length(marker2) != length(D))) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (alpha <= 0 | alpha >= 1) {
stop("ERROR: The level of significance, alpha, should be set between 0 and 1. A common choice is 0.05.")
} else if (sum(is.na(marker1)) > 0 | sum(is.na(marker2)) > 0 | sum(is.na(D)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if ((sum(marker1 < 0) > 0) | (sum(marker2 < 2)) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
erf <- function (x) 2 * pnorm(x * sqrt(2)) - 1
erfc <- function (x) 2 * pnorm(x * sqrt(2), lower.tail = FALSE)
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower.tail = FALSE)/sqrt(2)
pmalpha=qnorm(1-alpha/2);
if (plots!="on"){plots="off"}
W1a=marker1[D==0]
W1b=marker1[D==1]
W2a=marker2[D==0]
W2b=marker2[D==1]
Wa=cbind(W1a,W2a);
Wb=cbind(W1b,W2b);
na=length(W1a)
nb=length(W1b)
likbox2D<-function(W1a,W1b,W2a,W2b,lam){
na=length(W1a);
nb=length(W1b);
out=c();
#for (i in 1:length(h)){
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
#cova=1/na*sum((W1alam-mean(W1alam))*(W2alam-mean(W2alam)))
#covb=1/nb*sum((W1blam-mean(W1blam))*(W2blam-mean(W2blam)))
cova= cov(Walam)*(na-1)/na;
covb= cov(Wblam)*(nb-1)/nb;
mualam=c(mean(W1alam), mean(W2alam))
mublam=c(mean(W1blam), mean(W2blam))
#dmvn(c(0,0), mu, mcov, log = F)
# out= (-sum(log(dmvn(Walam,mualam,cova)))-sum(log(dmvn(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b)));
out= -sum(log(dmvnorm(Walam,mualam,cova)))-sum(log(dmvnorm(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b));
#out1=-sum(log(dmvnorm(Walam,mualam,cova)))
#out2=-sum(log(dmvnorm(Wblam,mublam,covb)))
#out3=-(lam[1]-1)*sum(log(W1a))
#out4=-(lam[2]-1)*sum(log(W2a))
#out5=-(lam[1]-1)*sum(log(W1b))
#out6=-(lam[2]-1)*sum(log(W2b))
#out=c(out,out1,out2,out3,out4,out5,out6, cova, covb)
#out= (-sum(log(mvnpdf2([W1alam(g(1)) W2alam(g(2))],[mean(W1alam(g(1))) mean(W2alam(g(2)))],[std(W1alam(g(1)),1).^2 cova;cova (std(W2alam(g(2)),1)).^2])))+...
# -sum(log(mvnpdf2([W1blam(g(1)) W2blam(g(2))],[mean(W1blam(g(1))) mean(W2blam(g(2)))],[std(W1blam(g(1)),1).^2 covb;covb (std(W2blam(g(2)),1)).^2])))+...
# -(g(1)-1).*sum(log(W1a))-(g(2)-1).*sum(log(W2a))-(g(1)-1).*sum(log(W1b))-(g(2)-1).*sum(log(W2b)));
return(out)
}
lam=c(2,2)
likbox2D(W1a,W1b,W2a,W2b,lam)
logL<-function(h){
likbox2D(W1a,W1b,W2a,W2b,h)
}
# init=c(0.8,0.8)
# lam=fminsearch(logL,init)
# lam=c(lam$optbase$xopt)
lam<- optim(c(1,1),logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
lam
#init=c(-10,10)
#lam=optimize(logL,init)
#lam
#f <- function (x, a) (x - a)^2
#xmin <- optimize(f, c(0, 1), tol = 0.0001, a = 1/3)
#xmin
out <- optim(c(1,1), logL, method = "Nelder-Mead")
lam = out$par
###################################
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
m1ahat=mean(W1alam)
m1bhat=mean(W1blam)
m2ahat=mean(W2alam)
m2bhat=mean(W2blam)
s1ahat=sqrt( var(W1alam)*(na-1)/na )
s1bhat=sqrt( var(W1blam)*(nb-1)/nb )
s2ahat=sqrt( var(W2alam)*(na-1)/na )
s2bhat=sqrt( var(W2blam)*(nb-1)/nb )
cova= cov(Walam)*(na-1)/na;cova=cova[1,2];
covb= cov(Wblam)*(nb-1)/nb;covb=covb[1,2];
lam1=lam[1]
lam2=lam[2]
##################################
h1_1=sum(-2/(2*s1ahat^2 - (2*cova^2)/s2ahat^2))
h1_2=sum(-(2*s1ahat*s2ahat^2*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_3=0
h1_4=0;
h1_5=sum(cova/(- cova^2 + s1ahat^2*s2ahat^2));
h1_6=sum((s2ahat*(2*cova^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat) - 2*cova*s1ahat^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_7=0
h1_8=0
h1_9=sum((cova^2*(lam2*m2ahat - W2a^lam2 + 1) + s2ahat^2*((lam2*m2ahat - W2a^lam2 + 1)*s1ahat^2 - 2*cova*lam2*m1ahat))/(lam2*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s2ahat^2*(2*lam2 - 2*W1a^lam1*lam2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_10=0
h1_11=sum((2*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h1_12=sum(-(cova*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h2_1=sum(-(2*s1ahat*s2ahat^2*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_2=sum(- (3*s1ahat^2*s2ahat^6 + cova^4*(lam1^2*m2ahat^2 + lam1^2*s2ahat^2) - cova^3*(2*lam1*m2ahat*s2ahat^2 - 2*W1a^lam1*lam1*m2ahat*s2ahat^2 + 2*lam1^2*m1ahat*m2ahat*s2ahat^2) - cova*(6*lam1*m2ahat*s1ahat^2*s2ahat^4 - 6*W1a^lam1*lam1*m2ahat*s1ahat^2*s2ahat^4 + 6*lam1^2*m1ahat*m2ahat*s1ahat^2*s2ahat^4) + cova^2*(W1a^(2*lam1)*s2ahat^4 - 2*W1a^lam1*s2ahat^4 + s2ahat^4 + 2*lam1*m1ahat*s2ahat^4 + lam1^2*m1ahat^2*s2ahat^4 - 2*W1a^lam1*lam1*m1ahat*s2ahat^4 + 3*lam1^2*m2ahat^2*s1ahat^2*s2ahat^2) - 6*W1a^lam1*s1ahat^2*s2ahat^6 + 3*W1a^(2*lam1)*s1ahat^2*s2ahat^6 - lam1^2*s1ahat^4*s2ahat^6 + 3*lam1^2*m1ahat^2*s1ahat^2*s2ahat^6 + 6*lam1*m1ahat*s1ahat^2*s2ahat^6 - 6*W1a^lam1*lam1*m1ahat*s1ahat^2*s2ahat^6)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^4*(W2a^(2*lam2)*lam1^2 - 2*W2a^lam2*lam1^2 + lam1^2) + cova^2*(3*lam1^2*s1ahat^2*s2ahat^2 + 3*W2a^(2*lam2)*lam1^2*s1ahat^2*s2ahat^2 - 6*W2a^lam2*lam1^2*s1ahat^2*s2ahat^2) - lam2*((2*W2a^lam2*lam1^2*m2ahat - 2*lam1^2*m2ahat)*cova^4 + (2*lam1*s2ahat^2 + 2*lam1^2*m1ahat*s2ahat^2 - 2*W1a^lam1*lam1*s2ahat^2 - 2*W2a^lam2*lam1*s2ahat^2 - 2*W2a^lam2*lam1^2*m1ahat*s2ahat^2 + 2*W1a^lam1*W2a^lam2*lam1*s2ahat^2)*cova^3 + (6*W2a^lam2*lam1^2*m2ahat*s1ahat^2*s2ahat^2 - 6*lam1^2*m2ahat*s1ahat^2*s2ahat^2)*cova^2 + (6*lam1*s1ahat^2*s2ahat^4 + 6*lam1^2*m1ahat*s1ahat^2*s2ahat^4 - 6*W1a^lam1*lam1*s1ahat^2*s2ahat^4 - 6*W2a^lam2*lam1*s1ahat^2*s2ahat^4 - 6*W2a^lam2*lam1^2*m1ahat*s1ahat^2*s2ahat^4 + 6*W1a^lam1*W2a^lam2*lam1*s1ahat^2*s2ahat^4)*cova))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_3=sum(0)
h2_4=sum(0)
h2_5=sum((s1ahat*(2*cova^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat) - 2*cova*s2ahat^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_6=sum((lam2*((2*cova*s2ahat^3*(2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat) - 2*cova*s2ahat*(4*cova*lam1^2*m2ahat - 4*W2a^lam2*cova*lam1^2*m2ahat))*s1ahat^3 + 2*cova*s2ahat*(2*cova^2*lam1 + 2*cova^2*lam1^2*m1ahat - 2*W1a^lam1*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1^2*m1ahat + 2*W1a^lam1*W2a^lam2*cova^2*lam1)*s1ahat) - 2*cova*s1ahat^3*s2ahat*(2*cova*lam1^2 + 2*W2a^(2*lam2)*cova*lam1^2 - 4*W2a^lam2*cova*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - ((4*cova^2*lam1^2*m2ahat^2*s2ahat - 2*cova*s2ahat^3*(2*lam1*m2ahat + cova*lam1^2 + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat))*s1ahat^3 + (2*cova*(2*cova + 2*W1a^(2*lam1)*cova - 4*W1a^lam1*cova + 2*cova*lam1^2*m1ahat^2 + 4*cova*lam1*m1ahat - 4*W1a^lam1*cova*lam1*m1ahat)*s2ahat^3 + 2*cova*(cova^3*lam1^2 - 2*cova^2*lam1*m2ahat + 2*W1a^lam1*cova^2*lam1*m2ahat - 2*cova^2*lam1^2*m1ahat*m2ahat)*s2ahat)*s1ahat)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_7=sum(0)
h2_8=sum(0)
h2_9=sum(- (cova*(s2ahat^4*(- 4*lam2^2*m1ahat^2*s1ahat + 2*lam2^2*s1ahat^3) - s1ahat^3*s2ahat^2*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2)) - cova^3*(s1ahat*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2) + 2*lam2^2*s1ahat*s2ahat^2) + s1ahat^3*s2ahat^4*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + cova^2*s1ahat*s2ahat^2*(6*lam2*m1ahat + 6*lam2^2*m1ahat*m2ahat - 6*W2a^lam2*lam2*m1ahat))/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2ahat - 6*W1a^lam1*lam2 - 6*W2a^lam2*lam2 + 6*W1a^lam1*W2a^lam2*lam2 - 6*W1a^lam1*lam2^2*m2ahat)*cova^2*s1ahat*s2ahat^2 + (8*W1a^lam1*lam2^2*m1ahat - 8*lam2^2*m1ahat)*cova*s1ahat*s2ahat^4 + (2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat)*s1ahat^3*s2ahat^4) - cova*s1ahat*s2ahat^4*(4*W1a^(2*lam1)*lam2^2 - 8*W1a^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_10=sum(0)
h2_11=sum((2*s1ahat*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_12=sum(-(2*cova*s1ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h3_1=sum(0)
h3_2=sum(0)
h3_3=sum(-2/(2*s1bhat^2 - (2*covb^2)/s2bhat^2))
h3_4=sum(-(2*s1bhat*s2bhat^2*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_5=sum(0)
h3_6=sum(0)
h3_7=sum(covb/(- covb^2 + s1bhat^2*s2bhat^2))
h3_8=sum((s2bhat*(2*covb^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat) - 2*covb*s1bhat^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_9=sum(0)
h3_10=sum((covb^2*(lam2*m2bhat - W2b^lam2 + 1) + s2bhat^2*((lam2*m2bhat - W2b^lam2 + 1)*s1bhat^2 - 2*covb*lam2*m1bhat))/(lam2*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s2bhat^2*(2*lam2 - 2*W1b^lam1*lam2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_11=sum((2*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h3_12=sum(-(covb*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h4_1=sum(0)
h4_2=sum(0)
h4_3=sum(-(2*s1bhat*s2bhat^2*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_4=sum(- (3*s1bhat^2*s2bhat^6 + covb^4*(lam1^2*m2bhat^2 + lam1^2*s2bhat^2) - covb^3*(2*lam1*m2bhat*s2bhat^2 - 2*W1b^lam1*lam1*m2bhat*s2bhat^2 + 2*lam1^2*m1bhat*m2bhat*s2bhat^2) - covb*(6*lam1*m2bhat*s1bhat^2*s2bhat^4 - 6*W1b^lam1*lam1*m2bhat*s1bhat^2*s2bhat^4 + 6*lam1^2*m1bhat*m2bhat*s1bhat^2*s2bhat^4) + covb^2*(W1b^(2*lam1)*s2bhat^4 - 2*W1b^lam1*s2bhat^4 + s2bhat^4 + 2*lam1*m1bhat*s2bhat^4 + lam1^2*m1bhat^2*s2bhat^4 - 2*W1b^lam1*lam1*m1bhat*s2bhat^4 + 3*lam1^2*m2bhat^2*s1bhat^2*s2bhat^2) - 6*W1b^lam1*s1bhat^2*s2bhat^6 + 3*W1b^(2*lam1)*s1bhat^2*s2bhat^6 - lam1^2*s1bhat^4*s2bhat^6 + 3*lam1^2*m1bhat^2*s1bhat^2*s2bhat^6 + 6*lam1*m1bhat*s1bhat^2*s2bhat^6 - 6*W1b^lam1*lam1*m1bhat*s1bhat^2*s2bhat^6)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^4*(W2b^(2*lam2)*lam1^2 - 2*W2b^lam2*lam1^2 + lam1^2) + covb^2*(3*lam1^2*s1bhat^2*s2bhat^2 + 3*W2b^(2*lam2)*lam1^2*s1bhat^2*s2bhat^2 - 6*W2b^lam2*lam1^2*s1bhat^2*s2bhat^2) - lam2*((2*W2b^lam2*lam1^2*m2bhat - 2*lam1^2*m2bhat)*covb^4 + (2*lam1*s2bhat^2 + 2*lam1^2*m1bhat*s2bhat^2 - 2*W1b^lam1*lam1*s2bhat^2 - 2*W2b^lam2*lam1*s2bhat^2 - 2*W2b^lam2*lam1^2*m1bhat*s2bhat^2 + 2*W1b^lam1*W2b^lam2*lam1*s2bhat^2)*covb^3 + (6*W2b^lam2*lam1^2*m2bhat*s1bhat^2*s2bhat^2 - 6*lam1^2*m2bhat*s1bhat^2*s2bhat^2)*covb^2 + (6*lam1*s1bhat^2*s2bhat^4 + 6*lam1^2*m1bhat*s1bhat^2*s2bhat^4 - 6*W1b^lam1*lam1*s1bhat^2*s2bhat^4 - 6*W2b^lam2*lam1*s1bhat^2*s2bhat^4 - 6*W2b^lam2*lam1^2*m1bhat*s1bhat^2*s2bhat^4 + 6*W1b^lam1*W2b^lam2*lam1*s1bhat^2*s2bhat^4)*covb))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_5=sum(0)
h4_6=sum(0)
h4_7=sum((s1bhat*(2*covb^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat) - 2*covb*s2bhat^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_8=sum((lam2*((2*covb*s2bhat^3*(2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat) - 2*covb*s2bhat*(4*covb*lam1^2*m2bhat - 4*W2b^lam2*covb*lam1^2*m2bhat))*s1bhat^3 + 2*covb*s2bhat*(2*covb^2*lam1 + 2*covb^2*lam1^2*m1bhat - 2*W1b^lam1*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1^2*m1bhat + 2*W1b^lam1*W2b^lam2*covb^2*lam1)*s1bhat) - 2*covb*s1bhat^3*s2bhat*(2*covb*lam1^2 + 2*W2b^(2*lam2)*covb*lam1^2 - 4*W2b^lam2*covb*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - ((4*covb^2*lam1^2*m2bhat^2*s2bhat - 2*covb*s2bhat^3*(2*lam1*m2bhat + covb*lam1^2 + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat))*s1bhat^3 + (2*covb*(2*covb + 2*W1b^(2*lam1)*covb - 4*W1b^lam1*covb + 2*covb*lam1^2*m1bhat^2 + 4*covb*lam1*m1bhat - 4*W1b^lam1*covb*lam1*m1bhat)*s2bhat^3 + 2*covb*(covb^3*lam1^2 - 2*covb^2*lam1*m2bhat + 2*W1b^lam1*covb^2*lam1*m2bhat - 2*covb^2*lam1^2*m1bhat*m2bhat)*s2bhat)*s1bhat)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_9=sum(0)
h4_10=sum(- (covb*(s2bhat^4*(- 4*lam2^2*m1bhat^2*s1bhat + 2*lam2^2*s1bhat^3) - s1bhat^3*s2bhat^2*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2)) - covb^3*(s1bhat*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2) + 2*lam2^2*s1bhat*s2bhat^2) + s1bhat^3*s2bhat^4*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + covb^2*s1bhat*s2bhat^2*(6*lam2*m1bhat + 6*lam2^2*m1bhat*m2bhat - 6*W2b^lam2*lam2*m1bhat))/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2bhat - 6*W1b^lam1*lam2 - 6*W2b^lam2*lam2 + 6*W1b^lam1*W2b^lam2*lam2 - 6*W1b^lam1*lam2^2*m2bhat)*covb^2*s1bhat*s2bhat^2 + (8*W1b^lam1*lam2^2*m1bhat - 8*lam2^2*m1bhat)*covb*s1bhat*s2bhat^4 + (2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat)*s1bhat^3*s2bhat^4) - covb*s1bhat*s2bhat^4*(4*W1b^(2*lam1)*lam2^2 - 8*W1b^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_11=sum((2*s1bhat*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_12=sum(-(2*covb*s1bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h5_1=sum(cova/(- cova^2 + s1ahat^2*s2ahat^2))
h5_2=sum((s1ahat*(2*cova^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat) - 2*cova*s2ahat^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_3=0
h5_4=0
h5_5=sum(-2/(2*s2ahat^2 - (2*cova^2)/s1ahat^2))
h5_6=sum(-(2*s1ahat^2*s2ahat*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_7=0
h5_8=0
h5_9=sum((cova^2*(lam1*m1ahat - W1a^lam1 + 1) + s1ahat^2*((lam1*m1ahat - W1a^lam1 + 1)*s2ahat^2 - 2*cova*lam1*m2ahat))/(lam1*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s1ahat^2*(2*lam1 - 2*W2a^lam2*lam1))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_10=0
h5_11=sum(-(cova*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h5_12=sum((2*s1ahat^2*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h6_1=sum((s2ahat*(2*cova^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat) - 2*cova*s1ahat^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_2=sum((lam2*((2*cova*s2ahat^3*(2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat) - 2*cova*s2ahat*(4*cova*lam1^2*m2ahat - 4*W2a^lam2*cova*lam1^2*m2ahat))*s1ahat^3 + 2*cova*s2ahat*(2*cova^2*lam1 + 2*cova^2*lam1^2*m1ahat - 2*W1a^lam1*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1^2*m1ahat + 2*W1a^lam1*W2a^lam2*cova^2*lam1)*s1ahat) - 2*cova*s1ahat^3*s2ahat*(2*cova*lam1^2 + 2*W2a^(2*lam2)*cova*lam1^2 - 4*W2a^lam2*cova*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - ((4*cova^2*lam1^2*m2ahat^2*s2ahat - 2*cova*s2ahat^3*(2*lam1*m2ahat + cova*lam1^2 + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat))*s1ahat^3 + (2*cova*(2*cova + 2*W1a^(2*lam1)*cova - 4*W1a^lam1*cova + 2*cova*lam1^2*m1ahat^2 + 4*cova*lam1*m1ahat - 4*W1a^lam1*cova*lam1*m1ahat)*s2ahat^3 + 2*cova*(cova^3*lam1^2 - 2*cova^2*lam1*m2ahat + 2*W1a^lam1*cova^2*lam1*m2ahat - 2*cova^2*lam1^2*m1ahat*m2ahat)*s2ahat)*s1ahat)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_3=0
h6_4=0
h6_5=sum(-(2*s1ahat^2*s2ahat*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_6=sum(- (3*s1ahat^6*s2ahat^2 + cova^4*(lam2^2*m1ahat^2 + lam2^2*s1ahat^2) - cova^3*(2*lam2*m1ahat*s1ahat^2 - 2*W2a^lam2*lam2*m1ahat*s1ahat^2 + 2*lam2^2*m1ahat*m2ahat*s1ahat^2) - cova*(6*lam2*m1ahat*s1ahat^4*s2ahat^2 - 6*W2a^lam2*lam2*m1ahat*s1ahat^4*s2ahat^2 + 6*lam2^2*m1ahat*m2ahat*s1ahat^4*s2ahat^2) + cova^2*(W2a^(2*lam2)*s1ahat^4 - 2*W2a^lam2*s1ahat^4 + s1ahat^4 + 2*lam2*m2ahat*s1ahat^4 + lam2^2*m2ahat^2*s1ahat^4 - 2*W2a^lam2*lam2*m2ahat*s1ahat^4 + 3*lam2^2*m1ahat^2*s1ahat^2*s2ahat^2) - 6*W2a^lam2*s1ahat^6*s2ahat^2 + 3*W2a^(2*lam2)*s1ahat^6*s2ahat^2 - lam2^2*s1ahat^6*s2ahat^4 + 3*lam2^2*m2ahat^2*s1ahat^6*s2ahat^2 + 6*lam2*m2ahat*s1ahat^6*s2ahat^2 - 6*W2a^lam2*lam2*m2ahat*s1ahat^6*s2ahat^2)/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^4*(W1a^(2*lam1)*lam2^2 - 2*W1a^lam1*lam2^2 + lam2^2) + cova^2*(3*lam2^2*s1ahat^2*s2ahat^2 + 3*W1a^(2*lam1)*lam2^2*s1ahat^2*s2ahat^2 - 6*W1a^lam1*lam2^2*s1ahat^2*s2ahat^2) - lam1*((2*W1a^lam1*lam2^2*m1ahat - 2*lam2^2*m1ahat)*cova^4 + (2*lam2*s1ahat^2 + 2*lam2^2*m2ahat*s1ahat^2 - 2*W1a^lam1*lam2*s1ahat^2 - 2*W2a^lam2*lam2*s1ahat^2 - 2*W1a^lam1*lam2^2*m2ahat*s1ahat^2 + 2*W1a^lam1*W2a^lam2*lam2*s1ahat^2)*cova^3 + (6*W1a^lam1*lam2^2*m1ahat*s1ahat^2*s2ahat^2 - 6*lam2^2*m1ahat*s1ahat^2*s2ahat^2)*cova^2 + (6*lam2*s1ahat^4*s2ahat^2 + 6*lam2^2*m2ahat*s1ahat^4*s2ahat^2 - 6*W1a^lam1*lam2*s1ahat^4*s2ahat^2 - 6*W2a^lam2*lam2*s1ahat^4*s2ahat^2 - 6*W1a^lam1*lam2^2*m2ahat*s1ahat^4*s2ahat^2 + 6*W1a^lam1*W2a^lam2*lam2*s1ahat^4*s2ahat^2)*cova))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_7=0
h6_8=0
h6_9=sum(- (cova*(s1ahat^4*(- 4*lam1^2*m2ahat^2*s2ahat + 2*lam1^2*s2ahat^3) - s1ahat^2*s2ahat^3*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2)) - cova^3*(s2ahat*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2) + 2*lam1^2*s1ahat^2*s2ahat) + s1ahat^4*s2ahat^3*(2*lam1*m2ahat + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat) + cova^2*s1ahat^2*s2ahat*(6*lam1*m2ahat + 6*lam1^2*m1ahat*m2ahat - 6*W1a^lam1*lam1*m2ahat))/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1ahat - 6*W1a^lam1*lam1 - 6*W2a^lam2*lam1 + 6*W1a^lam1*W2a^lam2*lam1 - 6*W2a^lam2*lam1^2*m1ahat)*cova^2*s1ahat^2*s2ahat + (8*W2a^lam2*lam1^2*m2ahat - 8*lam1^2*m2ahat)*cova*s1ahat^4*s2ahat + (2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat)*s1ahat^4*s2ahat^3) - cova*s1ahat^4*s2ahat*(4*W2a^(2*lam2)*lam1^2 - 8*W2a^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_10=0
h6_11=sum(-(2*cova*s2ahat*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_12=sum((2*s1ahat^2*s2ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h7_1=0
h7_2=0
h7_3=sum(covb/(- covb^2 + s1bhat^2*s2bhat^2))
h7_4=sum((s1bhat*(2*covb^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat) - 2*covb*s2bhat^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_5=0
h7_6=0
h7_7=sum(-2/(2*s2bhat^2 - (2*covb^2)/s1bhat^2))
h7_8=sum(-(2*s1bhat^2*s2bhat*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_9=0
h7_10=sum((covb^2*(lam1*m1bhat - W1b^lam1 + 1) + s1bhat^2*((lam1*m1bhat - W1b^lam1 + 1)*s2bhat^2 - 2*covb*lam1*m2bhat))/(lam1*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s1bhat^2*(2*lam1 - 2*W2b^lam2*lam1))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_11=sum(-(covb*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h7_12=sum((2*s1bhat^2*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h8_1=0
h8_2=0
h8_3=sum((s2bhat*(2*covb^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat) - 2*covb*s1bhat^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_4=sum((lam2*((2*covb*s2bhat^3*(2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat) - 2*covb*s2bhat*(4*covb*lam1^2*m2bhat - 4*W2b^lam2*covb*lam1^2*m2bhat))*s1bhat^3 + 2*covb*s2bhat*(2*covb^2*lam1 + 2*covb^2*lam1^2*m1bhat - 2*W1b^lam1*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1^2*m1bhat + 2*W1b^lam1*W2b^lam2*covb^2*lam1)*s1bhat) - 2*covb*s1bhat^3*s2bhat*(2*covb*lam1^2 + 2*W2b^(2*lam2)*covb*lam1^2 - 4*W2b^lam2*covb*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - ((4*covb^2*lam1^2*m2bhat^2*s2bhat - 2*covb*s2bhat^3*(2*lam1*m2bhat + covb*lam1^2 + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat))*s1bhat^3 + (2*covb*(2*covb + 2*W1b^(2*lam1)*covb - 4*W1b^lam1*covb + 2*covb*lam1^2*m1bhat^2 + 4*covb*lam1*m1bhat - 4*W1b^lam1*covb*lam1*m1bhat)*s2bhat^3 + 2*covb*(covb^3*lam1^2 - 2*covb^2*lam1*m2bhat + 2*W1b^lam1*covb^2*lam1*m2bhat - 2*covb^2*lam1^2*m1bhat*m2bhat)*s2bhat)*s1bhat)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_5=0
h8_6=0
h8_7=sum(-(2*s1bhat^2*s2bhat*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_8=sum(- (3*s1bhat^6*s2bhat^2 + covb^4*(lam2^2*m1bhat^2 + lam2^2*s1bhat^2) - covb^3*(2*lam2*m1bhat*s1bhat^2 - 2*W2b^lam2*lam2*m1bhat*s1bhat^2 + 2*lam2^2*m1bhat*m2bhat*s1bhat^2) - covb*(6*lam2*m1bhat*s1bhat^4*s2bhat^2 - 6*W2b^lam2*lam2*m1bhat*s1bhat^4*s2bhat^2 + 6*lam2^2*m1bhat*m2bhat*s1bhat^4*s2bhat^2) + covb^2*(W2b^(2*lam2)*s1bhat^4 - 2*W2b^lam2*s1bhat^4 + s1bhat^4 + 2*lam2*m2bhat*s1bhat^4 + lam2^2*m2bhat^2*s1bhat^4 - 2*W2b^lam2*lam2*m2bhat*s1bhat^4 + 3*lam2^2*m1bhat^2*s1bhat^2*s2bhat^2) - 6*W2b^lam2*s1bhat^6*s2bhat^2 + 3*W2b^(2*lam2)*s1bhat^6*s2bhat^2 - lam2^2*s1bhat^6*s2bhat^4 + 3*lam2^2*m2bhat^2*s1bhat^6*s2bhat^2 + 6*lam2*m2bhat*s1bhat^6*s2bhat^2 - 6*W2b^lam2*lam2*m2bhat*s1bhat^6*s2bhat^2)/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^4*(W1b^(2*lam1)*lam2^2 - 2*W1b^lam1*lam2^2 + lam2^2) + covb^2*(3*lam2^2*s1bhat^2*s2bhat^2 + 3*W1b^(2*lam1)*lam2^2*s1bhat^2*s2bhat^2 - 6*W1b^lam1*lam2^2*s1bhat^2*s2bhat^2) - lam1*((2*W1b^lam1*lam2^2*m1bhat - 2*lam2^2*m1bhat)*covb^4 + (2*lam2*s1bhat^2 + 2*lam2^2*m2bhat*s1bhat^2 - 2*W1b^lam1*lam2*s1bhat^2 - 2*W2b^lam2*lam2*s1bhat^2 - 2*W1b^lam1*lam2^2*m2bhat*s1bhat^2 + 2*W1b^lam1*W2b^lam2*lam2*s1bhat^2)*covb^3 + (6*W1b^lam1*lam2^2*m1bhat*s1bhat^2*s2bhat^2 - 6*lam2^2*m1bhat*s1bhat^2*s2bhat^2)*covb^2 + (6*lam2*s1bhat^4*s2bhat^2 + 6*lam2^2*m2bhat*s1bhat^4*s2bhat^2 - 6*W1b^lam1*lam2*s1bhat^4*s2bhat^2 - 6*W2b^lam2*lam2*s1bhat^4*s2bhat^2 - 6*W1b^lam1*lam2^2*m2bhat*s1bhat^4*s2bhat^2 + 6*W1b^lam1*W2b^lam2*lam2*s1bhat^4*s2bhat^2)*covb))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_9=0
h8_10=sum(- (covb*(s1bhat^4*(- 4*lam1^2*m2bhat^2*s2bhat + 2*lam1^2*s2bhat^3) - s1bhat^2*s2bhat^3*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2)) - covb^3*(s2bhat*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2) + 2*lam1^2*s1bhat^2*s2bhat) + s1bhat^4*s2bhat^3*(2*lam1*m2bhat + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat) + covb^2*s1bhat^2*s2bhat*(6*lam1*m2bhat + 6*lam1^2*m1bhat*m2bhat - 6*W1b^lam1*lam1*m2bhat))/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1bhat - 6*W1b^lam1*lam1 - 6*W2b^lam2*lam1 + 6*W1b^lam1*W2b^lam2*lam1 - 6*W2b^lam2*lam1^2*m1bhat)*covb^2*s1bhat^2*s2bhat + (8*W2b^lam2*lam1^2*m2bhat - 8*lam1^2*m2bhat)*covb*s1bhat^4*s2bhat + (2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat)*s1bhat^4*s2bhat^3) - covb*s1bhat^4*s2bhat*(4*W2b^(2*lam2)*lam1^2 - 8*W2b^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_11=sum(-(2*covb*s2bhat*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_12=sum((2*s1bhat^2*s2bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h9_1=sum((cova^2*(lam2*m2ahat - W2a^lam2 + 1) + s2ahat^2*((lam2*m2ahat - W2a^lam2 + 1)*s1ahat^2 - 2*cova*lam2*m1ahat))/(lam2*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s2ahat^2*(2*lam2 - 2*W1a^lam1*lam2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_2=sum(- (cova*(s2ahat^4*(- 4*lam2^2*m1ahat^2*s1ahat + 2*lam2^2*s1ahat^3) - s1ahat^3*s2ahat^2*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2)) - cova^3*(s1ahat*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2) + 2*lam2^2*s1ahat*s2ahat^2) + s1ahat^3*s2ahat^4*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + cova^2*s1ahat*s2ahat^2*(6*lam2*m1ahat + 6*lam2^2*m1ahat*m2ahat - 6*W2a^lam2*lam2*m1ahat))/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2ahat - 6*W1a^lam1*lam2 - 6*W2a^lam2*lam2 + 6*W1a^lam1*W2a^lam2*lam2 - 6*W1a^lam1*lam2^2*m2ahat)*cova^2*s1ahat*s2ahat^2 + (8*W1a^lam1*lam2^2*m1ahat - 8*lam2^2*m1ahat)*cova*s1ahat*s2ahat^4 + (2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat)*s1ahat^3*s2ahat^4) - cova*s1ahat*s2ahat^4*(4*W1a^(2*lam1)*lam2^2 - 8*W1a^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_3=0
h9_4=0
h9_5=sum((cova^2*(lam1*m1ahat - W1a^lam1 + 1) + s1ahat^2*((lam1*m1ahat - W1a^lam1 + 1)*s2ahat^2 - 2*cova*lam1*m2ahat))/(lam1*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s1ahat^2*(2*lam1 - 2*W2a^lam2*lam1))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_6=sum(- (cova*(s1ahat^4*(- 4*lam1^2*m2ahat^2*s2ahat + 2*lam1^2*s2ahat^3) - s1ahat^2*s2ahat^3*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2)) - cova^3*(s2ahat*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2) + 2*lam1^2*s1ahat^2*s2ahat) + s1ahat^4*s2ahat^3*(2*lam1*m2ahat + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat) + cova^2*s1ahat^2*s2ahat*(6*lam1*m2ahat + 6*lam1^2*m1ahat*m2ahat - 6*W1a^lam1*lam1*m2ahat))/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1ahat - 6*W1a^lam1*lam1 - 6*W2a^lam2*lam1 + 6*W1a^lam1*W2a^lam2*lam1 - 6*W2a^lam2*lam1^2*m1ahat)*cova^2*s1ahat^2*s2ahat + (8*W2a^lam2*lam1^2*m2ahat - 8*lam1^2*m2ahat)*cova*s1ahat^4*s2ahat + (2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat)*s1ahat^4*s2ahat^3) - cova*s1ahat^4*s2ahat*(4*W2a^(2*lam2)*lam1^2 - 8*W2a^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_7=0
h9_8=0
h9_9=sum(- (s1ahat^2*(cova^2*(6*lam2*m2ahat - 6*W2a^lam2 + 3*W2a^(2*lam2) + 3*lam2^2*m2ahat^2 - 6*W2a^lam2*lam2*m2ahat + 3) - cova*(6*lam2*m1ahat*s2ahat^2 - 6*W2a^lam2*lam2*m1ahat*s2ahat^2 + 6*lam2^2*m1ahat*m2ahat*s2ahat^2) + lam2^2*m1ahat^2*s2ahat^4) - cova^3*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + s1ahat^4*(W2a^(2*lam2)*s2ahat^2 - 2*W2a^lam2*s2ahat^2 + s2ahat^2 - lam2^2*s2ahat^4 + 2*lam2*m2ahat*s2ahat^2 + lam2^2*m2ahat^2*s2ahat^2 - 2*W2a^lam2*lam2*m2ahat*s2ahat^2) + cova^4*lam2^2 + 3*cova^2*lam2^2*m1ahat^2*s2ahat^2)/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^2*(3*lam2^2*s2ahat^2 - 6*W1a^lam1*lam2^2*s2ahat^2 + 3*W1a^(2*lam1)*lam2^2*s2ahat^2) + s1ahat^2*(lam2^2*s2ahat^4 - 2*W1a^lam1*lam2^2*s2ahat^4 + W1a^(2*lam1)*lam2^2*s2ahat^4) - lam1*(cova^3*(2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat) - cova^2*(6*lam2^2*m1ahat*s2ahat^2 - 6*W1a^lam1*lam2^2*m1ahat*s2ahat^2) + s1ahat^2*(cova*(6*lam2*s2ahat^2 + 6*lam2^2*m2ahat*s2ahat^2 - 6*W1a^lam1*lam2*s2ahat^2 - 6*W2a^lam2*lam2*s2ahat^2 - 6*W1a^lam1*lam2^2*m2ahat*s2ahat^2 + 6*W1a^lam1*W2a^lam2*lam2*s2ahat^2) - 2*lam2^2*m1ahat*s2ahat^4 + 2*W1a^lam1*lam2^2*m1ahat*s2ahat^4)))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_10=0
h9_11=sum(-((W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova^2*lam1 - 2*cova*lam2*s2ahat^2 + lam1*s1ahat^2*s2ahat^2 - W2a^lam2*cova^2*lam1 + 2*W1a^lam1*cova*lam2*s2ahat^2 + cova^2*lam1*lam2*m2ahat - W2a^lam2*lam1*s1ahat^2*s2ahat^2 + lam1*lam2*m2ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_12=sum(-((W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova^2*lam2 - 2*cova*lam1*s1ahat^2 + lam2*s1ahat^2*s2ahat^2 - W1a^lam1*cova^2*lam2 + 2*W2a^lam2*cova*lam1*s1ahat^2 + cova^2*lam1*lam2*m1ahat - W1a^lam1*lam2*s1ahat^2*s2ahat^2 + lam1*lam2*m1ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h10_1=0
h10_2=0
h10_3=sum((covb^2*(lam2*m2bhat - W2b^lam2 + 1) + s2bhat^2*((lam2*m2bhat - W2b^lam2 + 1)*s1bhat^2 - 2*covb*lam2*m1bhat))/(lam2*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s2bhat^2*(2*lam2 - 2*W1b^lam1*lam2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_4=sum(- (covb*(s2bhat^4*(- 4*lam2^2*m1bhat^2*s1bhat + 2*lam2^2*s1bhat^3) - s1bhat^3*s2bhat^2*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2)) - covb^3*(s1bhat*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2) + 2*lam2^2*s1bhat*s2bhat^2) + s1bhat^3*s2bhat^4*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + covb^2*s1bhat*s2bhat^2*(6*lam2*m1bhat + 6*lam2^2*m1bhat*m2bhat - 6*W2b^lam2*lam2*m1bhat))/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2bhat - 6*W1b^lam1*lam2 - 6*W2b^lam2*lam2 + 6*W1b^lam1*W2b^lam2*lam2 - 6*W1b^lam1*lam2^2*m2bhat)*covb^2*s1bhat*s2bhat^2 + (8*W1b^lam1*lam2^2*m1bhat - 8*lam2^2*m1bhat)*covb*s1bhat*s2bhat^4 + (2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat)*s1bhat^3*s2bhat^4) - covb*s1bhat*s2bhat^4*(4*W1b^(2*lam1)*lam2^2 - 8*W1b^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_5=0
h10_6=0
h10_7=sum((covb^2*(lam1*m1bhat - W1b^lam1 + 1) + s1bhat^2*((lam1*m1bhat - W1b^lam1 + 1)*s2bhat^2 - 2*covb*lam1*m2bhat))/(lam1*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s1bhat^2*(2*lam1 - 2*W2b^lam2*lam1))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_8=sum(- (covb*(s1bhat^4*(- 4*lam1^2*m2bhat^2*s2bhat + 2*lam1^2*s2bhat^3) - s1bhat^2*s2bhat^3*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2)) - covb^3*(s2bhat*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2) + 2*lam1^2*s1bhat^2*s2bhat) + s1bhat^4*s2bhat^3*(2*lam1*m2bhat + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat) + covb^2*s1bhat^2*s2bhat*(6*lam1*m2bhat + 6*lam1^2*m1bhat*m2bhat - 6*W1b^lam1*lam1*m2bhat))/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1bhat - 6*W1b^lam1*lam1 - 6*W2b^lam2*lam1 + 6*W1b^lam1*W2b^lam2*lam1 - 6*W2b^lam2*lam1^2*m1bhat)*covb^2*s1bhat^2*s2bhat + (8*W2b^lam2*lam1^2*m2bhat - 8*lam1^2*m2bhat)*covb*s1bhat^4*s2bhat + (2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat)*s1bhat^4*s2bhat^3) - covb*s1bhat^4*s2bhat*(4*W2b^(2*lam2)*lam1^2 - 8*W2b^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_9=0
h10_10=sum(- (s1bhat^2*(covb^2*(6*lam2*m2bhat - 6*W2b^lam2 + 3*W2b^(2*lam2) + 3*lam2^2*m2bhat^2 - 6*W2b^lam2*lam2*m2bhat + 3) - covb*(6*lam2*m1bhat*s2bhat^2 - 6*W2b^lam2*lam2*m1bhat*s2bhat^2 + 6*lam2^2*m1bhat*m2bhat*s2bhat^2) + lam2^2*m1bhat^2*s2bhat^4) - covb^3*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + s1bhat^4*(W2b^(2*lam2)*s2bhat^2 - 2*W2b^lam2*s2bhat^2 + s2bhat^2 - lam2^2*s2bhat^4 + 2*lam2*m2bhat*s2bhat^2 + lam2^2*m2bhat^2*s2bhat^2 - 2*W2b^lam2*lam2*m2bhat*s2bhat^2) + covb^4*lam2^2 + 3*covb^2*lam2^2*m1bhat^2*s2bhat^2)/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^2*(3*lam2^2*s2bhat^2 - 6*W1b^lam1*lam2^2*s2bhat^2 + 3*W1b^(2*lam1)*lam2^2*s2bhat^2) + s1bhat^2*(lam2^2*s2bhat^4 - 2*W1b^lam1*lam2^2*s2bhat^4 + W1b^(2*lam1)*lam2^2*s2bhat^4) - lam1*(covb^3*(2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat) - covb^2*(6*lam2^2*m1bhat*s2bhat^2 - 6*W1b^lam1*lam2^2*m1bhat*s2bhat^2) + s1bhat^2*(covb*(6*lam2*s2bhat^2 + 6*lam2^2*m2bhat*s2bhat^2 - 6*W1b^lam1*lam2*s2bhat^2 - 6*W2b^lam2*lam2*s2bhat^2 - 6*W1b^lam1*lam2^2*m2bhat*s2bhat^2 + 6*W1b^lam1*W2b^lam2*lam2*s2bhat^2) - 2*lam2^2*m1bhat*s2bhat^4 + 2*W1b^lam1*lam2^2*m1bhat*s2bhat^4)))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_11=sum(-((W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb^2*lam1 - 2*covb*lam2*s2bhat^2 + lam1*s1bhat^2*s2bhat^2 - W2b^lam2*covb^2*lam1 + 2*W1b^lam1*covb*lam2*s2bhat^2 + covb^2*lam1*lam2*m2bhat - W2b^lam2*lam1*s1bhat^2*s2bhat^2 + lam1*lam2*m2bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_12=sum(-((W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb^2*lam2 - 2*covb*lam1*s1bhat^2 + lam2*s1bhat^2*s2bhat^2 - W1b^lam1*covb^2*lam2 + 2*W2b^lam2*covb*lam1*s1bhat^2 + covb^2*lam1*lam2*m1bhat - W1b^lam1*lam2*s1bhat^2*s2bhat^2 + lam1*lam2*m1bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_1=sum((2*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h11_2=sum((2*s1ahat*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_3=sum((2*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h11_4=sum((2*s1bhat*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_5=sum(-(cova*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h11_6=sum(-(2*cova*s2ahat*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_7=sum(-(covb*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h11_8=sum(-(2*covb*s2bhat*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_9=sum(-((W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova^2*lam1 - 2*cova*lam2*s2ahat^2 + lam1*s1ahat^2*s2ahat^2 - W2a^lam2*cova^2*lam1 + 2*W1a^lam1*cova*lam2*s2ahat^2 + cova^2*lam1*lam2*m2ahat - W2a^lam2*lam1*s1ahat^2*s2ahat^2 + lam1*lam2*m2ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_10=sum(-((W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb^2*lam1 - 2*covb*lam2*s2bhat^2 + lam1*s1bhat^2*s2bhat^2 - W2b^lam2*covb^2*lam1 + 2*W1b^lam1*covb*lam2*s2bhat^2 + covb^2*lam1*lam2*m2bhat - W2b^lam2*lam1*s1bhat^2*s2bhat^2 + lam1*lam2*m2bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_11=sum(((2*((W1a^lam1-1)/lam1^2-(W1a^lam1*log(W1a))/lam1)^2)/s1ahat^2-(2*(m1ahat-(W1a^lam1-1)/lam1)*((2*W1a^lam1-2)/lam1^3-(2*W1a^lam1*log(W1a))/lam1^2+(W1a^lam1*log(W1a)^2)/lam1))/s1ahat^2+(2*cova*(m2ahat-(W2a^lam2-1)/lam2)*((2*W1a^lam1-2)/lam1^3-(2*W1a^lam1*log(W1a))/lam1^2+(W1a^lam1*log(W1a)^2)/lam1))/(s1ahat^2*s2ahat^2))/((2*cova^2)/(s1ahat^2*s2ahat^2)-2))+sum(((2*((W1b^lam1-1)/lam1^2-(W1b^lam1*log(W1b))/lam1)^2)/s1bhat^2-(2*(m1bhat-(W1b^lam1-1)/lam1)*((2*W1b^lam1-2)/lam1^3-(2*W1b^lam1*log(W1b))/lam1^2+(W1b^lam1*log(W1b)^2)/lam1))/s1bhat^2+(2*covb*(m2bhat-(W2b^lam2-1)/lam2)*((2*W1b^lam1-2)/lam1^3-(2*W1b^lam1*log(W1b))/lam1^2+(W1b^lam1*log(W1b)^2)/lam1))/(s1bhat^2*s2bhat^2))/((2*covb^2)/(s1bhat^2*s2bhat^2)-2))
h11_12=sum((2*cova*(W1a^lam1*lam1*log(W1a)-W1a^lam1+1)*(W2a^lam2*lam2*log(W2a)-W2a^lam2+1))/(lam1^2*lam2^2*(-2*cova^2+2*s1ahat^2*s2ahat^2)))+sum((2*covb*(W1b^lam1*lam1*log(W1b)-W1b^lam1+1)*(W2b^lam2*lam2*log(W2b)-W2b^lam2+1))/(lam1^2*lam2^2*(-2*covb^2+2*s1bhat^2*s2bhat^2)))
h12_1=sum(-(cova*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h12_2=sum(-(2*cova*s1ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_3=sum(-(covb*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h12_4=sum(-(2*covb*s1bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_5=sum((2*s1ahat^2*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h12_6=sum((2*s1ahat^2*s2ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_7=sum((2*s1bhat^2*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h12_8=sum((2*s1bhat^2*s2bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_9=sum(-((W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova^2*lam2 - 2*cova*lam1*s1ahat^2 + lam2*s1ahat^2*s2ahat^2 - W1a^lam1*cova^2*lam2 + 2*W2a^lam2*cova*lam1*s1ahat^2 + cova^2*lam1*lam2*m1ahat - W1a^lam1*lam2*s1ahat^2*s2ahat^2 + lam1*lam2*m1ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_10=sum(-((W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb^2*lam2 - 2*covb*lam1*s1bhat^2 + lam2*s1bhat^2*s2bhat^2 - W1b^lam1*covb^2*lam2 + 2*W2b^lam2*covb*lam1*s1bhat^2 + covb^2*lam1*lam2*m1bhat - W1b^lam1*lam2*s1bhat^2*s2bhat^2 + lam1*lam2*m1bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_11=sum((2*cova*(W1a^lam1*lam1*log(W1a)-W1a^lam1+1)*(W2a^lam2*lam2*log(W2a)-W2a^lam2+1))/(lam1^2*lam2^2*(-2*cova^2+2*s1ahat^2*s2ahat^2)))+sum((2*covb*(W1b^lam1*lam1*log(W1b)-W1b^lam1+1)*(W2b^lam2*lam2*log(W2b)-W2b^lam2+1))/(lam1^2*lam2^2*(-2*covb^2+2*s1bhat^2*s2bhat^2)))
h12_12=sum(((2*((W2a^lam2-1)/lam2^2-(W2a^lam2*log(W2a))/lam2)^2)/s2ahat^2-(2*(m2ahat-(W2a^lam2-1)/lam2)*((2*W2a^lam2-2)/lam2^3-(2*W2a^lam2*log(W2a))/lam2^2+(W2a^lam2*log(W2a)^2)/lam2))/s2ahat^2+(2*cova*(m1ahat-(W1a^lam1-1)/lam1)*((2*W2a^lam2-2)/lam2^3-(2*W2a^lam2*log(W2a))/lam2^2+(W2a^lam2*log(W2a)^2)/lam2))/(s1ahat^2*s2ahat^2))/((2*cova^2)/(s1ahat^2*s2ahat^2)-2))+sum(((2*((W2b^lam2-1)/lam2^2-(W2b^lam2*log(W2b))/lam2)^2)/s2bhat^2-(2*(m2bhat-(W2b^lam2-1)/lam2)*((2*W2b^lam2-2)/lam2^3-(2*W2b^lam2*log(W2b))/lam2^2+(W2b^lam2*log(W2b)^2)/lam2))/s2bhat^2+(2*covb*(m1bhat-(W1b^lam1-1)/lam1)*((2*W2b^lam2-2)/lam2^3-(2*W2b^lam2*log(W2b))/lam2^2+(W2b^lam2*log(W2b)^2)/lam2))/(s1bhat^2*s2bhat^2))/((2*covb^2)/(s1bhat^2*s2bhat^2)-2))
h1r=c(h1_1,h1_2,h1_3,h1_4,h1_5,h1_6,h1_7,h1_8,h1_9,h1_10,h1_11,h1_12)
h2r=c(h2_1,h2_2,h2_3,h2_4,h2_5,h2_6,h2_7,h2_8,h2_9,h2_10,h2_11,h2_12)
h3r=c(h3_1,h3_2,h3_3,h3_4,h3_5,h3_6,h3_7,h3_8,h3_9,h3_10,h3_11,h3_12)
h4r=c(h4_1,h4_2,h4_3,h4_4,h4_5,h4_6,h4_7,h4_8,h4_9,h4_10,h4_11,h4_12)
h5r=c(h5_1,h5_2,h5_3,h5_4,h5_5,h5_6,h5_7,h5_8,h5_9,h5_10,h5_11,h5_12)
h6r=c(h6_1,h6_2,h6_3,h6_4,h6_5,h6_6,h6_7,h6_8,h6_9,h6_10,h6_11,h6_12)
h7r=c(h7_1,h7_2,h7_3,h7_4,h7_5,h7_6,h7_7,h7_8,h7_9,h7_10,h7_11,h7_12)
h8r=c(h8_1,h8_2,h8_3,h8_4,h8_5,h8_6,h8_7,h8_8,h8_9,h8_10,h8_11,h8_12)
h9r=c(h9_1,h9_2,h9_3,h9_4,h9_5,h9_6,h9_7,h9_8,h9_9,h9_10,h9_11,h9_12)
h10r=c(h10_1,h10_2,h10_3,h10_4,h10_5,h10_6,h10_7,h10_8,h10_9,h10_10,h10_11,h10_12)
h11r=c(h11_1,h11_2,h11_3,h11_4,h11_5,h11_6,h11_7,h11_8,h11_9,h11_10,h11_11,h11_12)
h12r=c(h12_1,h12_2,h12_3,h12_4,h12_5,h12_6,h12_7,h12_8,h12_9,h12_10,h12_11,h12_12)
HCF=rbind(h1r,h2r,h3r,h4r,h5r,h6r,h7r,h8r,h9r,h10r,h11r,h12r)
HCF[1,2]=0;HCF[1,6]=0;HCF[1,9]=0;
HCF[2,1]=0;HCF[2,5]=0;
HCF[3,4]=0;HCF[3,8]=0;HCF[3,10]=0;HCF[4,3]=0;
HCF[4,7]=0;
HCF[5,2]=0;HCF[5,6]=0;HCF[5,9]=0;
HCF[6,1]=0;HCF[6,5]=0;
HCF[7,4]=0;HCF[7,8]=0;HCF[7,10]=0;
HCF[8,3]=0;HCF[8,7]=0;
HCF[9,1]=0;HCF[9,5]=0;
HCF[10,3]=0;
HCF[10,7]=0;
HCF[1,1]=na*HCF[1,1];
HCF[1,5]=na*HCF[1,5];
HCF[5,1]=na*HCF[5,1];
HCF[3,3]=nb*HCF[3,3];
HCF[3,7]=nb*HCF[3,7];
HCF[7,3]=nb*HCF[7,3];
HCF[7,7]=nb*HCF[7,7];
HCF[5,5]=na*HCF[5,5];
VV=inv(-HCF)
#############################################################################
#DEALING WITH ORIGINAL AUCS
#############################################################################
AUC1 = erfc((2^(1/2)*(m1ahat - m1bhat))/(2*(s1ahat^2 + s1bhat^2)^(1/2)))/2
dm1a = -(2^(1/2)*exp(-(m1ahat - m1bhat)^2/(2*(s1ahat^2 + s1bhat^2))))/(2*pi^(1/2)*(s1ahat^2 + s1bhat^2)^(1/2))
dm1b = (2^(1/2)*exp(-(m1ahat - m1bhat)^2/(2*(s1ahat^2 + s1bhat^2))))/(2*pi^(1/2)*(s1ahat^2 + s1bhat^2)^(1/2))
ds1a = (2^(1/2)*s1ahat*exp(-(m1ahat - m1bhat)^2/(2*(s1ahat^2 + s1bhat^2)))*(m1ahat - m1bhat))/(2*pi^(1/2)*(s1ahat^2 + s1bhat^2)^(3/2))
ds1b = (2^(1/2)*s1bhat*exp(-(m1ahat - m1bhat)^2/(2*(s1ahat^2 + s1bhat^2)))*(m1ahat - m1bhat))/(2*pi^(1/2)*(s1ahat^2 + s1bhat^2)^(3/2))
AUC2 = erfc((2^(1/2)*(m2ahat - m2bhat))/(2*(s2ahat^2 + s2bhat^2)^(1/2)))/2
dm2a = -(2^(1/2)*exp(-(m2ahat - m2bhat)^2/(2*(s2ahat^2 + s2bhat^2))))/(2*pi^(1/2)*(s2ahat^2 + s2bhat^2)^(1/2))
dm2b = (2^(1/2)*exp(-(m2ahat - m2bhat)^2/(2*(s2ahat^2 + s2bhat^2))))/(2*pi^(1/2)*(s2ahat^2 + s2bhat^2)^(1/2))
ds2a = (2^(1/2)*s2ahat*exp(-(m2ahat - m2bhat)^2/(2*(s2ahat^2 + s2bhat^2)))*(m2ahat - m2bhat))/(2*pi^(1/2)*(s2ahat^2 + s2bhat^2)^(3/2))
ds2b = (2^(1/2)*s2bhat*exp(-(m2ahat - m2bhat)^2/(2*(s2ahat^2 + s2bhat^2)))*(m2ahat - m2bhat))/(2*pi^(1/2)*(s2ahat^2 + s2bhat^2)^(3/2))
varAUC1=c(dm1a, ds1a, dm1b, ds1b)%*%VV[1:4,1:4]%*%t(t(c(dm1a, ds1a, dm1b, ds1b)));
varAUC2=c(dm2a, ds2a, dm2b, ds2b)%*%VV[5:8,5:8]%*%t(t(c(dm2a, ds2a, dm2b, ds2b)));
varAUC1=c(varAUC1)
varAUC2=c(varAUC2)
V=zeros(8,8);
covAUC=c(dm1a, ds1a, dm1b, ds1b, 0, 0, 0, 0)%*%VV[1:8,1:8]%*%t(t(c(0, 0, 0, 0, dm2a, ds2a, dm2b, ds2b)))
covAUC=c(covAUC)
ZdAUC=(AUC2-AUC1)/(sqrt(varAUC1+varAUC2-2*covAUC))
SEdAUC=sqrt(varAUC1+varAUC2-2*covAUC)
CIdiffdAUC=c(AUC2-AUC1-pmalpha*SEdAUC, AUC2-AUC1+pmalpha*SEdAUC)
pval2tdAUC=2*pnorm(-abs(ZdAUC))
#############################################################################
#PROBIT VERSIONS
#############################################################################
#norminv versions
AUC1= ((m1bhat-m1ahat)/s1bhat)/(sqrt(1+(s1ahat/s1bhat)^2));
AUC2= ((m2bhat-m2ahat)/s2bhat)/(sqrt(1+(s2ahat/s2bhat)^2));
AUC1MC=AUC1
AUC2MC=AUC2
AUC1original=pnorm(AUC1)
AUC2original=pnorm(AUC2)
#====================norminv AUC1==============================
dm1a =-1/(s1bhat*sqrt(1+(s1ahat/s1bhat)^2));
ds1a = (s1ahat*(m1ahat-m1bhat))/(s1bhat^3*(1+(s1ahat/s1bhat)^2)^(3/2))
dm1b =1/(s1bhat*sqrt(1+(s1ahat/s1bhat)^2));
ds1b =((m1ahat-m1bhat))/(s1bhat^2*(1+(s1ahat/s1bhat)^2)^(3/2))
varAUC1=c(dm1a, ds1a, dm1b, ds1b)%*%VV[1:4,1:4]%*%t(t(c(dm1a, ds1a, dm1b, ds1b)))
#====================norminv AUC2==============================
dm2a =-1/(s2bhat*sqrt(1+(s2ahat/s2bhat)^2));
ds2a = (s2ahat*(m2ahat-m2bhat))/(s2bhat^3*(1+(s2ahat/s2bhat)^2)^(3/2))
dm2b =1/(s2bhat*sqrt(1+(s2ahat/s2bhat)^2));
ds2b =((m2ahat-m2bhat))/(s2bhat^2*(1+(s2ahat/s2bhat)^2)^(3/2))
varAUC2=c(dm2a, ds2a, dm2b, ds2b)%*%VV[5:8,5:8]%*%t(t(c(dm2a, ds2a, dm2b, ds2b)));
#====================norminv cov(AUC1,AUC2)==============================
V=zeros(8,8);
covAUC=c(dm1a, ds1a, dm1b, ds1b, 0, 0, 0, 0)%*%VV[1:8,1:8]%*%t(t(c(0, 0, 0, 0, dm2a, ds2a, dm2b, ds2b)))
covAUC=c(covAUC)
Z=(AUC2MC-AUC1MC)/(sqrt(varAUC1+varAUC2-2*covAUC))
SE=sqrt(varAUC1+varAUC2-2*covAUC)
CIdiff=c(AUC2MC-AUC1MC-pmalpha*SE, AUC2MC-AUC1MC+pmalpha*SE)
pval2t=2*pnorm(-abs(Z))
#################### IF PLOTS ARE REQUESTED #################################
#====TWO ROC FUNCTIONS: ONE IS THE BOXCOX AND THE OTHER THE REFERENCE LINE================
roc1<-function(t){
na = length(W1alam)
nb = length(W1blam)
s1alam=sqrt( var(W1alam)*(na-1)/na )
s1blam=sqrt( var(W1blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W1alam),
sd=s1alam),
mean=mean(W1blam),
sd=s1blam)
}
roc2<-function(t){
na = length(W2alam)
nb = length(W2blam)
s2alam=sqrt( var(W2alam)*(na-1)/na )
s2blam=sqrt( var(W2blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W2alam),
sd=s2alam),
mean=mean(W2blam),
sd=s2blam)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
#================IF PLOTS ARE REQUESTED PLOT THE ROCS==
if (plots=="on") {
# x11()
plot(linspace(0,1,1000),roc1(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="red")
lines(linspace(0,1,1000),roc2(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="black")
lines(linspace(0,1,1000),linspace(0,1,1000),type="l", lty=2)
legend("bottomright", legend=c(paste("ROC for Marker 1 with AUC =", formattable(AUC1original, digits = 4, format = "f")),
paste("ROC for Marker 2 with AUC =", formattable(AUC2original, digits = 4, format = "f")),
paste("P-value for the probit AUC difference:", formattable(pval2t, digits = 4, format = "f"), " ")),
col=c("red", "black", "white"), lty=c(1, 1, NA), pch = c(NA, NA, NA), cex=0.8)
}
###############################################################################
###############################################################################
#================OUTPUT ARGUMENTS======================
res <- data.frame(AUC1 = formattable(AUC1original, digits = 4, format = "f"),
AUC2 = formattable(AUC2original, digits = 4, format = "f"),
diff = formattable(AUC2original - AUC1original, digits = 4, format = "f"),
p_value_probit = formattable(pval2t, digits = 4, format = "f"),
p_value = formattable(pval2tdAUC, digits = 4, format = "f"),
ci_ll = formattable(CIdiffdAUC[1], digits = 4, format = "f"),
ci_ul = formattable(CIdiffdAUC[2], digits = 4, format = "f"))
rownames(res) <- c("Estimates:")
colnames(res) <- c("AUC 1", "AUC 2", "Diff", "P-Val (Probit)", "P-Val", "CI (LL)", "CI (UL)")
res <- formattable(as.matrix(res), digits = 4, format = "f")
res
#return(list(AUCmarker1=AUC1original,AUCmarker2=AUC2original, pvalue_difference= pval2t, CI_difference= pnorm(CIdiff), rocbc1=roc1, rocbc2=roc2))
return(list(resultstable=res,
AUCmarker1=AUC1original,
AUCmarker2=AUC2original,
pvalue_probit_difference= pval2t,
pvalue_difference= pval2tdAUC,
CI_difference= CIdiffdAUC,
roc1=roc1,
roc2=roc2,
transx1=W1alam,
transy1=W1blam,
transformation.parameter.1 = lam[1],
transx2=W2alam,
transy2=W2blam,
transformation.parameter.2 = lam[2]))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
comparebcJ <-function (marker1, marker2, D, alpha, plots){
if ((length(marker1) != length(D)) | (length(marker2) != length(D))) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (alpha <= 0 | alpha >= 1) {
stop("ERROR: The level of significance, alpha, should be set between 0 and 1. A common choice is 0.05.")
} else if (sum(is.na(marker1)) > 0 | sum(is.na(marker2)) > 0 | sum(is.na(D)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if ((sum(marker1 < 0) > 0) | (sum(marker2 < 2)) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
erf <- function (x) 2 * pnorm(x * sqrt(2)) - 1
erfc <- function (x) 2 * pnorm(x * sqrt(2), lower.tail = FALSE)
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower.tail = FALSE)/sqrt(2)
pmalpha=qnorm(1-alpha/2);
if (plots!="on"){plots="off"}
W1a=marker1[D==0]
W1b=marker1[D==1]
W2a=marker2[D==0]
W2b=marker2[D==1]
Wa=cbind(W1a,W2a);
Wb=cbind(W1b,W2b);
na=length(W1a)
nb=length(W1b)
likbox2D<-function(W1a,W1b,W2a,W2b,lam){
na=length(W1a);
nb=length(W1b);
out=c();
#for (i in 1:length(h)){
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
#cova=1/na*sum((W1alam-mean(W1alam))*(W2alam-mean(W2alam)))
#covb=1/nb*sum((W1blam-mean(W1blam))*(W2blam-mean(W2blam)))
cova= cov(Walam)*(na-1)/na;
covb= cov(Wblam)*(nb-1)/nb;
mualam=c(mean(W1alam), mean(W2alam))
mublam=c(mean(W1blam), mean(W2blam))
#dmvn(c(0,0), mu, mcov, log = F)
# out= (-sum(log(dmvn(Walam,mualam,cova)))-sum(log(dmvn(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b)));
out= -sum(log(dmvnorm(Walam,mualam,cova)))-sum(log(dmvnorm(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b));
#out1=-sum(log(dmvnorm(Walam,mualam,cova)))
#out2=-sum(log(dmvnorm(Wblam,mublam,covb)))
#out3=-(lam[1]-1)*sum(log(W1a))
#out4=-(lam[2]-1)*sum(log(W2a))
#out5=-(lam[1]-1)*sum(log(W1b))
#out6=-(lam[2]-1)*sum(log(W2b))
#out=c(out,out1,out2,out3,out4,out5,out6, cova, covb)
#out= (-sum(log(mvnpdf2([W1alam(g(1)) W2alam(g(2))],[mean(W1alam(g(1))) mean(W2alam(g(2)))],[std(W1alam(g(1)),1).^2 cova;cova (std(W2alam(g(2)),1)).^2])))+...
# -sum(log(mvnpdf2([W1blam(g(1)) W2blam(g(2))],[mean(W1blam(g(1))) mean(W2blam(g(2)))],[std(W1blam(g(1)),1).^2 covb;covb (std(W2blam(g(2)),1)).^2])))+...
# -(g(1)-1).*sum(log(W1a))-(g(2)-1).*sum(log(W2a))-(g(1)-1).*sum(log(W1b))-(g(2)-1).*sum(log(W2b)));
return(out)
}
lam=c(2,2)
likbox2D(W1a,W1b,W2a,W2b,lam)
logL<-function(h){
likbox2D(W1a,W1b,W2a,W2b,h)
}
# init=c(0.8,0.8)
# lam=fminsearch(logL,init)
# lam=c(lam$optbase$xopt)
lam<- optim(c(1,1),logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
lam
#init=c(-10,10)
#lam=optimize(logL,init)
#lam
#f <- function (x, a) (x - a)^2
#xmin <- optimize(f, c(0, 1), tol = 0.0001, a = 1/3)
#xmin
out <- optim(c(1,1), logL, method = "Nelder-Mead")
lam = out$par
###################################
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
m1ahat=mean(W1alam)
m1bhat=mean(W1blam)
m2ahat=mean(W2alam)
m2bhat=mean(W2blam)
s1ahat=sqrt( var(W1alam)*(na-1)/na )
s1bhat=sqrt( var(W1blam)*(nb-1)/nb )
s2ahat=sqrt( var(W2alam)*(na-1)/na )
s2bhat=sqrt( var(W2blam)*(nb-1)/nb )
cova= cov(Walam)*(na-1)/na;cova=cova[1,2];
covb= cov(Wblam)*(nb-1)/nb;covb=covb[1,2];
lam1=lam[1]
lam2=lam[2]
##################################
h1_1=sum(-2/(2*s1ahat^2 - (2*cova^2)/s2ahat^2))
h1_2=sum(-(2*s1ahat*s2ahat^2*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_3=0
h1_4=0;
h1_5=sum(cova/(- cova^2 + s1ahat^2*s2ahat^2));
h1_6=sum((s2ahat*(2*cova^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat) - 2*cova*s1ahat^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_7=0
h1_8=0
h1_9=sum((cova^2*(lam2*m2ahat - W2a^lam2 + 1) + s2ahat^2*((lam2*m2ahat - W2a^lam2 + 1)*s1ahat^2 - 2*cova*lam2*m1ahat))/(lam2*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s2ahat^2*(2*lam2 - 2*W1a^lam1*lam2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_10=0
h1_11=sum((2*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h1_12=sum(-(cova*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h2_1=sum(-(2*s1ahat*s2ahat^2*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_2=sum(- (3*s1ahat^2*s2ahat^6 + cova^4*(lam1^2*m2ahat^2 + lam1^2*s2ahat^2) - cova^3*(2*lam1*m2ahat*s2ahat^2 - 2*W1a^lam1*lam1*m2ahat*s2ahat^2 + 2*lam1^2*m1ahat*m2ahat*s2ahat^2) - cova*(6*lam1*m2ahat*s1ahat^2*s2ahat^4 - 6*W1a^lam1*lam1*m2ahat*s1ahat^2*s2ahat^4 + 6*lam1^2*m1ahat*m2ahat*s1ahat^2*s2ahat^4) + cova^2*(W1a^(2*lam1)*s2ahat^4 - 2*W1a^lam1*s2ahat^4 + s2ahat^4 + 2*lam1*m1ahat*s2ahat^4 + lam1^2*m1ahat^2*s2ahat^4 - 2*W1a^lam1*lam1*m1ahat*s2ahat^4 + 3*lam1^2*m2ahat^2*s1ahat^2*s2ahat^2) - 6*W1a^lam1*s1ahat^2*s2ahat^6 + 3*W1a^(2*lam1)*s1ahat^2*s2ahat^6 - lam1^2*s1ahat^4*s2ahat^6 + 3*lam1^2*m1ahat^2*s1ahat^2*s2ahat^6 + 6*lam1*m1ahat*s1ahat^2*s2ahat^6 - 6*W1a^lam1*lam1*m1ahat*s1ahat^2*s2ahat^6)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^4*(W2a^(2*lam2)*lam1^2 - 2*W2a^lam2*lam1^2 + lam1^2) + cova^2*(3*lam1^2*s1ahat^2*s2ahat^2 + 3*W2a^(2*lam2)*lam1^2*s1ahat^2*s2ahat^2 - 6*W2a^lam2*lam1^2*s1ahat^2*s2ahat^2) - lam2*((2*W2a^lam2*lam1^2*m2ahat - 2*lam1^2*m2ahat)*cova^4 + (2*lam1*s2ahat^2 + 2*lam1^2*m1ahat*s2ahat^2 - 2*W1a^lam1*lam1*s2ahat^2 - 2*W2a^lam2*lam1*s2ahat^2 - 2*W2a^lam2*lam1^2*m1ahat*s2ahat^2 + 2*W1a^lam1*W2a^lam2*lam1*s2ahat^2)*cova^3 + (6*W2a^lam2*lam1^2*m2ahat*s1ahat^2*s2ahat^2 - 6*lam1^2*m2ahat*s1ahat^2*s2ahat^2)*cova^2 + (6*lam1*s1ahat^2*s2ahat^4 + 6*lam1^2*m1ahat*s1ahat^2*s2ahat^4 - 6*W1a^lam1*lam1*s1ahat^2*s2ahat^4 - 6*W2a^lam2*lam1*s1ahat^2*s2ahat^4 - 6*W2a^lam2*lam1^2*m1ahat*s1ahat^2*s2ahat^4 + 6*W1a^lam1*W2a^lam2*lam1*s1ahat^2*s2ahat^4)*cova))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_3=sum(0)
h2_4=sum(0)
h2_5=sum((s1ahat*(2*cova^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat) - 2*cova*s2ahat^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_6=sum((lam2*((2*cova*s2ahat^3*(2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat) - 2*cova*s2ahat*(4*cova*lam1^2*m2ahat - 4*W2a^lam2*cova*lam1^2*m2ahat))*s1ahat^3 + 2*cova*s2ahat*(2*cova^2*lam1 + 2*cova^2*lam1^2*m1ahat - 2*W1a^lam1*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1^2*m1ahat + 2*W1a^lam1*W2a^lam2*cova^2*lam1)*s1ahat) - 2*cova*s1ahat^3*s2ahat*(2*cova*lam1^2 + 2*W2a^(2*lam2)*cova*lam1^2 - 4*W2a^lam2*cova*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - ((4*cova^2*lam1^2*m2ahat^2*s2ahat - 2*cova*s2ahat^3*(2*lam1*m2ahat + cova*lam1^2 + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat))*s1ahat^3 + (2*cova*(2*cova + 2*W1a^(2*lam1)*cova - 4*W1a^lam1*cova + 2*cova*lam1^2*m1ahat^2 + 4*cova*lam1*m1ahat - 4*W1a^lam1*cova*lam1*m1ahat)*s2ahat^3 + 2*cova*(cova^3*lam1^2 - 2*cova^2*lam1*m2ahat + 2*W1a^lam1*cova^2*lam1*m2ahat - 2*cova^2*lam1^2*m1ahat*m2ahat)*s2ahat)*s1ahat)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_7=sum(0)
h2_8=sum(0)
h2_9=sum(- (cova*(s2ahat^4*(- 4*lam2^2*m1ahat^2*s1ahat + 2*lam2^2*s1ahat^3) - s1ahat^3*s2ahat^2*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2)) - cova^3*(s1ahat*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2) + 2*lam2^2*s1ahat*s2ahat^2) + s1ahat^3*s2ahat^4*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + cova^2*s1ahat*s2ahat^2*(6*lam2*m1ahat + 6*lam2^2*m1ahat*m2ahat - 6*W2a^lam2*lam2*m1ahat))/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2ahat - 6*W1a^lam1*lam2 - 6*W2a^lam2*lam2 + 6*W1a^lam1*W2a^lam2*lam2 - 6*W1a^lam1*lam2^2*m2ahat)*cova^2*s1ahat*s2ahat^2 + (8*W1a^lam1*lam2^2*m1ahat - 8*lam2^2*m1ahat)*cova*s1ahat*s2ahat^4 + (2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat)*s1ahat^3*s2ahat^4) - cova*s1ahat*s2ahat^4*(4*W1a^(2*lam1)*lam2^2 - 8*W1a^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_10=sum(0)
h2_11=sum((2*s1ahat*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_12=sum(-(2*cova*s1ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h3_1=sum(0)
h3_2=sum(0)
h3_3=sum(-2/(2*s1bhat^2 - (2*covb^2)/s2bhat^2))
h3_4=sum(-(2*s1bhat*s2bhat^2*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_5=sum(0)
h3_6=sum(0)
h3_7=sum(covb/(- covb^2 + s1bhat^2*s2bhat^2))
h3_8=sum((s2bhat*(2*covb^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat) - 2*covb*s1bhat^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_9=sum(0)
h3_10=sum((covb^2*(lam2*m2bhat - W2b^lam2 + 1) + s2bhat^2*((lam2*m2bhat - W2b^lam2 + 1)*s1bhat^2 - 2*covb*lam2*m1bhat))/(lam2*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s2bhat^2*(2*lam2 - 2*W1b^lam1*lam2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_11=sum((2*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h3_12=sum(-(covb*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h4_1=sum(0)
h4_2=sum(0)
h4_3=sum(-(2*s1bhat*s2bhat^2*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_4=sum(- (3*s1bhat^2*s2bhat^6 + covb^4*(lam1^2*m2bhat^2 + lam1^2*s2bhat^2) - covb^3*(2*lam1*m2bhat*s2bhat^2 - 2*W1b^lam1*lam1*m2bhat*s2bhat^2 + 2*lam1^2*m1bhat*m2bhat*s2bhat^2) - covb*(6*lam1*m2bhat*s1bhat^2*s2bhat^4 - 6*W1b^lam1*lam1*m2bhat*s1bhat^2*s2bhat^4 + 6*lam1^2*m1bhat*m2bhat*s1bhat^2*s2bhat^4) + covb^2*(W1b^(2*lam1)*s2bhat^4 - 2*W1b^lam1*s2bhat^4 + s2bhat^4 + 2*lam1*m1bhat*s2bhat^4 + lam1^2*m1bhat^2*s2bhat^4 - 2*W1b^lam1*lam1*m1bhat*s2bhat^4 + 3*lam1^2*m2bhat^2*s1bhat^2*s2bhat^2) - 6*W1b^lam1*s1bhat^2*s2bhat^6 + 3*W1b^(2*lam1)*s1bhat^2*s2bhat^6 - lam1^2*s1bhat^4*s2bhat^6 + 3*lam1^2*m1bhat^2*s1bhat^2*s2bhat^6 + 6*lam1*m1bhat*s1bhat^2*s2bhat^6 - 6*W1b^lam1*lam1*m1bhat*s1bhat^2*s2bhat^6)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^4*(W2b^(2*lam2)*lam1^2 - 2*W2b^lam2*lam1^2 + lam1^2) + covb^2*(3*lam1^2*s1bhat^2*s2bhat^2 + 3*W2b^(2*lam2)*lam1^2*s1bhat^2*s2bhat^2 - 6*W2b^lam2*lam1^2*s1bhat^2*s2bhat^2) - lam2*((2*W2b^lam2*lam1^2*m2bhat - 2*lam1^2*m2bhat)*covb^4 + (2*lam1*s2bhat^2 + 2*lam1^2*m1bhat*s2bhat^2 - 2*W1b^lam1*lam1*s2bhat^2 - 2*W2b^lam2*lam1*s2bhat^2 - 2*W2b^lam2*lam1^2*m1bhat*s2bhat^2 + 2*W1b^lam1*W2b^lam2*lam1*s2bhat^2)*covb^3 + (6*W2b^lam2*lam1^2*m2bhat*s1bhat^2*s2bhat^2 - 6*lam1^2*m2bhat*s1bhat^2*s2bhat^2)*covb^2 + (6*lam1*s1bhat^2*s2bhat^4 + 6*lam1^2*m1bhat*s1bhat^2*s2bhat^4 - 6*W1b^lam1*lam1*s1bhat^2*s2bhat^4 - 6*W2b^lam2*lam1*s1bhat^2*s2bhat^4 - 6*W2b^lam2*lam1^2*m1bhat*s1bhat^2*s2bhat^4 + 6*W1b^lam1*W2b^lam2*lam1*s1bhat^2*s2bhat^4)*covb))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_5=sum(0)
h4_6=sum(0)
h4_7=sum((s1bhat*(2*covb^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat) - 2*covb*s2bhat^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_8=sum((lam2*((2*covb*s2bhat^3*(2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat) - 2*covb*s2bhat*(4*covb*lam1^2*m2bhat - 4*W2b^lam2*covb*lam1^2*m2bhat))*s1bhat^3 + 2*covb*s2bhat*(2*covb^2*lam1 + 2*covb^2*lam1^2*m1bhat - 2*W1b^lam1*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1^2*m1bhat + 2*W1b^lam1*W2b^lam2*covb^2*lam1)*s1bhat) - 2*covb*s1bhat^3*s2bhat*(2*covb*lam1^2 + 2*W2b^(2*lam2)*covb*lam1^2 - 4*W2b^lam2*covb*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - ((4*covb^2*lam1^2*m2bhat^2*s2bhat - 2*covb*s2bhat^3*(2*lam1*m2bhat + covb*lam1^2 + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat))*s1bhat^3 + (2*covb*(2*covb + 2*W1b^(2*lam1)*covb - 4*W1b^lam1*covb + 2*covb*lam1^2*m1bhat^2 + 4*covb*lam1*m1bhat - 4*W1b^lam1*covb*lam1*m1bhat)*s2bhat^3 + 2*covb*(covb^3*lam1^2 - 2*covb^2*lam1*m2bhat + 2*W1b^lam1*covb^2*lam1*m2bhat - 2*covb^2*lam1^2*m1bhat*m2bhat)*s2bhat)*s1bhat)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_9=sum(0)
h4_10=sum(- (covb*(s2bhat^4*(- 4*lam2^2*m1bhat^2*s1bhat + 2*lam2^2*s1bhat^3) - s1bhat^3*s2bhat^2*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2)) - covb^3*(s1bhat*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2) + 2*lam2^2*s1bhat*s2bhat^2) + s1bhat^3*s2bhat^4*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + covb^2*s1bhat*s2bhat^2*(6*lam2*m1bhat + 6*lam2^2*m1bhat*m2bhat - 6*W2b^lam2*lam2*m1bhat))/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2bhat - 6*W1b^lam1*lam2 - 6*W2b^lam2*lam2 + 6*W1b^lam1*W2b^lam2*lam2 - 6*W1b^lam1*lam2^2*m2bhat)*covb^2*s1bhat*s2bhat^2 + (8*W1b^lam1*lam2^2*m1bhat - 8*lam2^2*m1bhat)*covb*s1bhat*s2bhat^4 + (2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat)*s1bhat^3*s2bhat^4) - covb*s1bhat*s2bhat^4*(4*W1b^(2*lam1)*lam2^2 - 8*W1b^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_11=sum((2*s1bhat*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_12=sum(-(2*covb*s1bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h5_1=sum(cova/(- cova^2 + s1ahat^2*s2ahat^2))
h5_2=sum((s1ahat*(2*cova^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat) - 2*cova*s2ahat^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_3=0
h5_4=0
h5_5=sum(-2/(2*s2ahat^2 - (2*cova^2)/s1ahat^2))
h5_6=sum(-(2*s1ahat^2*s2ahat*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_7=0
h5_8=0
h5_9=sum((cova^2*(lam1*m1ahat - W1a^lam1 + 1) + s1ahat^2*((lam1*m1ahat - W1a^lam1 + 1)*s2ahat^2 - 2*cova*lam1*m2ahat))/(lam1*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s1ahat^2*(2*lam1 - 2*W2a^lam2*lam1))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_10=0
h5_11=sum(-(cova*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h5_12=sum((2*s1ahat^2*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h6_1=sum((s2ahat*(2*cova^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat) - 2*cova*s1ahat^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_2=sum((lam2*((2*cova*s2ahat^3*(2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat) - 2*cova*s2ahat*(4*cova*lam1^2*m2ahat - 4*W2a^lam2*cova*lam1^2*m2ahat))*s1ahat^3 + 2*cova*s2ahat*(2*cova^2*lam1 + 2*cova^2*lam1^2*m1ahat - 2*W1a^lam1*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1^2*m1ahat + 2*W1a^lam1*W2a^lam2*cova^2*lam1)*s1ahat) - 2*cova*s1ahat^3*s2ahat*(2*cova*lam1^2 + 2*W2a^(2*lam2)*cova*lam1^2 - 4*W2a^lam2*cova*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - ((4*cova^2*lam1^2*m2ahat^2*s2ahat - 2*cova*s2ahat^3*(2*lam1*m2ahat + cova*lam1^2 + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat))*s1ahat^3 + (2*cova*(2*cova + 2*W1a^(2*lam1)*cova - 4*W1a^lam1*cova + 2*cova*lam1^2*m1ahat^2 + 4*cova*lam1*m1ahat - 4*W1a^lam1*cova*lam1*m1ahat)*s2ahat^3 + 2*cova*(cova^3*lam1^2 - 2*cova^2*lam1*m2ahat + 2*W1a^lam1*cova^2*lam1*m2ahat - 2*cova^2*lam1^2*m1ahat*m2ahat)*s2ahat)*s1ahat)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_3=0
h6_4=0
h6_5=sum(-(2*s1ahat^2*s2ahat*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_6=sum(- (3*s1ahat^6*s2ahat^2 + cova^4*(lam2^2*m1ahat^2 + lam2^2*s1ahat^2) - cova^3*(2*lam2*m1ahat*s1ahat^2 - 2*W2a^lam2*lam2*m1ahat*s1ahat^2 + 2*lam2^2*m1ahat*m2ahat*s1ahat^2) - cova*(6*lam2*m1ahat*s1ahat^4*s2ahat^2 - 6*W2a^lam2*lam2*m1ahat*s1ahat^4*s2ahat^2 + 6*lam2^2*m1ahat*m2ahat*s1ahat^4*s2ahat^2) + cova^2*(W2a^(2*lam2)*s1ahat^4 - 2*W2a^lam2*s1ahat^4 + s1ahat^4 + 2*lam2*m2ahat*s1ahat^4 + lam2^2*m2ahat^2*s1ahat^4 - 2*W2a^lam2*lam2*m2ahat*s1ahat^4 + 3*lam2^2*m1ahat^2*s1ahat^2*s2ahat^2) - 6*W2a^lam2*s1ahat^6*s2ahat^2 + 3*W2a^(2*lam2)*s1ahat^6*s2ahat^2 - lam2^2*s1ahat^6*s2ahat^4 + 3*lam2^2*m2ahat^2*s1ahat^6*s2ahat^2 + 6*lam2*m2ahat*s1ahat^6*s2ahat^2 - 6*W2a^lam2*lam2*m2ahat*s1ahat^6*s2ahat^2)/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^4*(W1a^(2*lam1)*lam2^2 - 2*W1a^lam1*lam2^2 + lam2^2) + cova^2*(3*lam2^2*s1ahat^2*s2ahat^2 + 3*W1a^(2*lam1)*lam2^2*s1ahat^2*s2ahat^2 - 6*W1a^lam1*lam2^2*s1ahat^2*s2ahat^2) - lam1*((2*W1a^lam1*lam2^2*m1ahat - 2*lam2^2*m1ahat)*cova^4 + (2*lam2*s1ahat^2 + 2*lam2^2*m2ahat*s1ahat^2 - 2*W1a^lam1*lam2*s1ahat^2 - 2*W2a^lam2*lam2*s1ahat^2 - 2*W1a^lam1*lam2^2*m2ahat*s1ahat^2 + 2*W1a^lam1*W2a^lam2*lam2*s1ahat^2)*cova^3 + (6*W1a^lam1*lam2^2*m1ahat*s1ahat^2*s2ahat^2 - 6*lam2^2*m1ahat*s1ahat^2*s2ahat^2)*cova^2 + (6*lam2*s1ahat^4*s2ahat^2 + 6*lam2^2*m2ahat*s1ahat^4*s2ahat^2 - 6*W1a^lam1*lam2*s1ahat^4*s2ahat^2 - 6*W2a^lam2*lam2*s1ahat^4*s2ahat^2 - 6*W1a^lam1*lam2^2*m2ahat*s1ahat^4*s2ahat^2 + 6*W1a^lam1*W2a^lam2*lam2*s1ahat^4*s2ahat^2)*cova))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_7=0
h6_8=0
h6_9=sum(- (cova*(s1ahat^4*(- 4*lam1^2*m2ahat^2*s2ahat + 2*lam1^2*s2ahat^3) - s1ahat^2*s2ahat^3*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2)) - cova^3*(s2ahat*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2) + 2*lam1^2*s1ahat^2*s2ahat) + s1ahat^4*s2ahat^3*(2*lam1*m2ahat + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat) + cova^2*s1ahat^2*s2ahat*(6*lam1*m2ahat + 6*lam1^2*m1ahat*m2ahat - 6*W1a^lam1*lam1*m2ahat))/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1ahat - 6*W1a^lam1*lam1 - 6*W2a^lam2*lam1 + 6*W1a^lam1*W2a^lam2*lam1 - 6*W2a^lam2*lam1^2*m1ahat)*cova^2*s1ahat^2*s2ahat + (8*W2a^lam2*lam1^2*m2ahat - 8*lam1^2*m2ahat)*cova*s1ahat^4*s2ahat + (2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat)*s1ahat^4*s2ahat^3) - cova*s1ahat^4*s2ahat*(4*W2a^(2*lam2)*lam1^2 - 8*W2a^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_10=0
h6_11=sum(-(2*cova*s2ahat*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_12=sum((2*s1ahat^2*s2ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h7_1=0
h7_2=0
h7_3=sum(covb/(- covb^2 + s1bhat^2*s2bhat^2))
h7_4=sum((s1bhat*(2*covb^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat) - 2*covb*s2bhat^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_5=0
h7_6=0
h7_7=sum(-2/(2*s2bhat^2 - (2*covb^2)/s1bhat^2))
h7_8=sum(-(2*s1bhat^2*s2bhat*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_9=0
h7_10=sum((covb^2*(lam1*m1bhat - W1b^lam1 + 1) + s1bhat^2*((lam1*m1bhat - W1b^lam1 + 1)*s2bhat^2 - 2*covb*lam1*m2bhat))/(lam1*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s1bhat^2*(2*lam1 - 2*W2b^lam2*lam1))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_11=sum(-(covb*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h7_12=sum((2*s1bhat^2*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h8_1=0
h8_2=0
h8_3=sum((s2bhat*(2*covb^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat) - 2*covb*s1bhat^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_4=sum((lam2*((2*covb*s2bhat^3*(2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat) - 2*covb*s2bhat*(4*covb*lam1^2*m2bhat - 4*W2b^lam2*covb*lam1^2*m2bhat))*s1bhat^3 + 2*covb*s2bhat*(2*covb^2*lam1 + 2*covb^2*lam1^2*m1bhat - 2*W1b^lam1*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1^2*m1bhat + 2*W1b^lam1*W2b^lam2*covb^2*lam1)*s1bhat) - 2*covb*s1bhat^3*s2bhat*(2*covb*lam1^2 + 2*W2b^(2*lam2)*covb*lam1^2 - 4*W2b^lam2*covb*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - ((4*covb^2*lam1^2*m2bhat^2*s2bhat - 2*covb*s2bhat^3*(2*lam1*m2bhat + covb*lam1^2 + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat))*s1bhat^3 + (2*covb*(2*covb + 2*W1b^(2*lam1)*covb - 4*W1b^lam1*covb + 2*covb*lam1^2*m1bhat^2 + 4*covb*lam1*m1bhat - 4*W1b^lam1*covb*lam1*m1bhat)*s2bhat^3 + 2*covb*(covb^3*lam1^2 - 2*covb^2*lam1*m2bhat + 2*W1b^lam1*covb^2*lam1*m2bhat - 2*covb^2*lam1^2*m1bhat*m2bhat)*s2bhat)*s1bhat)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_5=0
h8_6=0
h8_7=sum(-(2*s1bhat^2*s2bhat*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_8=sum(- (3*s1bhat^6*s2bhat^2 + covb^4*(lam2^2*m1bhat^2 + lam2^2*s1bhat^2) - covb^3*(2*lam2*m1bhat*s1bhat^2 - 2*W2b^lam2*lam2*m1bhat*s1bhat^2 + 2*lam2^2*m1bhat*m2bhat*s1bhat^2) - covb*(6*lam2*m1bhat*s1bhat^4*s2bhat^2 - 6*W2b^lam2*lam2*m1bhat*s1bhat^4*s2bhat^2 + 6*lam2^2*m1bhat*m2bhat*s1bhat^4*s2bhat^2) + covb^2*(W2b^(2*lam2)*s1bhat^4 - 2*W2b^lam2*s1bhat^4 + s1bhat^4 + 2*lam2*m2bhat*s1bhat^4 + lam2^2*m2bhat^2*s1bhat^4 - 2*W2b^lam2*lam2*m2bhat*s1bhat^4 + 3*lam2^2*m1bhat^2*s1bhat^2*s2bhat^2) - 6*W2b^lam2*s1bhat^6*s2bhat^2 + 3*W2b^(2*lam2)*s1bhat^6*s2bhat^2 - lam2^2*s1bhat^6*s2bhat^4 + 3*lam2^2*m2bhat^2*s1bhat^6*s2bhat^2 + 6*lam2*m2bhat*s1bhat^6*s2bhat^2 - 6*W2b^lam2*lam2*m2bhat*s1bhat^6*s2bhat^2)/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^4*(W1b^(2*lam1)*lam2^2 - 2*W1b^lam1*lam2^2 + lam2^2) + covb^2*(3*lam2^2*s1bhat^2*s2bhat^2 + 3*W1b^(2*lam1)*lam2^2*s1bhat^2*s2bhat^2 - 6*W1b^lam1*lam2^2*s1bhat^2*s2bhat^2) - lam1*((2*W1b^lam1*lam2^2*m1bhat - 2*lam2^2*m1bhat)*covb^4 + (2*lam2*s1bhat^2 + 2*lam2^2*m2bhat*s1bhat^2 - 2*W1b^lam1*lam2*s1bhat^2 - 2*W2b^lam2*lam2*s1bhat^2 - 2*W1b^lam1*lam2^2*m2bhat*s1bhat^2 + 2*W1b^lam1*W2b^lam2*lam2*s1bhat^2)*covb^3 + (6*W1b^lam1*lam2^2*m1bhat*s1bhat^2*s2bhat^2 - 6*lam2^2*m1bhat*s1bhat^2*s2bhat^2)*covb^2 + (6*lam2*s1bhat^4*s2bhat^2 + 6*lam2^2*m2bhat*s1bhat^4*s2bhat^2 - 6*W1b^lam1*lam2*s1bhat^4*s2bhat^2 - 6*W2b^lam2*lam2*s1bhat^4*s2bhat^2 - 6*W1b^lam1*lam2^2*m2bhat*s1bhat^4*s2bhat^2 + 6*W1b^lam1*W2b^lam2*lam2*s1bhat^4*s2bhat^2)*covb))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_9=0
h8_10=sum(- (covb*(s1bhat^4*(- 4*lam1^2*m2bhat^2*s2bhat + 2*lam1^2*s2bhat^3) - s1bhat^2*s2bhat^3*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2)) - covb^3*(s2bhat*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2) + 2*lam1^2*s1bhat^2*s2bhat) + s1bhat^4*s2bhat^3*(2*lam1*m2bhat + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat) + covb^2*s1bhat^2*s2bhat*(6*lam1*m2bhat + 6*lam1^2*m1bhat*m2bhat - 6*W1b^lam1*lam1*m2bhat))/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1bhat - 6*W1b^lam1*lam1 - 6*W2b^lam2*lam1 + 6*W1b^lam1*W2b^lam2*lam1 - 6*W2b^lam2*lam1^2*m1bhat)*covb^2*s1bhat^2*s2bhat + (8*W2b^lam2*lam1^2*m2bhat - 8*lam1^2*m2bhat)*covb*s1bhat^4*s2bhat + (2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat)*s1bhat^4*s2bhat^3) - covb*s1bhat^4*s2bhat*(4*W2b^(2*lam2)*lam1^2 - 8*W2b^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_11=sum(-(2*covb*s2bhat*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_12=sum((2*s1bhat^2*s2bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h9_1=sum((cova^2*(lam2*m2ahat - W2a^lam2 + 1) + s2ahat^2*((lam2*m2ahat - W2a^lam2 + 1)*s1ahat^2 - 2*cova*lam2*m1ahat))/(lam2*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s2ahat^2*(2*lam2 - 2*W1a^lam1*lam2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_2=sum(- (cova*(s2ahat^4*(- 4*lam2^2*m1ahat^2*s1ahat + 2*lam2^2*s1ahat^3) - s1ahat^3*s2ahat^2*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2)) - cova^3*(s1ahat*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2) + 2*lam2^2*s1ahat*s2ahat^2) + s1ahat^3*s2ahat^4*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + cova^2*s1ahat*s2ahat^2*(6*lam2*m1ahat + 6*lam2^2*m1ahat*m2ahat - 6*W2a^lam2*lam2*m1ahat))/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2ahat - 6*W1a^lam1*lam2 - 6*W2a^lam2*lam2 + 6*W1a^lam1*W2a^lam2*lam2 - 6*W1a^lam1*lam2^2*m2ahat)*cova^2*s1ahat*s2ahat^2 + (8*W1a^lam1*lam2^2*m1ahat - 8*lam2^2*m1ahat)*cova*s1ahat*s2ahat^4 + (2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat)*s1ahat^3*s2ahat^4) - cova*s1ahat*s2ahat^4*(4*W1a^(2*lam1)*lam2^2 - 8*W1a^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_3=0
h9_4=0
h9_5=sum((cova^2*(lam1*m1ahat - W1a^lam1 + 1) + s1ahat^2*((lam1*m1ahat - W1a^lam1 + 1)*s2ahat^2 - 2*cova*lam1*m2ahat))/(lam1*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s1ahat^2*(2*lam1 - 2*W2a^lam2*lam1))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_6=sum(- (cova*(s1ahat^4*(- 4*lam1^2*m2ahat^2*s2ahat + 2*lam1^2*s2ahat^3) - s1ahat^2*s2ahat^3*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2)) - cova^3*(s2ahat*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2) + 2*lam1^2*s1ahat^2*s2ahat) + s1ahat^4*s2ahat^3*(2*lam1*m2ahat + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat) + cova^2*s1ahat^2*s2ahat*(6*lam1*m2ahat + 6*lam1^2*m1ahat*m2ahat - 6*W1a^lam1*lam1*m2ahat))/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1ahat - 6*W1a^lam1*lam1 - 6*W2a^lam2*lam1 + 6*W1a^lam1*W2a^lam2*lam1 - 6*W2a^lam2*lam1^2*m1ahat)*cova^2*s1ahat^2*s2ahat + (8*W2a^lam2*lam1^2*m2ahat - 8*lam1^2*m2ahat)*cova*s1ahat^4*s2ahat + (2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat)*s1ahat^4*s2ahat^3) - cova*s1ahat^4*s2ahat*(4*W2a^(2*lam2)*lam1^2 - 8*W2a^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_7=0
h9_8=0
h9_9=sum(- (s1ahat^2*(cova^2*(6*lam2*m2ahat - 6*W2a^lam2 + 3*W2a^(2*lam2) + 3*lam2^2*m2ahat^2 - 6*W2a^lam2*lam2*m2ahat + 3) - cova*(6*lam2*m1ahat*s2ahat^2 - 6*W2a^lam2*lam2*m1ahat*s2ahat^2 + 6*lam2^2*m1ahat*m2ahat*s2ahat^2) + lam2^2*m1ahat^2*s2ahat^4) - cova^3*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + s1ahat^4*(W2a^(2*lam2)*s2ahat^2 - 2*W2a^lam2*s2ahat^2 + s2ahat^2 - lam2^2*s2ahat^4 + 2*lam2*m2ahat*s2ahat^2 + lam2^2*m2ahat^2*s2ahat^2 - 2*W2a^lam2*lam2*m2ahat*s2ahat^2) + cova^4*lam2^2 + 3*cova^2*lam2^2*m1ahat^2*s2ahat^2)/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^2*(3*lam2^2*s2ahat^2 - 6*W1a^lam1*lam2^2*s2ahat^2 + 3*W1a^(2*lam1)*lam2^2*s2ahat^2) + s1ahat^2*(lam2^2*s2ahat^4 - 2*W1a^lam1*lam2^2*s2ahat^4 + W1a^(2*lam1)*lam2^2*s2ahat^4) - lam1*(cova^3*(2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat) - cova^2*(6*lam2^2*m1ahat*s2ahat^2 - 6*W1a^lam1*lam2^2*m1ahat*s2ahat^2) + s1ahat^2*(cova*(6*lam2*s2ahat^2 + 6*lam2^2*m2ahat*s2ahat^2 - 6*W1a^lam1*lam2*s2ahat^2 - 6*W2a^lam2*lam2*s2ahat^2 - 6*W1a^lam1*lam2^2*m2ahat*s2ahat^2 + 6*W1a^lam1*W2a^lam2*lam2*s2ahat^2) - 2*lam2^2*m1ahat*s2ahat^4 + 2*W1a^lam1*lam2^2*m1ahat*s2ahat^4)))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_10=0
h9_11=sum(-((W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova^2*lam1 - 2*cova*lam2*s2ahat^2 + lam1*s1ahat^2*s2ahat^2 - W2a^lam2*cova^2*lam1 + 2*W1a^lam1*cova*lam2*s2ahat^2 + cova^2*lam1*lam2*m2ahat - W2a^lam2*lam1*s1ahat^2*s2ahat^2 + lam1*lam2*m2ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_12=sum(-((W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova^2*lam2 - 2*cova*lam1*s1ahat^2 + lam2*s1ahat^2*s2ahat^2 - W1a^lam1*cova^2*lam2 + 2*W2a^lam2*cova*lam1*s1ahat^2 + cova^2*lam1*lam2*m1ahat - W1a^lam1*lam2*s1ahat^2*s2ahat^2 + lam1*lam2*m1ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h10_1=0
h10_2=0
h10_3=sum((covb^2*(lam2*m2bhat - W2b^lam2 + 1) + s2bhat^2*((lam2*m2bhat - W2b^lam2 + 1)*s1bhat^2 - 2*covb*lam2*m1bhat))/(lam2*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s2bhat^2*(2*lam2 - 2*W1b^lam1*lam2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_4=sum(- (covb*(s2bhat^4*(- 4*lam2^2*m1bhat^2*s1bhat + 2*lam2^2*s1bhat^3) - s1bhat^3*s2bhat^2*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2)) - covb^3*(s1bhat*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2) + 2*lam2^2*s1bhat*s2bhat^2) + s1bhat^3*s2bhat^4*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + covb^2*s1bhat*s2bhat^2*(6*lam2*m1bhat + 6*lam2^2*m1bhat*m2bhat - 6*W2b^lam2*lam2*m1bhat))/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2bhat - 6*W1b^lam1*lam2 - 6*W2b^lam2*lam2 + 6*W1b^lam1*W2b^lam2*lam2 - 6*W1b^lam1*lam2^2*m2bhat)*covb^2*s1bhat*s2bhat^2 + (8*W1b^lam1*lam2^2*m1bhat - 8*lam2^2*m1bhat)*covb*s1bhat*s2bhat^4 + (2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat)*s1bhat^3*s2bhat^4) - covb*s1bhat*s2bhat^4*(4*W1b^(2*lam1)*lam2^2 - 8*W1b^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_5=0
h10_6=0
h10_7=sum((covb^2*(lam1*m1bhat - W1b^lam1 + 1) + s1bhat^2*((lam1*m1bhat - W1b^lam1 + 1)*s2bhat^2 - 2*covb*lam1*m2bhat))/(lam1*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s1bhat^2*(2*lam1 - 2*W2b^lam2*lam1))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_8=sum(- (covb*(s1bhat^4*(- 4*lam1^2*m2bhat^2*s2bhat + 2*lam1^2*s2bhat^3) - s1bhat^2*s2bhat^3*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2)) - covb^3*(s2bhat*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2) + 2*lam1^2*s1bhat^2*s2bhat) + s1bhat^4*s2bhat^3*(2*lam1*m2bhat + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat) + covb^2*s1bhat^2*s2bhat*(6*lam1*m2bhat + 6*lam1^2*m1bhat*m2bhat - 6*W1b^lam1*lam1*m2bhat))/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1bhat - 6*W1b^lam1*lam1 - 6*W2b^lam2*lam1 + 6*W1b^lam1*W2b^lam2*lam1 - 6*W2b^lam2*lam1^2*m1bhat)*covb^2*s1bhat^2*s2bhat + (8*W2b^lam2*lam1^2*m2bhat - 8*lam1^2*m2bhat)*covb*s1bhat^4*s2bhat + (2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat)*s1bhat^4*s2bhat^3) - covb*s1bhat^4*s2bhat*(4*W2b^(2*lam2)*lam1^2 - 8*W2b^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_9=0
h10_10=sum(- (s1bhat^2*(covb^2*(6*lam2*m2bhat - 6*W2b^lam2 + 3*W2b^(2*lam2) + 3*lam2^2*m2bhat^2 - 6*W2b^lam2*lam2*m2bhat + 3) - covb*(6*lam2*m1bhat*s2bhat^2 - 6*W2b^lam2*lam2*m1bhat*s2bhat^2 + 6*lam2^2*m1bhat*m2bhat*s2bhat^2) + lam2^2*m1bhat^2*s2bhat^4) - covb^3*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + s1bhat^4*(W2b^(2*lam2)*s2bhat^2 - 2*W2b^lam2*s2bhat^2 + s2bhat^2 - lam2^2*s2bhat^4 + 2*lam2*m2bhat*s2bhat^2 + lam2^2*m2bhat^2*s2bhat^2 - 2*W2b^lam2*lam2*m2bhat*s2bhat^2) + covb^4*lam2^2 + 3*covb^2*lam2^2*m1bhat^2*s2bhat^2)/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^2*(3*lam2^2*s2bhat^2 - 6*W1b^lam1*lam2^2*s2bhat^2 + 3*W1b^(2*lam1)*lam2^2*s2bhat^2) + s1bhat^2*(lam2^2*s2bhat^4 - 2*W1b^lam1*lam2^2*s2bhat^4 + W1b^(2*lam1)*lam2^2*s2bhat^4) - lam1*(covb^3*(2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat) - covb^2*(6*lam2^2*m1bhat*s2bhat^2 - 6*W1b^lam1*lam2^2*m1bhat*s2bhat^2) + s1bhat^2*(covb*(6*lam2*s2bhat^2 + 6*lam2^2*m2bhat*s2bhat^2 - 6*W1b^lam1*lam2*s2bhat^2 - 6*W2b^lam2*lam2*s2bhat^2 - 6*W1b^lam1*lam2^2*m2bhat*s2bhat^2 + 6*W1b^lam1*W2b^lam2*lam2*s2bhat^2) - 2*lam2^2*m1bhat*s2bhat^4 + 2*W1b^lam1*lam2^2*m1bhat*s2bhat^4)))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_11=sum(-((W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb^2*lam1 - 2*covb*lam2*s2bhat^2 + lam1*s1bhat^2*s2bhat^2 - W2b^lam2*covb^2*lam1 + 2*W1b^lam1*covb*lam2*s2bhat^2 + covb^2*lam1*lam2*m2bhat - W2b^lam2*lam1*s1bhat^2*s2bhat^2 + lam1*lam2*m2bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_12=sum(-((W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb^2*lam2 - 2*covb*lam1*s1bhat^2 + lam2*s1bhat^2*s2bhat^2 - W1b^lam1*covb^2*lam2 + 2*W2b^lam2*covb*lam1*s1bhat^2 + covb^2*lam1*lam2*m1bhat - W1b^lam1*lam2*s1bhat^2*s2bhat^2 + lam1*lam2*m1bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_1=sum((2*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h11_2=sum((2*s1ahat*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_3=sum((2*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h11_4=sum((2*s1bhat*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_5=sum(-(cova*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h11_6=sum(-(2*cova*s2ahat*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_7=sum(-(covb*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h11_8=sum(-(2*covb*s2bhat*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_9=sum(-((W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova^2*lam1 - 2*cova*lam2*s2ahat^2 + lam1*s1ahat^2*s2ahat^2 - W2a^lam2*cova^2*lam1 + 2*W1a^lam1*cova*lam2*s2ahat^2 + cova^2*lam1*lam2*m2ahat - W2a^lam2*lam1*s1ahat^2*s2ahat^2 + lam1*lam2*m2ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_10=sum(-((W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb^2*lam1 - 2*covb*lam2*s2bhat^2 + lam1*s1bhat^2*s2bhat^2 - W2b^lam2*covb^2*lam1 + 2*W1b^lam1*covb*lam2*s2bhat^2 + covb^2*lam1*lam2*m2bhat - W2b^lam2*lam1*s1bhat^2*s2bhat^2 + lam1*lam2*m2bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_11=sum(((2*((W1a^lam1-1)/lam1^2-(W1a^lam1*log(W1a))/lam1)^2)/s1ahat^2-(2*(m1ahat-(W1a^lam1-1)/lam1)*((2*W1a^lam1-2)/lam1^3-(2*W1a^lam1*log(W1a))/lam1^2+(W1a^lam1*log(W1a)^2)/lam1))/s1ahat^2+(2*cova*(m2ahat-(W2a^lam2-1)/lam2)*((2*W1a^lam1-2)/lam1^3-(2*W1a^lam1*log(W1a))/lam1^2+(W1a^lam1*log(W1a)^2)/lam1))/(s1ahat^2*s2ahat^2))/((2*cova^2)/(s1ahat^2*s2ahat^2)-2))+sum(((2*((W1b^lam1-1)/lam1^2-(W1b^lam1*log(W1b))/lam1)^2)/s1bhat^2-(2*(m1bhat-(W1b^lam1-1)/lam1)*((2*W1b^lam1-2)/lam1^3-(2*W1b^lam1*log(W1b))/lam1^2+(W1b^lam1*log(W1b)^2)/lam1))/s1bhat^2+(2*covb*(m2bhat-(W2b^lam2-1)/lam2)*((2*W1b^lam1-2)/lam1^3-(2*W1b^lam1*log(W1b))/lam1^2+(W1b^lam1*log(W1b)^2)/lam1))/(s1bhat^2*s2bhat^2))/((2*covb^2)/(s1bhat^2*s2bhat^2)-2))
h11_12=sum((2*cova*(W1a^lam1*lam1*log(W1a)-W1a^lam1+1)*(W2a^lam2*lam2*log(W2a)-W2a^lam2+1))/(lam1^2*lam2^2*(-2*cova^2+2*s1ahat^2*s2ahat^2)))+sum((2*covb*(W1b^lam1*lam1*log(W1b)-W1b^lam1+1)*(W2b^lam2*lam2*log(W2b)-W2b^lam2+1))/(lam1^2*lam2^2*(-2*covb^2+2*s1bhat^2*s2bhat^2)))
h12_1=sum(-(cova*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h12_2=sum(-(2*cova*s1ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_3=sum(-(covb*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h12_4=sum(-(2*covb*s1bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_5=sum((2*s1ahat^2*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h12_6=sum((2*s1ahat^2*s2ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_7=sum((2*s1bhat^2*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h12_8=sum((2*s1bhat^2*s2bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_9=sum(-((W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova^2*lam2 - 2*cova*lam1*s1ahat^2 + lam2*s1ahat^2*s2ahat^2 - W1a^lam1*cova^2*lam2 + 2*W2a^lam2*cova*lam1*s1ahat^2 + cova^2*lam1*lam2*m1ahat - W1a^lam1*lam2*s1ahat^2*s2ahat^2 + lam1*lam2*m1ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_10=sum(-((W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb^2*lam2 - 2*covb*lam1*s1bhat^2 + lam2*s1bhat^2*s2bhat^2 - W1b^lam1*covb^2*lam2 + 2*W2b^lam2*covb*lam1*s1bhat^2 + covb^2*lam1*lam2*m1bhat - W1b^lam1*lam2*s1bhat^2*s2bhat^2 + lam1*lam2*m1bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_11=sum((2*cova*(W1a^lam1*lam1*log(W1a)-W1a^lam1+1)*(W2a^lam2*lam2*log(W2a)-W2a^lam2+1))/(lam1^2*lam2^2*(-2*cova^2+2*s1ahat^2*s2ahat^2)))+sum((2*covb*(W1b^lam1*lam1*log(W1b)-W1b^lam1+1)*(W2b^lam2*lam2*log(W2b)-W2b^lam2+1))/(lam1^2*lam2^2*(-2*covb^2+2*s1bhat^2*s2bhat^2)))
h12_12=sum(((2*((W2a^lam2-1)/lam2^2-(W2a^lam2*log(W2a))/lam2)^2)/s2ahat^2-(2*(m2ahat-(W2a^lam2-1)/lam2)*((2*W2a^lam2-2)/lam2^3-(2*W2a^lam2*log(W2a))/lam2^2+(W2a^lam2*log(W2a)^2)/lam2))/s2ahat^2+(2*cova*(m1ahat-(W1a^lam1-1)/lam1)*((2*W2a^lam2-2)/lam2^3-(2*W2a^lam2*log(W2a))/lam2^2+(W2a^lam2*log(W2a)^2)/lam2))/(s1ahat^2*s2ahat^2))/((2*cova^2)/(s1ahat^2*s2ahat^2)-2))+sum(((2*((W2b^lam2-1)/lam2^2-(W2b^lam2*log(W2b))/lam2)^2)/s2bhat^2-(2*(m2bhat-(W2b^lam2-1)/lam2)*((2*W2b^lam2-2)/lam2^3-(2*W2b^lam2*log(W2b))/lam2^2+(W2b^lam2*log(W2b)^2)/lam2))/s2bhat^2+(2*covb*(m1bhat-(W1b^lam1-1)/lam1)*((2*W2b^lam2-2)/lam2^3-(2*W2b^lam2*log(W2b))/lam2^2+(W2b^lam2*log(W2b)^2)/lam2))/(s1bhat^2*s2bhat^2))/((2*covb^2)/(s1bhat^2*s2bhat^2)-2))
h1r=c(h1_1,h1_2,h1_3,h1_4,h1_5,h1_6,h1_7,h1_8,h1_9,h1_10,h1_11,h1_12)
h2r=c(h2_1,h2_2,h2_3,h2_4,h2_5,h2_6,h2_7,h2_8,h2_9,h2_10,h2_11,h2_12)
h3r=c(h3_1,h3_2,h3_3,h3_4,h3_5,h3_6,h3_7,h3_8,h3_9,h3_10,h3_11,h3_12)
h4r=c(h4_1,h4_2,h4_3,h4_4,h4_5,h4_6,h4_7,h4_8,h4_9,h4_10,h4_11,h4_12)
h5r=c(h5_1,h5_2,h5_3,h5_4,h5_5,h5_6,h5_7,h5_8,h5_9,h5_10,h5_11,h5_12)
h6r=c(h6_1,h6_2,h6_3,h6_4,h6_5,h6_6,h6_7,h6_8,h6_9,h6_10,h6_11,h6_12)
h7r=c(h7_1,h7_2,h7_3,h7_4,h7_5,h7_6,h7_7,h7_8,h7_9,h7_10,h7_11,h7_12)
h8r=c(h8_1,h8_2,h8_3,h8_4,h8_5,h8_6,h8_7,h8_8,h8_9,h8_10,h8_11,h8_12)
h9r=c(h9_1,h9_2,h9_3,h9_4,h9_5,h9_6,h9_7,h9_8,h9_9,h9_10,h9_11,h9_12)
h10r=c(h10_1,h10_2,h10_3,h10_4,h10_5,h10_6,h10_7,h10_8,h10_9,h10_10,h10_11,h10_12)
h11r=c(h11_1,h11_2,h11_3,h11_4,h11_5,h11_6,h11_7,h11_8,h11_9,h11_10,h11_11,h11_12)
h12r=c(h12_1,h12_2,h12_3,h12_4,h12_5,h12_6,h12_7,h12_8,h12_9,h12_10,h12_11,h12_12)
HCF=rbind(h1r,h2r,h3r,h4r,h5r,h6r,h7r,h8r,h9r,h10r,h11r,h12r)
HCF[1,2]=0;HCF[1,6]=0;HCF[1,9]=0;
HCF[2,1]=0;HCF[2,5]=0;
HCF[3,4]=0;HCF[3,8]=0;HCF[3,10]=0;HCF[4,3]=0;
HCF[4,7]=0;
HCF[5,2]=0;HCF[5,6]=0;HCF[5,9]=0;
HCF[6,1]=0;HCF[6,5]=0;
HCF[7,4]=0;HCF[7,8]=0;HCF[7,10]=0;
HCF[8,3]=0;HCF[8,7]=0;
HCF[9,1]=0;HCF[9,5]=0;
HCF[10,3]=0;
HCF[10,7]=0;
HCF[1,1]=na*HCF[1,1];
HCF[1,5]=na*HCF[1,5];
HCF[5,1]=na*HCF[5,1];
HCF[3,3]=nb*HCF[3,3];
HCF[3,7]=nb*HCF[3,7];
HCF[7,3]=nb*HCF[7,3];
HCF[7,7]=nb*HCF[7,7];
HCF[5,5]=na*HCF[5,5];
VV=inv(-HCF)
#############################################################################
#############################################################################
#================ DIFFERENCES OF THE YOUDEN INDICES=======================
m1a=m1ahat;
s1a=s1ahat;
m1b=m1bhat;
s1b=s1bhat;
m2a=m2ahat;
s2a=s2ahat;
m2b=m2bhat;
s2b=s2bhat;
a1=m1b-m1a;
b1=s1b/s1a;
c1=(m1a*(b1^2-1)-a1+b1*sqrt(a1^2+(b1^2-1)*s1a^2*log(b1^2)))/(b1^2-1);
J1=pnorm((m1b-c1)/s1b)+pnorm((c1-m1a)/s1a)-1
J1BCCC=J1
a2=m2b-m2a;
b2=s2b/s2a;
c2=(m2a*(b2^2-1)-a2+b2*sqrt(a2^2+(b2^2-1)*s2a^2*log(b2^2)))/(b2^2-1);
J2=pnorm((m2b-c2)/s2b)+pnorm((c2-m2a)/s2a)-1
J2BCCC=J2
dJ1_dm1a = (2^(1/2)*exp(-(m1a - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1a^2))*((s1b^2/s1a^2 + (s1b*(2*m1a - 2*m1b))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)))/(s1b^2/s1a^2 - 1) - 1))/(2*pi^(1/2)*s1a) - (2^(1/2)*exp(-(m1b - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1b^2))*(s1b^2/s1a^2 + (s1b*(2*m1a - 2*m1b))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))))/(2*pi^(1/2)*s1b*(s1b^2/s1a^2 - 1));
dJ1_ds1a = (2^(1/2)*exp(-(m1b - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1b^2))*(((s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a^2 + (2*m1a*s1b^2)/s1a^3 + (s1b*(2*s1a*(s1b^2/s1a^2 - 1) + (2*s1b^2*log(s1b^2/s1a^2))/s1a - 2*s1a*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1)))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)))/(s1b^2/s1a^2 - 1) - (2*s1b^2*(m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a))/(s1a^3*(s1b^2/s1a^2 - 1)^2)))/(2*pi^(1/2)*s1b) - (exp(-(m1a - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1a^2))*((2^(1/2)*(((s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a^2 + (2*m1a*s1b^2)/s1a^3 + (s1b*(2*s1a*(s1b^2/s1a^2 - 1) + (2*s1b^2*log(s1b^2/s1a^2))/s1a - 2*s1a*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1)))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)))/(s1b^2/s1a^2 - 1) - (2*s1b^2*(m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a))/(s1a^3*(s1b^2/s1a^2 - 1)^2)))/(2*s1a) - (2^(1/2)*(m1a - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1)))/(2*s1a^2)))/pi^(1/2);
dJ1_dm1b = (2^(1/2)*exp(-(m1b - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1b^2))*(((s1b*(2*m1a - 2*m1b))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)) + 1)/(s1b^2/s1a^2 - 1) + 1))/(2*pi^(1/2)*s1b) - (2^(1/2)*exp(-(m1a - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1a^2))*((s1b*(2*m1a - 2*m1b))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)) + 1))/(2*pi^(1/2)*s1a*(s1b^2/s1a^2 - 1));
dJ1_ds1b = (2^(1/2)*exp(-(m1a - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1a^2))*((((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)/s1a + (2*m1a*s1b)/s1a^2 + (s1b*(2*s1b*log(s1b^2/s1a^2) + (2*s1a^2*(s1b^2/s1a^2 - 1))/s1b))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)))/(s1b^2/s1a^2 - 1) - (2*s1b*(m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a))/(s1a^2*(s1b^2/s1a^2 - 1)^2)))/(2*pi^(1/2)*s1a) - (exp(-(m1b - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1b^2))*((2^(1/2)*((((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)/s1a + (2*m1a*s1b)/s1a^2 + (s1b*(2*s1b*log(s1b^2/s1a^2) + (2*s1a^2*(s1b^2/s1a^2 - 1))/s1b))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)))/(s1b^2/s1a^2 - 1) - (2*s1b*(m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a))/(s1a^2*(s1b^2/s1a^2 - 1)^2)))/(2*s1b) + (2^(1/2)*(m1b - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1)))/(2*s1b^2)))/pi^(1/2);
dJ1_dm2a =0;
dJ1_ds2a =0;
dJ1_dm2b =0;
dJ1_ds2b =0;
dJ2_dm1a =0;
dJ2_ds1a =0;
dJ2_dm1b =0;
dJ2_ds1b =0;
dJ2_dm2a = (2^(1/2)*exp(-(m2a - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2a^2))*((s2b^2/s2a^2 + (s2b*(2*m2a - 2*m2b))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)))/(s2b^2/s2a^2 - 1) - 1))/(2*pi^(1/2)*s2a) - (2^(1/2)*exp(-(m2b - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2b^2))*(s2b^2/s2a^2 + (s2b*(2*m2a - 2*m2b))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))))/(2*pi^(1/2)*s2b*(s2b^2/s2a^2 - 1));
dJ2_ds2a = (2^(1/2)*exp(-(m2b - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2b^2))*(((s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a^2 + (2*m2a*s2b^2)/s2a^3 + (s2b*(2*s2a*(s2b^2/s2a^2 - 1) + (2*s2b^2*log(s2b^2/s2a^2))/s2a - 2*s2a*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1)))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)))/(s2b^2/s2a^2 - 1) - (2*s2b^2*(m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a))/(s2a^3*(s2b^2/s2a^2 - 1)^2)))/(2*pi^(1/2)*s2b) - (exp(-(m2a - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2a^2))*((2^(1/2)*(((s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a^2 + (2*m2a*s2b^2)/s2a^3 + (s2b*(2*s2a*(s2b^2/s2a^2 - 1) + (2*s2b^2*log(s2b^2/s2a^2))/s2a - 2*s2a*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1)))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)))/(s2b^2/s2a^2 - 1) - (2*s2b^2*(m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a))/(s2a^3*(s2b^2/s2a^2 - 1)^2)))/(2*s2a) - (2^(1/2)*(m2a - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1)))/(2*s2a^2)))/pi^(1/2);
dJ2_dm2b = (2^(1/2)*exp(-(m2b - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2b^2))*(((s2b*(2*m2a - 2*m2b))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)) + 1)/(s2b^2/s2a^2 - 1) + 1))/(2*pi^(1/2)*s2b) - (2^(1/2)*exp(-(m2a - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2a^2))*((s2b*(2*m2a - 2*m2b))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)) + 1))/(2*pi^(1/2)*s2a*(s2b^2/s2a^2 - 1));
dJ2_ds2b = (2^(1/2)*exp(-(m2a - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2a^2))*((((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)/s2a + (2*m2a*s2b)/s2a^2 + (s2b*(2*s2b*log(s2b^2/s2a^2) + (2*s2a^2*(s2b^2/s2a^2 - 1))/s2b))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)))/(s2b^2/s2a^2 - 1) - (2*s2b*(m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a))/(s2a^2*(s2b^2/s2a^2 - 1)^2)))/(2*pi^(1/2)*s2a) - (exp(-(m2b - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2b^2))*((2^(1/2)*((((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)/s2a + (2*m2a*s2b)/s2a^2 + (s2b*(2*s2b*log(s2b^2/s2a^2) + (2*s2a^2*(s2b^2/s2a^2 - 1))/s2b))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)))/(s2b^2/s2a^2 - 1) - (2*s2b*(m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a))/(s2a^2*(s2b^2/s2a^2 - 1)^2)))/(2*s2b) + (2^(1/2)*(m2b - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1)))/(2*s2b^2)))/pi^(1/2);
VVpars=VV[1:8,1:8];
VJ1=c(dJ1_dm1a, dJ1_ds1a, dJ1_dm1b, dJ1_ds1b, dJ1_dm2a, dJ1_ds2a, dJ1_dm2b, dJ1_ds2b)%*%VVpars%*%t(t(c(dJ1_dm1a, dJ1_ds1a, dJ1_dm1b, dJ1_ds1b, dJ1_dm2a, dJ1_ds2a, dJ1_dm2b, dJ1_ds2b)));
VJ2=c(dJ2_dm1a, dJ2_ds1a, dJ2_dm1b, dJ2_ds1b, dJ2_dm2a, dJ2_ds2a, dJ2_dm2b, dJ2_ds2b)%*%VVpars%*%t(t(c(dJ2_dm1a, dJ2_ds1a, dJ2_dm1b, dJ2_ds1b, dJ2_dm2a, dJ2_ds2a, dJ2_dm2b, dJ2_ds2b)));
COVJ12=c(dJ1_dm1a, dJ1_ds1a, dJ1_dm1b, dJ1_ds1b, dJ1_dm2a, dJ1_ds2a, dJ1_dm2b, dJ1_ds2b)%*%VVpars%*%t(t(c(dJ2_dm1a, dJ2_ds1a, dJ2_dm1b, dJ2_ds1b, dJ2_dm2a, dJ2_ds2a, dJ2_dm2b, dJ2_ds2b)));
kk1=c(VJ1, COVJ12)
kk2=c(COVJ12, VJ2)
V12=rbind(kk1,kk2)
Z=(J2-J1)/(sqrt(VJ1+VJ2-2*COVJ12));
J1original=J1;
J2original=J2;
SE=sqrt(VJ1+VJ2-2*COVJ12);
CIoriginal=c((J2-J1)-pmalpha*SE, (J2-J1)+pmalpha*SE);
pval2tJ=2*pnorm(-abs(Z))
#JT2=log(0.5*(J2+1)/(1-0.5*(J2+1)));
#JT1=log(0.5*(J1+1)/(1-0.5*(J1+1)));
#VJT1=4/(J1^2-1)^2*VJ1;
#VJT2=4/(J2^2-1)^2*VJ2;
#COVJT12=c(-2/(J1^2-1), 0)%*%V12%*%t(t(c(0, -2/(J2^2-1))))
#Zstar=(JT2-JT1)/(sqrt(VJT1+VJT2-2*COVJT12))
#SE=(sqrt(VJT1+VJT2-2*COVJT12))
#CIZstar=c((JT2-JT1)-pmalpha*SE, (JT2-JT1)+pmalpha*SE)
#CIZstar
JT2=qnorm(J2);
JT1=qnorm(J1);
VJT1=(1/(dnorm(qnorm(J1))))^2*VJ1;
VJT2=(1/(dnorm(qnorm(J2))))^2*VJ2;
COVJT12=c((1/(dnorm(qnorm(J1))))*VJ1, 0)%*%V12%*%t(t(c(0, (1/(dnorm(qnorm(J2))))*VJ2)))
Zstar=(JT2-JT1)/(sqrt(VJT1+VJT2-2*COVJT12))
SE=(sqrt(VJT1+VJT2-2*COVJT12))
CIZstar=c((JT2-JT1)-pmalpha*SE, (JT2-JT1)+pmalpha*SE)
CIZstar
pval2t=2*pnorm(-abs(Zstar))
#CIoriginal=c(2*(exp(CIZstar[1])/(1+exp(CIZstar[1])))-1 , 2*(exp(CIZstar[2])/(1+exp(CIZstar[2])))-1)
#====TWO ROC FUNCTIONS: ONE IS THE BOXCOX AND THE OTHER THE REFERENCE LINE================
roc1<-function(t){
na = length(W1alam)
nb = length(W1blam)
s1alam=sqrt( var(W1alam)*(na-1)/na )
s1blam=sqrt( var(W1blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W1alam),
sd=s1alam),
mean=mean(W1blam),
sd=s1blam)
}
roc2<-function(t){
na = length(W2alam)
nb = length(W2blam)
s2alam=sqrt( var(W2alam)*(na-1)/na )
s2blam=sqrt( var(W2blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W2alam),
sd=s2alam),
mean=mean(W2blam),
sd=s2blam)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
Sens1=1-pnorm(c1, mean=mean(W1blam), sd=sqrt(var(W1blam)*(nb-1)/nb))
Spec1= pnorm(c1, mean=mean(W1alam), sd=sqrt(var(W1alam)*(na-1)/na))
Sens2=1-pnorm(c2, mean=mean(W2blam), sd=sqrt(var(W2blam)*(nb-1)/nb))
Spec2= pnorm(c2, mean=mean(W2alam), sd=sqrt(var(W2alam)*(na-1)/na))
#================IF PLOTS ARE REQUESTED PLOT THE ROCS==
if (plots=="on") {
# x11()
plot(linspace(0,1,1000),roc1(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="red")
lines(linspace(0,1,1000),roc2(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="black")
lines(linspace(0,1,1000),linspace(0,1,1000),type="l", lty=2)
points(1-Spec1,Sens1, col = "red")
points(1-Spec2,Sens2, col = "black")
#line for the Youden:
lines(c(1-Spec1,1-Spec1),c(rocuseless(1-Spec1),Sens1),col="blue")
lines(c(1-Spec2,1-Spec2),c(rocuseless(1-Spec2),Sens2),col="green")
legend("bottomright", legend=c(paste("ROC for Marker 1 with J =", formattable(J1original, digits = 4, format = "f"), " "),
paste("ROC for Marker 2 with J =", formattable(J2original, digits = 4, format = "f"), " "),
paste("P-value for the difference:", formattable(pval2t, digits = 4, format = "f"), " ")),
col=c("red", "black", "white"), lty=c(1, 1, NA), pch = c(NA, NA, NA), cex=0.8)
}
# if (plots=="on"){
# points(1-Spec1,Sens1, col = "red")
# points(1-Spec2,Sens2, col = "black")
#
# #line for the Youden:
# lines(c(1-Spec1,1-Spec1),c(rocuseless(1-Spec1),Sens1),col="blue")
# lines(c(1-Spec2,1-Spec2),c(rocuseless(1-Spec2),Sens2),col="green")
#
# }
#======================================================
res <- data.frame(J1 = formattable(J1original, digits = 4, format = "f"),
J2 = formattable(J2original, digits = 4, format = "f"),
diff = formattable(J2original - J1original, digits = 4, format = "f"),
p_value_probit = formattable(pval2t, digits = 4, format = "f"),
p_value = formattable(pval2tJ, digits = 4, format = "f"),
ci_ll = formattable(CIoriginal[1], digits = 4, format = "f"),
ci_ul = formattable(CIoriginal[2], digits = 4, format = "f"))
rownames(res) <- c("Estimates:")
colnames(res) <- c("J 1", "J 2", "Diff", "P-Val (Probit)", "P-Val", "CI (LL)", "CI (UL)")
res <- formattable(as.matrix(res), digits = 4, format = "f")
res
#return(list(AUCmarker1=AUC1original,AUCmarker2=AUC2original, pvalue_difference= pval2t, CI_difference= pnorm(CIdiff), rocbc1=roc1, rocbc2=roc2))
#return(list(resultstable=res,J1=J1original,J2=J2original, pvalue_difference= pval2t, CI_difference= CIoriginal, rocbc1=roc1, rocbc2=roc2))
return(list(resultstable=res,
J1=J1original,
J2=J2original,
pvalue_probit_difference= pval2t,
pvalue_difference= pval2tJ,
CI_difference= CIoriginal,
roc1=roc1, roc2=roc2,
transx1=W1alam,
transy1=W1blam,
transformation.parameter.1 = lam[1],
transx2=W2alam,
transy2=W2blam,
transformation.parameter.2 = lam[2]))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
comparebcSens <-function (marker1, marker2, D, atSpec, alpha, plots){
if ((length(marker1) != length(D)) | (length(marker2) != length(D))) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (alpha <= 0 | alpha >= 1) {
stop("ERROR: The level of significance, alpha, should be set between 0 and 1. A common choice is 0.05.")
} else if (sum(is.na(marker1)) > 0 | sum(is.na(marker2)) > 0 | sum(is.na(D)) > 0 | sum(is.na(atSpec)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if ((!is.numeric(atSpec)) | (length(atSpec) != 1)) {
stop("ERROR: 'atSpec' must be a single numeric value.")
} else if ((sum(marker1 < 0) > 0) | (sum(marker2 < 2)) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
erf <- function (x) 2 * pnorm(x * sqrt(2)) - 1
erfc <- function (x) 2 * pnorm(x * sqrt(2), lower.tail = FALSE)
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower.tail = FALSE)/sqrt(2)
pmalpha=qnorm(1-alpha/2);
if (plots!="on"){plots="off"}
W1a=marker1[D==0]
W1b=marker1[D==1]
W2a=marker2[D==0]
W2b=marker2[D==1]
Wa=cbind(W1a,W2a);
Wb=cbind(W1b,W2b);
na=length(W1a)
nb=length(W1b)
likbox2D<-function(W1a,W1b,W2a,W2b,lam){
na=length(W1a);
nb=length(W1b);
out=c();
#for (i in 1:length(h)){
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
cova= cov(Walam)*(na-1)/na;
covb= cov(Wblam)*(nb-1)/nb;
mualam=c(mean(W1alam), mean(W2alam))
mublam=c(mean(W1blam), mean(W2blam))
out= -sum(log(dmvnorm(Walam,mualam,cova)))-sum(log(dmvnorm(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b));
return(out)
}
lam=c(2,2)
likbox2D(W1a,W1b,W2a,W2b,lam)
logL<-function(h){
likbox2D(W1a,W1b,W2a,W2b,h)
}
# init=c(0.8,0.8)
# lam=fminsearch(logL,init)
# lam=c(lam$optbase$xopt)
lam<- optim(c(1,1),logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
lam
out <- optim(c(1,1), logL, method = "Nelder-Mead")
lam = out$par
###################################
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
m1ahat=mean(W1alam)
m1bhat=mean(W1blam)
m2ahat=mean(W2alam)
m2bhat=mean(W2blam)
s1ahat=sqrt( var(W1alam)*(na-1)/na )
s1bhat=sqrt( var(W1blam)*(nb-1)/nb )
s2ahat=sqrt( var(W2alam)*(na-1)/na )
s2bhat=sqrt( var(W2blam)*(nb-1)/nb )
cova= cov(Walam)*(na-1)/na;cova=cova[1,2];
covb= cov(Wblam)*(nb-1)/nb;covb=covb[1,2];
lam1=lam[1]
lam2=lam[2]
##################################
h1_1=sum(-2/(2*s1ahat^2 - (2*cova^2)/s2ahat^2))
h1_2=sum(-(2*s1ahat*s2ahat^2*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_3=0
h1_4=0;
h1_5=sum(cova/(- cova^2 + s1ahat^2*s2ahat^2));
h1_6=sum((s2ahat*(2*cova^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat) - 2*cova*s1ahat^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_7=0
h1_8=0
h1_9=sum((cova^2*(lam2*m2ahat - W2a^lam2 + 1) + s2ahat^2*((lam2*m2ahat - W2a^lam2 + 1)*s1ahat^2 - 2*cova*lam2*m1ahat))/(lam2*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s2ahat^2*(2*lam2 - 2*W1a^lam1*lam2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_10=0
h1_11=sum((2*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h1_12=sum(-(cova*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h2_1=sum(-(2*s1ahat*s2ahat^2*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_2=sum(- (3*s1ahat^2*s2ahat^6 + cova^4*(lam1^2*m2ahat^2 + lam1^2*s2ahat^2) - cova^3*(2*lam1*m2ahat*s2ahat^2 - 2*W1a^lam1*lam1*m2ahat*s2ahat^2 + 2*lam1^2*m1ahat*m2ahat*s2ahat^2) - cova*(6*lam1*m2ahat*s1ahat^2*s2ahat^4 - 6*W1a^lam1*lam1*m2ahat*s1ahat^2*s2ahat^4 + 6*lam1^2*m1ahat*m2ahat*s1ahat^2*s2ahat^4) + cova^2*(W1a^(2*lam1)*s2ahat^4 - 2*W1a^lam1*s2ahat^4 + s2ahat^4 + 2*lam1*m1ahat*s2ahat^4 + lam1^2*m1ahat^2*s2ahat^4 - 2*W1a^lam1*lam1*m1ahat*s2ahat^4 + 3*lam1^2*m2ahat^2*s1ahat^2*s2ahat^2) - 6*W1a^lam1*s1ahat^2*s2ahat^6 + 3*W1a^(2*lam1)*s1ahat^2*s2ahat^6 - lam1^2*s1ahat^4*s2ahat^6 + 3*lam1^2*m1ahat^2*s1ahat^2*s2ahat^6 + 6*lam1*m1ahat*s1ahat^2*s2ahat^6 - 6*W1a^lam1*lam1*m1ahat*s1ahat^2*s2ahat^6)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^4*(W2a^(2*lam2)*lam1^2 - 2*W2a^lam2*lam1^2 + lam1^2) + cova^2*(3*lam1^2*s1ahat^2*s2ahat^2 + 3*W2a^(2*lam2)*lam1^2*s1ahat^2*s2ahat^2 - 6*W2a^lam2*lam1^2*s1ahat^2*s2ahat^2) - lam2*((2*W2a^lam2*lam1^2*m2ahat - 2*lam1^2*m2ahat)*cova^4 + (2*lam1*s2ahat^2 + 2*lam1^2*m1ahat*s2ahat^2 - 2*W1a^lam1*lam1*s2ahat^2 - 2*W2a^lam2*lam1*s2ahat^2 - 2*W2a^lam2*lam1^2*m1ahat*s2ahat^2 + 2*W1a^lam1*W2a^lam2*lam1*s2ahat^2)*cova^3 + (6*W2a^lam2*lam1^2*m2ahat*s1ahat^2*s2ahat^2 - 6*lam1^2*m2ahat*s1ahat^2*s2ahat^2)*cova^2 + (6*lam1*s1ahat^2*s2ahat^4 + 6*lam1^2*m1ahat*s1ahat^2*s2ahat^4 - 6*W1a^lam1*lam1*s1ahat^2*s2ahat^4 - 6*W2a^lam2*lam1*s1ahat^2*s2ahat^4 - 6*W2a^lam2*lam1^2*m1ahat*s1ahat^2*s2ahat^4 + 6*W1a^lam1*W2a^lam2*lam1*s1ahat^2*s2ahat^4)*cova))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_3=sum(0)
h2_4=sum(0)
h2_5=sum((s1ahat*(2*cova^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat) - 2*cova*s2ahat^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_6=sum((lam2*((2*cova*s2ahat^3*(2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat) - 2*cova*s2ahat*(4*cova*lam1^2*m2ahat - 4*W2a^lam2*cova*lam1^2*m2ahat))*s1ahat^3 + 2*cova*s2ahat*(2*cova^2*lam1 + 2*cova^2*lam1^2*m1ahat - 2*W1a^lam1*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1^2*m1ahat + 2*W1a^lam1*W2a^lam2*cova^2*lam1)*s1ahat) - 2*cova*s1ahat^3*s2ahat*(2*cova*lam1^2 + 2*W2a^(2*lam2)*cova*lam1^2 - 4*W2a^lam2*cova*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - ((4*cova^2*lam1^2*m2ahat^2*s2ahat - 2*cova*s2ahat^3*(2*lam1*m2ahat + cova*lam1^2 + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat))*s1ahat^3 + (2*cova*(2*cova + 2*W1a^(2*lam1)*cova - 4*W1a^lam1*cova + 2*cova*lam1^2*m1ahat^2 + 4*cova*lam1*m1ahat - 4*W1a^lam1*cova*lam1*m1ahat)*s2ahat^3 + 2*cova*(cova^3*lam1^2 - 2*cova^2*lam1*m2ahat + 2*W1a^lam1*cova^2*lam1*m2ahat - 2*cova^2*lam1^2*m1ahat*m2ahat)*s2ahat)*s1ahat)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_7=sum(0)
h2_8=sum(0)
h2_9=sum(- (cova*(s2ahat^4*(- 4*lam2^2*m1ahat^2*s1ahat + 2*lam2^2*s1ahat^3) - s1ahat^3*s2ahat^2*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2)) - cova^3*(s1ahat*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2) + 2*lam2^2*s1ahat*s2ahat^2) + s1ahat^3*s2ahat^4*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + cova^2*s1ahat*s2ahat^2*(6*lam2*m1ahat + 6*lam2^2*m1ahat*m2ahat - 6*W2a^lam2*lam2*m1ahat))/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2ahat - 6*W1a^lam1*lam2 - 6*W2a^lam2*lam2 + 6*W1a^lam1*W2a^lam2*lam2 - 6*W1a^lam1*lam2^2*m2ahat)*cova^2*s1ahat*s2ahat^2 + (8*W1a^lam1*lam2^2*m1ahat - 8*lam2^2*m1ahat)*cova*s1ahat*s2ahat^4 + (2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat)*s1ahat^3*s2ahat^4) - cova*s1ahat*s2ahat^4*(4*W1a^(2*lam1)*lam2^2 - 8*W1a^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_10=sum(0)
h2_11=sum((2*s1ahat*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_12=sum(-(2*cova*s1ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h3_1=sum(0)
h3_2=sum(0)
h3_3=sum(-2/(2*s1bhat^2 - (2*covb^2)/s2bhat^2))
h3_4=sum(-(2*s1bhat*s2bhat^2*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_5=sum(0)
h3_6=sum(0)
h3_7=sum(covb/(- covb^2 + s1bhat^2*s2bhat^2))
h3_8=sum((s2bhat*(2*covb^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat) - 2*covb*s1bhat^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_9=sum(0)
h3_10=sum((covb^2*(lam2*m2bhat - W2b^lam2 + 1) + s2bhat^2*((lam2*m2bhat - W2b^lam2 + 1)*s1bhat^2 - 2*covb*lam2*m1bhat))/(lam2*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s2bhat^2*(2*lam2 - 2*W1b^lam1*lam2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_11=sum((2*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h3_12=sum(-(covb*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h4_1=sum(0)
h4_2=sum(0)
h4_3=sum(-(2*s1bhat*s2bhat^2*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_4=sum(- (3*s1bhat^2*s2bhat^6 + covb^4*(lam1^2*m2bhat^2 + lam1^2*s2bhat^2) - covb^3*(2*lam1*m2bhat*s2bhat^2 - 2*W1b^lam1*lam1*m2bhat*s2bhat^2 + 2*lam1^2*m1bhat*m2bhat*s2bhat^2) - covb*(6*lam1*m2bhat*s1bhat^2*s2bhat^4 - 6*W1b^lam1*lam1*m2bhat*s1bhat^2*s2bhat^4 + 6*lam1^2*m1bhat*m2bhat*s1bhat^2*s2bhat^4) + covb^2*(W1b^(2*lam1)*s2bhat^4 - 2*W1b^lam1*s2bhat^4 + s2bhat^4 + 2*lam1*m1bhat*s2bhat^4 + lam1^2*m1bhat^2*s2bhat^4 - 2*W1b^lam1*lam1*m1bhat*s2bhat^4 + 3*lam1^2*m2bhat^2*s1bhat^2*s2bhat^2) - 6*W1b^lam1*s1bhat^2*s2bhat^6 + 3*W1b^(2*lam1)*s1bhat^2*s2bhat^6 - lam1^2*s1bhat^4*s2bhat^6 + 3*lam1^2*m1bhat^2*s1bhat^2*s2bhat^6 + 6*lam1*m1bhat*s1bhat^2*s2bhat^6 - 6*W1b^lam1*lam1*m1bhat*s1bhat^2*s2bhat^6)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^4*(W2b^(2*lam2)*lam1^2 - 2*W2b^lam2*lam1^2 + lam1^2) + covb^2*(3*lam1^2*s1bhat^2*s2bhat^2 + 3*W2b^(2*lam2)*lam1^2*s1bhat^2*s2bhat^2 - 6*W2b^lam2*lam1^2*s1bhat^2*s2bhat^2) - lam2*((2*W2b^lam2*lam1^2*m2bhat - 2*lam1^2*m2bhat)*covb^4 + (2*lam1*s2bhat^2 + 2*lam1^2*m1bhat*s2bhat^2 - 2*W1b^lam1*lam1*s2bhat^2 - 2*W2b^lam2*lam1*s2bhat^2 - 2*W2b^lam2*lam1^2*m1bhat*s2bhat^2 + 2*W1b^lam1*W2b^lam2*lam1*s2bhat^2)*covb^3 + (6*W2b^lam2*lam1^2*m2bhat*s1bhat^2*s2bhat^2 - 6*lam1^2*m2bhat*s1bhat^2*s2bhat^2)*covb^2 + (6*lam1*s1bhat^2*s2bhat^4 + 6*lam1^2*m1bhat*s1bhat^2*s2bhat^4 - 6*W1b^lam1*lam1*s1bhat^2*s2bhat^4 - 6*W2b^lam2*lam1*s1bhat^2*s2bhat^4 - 6*W2b^lam2*lam1^2*m1bhat*s1bhat^2*s2bhat^4 + 6*W1b^lam1*W2b^lam2*lam1*s1bhat^2*s2bhat^4)*covb))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_5=sum(0)
h4_6=sum(0)
h4_7=sum((s1bhat*(2*covb^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat) - 2*covb*s2bhat^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_8=sum((lam2*((2*covb*s2bhat^3*(2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat) - 2*covb*s2bhat*(4*covb*lam1^2*m2bhat - 4*W2b^lam2*covb*lam1^2*m2bhat))*s1bhat^3 + 2*covb*s2bhat*(2*covb^2*lam1 + 2*covb^2*lam1^2*m1bhat - 2*W1b^lam1*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1^2*m1bhat + 2*W1b^lam1*W2b^lam2*covb^2*lam1)*s1bhat) - 2*covb*s1bhat^3*s2bhat*(2*covb*lam1^2 + 2*W2b^(2*lam2)*covb*lam1^2 - 4*W2b^lam2*covb*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - ((4*covb^2*lam1^2*m2bhat^2*s2bhat - 2*covb*s2bhat^3*(2*lam1*m2bhat + covb*lam1^2 + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat))*s1bhat^3 + (2*covb*(2*covb + 2*W1b^(2*lam1)*covb - 4*W1b^lam1*covb + 2*covb*lam1^2*m1bhat^2 + 4*covb*lam1*m1bhat - 4*W1b^lam1*covb*lam1*m1bhat)*s2bhat^3 + 2*covb*(covb^3*lam1^2 - 2*covb^2*lam1*m2bhat + 2*W1b^lam1*covb^2*lam1*m2bhat - 2*covb^2*lam1^2*m1bhat*m2bhat)*s2bhat)*s1bhat)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_9=sum(0)
h4_10=sum(- (covb*(s2bhat^4*(- 4*lam2^2*m1bhat^2*s1bhat + 2*lam2^2*s1bhat^3) - s1bhat^3*s2bhat^2*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2)) - covb^3*(s1bhat*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2) + 2*lam2^2*s1bhat*s2bhat^2) + s1bhat^3*s2bhat^4*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + covb^2*s1bhat*s2bhat^2*(6*lam2*m1bhat + 6*lam2^2*m1bhat*m2bhat - 6*W2b^lam2*lam2*m1bhat))/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2bhat - 6*W1b^lam1*lam2 - 6*W2b^lam2*lam2 + 6*W1b^lam1*W2b^lam2*lam2 - 6*W1b^lam1*lam2^2*m2bhat)*covb^2*s1bhat*s2bhat^2 + (8*W1b^lam1*lam2^2*m1bhat - 8*lam2^2*m1bhat)*covb*s1bhat*s2bhat^4 + (2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat)*s1bhat^3*s2bhat^4) - covb*s1bhat*s2bhat^4*(4*W1b^(2*lam1)*lam2^2 - 8*W1b^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_11=sum((2*s1bhat*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_12=sum(-(2*covb*s1bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h5_1=sum(cova/(- cova^2 + s1ahat^2*s2ahat^2))
h5_2=sum((s1ahat*(2*cova^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat) - 2*cova*s2ahat^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_3=0
h5_4=0
h5_5=sum(-2/(2*s2ahat^2 - (2*cova^2)/s1ahat^2))
h5_6=sum(-(2*s1ahat^2*s2ahat*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_7=0
h5_8=0
h5_9=sum((cova^2*(lam1*m1ahat - W1a^lam1 + 1) + s1ahat^2*((lam1*m1ahat - W1a^lam1 + 1)*s2ahat^2 - 2*cova*lam1*m2ahat))/(lam1*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s1ahat^2*(2*lam1 - 2*W2a^lam2*lam1))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_10=0
h5_11=sum(-(cova*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h5_12=sum((2*s1ahat^2*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h6_1=sum((s2ahat*(2*cova^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat) - 2*cova*s1ahat^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_2=sum((lam2*((2*cova*s2ahat^3*(2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat) - 2*cova*s2ahat*(4*cova*lam1^2*m2ahat - 4*W2a^lam2*cova*lam1^2*m2ahat))*s1ahat^3 + 2*cova*s2ahat*(2*cova^2*lam1 + 2*cova^2*lam1^2*m1ahat - 2*W1a^lam1*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1^2*m1ahat + 2*W1a^lam1*W2a^lam2*cova^2*lam1)*s1ahat) - 2*cova*s1ahat^3*s2ahat*(2*cova*lam1^2 + 2*W2a^(2*lam2)*cova*lam1^2 - 4*W2a^lam2*cova*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - ((4*cova^2*lam1^2*m2ahat^2*s2ahat - 2*cova*s2ahat^3*(2*lam1*m2ahat + cova*lam1^2 + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat))*s1ahat^3 + (2*cova*(2*cova + 2*W1a^(2*lam1)*cova - 4*W1a^lam1*cova + 2*cova*lam1^2*m1ahat^2 + 4*cova*lam1*m1ahat - 4*W1a^lam1*cova*lam1*m1ahat)*s2ahat^3 + 2*cova*(cova^3*lam1^2 - 2*cova^2*lam1*m2ahat + 2*W1a^lam1*cova^2*lam1*m2ahat - 2*cova^2*lam1^2*m1ahat*m2ahat)*s2ahat)*s1ahat)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_3=0
h6_4=0
h6_5=sum(-(2*s1ahat^2*s2ahat*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_6=sum(- (3*s1ahat^6*s2ahat^2 + cova^4*(lam2^2*m1ahat^2 + lam2^2*s1ahat^2) - cova^3*(2*lam2*m1ahat*s1ahat^2 - 2*W2a^lam2*lam2*m1ahat*s1ahat^2 + 2*lam2^2*m1ahat*m2ahat*s1ahat^2) - cova*(6*lam2*m1ahat*s1ahat^4*s2ahat^2 - 6*W2a^lam2*lam2*m1ahat*s1ahat^4*s2ahat^2 + 6*lam2^2*m1ahat*m2ahat*s1ahat^4*s2ahat^2) + cova^2*(W2a^(2*lam2)*s1ahat^4 - 2*W2a^lam2*s1ahat^4 + s1ahat^4 + 2*lam2*m2ahat*s1ahat^4 + lam2^2*m2ahat^2*s1ahat^4 - 2*W2a^lam2*lam2*m2ahat*s1ahat^4 + 3*lam2^2*m1ahat^2*s1ahat^2*s2ahat^2) - 6*W2a^lam2*s1ahat^6*s2ahat^2 + 3*W2a^(2*lam2)*s1ahat^6*s2ahat^2 - lam2^2*s1ahat^6*s2ahat^4 + 3*lam2^2*m2ahat^2*s1ahat^6*s2ahat^2 + 6*lam2*m2ahat*s1ahat^6*s2ahat^2 - 6*W2a^lam2*lam2*m2ahat*s1ahat^6*s2ahat^2)/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^4*(W1a^(2*lam1)*lam2^2 - 2*W1a^lam1*lam2^2 + lam2^2) + cova^2*(3*lam2^2*s1ahat^2*s2ahat^2 + 3*W1a^(2*lam1)*lam2^2*s1ahat^2*s2ahat^2 - 6*W1a^lam1*lam2^2*s1ahat^2*s2ahat^2) - lam1*((2*W1a^lam1*lam2^2*m1ahat - 2*lam2^2*m1ahat)*cova^4 + (2*lam2*s1ahat^2 + 2*lam2^2*m2ahat*s1ahat^2 - 2*W1a^lam1*lam2*s1ahat^2 - 2*W2a^lam2*lam2*s1ahat^2 - 2*W1a^lam1*lam2^2*m2ahat*s1ahat^2 + 2*W1a^lam1*W2a^lam2*lam2*s1ahat^2)*cova^3 + (6*W1a^lam1*lam2^2*m1ahat*s1ahat^2*s2ahat^2 - 6*lam2^2*m1ahat*s1ahat^2*s2ahat^2)*cova^2 + (6*lam2*s1ahat^4*s2ahat^2 + 6*lam2^2*m2ahat*s1ahat^4*s2ahat^2 - 6*W1a^lam1*lam2*s1ahat^4*s2ahat^2 - 6*W2a^lam2*lam2*s1ahat^4*s2ahat^2 - 6*W1a^lam1*lam2^2*m2ahat*s1ahat^4*s2ahat^2 + 6*W1a^lam1*W2a^lam2*lam2*s1ahat^4*s2ahat^2)*cova))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_7=0
h6_8=0
h6_9=sum(- (cova*(s1ahat^4*(- 4*lam1^2*m2ahat^2*s2ahat + 2*lam1^2*s2ahat^3) - s1ahat^2*s2ahat^3*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2)) - cova^3*(s2ahat*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2) + 2*lam1^2*s1ahat^2*s2ahat) + s1ahat^4*s2ahat^3*(2*lam1*m2ahat + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat) + cova^2*s1ahat^2*s2ahat*(6*lam1*m2ahat + 6*lam1^2*m1ahat*m2ahat - 6*W1a^lam1*lam1*m2ahat))/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1ahat - 6*W1a^lam1*lam1 - 6*W2a^lam2*lam1 + 6*W1a^lam1*W2a^lam2*lam1 - 6*W2a^lam2*lam1^2*m1ahat)*cova^2*s1ahat^2*s2ahat + (8*W2a^lam2*lam1^2*m2ahat - 8*lam1^2*m2ahat)*cova*s1ahat^4*s2ahat + (2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat)*s1ahat^4*s2ahat^3) - cova*s1ahat^4*s2ahat*(4*W2a^(2*lam2)*lam1^2 - 8*W2a^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_10=0
h6_11=sum(-(2*cova*s2ahat*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_12=sum((2*s1ahat^2*s2ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h7_1=0
h7_2=0
h7_3=sum(covb/(- covb^2 + s1bhat^2*s2bhat^2))
h7_4=sum((s1bhat*(2*covb^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat) - 2*covb*s2bhat^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_5=0
h7_6=0
h7_7=sum(-2/(2*s2bhat^2 - (2*covb^2)/s1bhat^2))
h7_8=sum(-(2*s1bhat^2*s2bhat*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_9=0
h7_10=sum((covb^2*(lam1*m1bhat - W1b^lam1 + 1) + s1bhat^2*((lam1*m1bhat - W1b^lam1 + 1)*s2bhat^2 - 2*covb*lam1*m2bhat))/(lam1*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s1bhat^2*(2*lam1 - 2*W2b^lam2*lam1))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_11=sum(-(covb*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h7_12=sum((2*s1bhat^2*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h8_1=0
h8_2=0
h8_3=sum((s2bhat*(2*covb^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat) - 2*covb*s1bhat^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_4=sum((lam2*((2*covb*s2bhat^3*(2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat) - 2*covb*s2bhat*(4*covb*lam1^2*m2bhat - 4*W2b^lam2*covb*lam1^2*m2bhat))*s1bhat^3 + 2*covb*s2bhat*(2*covb^2*lam1 + 2*covb^2*lam1^2*m1bhat - 2*W1b^lam1*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1^2*m1bhat + 2*W1b^lam1*W2b^lam2*covb^2*lam1)*s1bhat) - 2*covb*s1bhat^3*s2bhat*(2*covb*lam1^2 + 2*W2b^(2*lam2)*covb*lam1^2 - 4*W2b^lam2*covb*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - ((4*covb^2*lam1^2*m2bhat^2*s2bhat - 2*covb*s2bhat^3*(2*lam1*m2bhat + covb*lam1^2 + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat))*s1bhat^3 + (2*covb*(2*covb + 2*W1b^(2*lam1)*covb - 4*W1b^lam1*covb + 2*covb*lam1^2*m1bhat^2 + 4*covb*lam1*m1bhat - 4*W1b^lam1*covb*lam1*m1bhat)*s2bhat^3 + 2*covb*(covb^3*lam1^2 - 2*covb^2*lam1*m2bhat + 2*W1b^lam1*covb^2*lam1*m2bhat - 2*covb^2*lam1^2*m1bhat*m2bhat)*s2bhat)*s1bhat)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_5=0
h8_6=0
h8_7=sum(-(2*s1bhat^2*s2bhat*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_8=sum(- (3*s1bhat^6*s2bhat^2 + covb^4*(lam2^2*m1bhat^2 + lam2^2*s1bhat^2) - covb^3*(2*lam2*m1bhat*s1bhat^2 - 2*W2b^lam2*lam2*m1bhat*s1bhat^2 + 2*lam2^2*m1bhat*m2bhat*s1bhat^2) - covb*(6*lam2*m1bhat*s1bhat^4*s2bhat^2 - 6*W2b^lam2*lam2*m1bhat*s1bhat^4*s2bhat^2 + 6*lam2^2*m1bhat*m2bhat*s1bhat^4*s2bhat^2) + covb^2*(W2b^(2*lam2)*s1bhat^4 - 2*W2b^lam2*s1bhat^4 + s1bhat^4 + 2*lam2*m2bhat*s1bhat^4 + lam2^2*m2bhat^2*s1bhat^4 - 2*W2b^lam2*lam2*m2bhat*s1bhat^4 + 3*lam2^2*m1bhat^2*s1bhat^2*s2bhat^2) - 6*W2b^lam2*s1bhat^6*s2bhat^2 + 3*W2b^(2*lam2)*s1bhat^6*s2bhat^2 - lam2^2*s1bhat^6*s2bhat^4 + 3*lam2^2*m2bhat^2*s1bhat^6*s2bhat^2 + 6*lam2*m2bhat*s1bhat^6*s2bhat^2 - 6*W2b^lam2*lam2*m2bhat*s1bhat^6*s2bhat^2)/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^4*(W1b^(2*lam1)*lam2^2 - 2*W1b^lam1*lam2^2 + lam2^2) + covb^2*(3*lam2^2*s1bhat^2*s2bhat^2 + 3*W1b^(2*lam1)*lam2^2*s1bhat^2*s2bhat^2 - 6*W1b^lam1*lam2^2*s1bhat^2*s2bhat^2) - lam1*((2*W1b^lam1*lam2^2*m1bhat - 2*lam2^2*m1bhat)*covb^4 + (2*lam2*s1bhat^2 + 2*lam2^2*m2bhat*s1bhat^2 - 2*W1b^lam1*lam2*s1bhat^2 - 2*W2b^lam2*lam2*s1bhat^2 - 2*W1b^lam1*lam2^2*m2bhat*s1bhat^2 + 2*W1b^lam1*W2b^lam2*lam2*s1bhat^2)*covb^3 + (6*W1b^lam1*lam2^2*m1bhat*s1bhat^2*s2bhat^2 - 6*lam2^2*m1bhat*s1bhat^2*s2bhat^2)*covb^2 + (6*lam2*s1bhat^4*s2bhat^2 + 6*lam2^2*m2bhat*s1bhat^4*s2bhat^2 - 6*W1b^lam1*lam2*s1bhat^4*s2bhat^2 - 6*W2b^lam2*lam2*s1bhat^4*s2bhat^2 - 6*W1b^lam1*lam2^2*m2bhat*s1bhat^4*s2bhat^2 + 6*W1b^lam1*W2b^lam2*lam2*s1bhat^4*s2bhat^2)*covb))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_9=0
h8_10=sum(- (covb*(s1bhat^4*(- 4*lam1^2*m2bhat^2*s2bhat + 2*lam1^2*s2bhat^3) - s1bhat^2*s2bhat^3*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2)) - covb^3*(s2bhat*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2) + 2*lam1^2*s1bhat^2*s2bhat) + s1bhat^4*s2bhat^3*(2*lam1*m2bhat + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat) + covb^2*s1bhat^2*s2bhat*(6*lam1*m2bhat + 6*lam1^2*m1bhat*m2bhat - 6*W1b^lam1*lam1*m2bhat))/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1bhat - 6*W1b^lam1*lam1 - 6*W2b^lam2*lam1 + 6*W1b^lam1*W2b^lam2*lam1 - 6*W2b^lam2*lam1^2*m1bhat)*covb^2*s1bhat^2*s2bhat + (8*W2b^lam2*lam1^2*m2bhat - 8*lam1^2*m2bhat)*covb*s1bhat^4*s2bhat + (2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat)*s1bhat^4*s2bhat^3) - covb*s1bhat^4*s2bhat*(4*W2b^(2*lam2)*lam1^2 - 8*W2b^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_11=sum(-(2*covb*s2bhat*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_12=sum((2*s1bhat^2*s2bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h9_1=sum((cova^2*(lam2*m2ahat - W2a^lam2 + 1) + s2ahat^2*((lam2*m2ahat - W2a^lam2 + 1)*s1ahat^2 - 2*cova*lam2*m1ahat))/(lam2*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s2ahat^2*(2*lam2 - 2*W1a^lam1*lam2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_2=sum(- (cova*(s2ahat^4*(- 4*lam2^2*m1ahat^2*s1ahat + 2*lam2^2*s1ahat^3) - s1ahat^3*s2ahat^2*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2)) - cova^3*(s1ahat*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2) + 2*lam2^2*s1ahat*s2ahat^2) + s1ahat^3*s2ahat^4*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + cova^2*s1ahat*s2ahat^2*(6*lam2*m1ahat + 6*lam2^2*m1ahat*m2ahat - 6*W2a^lam2*lam2*m1ahat))/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2ahat - 6*W1a^lam1*lam2 - 6*W2a^lam2*lam2 + 6*W1a^lam1*W2a^lam2*lam2 - 6*W1a^lam1*lam2^2*m2ahat)*cova^2*s1ahat*s2ahat^2 + (8*W1a^lam1*lam2^2*m1ahat - 8*lam2^2*m1ahat)*cova*s1ahat*s2ahat^4 + (2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat)*s1ahat^3*s2ahat^4) - cova*s1ahat*s2ahat^4*(4*W1a^(2*lam1)*lam2^2 - 8*W1a^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_3=0
h9_4=0
h9_5=sum((cova^2*(lam1*m1ahat - W1a^lam1 + 1) + s1ahat^2*((lam1*m1ahat - W1a^lam1 + 1)*s2ahat^2 - 2*cova*lam1*m2ahat))/(lam1*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s1ahat^2*(2*lam1 - 2*W2a^lam2*lam1))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_6=sum(- (cova*(s1ahat^4*(- 4*lam1^2*m2ahat^2*s2ahat + 2*lam1^2*s2ahat^3) - s1ahat^2*s2ahat^3*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2)) - cova^3*(s2ahat*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2) + 2*lam1^2*s1ahat^2*s2ahat) + s1ahat^4*s2ahat^3*(2*lam1*m2ahat + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat) + cova^2*s1ahat^2*s2ahat*(6*lam1*m2ahat + 6*lam1^2*m1ahat*m2ahat - 6*W1a^lam1*lam1*m2ahat))/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1ahat - 6*W1a^lam1*lam1 - 6*W2a^lam2*lam1 + 6*W1a^lam1*W2a^lam2*lam1 - 6*W2a^lam2*lam1^2*m1ahat)*cova^2*s1ahat^2*s2ahat + (8*W2a^lam2*lam1^2*m2ahat - 8*lam1^2*m2ahat)*cova*s1ahat^4*s2ahat + (2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat)*s1ahat^4*s2ahat^3) - cova*s1ahat^4*s2ahat*(4*W2a^(2*lam2)*lam1^2 - 8*W2a^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_7=0
h9_8=0
h9_9=sum(- (s1ahat^2*(cova^2*(6*lam2*m2ahat - 6*W2a^lam2 + 3*W2a^(2*lam2) + 3*lam2^2*m2ahat^2 - 6*W2a^lam2*lam2*m2ahat + 3) - cova*(6*lam2*m1ahat*s2ahat^2 - 6*W2a^lam2*lam2*m1ahat*s2ahat^2 + 6*lam2^2*m1ahat*m2ahat*s2ahat^2) + lam2^2*m1ahat^2*s2ahat^4) - cova^3*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + s1ahat^4*(W2a^(2*lam2)*s2ahat^2 - 2*W2a^lam2*s2ahat^2 + s2ahat^2 - lam2^2*s2ahat^4 + 2*lam2*m2ahat*s2ahat^2 + lam2^2*m2ahat^2*s2ahat^2 - 2*W2a^lam2*lam2*m2ahat*s2ahat^2) + cova^4*lam2^2 + 3*cova^2*lam2^2*m1ahat^2*s2ahat^2)/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^2*(3*lam2^2*s2ahat^2 - 6*W1a^lam1*lam2^2*s2ahat^2 + 3*W1a^(2*lam1)*lam2^2*s2ahat^2) + s1ahat^2*(lam2^2*s2ahat^4 - 2*W1a^lam1*lam2^2*s2ahat^4 + W1a^(2*lam1)*lam2^2*s2ahat^4) - lam1*(cova^3*(2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat) - cova^2*(6*lam2^2*m1ahat*s2ahat^2 - 6*W1a^lam1*lam2^2*m1ahat*s2ahat^2) + s1ahat^2*(cova*(6*lam2*s2ahat^2 + 6*lam2^2*m2ahat*s2ahat^2 - 6*W1a^lam1*lam2*s2ahat^2 - 6*W2a^lam2*lam2*s2ahat^2 - 6*W1a^lam1*lam2^2*m2ahat*s2ahat^2 + 6*W1a^lam1*W2a^lam2*lam2*s2ahat^2) - 2*lam2^2*m1ahat*s2ahat^4 + 2*W1a^lam1*lam2^2*m1ahat*s2ahat^4)))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_10=0
h9_11=sum(-((W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova^2*lam1 - 2*cova*lam2*s2ahat^2 + lam1*s1ahat^2*s2ahat^2 - W2a^lam2*cova^2*lam1 + 2*W1a^lam1*cova*lam2*s2ahat^2 + cova^2*lam1*lam2*m2ahat - W2a^lam2*lam1*s1ahat^2*s2ahat^2 + lam1*lam2*m2ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_12=sum(-((W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova^2*lam2 - 2*cova*lam1*s1ahat^2 + lam2*s1ahat^2*s2ahat^2 - W1a^lam1*cova^2*lam2 + 2*W2a^lam2*cova*lam1*s1ahat^2 + cova^2*lam1*lam2*m1ahat - W1a^lam1*lam2*s1ahat^2*s2ahat^2 + lam1*lam2*m1ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h10_1=0
h10_2=0
h10_3=sum((covb^2*(lam2*m2bhat - W2b^lam2 + 1) + s2bhat^2*((lam2*m2bhat - W2b^lam2 + 1)*s1bhat^2 - 2*covb*lam2*m1bhat))/(lam2*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s2bhat^2*(2*lam2 - 2*W1b^lam1*lam2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_4=sum(- (covb*(s2bhat^4*(- 4*lam2^2*m1bhat^2*s1bhat + 2*lam2^2*s1bhat^3) - s1bhat^3*s2bhat^2*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2)) - covb^3*(s1bhat*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2) + 2*lam2^2*s1bhat*s2bhat^2) + s1bhat^3*s2bhat^4*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + covb^2*s1bhat*s2bhat^2*(6*lam2*m1bhat + 6*lam2^2*m1bhat*m2bhat - 6*W2b^lam2*lam2*m1bhat))/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2bhat - 6*W1b^lam1*lam2 - 6*W2b^lam2*lam2 + 6*W1b^lam1*W2b^lam2*lam2 - 6*W1b^lam1*lam2^2*m2bhat)*covb^2*s1bhat*s2bhat^2 + (8*W1b^lam1*lam2^2*m1bhat - 8*lam2^2*m1bhat)*covb*s1bhat*s2bhat^4 + (2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat)*s1bhat^3*s2bhat^4) - covb*s1bhat*s2bhat^4*(4*W1b^(2*lam1)*lam2^2 - 8*W1b^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_5=0
h10_6=0
h10_7=sum((covb^2*(lam1*m1bhat - W1b^lam1 + 1) + s1bhat^2*((lam1*m1bhat - W1b^lam1 + 1)*s2bhat^2 - 2*covb*lam1*m2bhat))/(lam1*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s1bhat^2*(2*lam1 - 2*W2b^lam2*lam1))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_8=sum(- (covb*(s1bhat^4*(- 4*lam1^2*m2bhat^2*s2bhat + 2*lam1^2*s2bhat^3) - s1bhat^2*s2bhat^3*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2)) - covb^3*(s2bhat*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2) + 2*lam1^2*s1bhat^2*s2bhat) + s1bhat^4*s2bhat^3*(2*lam1*m2bhat + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat) + covb^2*s1bhat^2*s2bhat*(6*lam1*m2bhat + 6*lam1^2*m1bhat*m2bhat - 6*W1b^lam1*lam1*m2bhat))/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1bhat - 6*W1b^lam1*lam1 - 6*W2b^lam2*lam1 + 6*W1b^lam1*W2b^lam2*lam1 - 6*W2b^lam2*lam1^2*m1bhat)*covb^2*s1bhat^2*s2bhat + (8*W2b^lam2*lam1^2*m2bhat - 8*lam1^2*m2bhat)*covb*s1bhat^4*s2bhat + (2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat)*s1bhat^4*s2bhat^3) - covb*s1bhat^4*s2bhat*(4*W2b^(2*lam2)*lam1^2 - 8*W2b^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_9=0
h10_10=sum(- (s1bhat^2*(covb^2*(6*lam2*m2bhat - 6*W2b^lam2 + 3*W2b^(2*lam2) + 3*lam2^2*m2bhat^2 - 6*W2b^lam2*lam2*m2bhat + 3) - covb*(6*lam2*m1bhat*s2bhat^2 - 6*W2b^lam2*lam2*m1bhat*s2bhat^2 + 6*lam2^2*m1bhat*m2bhat*s2bhat^2) + lam2^2*m1bhat^2*s2bhat^4) - covb^3*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + s1bhat^4*(W2b^(2*lam2)*s2bhat^2 - 2*W2b^lam2*s2bhat^2 + s2bhat^2 - lam2^2*s2bhat^4 + 2*lam2*m2bhat*s2bhat^2 + lam2^2*m2bhat^2*s2bhat^2 - 2*W2b^lam2*lam2*m2bhat*s2bhat^2) + covb^4*lam2^2 + 3*covb^2*lam2^2*m1bhat^2*s2bhat^2)/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^2*(3*lam2^2*s2bhat^2 - 6*W1b^lam1*lam2^2*s2bhat^2 + 3*W1b^(2*lam1)*lam2^2*s2bhat^2) + s1bhat^2*(lam2^2*s2bhat^4 - 2*W1b^lam1*lam2^2*s2bhat^4 + W1b^(2*lam1)*lam2^2*s2bhat^4) - lam1*(covb^3*(2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat) - covb^2*(6*lam2^2*m1bhat*s2bhat^2 - 6*W1b^lam1*lam2^2*m1bhat*s2bhat^2) + s1bhat^2*(covb*(6*lam2*s2bhat^2 + 6*lam2^2*m2bhat*s2bhat^2 - 6*W1b^lam1*lam2*s2bhat^2 - 6*W2b^lam2*lam2*s2bhat^2 - 6*W1b^lam1*lam2^2*m2bhat*s2bhat^2 + 6*W1b^lam1*W2b^lam2*lam2*s2bhat^2) - 2*lam2^2*m1bhat*s2bhat^4 + 2*W1b^lam1*lam2^2*m1bhat*s2bhat^4)))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_11=sum(-((W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb^2*lam1 - 2*covb*lam2*s2bhat^2 + lam1*s1bhat^2*s2bhat^2 - W2b^lam2*covb^2*lam1 + 2*W1b^lam1*covb*lam2*s2bhat^2 + covb^2*lam1*lam2*m2bhat - W2b^lam2*lam1*s1bhat^2*s2bhat^2 + lam1*lam2*m2bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_12=sum(-((W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb^2*lam2 - 2*covb*lam1*s1bhat^2 + lam2*s1bhat^2*s2bhat^2 - W1b^lam1*covb^2*lam2 + 2*W2b^lam2*covb*lam1*s1bhat^2 + covb^2*lam1*lam2*m1bhat - W1b^lam1*lam2*s1bhat^2*s2bhat^2 + lam1*lam2*m1bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_1=sum((2*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h11_2=sum((2*s1ahat*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_3=sum((2*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h11_4=sum((2*s1bhat*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_5=sum(-(cova*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h11_6=sum(-(2*cova*s2ahat*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_7=sum(-(covb*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h11_8=sum(-(2*covb*s2bhat*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_9=sum(-((W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova^2*lam1 - 2*cova*lam2*s2ahat^2 + lam1*s1ahat^2*s2ahat^2 - W2a^lam2*cova^2*lam1 + 2*W1a^lam1*cova*lam2*s2ahat^2 + cova^2*lam1*lam2*m2ahat - W2a^lam2*lam1*s1ahat^2*s2ahat^2 + lam1*lam2*m2ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_10=sum(-((W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb^2*lam1 - 2*covb*lam2*s2bhat^2 + lam1*s1bhat^2*s2bhat^2 - W2b^lam2*covb^2*lam1 + 2*W1b^lam1*covb*lam2*s2bhat^2 + covb^2*lam1*lam2*m2bhat - W2b^lam2*lam1*s1bhat^2*s2bhat^2 + lam1*lam2*m2bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_11=sum(((2*((W1a^lam1-1)/lam1^2-(W1a^lam1*log(W1a))/lam1)^2)/s1ahat^2-(2*(m1ahat-(W1a^lam1-1)/lam1)*((2*W1a^lam1-2)/lam1^3-(2*W1a^lam1*log(W1a))/lam1^2+(W1a^lam1*log(W1a)^2)/lam1))/s1ahat^2+(2*cova*(m2ahat-(W2a^lam2-1)/lam2)*((2*W1a^lam1-2)/lam1^3-(2*W1a^lam1*log(W1a))/lam1^2+(W1a^lam1*log(W1a)^2)/lam1))/(s1ahat^2*s2ahat^2))/((2*cova^2)/(s1ahat^2*s2ahat^2)-2))+sum(((2*((W1b^lam1-1)/lam1^2-(W1b^lam1*log(W1b))/lam1)^2)/s1bhat^2-(2*(m1bhat-(W1b^lam1-1)/lam1)*((2*W1b^lam1-2)/lam1^3-(2*W1b^lam1*log(W1b))/lam1^2+(W1b^lam1*log(W1b)^2)/lam1))/s1bhat^2+(2*covb*(m2bhat-(W2b^lam2-1)/lam2)*((2*W1b^lam1-2)/lam1^3-(2*W1b^lam1*log(W1b))/lam1^2+(W1b^lam1*log(W1b)^2)/lam1))/(s1bhat^2*s2bhat^2))/((2*covb^2)/(s1bhat^2*s2bhat^2)-2))
h11_12=sum((2*cova*(W1a^lam1*lam1*log(W1a)-W1a^lam1+1)*(W2a^lam2*lam2*log(W2a)-W2a^lam2+1))/(lam1^2*lam2^2*(-2*cova^2+2*s1ahat^2*s2ahat^2)))+sum((2*covb*(W1b^lam1*lam1*log(W1b)-W1b^lam1+1)*(W2b^lam2*lam2*log(W2b)-W2b^lam2+1))/(lam1^2*lam2^2*(-2*covb^2+2*s1bhat^2*s2bhat^2)))
h12_1=sum(-(cova*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h12_2=sum(-(2*cova*s1ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_3=sum(-(covb*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h12_4=sum(-(2*covb*s1bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_5=sum((2*s1ahat^2*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h12_6=sum((2*s1ahat^2*s2ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_7=sum((2*s1bhat^2*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h12_8=sum((2*s1bhat^2*s2bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_9=sum(-((W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova^2*lam2 - 2*cova*lam1*s1ahat^2 + lam2*s1ahat^2*s2ahat^2 - W1a^lam1*cova^2*lam2 + 2*W2a^lam2*cova*lam1*s1ahat^2 + cova^2*lam1*lam2*m1ahat - W1a^lam1*lam2*s1ahat^2*s2ahat^2 + lam1*lam2*m1ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_10=sum(-((W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb^2*lam2 - 2*covb*lam1*s1bhat^2 + lam2*s1bhat^2*s2bhat^2 - W1b^lam1*covb^2*lam2 + 2*W2b^lam2*covb*lam1*s1bhat^2 + covb^2*lam1*lam2*m1bhat - W1b^lam1*lam2*s1bhat^2*s2bhat^2 + lam1*lam2*m1bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_11=sum((2*cova*(W1a^lam1*lam1*log(W1a)-W1a^lam1+1)*(W2a^lam2*lam2*log(W2a)-W2a^lam2+1))/(lam1^2*lam2^2*(-2*cova^2+2*s1ahat^2*s2ahat^2)))+sum((2*covb*(W1b^lam1*lam1*log(W1b)-W1b^lam1+1)*(W2b^lam2*lam2*log(W2b)-W2b^lam2+1))/(lam1^2*lam2^2*(-2*covb^2+2*s1bhat^2*s2bhat^2)))
h12_12=sum(((2*((W2a^lam2-1)/lam2^2-(W2a^lam2*log(W2a))/lam2)^2)/s2ahat^2-(2*(m2ahat-(W2a^lam2-1)/lam2)*((2*W2a^lam2-2)/lam2^3-(2*W2a^lam2*log(W2a))/lam2^2+(W2a^lam2*log(W2a)^2)/lam2))/s2ahat^2+(2*cova*(m1ahat-(W1a^lam1-1)/lam1)*((2*W2a^lam2-2)/lam2^3-(2*W2a^lam2*log(W2a))/lam2^2+(W2a^lam2*log(W2a)^2)/lam2))/(s1ahat^2*s2ahat^2))/((2*cova^2)/(s1ahat^2*s2ahat^2)-2))+sum(((2*((W2b^lam2-1)/lam2^2-(W2b^lam2*log(W2b))/lam2)^2)/s2bhat^2-(2*(m2bhat-(W2b^lam2-1)/lam2)*((2*W2b^lam2-2)/lam2^3-(2*W2b^lam2*log(W2b))/lam2^2+(W2b^lam2*log(W2b)^2)/lam2))/s2bhat^2+(2*covb*(m1bhat-(W1b^lam1-1)/lam1)*((2*W2b^lam2-2)/lam2^3-(2*W2b^lam2*log(W2b))/lam2^2+(W2b^lam2*log(W2b)^2)/lam2))/(s1bhat^2*s2bhat^2))/((2*covb^2)/(s1bhat^2*s2bhat^2)-2))
h1r=c(h1_1,h1_2,h1_3,h1_4,h1_5,h1_6,h1_7,h1_8,h1_9,h1_10,h1_11,h1_12)
h2r=c(h2_1,h2_2,h2_3,h2_4,h2_5,h2_6,h2_7,h2_8,h2_9,h2_10,h2_11,h2_12)
h3r=c(h3_1,h3_2,h3_3,h3_4,h3_5,h3_6,h3_7,h3_8,h3_9,h3_10,h3_11,h3_12)
h4r=c(h4_1,h4_2,h4_3,h4_4,h4_5,h4_6,h4_7,h4_8,h4_9,h4_10,h4_11,h4_12)
h5r=c(h5_1,h5_2,h5_3,h5_4,h5_5,h5_6,h5_7,h5_8,h5_9,h5_10,h5_11,h5_12)
h6r=c(h6_1,h6_2,h6_3,h6_4,h6_5,h6_6,h6_7,h6_8,h6_9,h6_10,h6_11,h6_12)
h7r=c(h7_1,h7_2,h7_3,h7_4,h7_5,h7_6,h7_7,h7_8,h7_9,h7_10,h7_11,h7_12)
h8r=c(h8_1,h8_2,h8_3,h8_4,h8_5,h8_6,h8_7,h8_8,h8_9,h8_10,h8_11,h8_12)
h9r=c(h9_1,h9_2,h9_3,h9_4,h9_5,h9_6,h9_7,h9_8,h9_9,h9_10,h9_11,h9_12)
h10r=c(h10_1,h10_2,h10_3,h10_4,h10_5,h10_6,h10_7,h10_8,h10_9,h10_10,h10_11,h10_12)
h11r=c(h11_1,h11_2,h11_3,h11_4,h11_5,h11_6,h11_7,h11_8,h11_9,h11_10,h11_11,h11_12)
h12r=c(h12_1,h12_2,h12_3,h12_4,h12_5,h12_6,h12_7,h12_8,h12_9,h12_10,h12_11,h12_12)
HCF=rbind(h1r,h2r,h3r,h4r,h5r,h6r,h7r,h8r,h9r,h10r,h11r,h12r)
HCF[1,2]=0;HCF[1,6]=0;HCF[1,9]=0;
HCF[2,1]=0;HCF[2,5]=0;
HCF[3,4]=0;HCF[3,8]=0;HCF[3,10]=0;HCF[4,3]=0;
HCF[4,7]=0;
HCF[5,2]=0;HCF[5,6]=0;HCF[5,9]=0;
HCF[6,1]=0;HCF[6,5]=0;
HCF[7,4]=0;HCF[7,8]=0;HCF[7,10]=0;
HCF[8,3]=0;HCF[8,7]=0;
HCF[9,1]=0;HCF[9,5]=0;
HCF[10,3]=0;
HCF[10,7]=0;
HCF[1,1]=na*HCF[1,1];
HCF[1,5]=na*HCF[1,5];
HCF[5,1]=na*HCF[5,1];
HCF[3,3]=nb*HCF[3,3];
HCF[3,7]=nb*HCF[3,7];
HCF[7,3]=nb*HCF[7,3];
HCF[7,7]=nb*HCF[7,7];
HCF[5,5]=na*HCF[5,5];
VV=inv(-HCF)
att=1-atSpec;
t=att;
s1alam=sqrt( var(W1alam)*(na-1)/na )
s1blam=sqrt( var(W1blam)*(nb-1)/nb )
s2alam=sqrt( var(W2alam)*(na-1)/na )
s2blam=sqrt( var(W2blam)*(nb-1)/nb )
m1alam=mean(W1alam)
m1blam=mean(W1blam)
m2alam=mean(W2alam)
m2blam=mean(W2blam)
#############################################################################
#############################################################################
#==================AT A GIVEN T FOR THE ORIGINAL===================
ROC1or= (1-pnorm(qnorm(1-t,m1alam,s1alam),m1blam,s1blam));
ROC2or= (1-pnorm(qnorm(1-t,m2alam,s2alam),m2blam,s2blam));
dm1a = -(2^(1/2)*exp(-((m1ahat - m1bhat)/s1bhat + (2^(1/2)*s1ahat*erfcinv(2*t))/s1bhat)^2/2))/(2*s1bhat*pi^(1/2))
dm1b = (2^(1/2)*exp(-((m1ahat - m1bhat)/s1bhat + (2^(1/2)*s1ahat*erfcinv(2*t))/s1bhat)^2/2))/(2*s1bhat*pi^(1/2))
ds1a = -(erfcinv(2*t)*exp(-((m1ahat - m1bhat)/s1bhat + (2^(1/2)*s1ahat*erfcinv(2*t))/s1bhat)^2/2))/(s1bhat*pi^(1/2))
ds1b = (2^(1/2)*exp(-((m1ahat - m1bhat)/s1bhat + (2^(1/2)*s1ahat*erfcinv(2*t))/s1bhat)^2/2)*((m1ahat - m1bhat)/s1bhat^2 + (2^(1/2)*s1ahat*erfcinv(2*t))/s1bhat^2))/(2*pi^(1/2))
der1=c(dm1a, ds1a, dm1b, ds1b);
vROC1=c(der1)%*%VV[1:4,1:4]%*%t(t(c(der1)))
vROC1=c(vROC1)
dm2a = -(2^(1/2)*exp(-((m2ahat - m2bhat)/s2bhat + (2^(1/2)*s2ahat*erfcinv(2*t))/s2bhat)^2/2))/(2*s2bhat*pi^(1/2))
dm2b = (2^(1/2)*exp(-((m2ahat - m2bhat)/s2bhat + (2^(1/2)*s2ahat*erfcinv(2*t))/s2bhat)^2/2))/(2*s2bhat*pi^(1/2))
ds2a = -(erfcinv(2*t)*exp(-((m2ahat - m2bhat)/s2bhat + (2^(1/2)*s2ahat*erfcinv(2*t))/s2bhat)^2/2))/(s2bhat*pi^(1/2))
ds2b = (2^(1/2)*exp(-((m2ahat - m2bhat)/s2bhat + (2^(1/2)*s2ahat*erfcinv(2*t))/s2bhat)^2/2)*((m2ahat - m2bhat)/s2bhat^2 + (2^(1/2)*s2ahat*erfcinv(2*t))/s2bhat^2))/(2*pi^(1/2))
der2=c(dm2a, ds2a, dm2b, ds2b);
vROC2=c(der2)%*%VV[5:8,5:8]%*%t(t(c(der2)))
covROC=c(der1, 0, 0, 0, 0)%*%VV[1:8,1:8]%*%t(t(c(0, 0, 0, 0, der2)))
Z=(ROC2or-ROC1or)/(sqrt(vROC2+vROC1-2*covROC))
SEZ=sqrt(vROC1+vROC2-2*covROC)
pval2tZ=2*pnorm(-abs(Z));
CIoriginal=c((ROC2or-ROC1or)-pmalpha*SEZ, (ROC2or-ROC1or)+pmalpha*SEZ)
CIoriginal
#############################################################################
#############################################################################
#==================AT A GIVEN T FOR THE TRANSORMED===================
ROC1MC= qnorm(1-pnorm(qnorm(1-t,m1alam,s1alam),m1blam,s1blam));
ROC2MC= qnorm(1-pnorm(qnorm(1-t,m2alam,s2alam),m2blam,s2blam));
dm1a =-1/s1blam;
ds1a =-(2^(1/2)*erfcinv(2*t))/s1blam;
dm1b =1/s1blam;
ds1b =(m1alam - m1blam)/s1blam^2 + (2^(1/2)*s1alam*erfcinv(2*t))/s1blam^2;
der1=c(dm1a, ds1a, dm1b, ds1b);
vROC1=c(der1)%*%VV[1:4,1:4]%*%t(t(c(der1)))
vROC1=c(vROC1)
dm2a =-1/s2blam;
ds2a =-(2^(1/2)*erfcinv(2*t))/s2blam;
dm2b =1/s2blam;
ds2b =(m2alam - m2blam)/s2blam^2 + (2^(1/2)*s2alam*erfcinv(2*t))/s2blam^2;
der2=c(dm2a, ds2a, dm2b, ds2b);
vROC2=c(der2)%*%VV[5:8,5:8]%*%t(t(c(der2)))
vROC2=c(vROC2)
covROC=c(der1, 0, 0, 0, 0)%*%VV[1:8,1:8]%*%t(t(c(0, 0, 0, 0, der2)))
covROC=c(covROC)
Zroct=(ROC2MC-ROC1MC)/(sqrt(vROC2+vROC1-2*covROC))
SEroct=sqrt(vROC1+vROC2-2*covROC)
pval2t_roct=2*pnorm(-abs(Zroct));pval2t=pval2t_roct;
CIZstar=c((ROC2MC-ROC1MC)-pmalpha*SEroct, (ROC2MC-ROC1MC)+pmalpha*SEroct)
CIZstar
SE1=pnorm(ROC1MC)
SE2=pnorm(ROC2MC)
res <- data.frame(Sens1 = formattable(SE1, digits = 4, format = "f"),
Sens2 = formattable(SE2, digits = 4, format = "f"),
diff = formattable(SE2 - SE1, digits = 4, format = "f"),
p_value_probit = formattable(pval2t, digits = 4, format = "f"),
p_value = formattable(pval2tZ, digits = 4, format = "f"),
ci_ll = formattable(CIoriginal[1], digits = 4, format = "f"),
ci_ul = formattable(CIoriginal[2], digits = 4, format = "f"))
rownames(res) <- c("Estimates:")
colnames(res) <- c("Sens 1", "Sens 2", "Diff", "P-Val (Probit)", "P-Val", "CI (LL)", "CI (UL)")
res <- formattable(as.matrix(res), digits = 4, format = "f")
res
#====TWO ROC FUNCTIONS: ONE IS THE BOXCOX AND THE OTHER THE REFERENCE LINE================
roc1<-function(t){
na = length(W1alam)
nb = length(W1blam)
s1alam=sqrt( var(W1alam)*(na-1)/na )
s1blam=sqrt( var(W1blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W1alam),
sd=s1alam),
mean=mean(W1blam),
sd=s1blam)
}
roc2<-function(t){
na = length(W2alam)
nb = length(W2blam)
s2alam=sqrt( var(W2alam)*(na-1)/na )
s2blam=sqrt( var(W2blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W2alam),
sd=s2alam),
mean=mean(W2blam),
sd=s2blam)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
#================IF PLOTS ARE REQUESTED PLOT THE ROCS==
if (plots=="on") {
# x11()
plot(linspace(0,1,1000),roc1(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="red")
lines(linspace(0,1,1000),roc2(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="black")
lines(linspace(0,1,1000),linspace(0,1,1000),type="l", lty=2)
points(1-atSpec,SE1, col = "red")
points(1-atSpec,SE2, col = "black")
legend("bottomright", legend=c(paste("ROC for Marker 1 with Sens =", formattable(SE1, digits = 4, format = "f"), "at Spec =", formattable(atSpec, digits = 4, format = "f")),
paste("ROC for Marker 2 with Sens =", formattable(SE2, digits = 4, format = "f"), "at Spec =", formattable(atSpec, digits = 4, format = "f")),
paste("P-value for the Sens difference:", formattable(pval2t, digits = 4, format = "f"))),
col=c("red", "black", "white"), lty=c(1, 1, NA), pch = c(NA, NA, NA), cex=0.8)
}
return(list(resultstable=res,
Sens1=SE1,
Sens2=SE2,
pvalue_probit_difference= pval2t,
pvalue_difference= pval2tZ,
CI_difference= CIoriginal,
roc1=roc1,
roc2=roc2,
transx1=W1alam,
transy1=W1blam,
transformation.parameter.1 = lam[1],
transx2=W2alam,
transy2=W2blam,
transformation.parameter.2 = lam[2]))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
comparebcSpec <-function (marker1, marker2, D, atSens, alpha, plots){
if ((length(marker1) != length(D)) | (length(marker2) != length(D))) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (alpha <= 0 | alpha >= 1) {
stop("ERROR: The level of significance, alpha, should be set between 0 and 1. A common choice is 0.05.")
} else if (sum(is.na(marker1)) > 0 | sum(is.na(marker2)) > 0 | sum(is.na(D)) > 0 | sum(is.na(atSens)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if ((!is.numeric(atSens)) | (length(atSens) != 1)) {
stop("ERROR: 'atSens' must be a single numeric value.")
} else if ((sum(marker1 < 0) > 0) | (sum(marker2 < 2)) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
erf <- function (x) 2 * pnorm(x * sqrt(2)) - 1
erfc <- function (x) 2 * pnorm(x * sqrt(2), lower.tail = FALSE)
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower.tail = FALSE)/sqrt(2)
pmalpha=qnorm(1-alpha/2);
if (plots!="on"){plots="off"}
W1a=marker1[D==0]
W1b=marker1[D==1]
W2a=marker2[D==0]
W2b=marker2[D==1]
Dflip=abs(D-1);
atSpec=1-atSens;
#out=comparebcSens(marker1group1=W1b, marker1group2=W1a ,marker2group1=W2b, marker2group2=W2a, atSpec=atSens, plots="off")
out=comparebcSens(marker1, marker2, Dflip, atSpec=1-atSens, alpha, plots="off")
#return(list(resultstable=res,Sens1=SE1,Sens2=SE2, pvalue_probit_difference= pval2t, CI_probit_difference= CIZstar, pvalue_difference= pval2tZ, CI_difference= CIoriginal, roc1=roc1, roc2=roc2, transx1=W1alam, transy1=W1blam, transx2=W2alam, transy2=W2blam))
FPR1=out$Sens1
FPR2=out$Sens2
CIZstar=out$CI_probit_difference
CIoriginal=out$CI_difference
pval2t=out$pvalue_probit_difference
pval2tZ=out$pvalue_difference
W1alam=out$transx1
W1blam=out$transy1
W2alam=out$transx2
W2blam=out$transy2
#====TWO ROC FUNCTIONS: ONE IS THE BOXCOX AND THE OTHER THE REFERENCE LINE================
roct1<-function(t){
na = length(W1alam)
nb = length(W1blam)
s1alam=sqrt(var(W1alam)*(na-1)/na)
s1blam=sqrt(var(W1blam)*(nb-1)/nb)
1-pnorm(qnorm(1-t,
mean=mean(W1blam),
sd=s1blam),
mean=mean(W1alam),
sd=s1alam)
}
roct2<-function(t){
na = length(W2alam)
nb = length(W2blam)
s2alam=sqrt(var(W2alam)*(na-1)/na)
s2blam=sqrt(var(W2blam)*(nb-1)/nb)
1-pnorm(qnorm(1-t,
mean=mean(W2blam),
sd=s2blam),
mean=mean(W2alam),
sd=s2alam)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
#================IF PLOTS ARE REQUESTED PLOT THE ROCS==
if (plots=="on") {
# x11()
plot(linspace(0,1,1000),roct1(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="red")
lines(linspace(0,1,1000),roct2(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="black")
lines(linspace(0,1,1000),linspace(0,1,1000),type="l", lty=2)
points(FPR1, 1-atSpec, col = "red")
points(FPR2, 1-atSpec, col = "black")
#line for the Youden:
#lines(c(1-atSpec,1-SE1),c(1-atSpec,1-SE2),col="green")
legend("bottomright", legend=c(paste("ROC for Marker 1 with FPR =", formattable(FPR1, digits = 4, format = "f"), "at Sens =", formattable(atSens, digits = 4, format = "f")),
paste("ROC for Marker 2 with FPR =", formattable(FPR2, digits = 4, format = "f"), "at Sens =", formattable(atSens, digits = 4, format = "f")),
paste("P-value for the Spec difference:", formattable(pval2t, digits = 4, format = "f"))),
col=c("red", "black", "white"), lty=c(1, 1, NA), pch = c(NA, NA, NA), cex=0.8)
}
res <- data.frame(FPR1 = formattable(FPR1, digits = 4, format = "f"),
FPR2 = formattable(FPR2, digits = 4, format = "f"),
formattable(FPR2 - FPR1, digits = 4, format = "f"),
p_value_probit = formattable(pval2t, digits = 4, format = "f"),
p_value = formattable(pval2tZ, digits = 4, format = "f"),
ci_ll = formattable(CIoriginal[1], digits = 4, format = "f"),
ci_ul = formattable(CIoriginal[2], digits = 4, format = "f"))
rownames(res) <- c("Estimates:")
colnames(res) <- c("FPR 1", "FPR 2", "Diff", "P-Val (Probit)", "P-Val", "CI (LL)", "CI (UL)")
res <- formattable(as.matrix(res), digits = 4, format = "f")
res
#return(list(resultstable=res,Sens1=SE1,Sens2=SE2, pvalue_probit_difference= pval2t, CI_probit_difference= CIZstar, pvalue_difference= pval2tZ, CI_difference= CIoriginal, roc1=roc1, roc2=roc2, transx1=W1alam, transy1=W1blam, transx2=W2alam, transy2=W2blam))
return(list(resultstable=res,
FPR1=FPR1,
FPR2=FPR2,
pvalue_probit_difference= pval2t,
pvalue_difference= pval2tZ,
CI_difference= CIoriginal,
roc1=roct1,
roc2=roct2,
transx1=W1alam,
transy1=W1blam,
transformation.parameter.1 = out$transformation.parameter.1,
transx2=W2alam,
transy2=W2blam,
transformation.parameter.2 = out$transformation.parameter.2))
#return(list(FPR1=FPR1))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
threerocs2 <- function (marker1, marker2, D, plots) {
erf <- function (x) 2 * pnorm(x * sqrt(2)) - 1
erfc <- function (x) 2 * pnorm(x * sqrt(2), lower = FALSE)
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower = FALSE)/sqrt(2)
W1a=marker1[D==0]
W1b=marker1[D==1]
W2a=marker2[D==0]
W2b=marker2[D==1]
Wa=cbind(W1a,W2a);
Wb=cbind(W1b,W2b);
na=length(W1a)
nb=length(W1b)
likbox2D<-function(W1a,W1b,W2a,W2b,lam){
na=length(W1a);
nb=length(W1b);
out=c();
#for (i in 1:length(h)){
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
cova= cov(Walam)*(na-1)/na;
covb= cov(Wblam)*(nb-1)/nb;
mualam=c(mean(W1alam), mean(W2alam))
mublam=c(mean(W1blam), mean(W2blam))
out= -sum(log(dmvnorm(Walam,mualam,cova)))-sum(log(dmvnorm(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b));
return(out)
}
lam=c(2,2)
likbox2D(W1a,W1b,W2a,W2b,lam)
logL<-function(h){
likbox2D(W1a,W1b,W2a,W2b,h)
}
init=c(0.8,0.8)
lam=fminsearch(logL,init)
lam=c(lam$optbase$xopt)
lam
out <- optim(c(1,1), logL, method = "Nelder-Mead")
lam = out$par
###################################
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
lam1=lam[1]
lam2=lam[2]
cova= cov(Walam)*(na-1)/na;cova=cova[1,2];
covb= cov(Wblam)*(nb-1)/nb;covb=covb[1,2];
s1alam=sqrt( var(W1alam)*(na-1)/na )
s1blam=sqrt( var(W1blam)*(nb-1)/nb )
s2alam=sqrt( var(W2alam)*(na-1)/na )
s2blam=sqrt( var(W2blam)*(nb-1)/nb )
m1alam=mean(W1alam)
m1blam=mean(W1blam)
m2alam=mean(W2alam)
m2blam=mean(W2blam)
roc1bc<-function(t){
1-pnorm(qnorm(1-t,mean=mean(W1alam),sd=s1alam),mean=mean(W1blam),sd=s1blam)
}
roc2bc<-function(t){
1-pnorm(qnorm(1-t,mean=mean(W2alam),sd=s2alam),mean=mean(W2blam),sd=s2blam)
}
####################### MRM ###################
rating=c(marker1,marker2)
truth=c(D,D)
reader=c(linspace(1,1,length(marker1)),linspace(2,2,length(marker2)))
curves_binorm <- roc_curves(truth, rating,
groups = list(Reader = reader),
method = "binormal")
params_binorm <- parameters(curves_binorm)
t=linspace(0,1,1000);
roct_mrm_1=pnorm(c(params_binorm$a[1])+c(params_binorm$b[1])*qnorm(t))
roct_mrm_2=pnorm(c(params_binorm$a[2])+c(params_binorm$b[2])*qnorm(t))
# Plot the empirical ROC curve
roc_obj1 <- roc(D, marker1, quiet = TRUE)
auc_emp1 <- auc(roc_obj1)
tpr1=roc_obj1$sensitivities
fpr1=1-roc_obj1$specificities
roc_obj2 <- roc(D, marker2, quiet = TRUE)
auc_emp2 <- auc(roc_obj2)
tpr2=roc_obj2$sensitivities
fpr2=1-roc_obj2$specificities
AUC_empirical_marker1=auc_emp1
AUC_empirical_marker2=auc_emp2
if (plots == "on") {
plot(t, roc1bc(t), type="l", lty = "dotted", col="indianred2", lwd=1.5, ylab="TPR = Sensitivity", xlab="FPR = 1 - Sensitivity")
lines(t, roc2bc(t), type="l", lty = "dotted", col="red4", lwd=1.5)
lines(t, roct_mrm_1, type="l", lty = "dashed", lwd=1.5, col="green2")
lines(t, roct_mrm_2, type="l", lty = "dashed", lwd=1.5, col="darkgreen")
lines(fpr1, tpr1, type="l", lty = "solid", lwd=1, col="deepskyblue")
lines(fpr2, tpr2, type="l",lty = "solid", lwd=1, col="blue4")
}
#plot(fpr1, tpr1, type = "l", col = "magenta", lwd = 2,
# xlim = c(0, 1), ylim = c(0, 1), xlab = "False Positive Rate (FPR)",
# ylab = "True Positive Rate (TPR)", main = "Three Estimates of the ROC Curve")
AUC1bc=trapz(t,roc1bc(t))
AUC2bc=trapz(t,roc2bc(t))
AUCmetz1=trapz(t,roct_mrm_1)
AUCmetz2=trapz(t,roct_mrm_2)
AUCemp1=abs(trapz(fpr1,tpr1))
AUCemp2=abs(trapz(fpr2,tpr2))
#legend("bottomright", legend = c("Box-Cox marker 1", "Box-Cox marker 2", "Metz marker 1",, "Metz marker 2","Empirical marker 1","Empirical marker 2"),
# col = c("black","red","blue","green", "magenta", "cyan"), lty = c("solid", "solid", "dashed","dashed","dashed","dashed"),
# lwd = 2, bty = "n")
#CORRECT ONE:
if (plots == "on") {
legend("bottomright",
legend = c(paste("Box-Cox Marker 1 (AUC = ", formattable(AUC1bc, digits = 4, format = "f"), ")", sep = ""),
paste("Box-Cox Marker 2 (AUC = ", formattable(AUC2bc, digits = 4, format = "f"), ")", sep = ""),
paste("Metz Marker 1 (AUC = ", formattable(AUCmetz1, digits = 4, format = "f"), ")", sep = ""),
paste("Metz Marker 2 (AUC = ", formattable(AUCmetz2, digits = 4, format = "f"), ")", sep = ""),
paste("Empirical Marker 1 (AUC = ", formattable(AUCemp1, digits = 4, format = "f"), ")", sep = ""),
paste("Empirical Marker 2 (AUC = ", formattable(AUCemp2, digits = 4, format = "f"), ")", sep = "")),
col = c("indianred2", "red4", "green2", "darkgreen", "deepskyblue", "blue4"),
lty = c("dotted", "dotted", "dashed", "dashed", "solid", "solid"),
cex = 0.8, lwd = 2)
title(main = "Three Estimates of the ROC Curve (Two Markers)")
}
return(list(AUC_BoxCox1 = AUC1bc,
AUC_BoxCox2 = AUC2bc,
AUC_Metz1 = AUCmetz1,
AUC_Metz2 = AUCmetz2,
AUC_Empirical1 = AUCemp1,
AUC_Empirical2 = AUCemp2))
}
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
#DATA GENERATION
set.seed(123)
x=rgamma(100, shape=2, rate = 8) # Generates biomarker data from a gamma distribution
# for the healthy group.
y=rgamma(100, shape=2, rate = 4) # Generates biomarker data from a gamma distribution
# for the diseased group.
scores=c(x,y)
D=c(zeros(1,100), ones(1,100))
out=checkboxcox(marker=scores, D, plots="on", printShapiro = TRUE)
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
#DATA GENERATION
set.seed(123)
x <- rgamma(100, shape=2, rate = 8) # generates biomarker data from a gamma
# distribution for the healthy group.
y <- rgamma(100, shape=2, rate = 4) # generates biomarker data from a gamma
# distribution for the diseased group.
scores <- c(x,y)
D=c(pracma::zeros(1,100), pracma::ones(1,100))
out=threerocs(marker=scores, D, plots="on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
set.seed(123)
x=rgamma(100, shape=2, rate = 8)
y=rgamma(100, shape=2, rate = 4)
scores=c(x,y)
D=c(zeros(1,100), ones(1,100))
out=rocboxcox(marker=scores,D, 0.05, plots="on", printProgress=FALSE)
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
givenSP=c(0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9)
givenSE=NA
out=rocboxcoxCI(marker=scores ,D, givenSP=givenSP, givenSE=NA, alpha=0.05, plots="on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
#DATA GENERATION
set.seed(123)
nx <- 100
Sx <- matrix(c(1, 0.5, 0.5, 1), nrow = 2, ncol = 2)
mux <- c(X = 10, Y = 12)
X = mvtnorm::rmvnorm(nx, mean = mux, sigma = Sx)
ny <- 100
Sy <- matrix(c(1.1, 0.6, 0.6, 1.1), nrow = 2, ncol = 2)
muy <- c(X = 11, Y = 13.7)
Y = mvtnorm::rmvnorm(ny, mean = muy, sigma = Sy)
dx = pracma::zeros(nx,1)
dy = pracma::ones(ny,1)
markers = rbind(X,Y);
marker1 = markers[,1]
marker2 = markers[,2]
D = c(rbind(dx,dy))
out=checkboxcox2(marker1, marker2, D, plots = "on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
#GENERATE SOME BIVARIATE DATA===
set.seed(123)
nx <- 100
Sx <- matrix(c(1, 0.5,
0.5, 1),
nrow = 2, ncol = 2)
mux <- c(X = 10, Y = 12)
X=rmvnorm(nx, mean = mux, sigma = Sx)
ny <- 100
Sy <- matrix(c(1.1, 0.6,
0.6, 1.1),
nrow = 2, ncol = 2)
muy <- c(X = 11, Y = 13.7)
Y=rmvnorm(ny, mean = muy, sigma = Sy)
dx=zeros(nx,1)
dy=ones(ny,1)
markers=rbind(X,Y);
marker1=markers[,1]
marker2=markers[,2]
D=c(rbind(dx,dy))
#==DATA HAVE BEEN GENERATED====
#===COMPARE THE AUCs of Marker 1 vs Maker 2
out=comparebcAUC(marker1, marker2, D, alpha=0.05, plots="on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
#==DATA HAVE BEEN GENERATED====
#===COMPARE THE Js of Marker 1 vs Maker 2
out=comparebcJ(marker1, marker2, D, alpha=0.05, plots="on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
out=comparebcSens(marker1=marker1, marker2=marker2, D=D, alpha =0.05, atSpec=0.8, plots="on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
out=comparebcSpec(marker1=marker1, marker2=marker2, D=D, alpha =0.05, atSens=0.8, plots="on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
out=threerocs2(marker1, marker2, D, plots = "on")
summary(out)
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## ----eval=TRUE, error=FALSE, warning=FALSE, cache=FALSE, comment=FALSE, echo=FALSE, message=FALSE, results=FALSE----
# Hello, world!
#
# This is an example function named 'hello'
# which prints 'Hello, world!'.
#
# You can learn more about package authoring with RStudio at:
#
# http://r-pkgs.had.co.nz/
#
# Some useful keyboard shortcuts for package authoring:
#
# Install Package: 'Ctrl + Shift + B'
# Check Package: 'Ctrl + Shift + E'
# Test Package: 'Ctrl + Shift + T'
library(pracma)
library(clinfun)
library(splancs)
library(mvtnorm)
library(matlab)
library(formattable)
library(pROC)
library(MRMCaov)
## ----echo=FALSE, eval=TRUE----------------------------------------------------
rocboxcox<-function(marker, D, alpha, plots, printProgress = FALSE){
if (plots!="on"){plots="off"}
if (length(marker) != length(D)) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (sum(is.na(marker)) > 0 | sum(is.na(D)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if (alpha <= 0 | alpha >= 1) {
stop("ERROR: The level of significance, alpha, should be set between 0 and 1. A common choice is 0.05.")
} else if (min(marker<= 0)) {
stop("ERROR: All marker scores need to be positive")
} else if (sum(marker < 0) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
x=marker[D==0]
y=marker[D==1]
if (plots!="on"){plots="off"}
n1=length(x)
n2=length(y)
xor=x;yor=y;
Za=qnorm(1-alpha/2)
likbox<-function(x,y,h){
n=length(x);
m=length(y);
out=c();
for (i in 1:length(h)){
#print(i)
if (h[i]==0){
xh=log(x);
yh=log(y);
} else {
xh=((x^h[i])-1)/h[i];
yh=((y^h[i])-1)/h[i];
}
out[i]<-c( -n/2*log(sum((xh-sum(xh)/n)^2)/n) -m/2*log(sum((yh-sum(yh)/m)^2)/m) +(h[i]-1)*(sum(log(x))+sum(log(y))))
}
return(out)
}
boxcoxleo<-function(x,y){
init=1;
logL<-function(h){
-likbox(x,y,h)
}
#lam=fminsearch(logL,init)
#lam=c(lam$optbase$xopt)
lam<- optim(1,logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
transx=((x^lam)-1)/lam
transy=((y^lam)-1)/lam
#test1=print(shapiro.test(transx))
#test2=print(shapiro.test(transy))
return(list(transformation.parameter=lam,transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam ))
}
boxcoxleo2<-function(x,y){
init=1;
logL<-function(h){
-likbox(x,y,h)
}
lam<- optim(1,logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
#lam=fminsearch(logL,init)
#lam=c(lam$optbase$xopt)
transx=((x^lam)-1)/lam
transy=((y^lam)-1)/lam
return(list(transformation.parameter=lam,transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam ))
}
COVlam <-function (x,y){
#%Accepts X<Y Yields as an output the Variance Covariance matrix
#%of the mh, sh, md, sd, and lam, (with this order) where lam is the estimated boxcox
#%parameter Thus, the output will be a 5x5 matrix
#%---FOR TWO POPULATIONS:
n=length(x); #original scores
m=length(y);
# %mh-->g(1)
# %sh-->g(2)
# %md-->g(3)
# %sd-->g(4)
# %lam-->g(5)
# % logL=@(g) -n*log(sh)-1/(2*sh^2)*sum((xlam-mh)^2)+(lam-1)*sum(log(x)) -
# % m*log(sd)-1/(2*sd^2)*sum((ylam-md)^2)+(lam-1)*sum(log(y))
outt=boxcoxleo(x,y); #the original scores
transx=outt$transx
transy=outt$transy
lam=outt$transformation.parameter
mh=mean(transx); xlam=transx;
md=mean(transy); ylam=transy;
sh=sqrt(1/(length(transx))*sum((transx-mean(transx))^2));
sd=sqrt(1/(length(transy))*sum((transy-mean(transy))^2));
I=zeros(5,5);
I[1,1]=n/sh^2;
I[2,2]=-(n/sh^2-3/sh^4*sum((xlam-mh)^2));
I[3,3]=m/sd^2;
I[4,4]=-(m/sd^2-3/sd^4*sum((ylam-md)^2));
kk= sum(((mh - (x^lam - 1)/lam)*((2*(x^lam - 1))/lam^3 - (2*x^lam*log(x))/lam^2 + (x^lam*log(x)^2)/lam))/sh^2) + - sum(((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)^2/sd^2) + - sum(((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)^2/sh^2) + + sum(((md - (y^lam - 1)/lam)*((2*(y^lam - 1))/lam^3 - (2*y^lam*log(y))/lam^2 + (y^lam*log(y)^2)/lam))/sd^2);
I[5,5]=-kk ;
I[1,5]=sum(((2*(x^lam - 1))/lam^2 - (2*x^lam*log(x))/lam)/(2*sh^2));
I[2,5]=-sum((2*((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)*(mh - (x^lam - 1)/lam))/sh^3);
I[3,5]=-sum(-((2*(y^lam - 1))/lam^2 - (2*y^lam*log(y))/lam)/(2*sd^2));
I[4,5]=-sum((2*((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)*(md - (y^lam - 1)/lam))/sd^3);
I[5,1]=I[1,5];
I[5,2]=I[2,5];
I[5,3]=I[3,5];
I[5,4]=I[4,5];
S= inv(I); #%that's better
return(list(out=S))
}
cc=boxcoxleo(x,y)
transx=cc$transx
transy=cc$transy
#x11()
#qqnorm(x)
#title(main=" for X")
#x11()
#qqnorm(y)
#title(main=" for Y")
roc<-function(t,trans_x,trans_y){
na = length(trans_x)
nb = length(trans_y)
stransx=sqrt(var(trans_x)*(na-1)/na)
stransy=sqrt(var(trans_y)*(nb-1)/nb)
1-pnorm(qnorm(1-t,
mean=mean(trans_x),
sd=stransx),
mean=mean(trans_y),
sd=stransy)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
if (plots=="on"){
txt <- paste("Box-Cox Based ROC, alpha =", formattable(alpha, digits = 3))
plot(linspace(0,1,1000),roc(linspace(0,1,1000),transx,transy),main=" ",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="red")
lines(linspace(0,1,10),linspace(0,1,10),type="l", lty=2)
title(main=txt)
}
rocfun=function(t){1-pnorm(qnorm(1-t,mean=mean(transx),sd=sqrt(var(transx)*(length(transx)-1)/length(transx))),mean=mean(transy),sd=sqrt(var(transy)*(length(transy)-1)/length(transy)))}
m1hat=mean(transx)
m2hat=mean(transy)
s1hat=sqrt(var(transx)*(length(transx)-1)/length(transx))
s2hat=sqrt(var(transy)*(length(transy)-1)/length(transy))
n1=length(x)
n2=length(y)
#NOTE THAT: m1hat->m1
# m2hat->m2
# s1hat->s1hat
# s2hat->s2hat
#########################################
derde =c( (s2hat^2 + (s1hat*s2hat*(2*m1hat - 2*m2hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s2hat*(s1hat^2 - s2hat^2)), ((s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2) - 2*m2hat*s1hat + (s1hat*s2hat*(2*s1hat*log(s1hat^2/s2hat^2) + (2*(s1hat^2 - s2hat^2))/s1hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2) - (2*s1hat*(m1hat*s2hat^2 - m2hat*s1hat^2 + s1hat*s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2)^2)/s2hat, -((s1hat^2 + (s1hat*s2hat*(2*m1hat - 2*m2hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2) - 1)/s2hat, ((2*m1hat*s2hat + s1hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2) - (s1hat*s2hat*(2*s2hat*log(s1hat^2/s2hat^2) + (2*(s1hat^2 - s2hat^2))/s2hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2) + (2*s2hat*(m1hat*s2hat^2 - m2hat*s1hat^2 + s1hat*s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2)^2)/s2hat - (m2hat + (m1hat*s2hat^2 - m2hat*s1hat^2 + s1hat*s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2))/(s1hat^2 - s2hat^2))/s2hat^2);
derdp =c( -((s2hat^2 + (s1hat*s2hat*(2*m1hat - 2*m2hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2) + 1)/s1hat, (m1hat + (m1hat*s2hat^2 - m2hat*s1hat^2 + s1hat*s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2))/(s1hat^2 - s2hat^2))/s1hat^2 - ((s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2) - 2*m2hat*s1hat + (s1hat*s2hat*(2*s1hat*log(s1hat^2/s2hat^2) + (2*(s1hat^2 - s2hat^2))/s1hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2) - (2*s1hat*(m1hat*s2hat^2 - m2hat*s1hat^2 + s1hat*s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2)^2)/s1hat, (s1hat^2 + (s1hat*s2hat*(2*m1hat - 2*m2hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat*(s1hat^2 - s2hat^2)), -((2*m1hat*s2hat + s1hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2) - (s1hat*s2hat*(2*s2hat*log(s1hat^2/s2hat^2) + (2*(s1hat^2 - s2hat^2))/s2hat))/(2*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2) + (2*s2hat*(m1hat*s2hat^2 - m2hat*s1hat^2 + s1hat*s2hat*(log(s1hat^2/s2hat^2)*(s1hat^2 - s2hat^2) + (m1hat - m2hat)^2)^(1/2)))/(s1hat^2 - s2hat^2)^2)/s1hat);
outt=COVlam(xor,yor);
S=outt$out;
S=S[1:4,1:4];
# if (sum(is.infinite(S)) > 0 |
# sum(is.nan(S)) > 0 |
# sum(is.na(S)) > 0 |
# sum(diag(S) < 0) > 0) {
# stop("ERROR: The information matrix cannot be inverted - this might be due to the scale of the marker. Try rescaling the marker measurements by subtracting a constant or by dividing with a constant all marker measurements (of both groups) before you try again.")
# }
varde=t(derde)%*%S%*%t(t(derde));
vardp=t(derdp)%*%S%*%t(t(derdp));
covdedp=t(derde)%*%S%*%t(t(derdp));
##########################################
vardeltase=varde;
vardeltasp=vardp;
#########################
cutoff= (s1hat^2*m2hat-s2hat^2*m1hat-s1hat*s2hat*sqrt((m1hat-m2hat)^2+(s1hat^2-s2hat^2)*log(s1hat^2/s2hat^2)))/(s1hat^2-s2hat^2)
deltasp= (((s1hat^2*m2hat-s2hat^2*m1hat-s1hat*s2hat*sqrt((m1hat-m2hat)^2+(s1hat^2-s2hat^2)*log(s1hat^2./s2hat^2)))/(s1hat^2-s2hat^2))-m1hat)/(s1hat)
deltase= (m2hat-((s1hat^2*m2hat-s2hat^2*m1hat-s1hat*s2hat*sqrt((m1hat-m2hat)^2+(s1hat^2-s2hat^2)*log(s1hat^2/s2hat^2)))/(s1hat^2-s2hat^2)))/s2hat
Se=pnorm(deltase)
Sp=pnorm(deltasp)
Se
Sp
if (plots=="on"){
points(1-Sp,Se, col = "red")
}
covdeltas = matrix( c(vardeltasp, covdedp, covdedp, vardeltase), nrow=2, ncol=2, byrow = TRUE) # fill m
# if (sum(is.infinite(covdeltas)) > 0 |
# sum(is.nan(covdeltas)) > 0) {
# stop("ERROR: The information matrix cannot be inverted - this might be due to the scale of the marker. Try rescaling the marker measurements by subtracting a constant or by dividing with a constant all marker measurements (of both groups) before you try again.")
# }
svdA=svd(inv(covdeltas))
a=sqrt(qchisq(1-alpha,2))/sqrt(svdA$d[1])
b=sqrt(qchisq(1-alpha,2))/sqrt(svdA$d[2])
theta=seq(from=0,to=2*pi+1/40,by=1/40)
state1=a*cos(theta)
state2=b*sin(theta)
states=rbind(state1,state2)
X=svdA$v %*%states
x1=X[1,]+deltasp
x2=X[2,]+deltase
px1=pnorm(x1);
px2=pnorm(x2);
yegg=px2
xegg=1-px1
if (plots=="on"){
lines(xegg,yegg,col="green")
}
#----Area egg---------------
bxegg=c(xegg,xegg[1])
byegg=c(yegg,yegg[1])
ell <- cbind(bxegg, byegg)
areaegg=areapl(ell)
#----End Area egg---------------
margcisp=c(pnorm(deltasp-qnorm(1-alpha/2)*sqrt(vardeltasp)), pnorm(deltasp+qnorm(1-alpha/2)*sqrt(vardeltasp)))
margcise=c(pnorm(deltase-qnorm(1-alpha/2)*sqrt(vardeltase)), pnorm(deltase+qnorm(1-alpha/2)*sqrt(vardeltase)))
#-----Now deal with the rectangle-----
cisp=c(deltasp-qnorm(1-alpha/4)*sqrt(vardeltasp), deltasp+qnorm(1-alpha/4)*sqrt(vardeltasp))
cise=c(deltase-qnorm(1-alpha/4)*sqrt(vardeltase), deltase+qnorm(1-alpha/4)*sqrt(vardeltase))
bsenslb=pnorm(cise[1])#exp(cibootsse[1])/(1+exp(cibootsse[1])) #NOW trans back, going to construct a 95% rectangle for boots
bsensub=pnorm(cise[2])#exp(cibootsse[2])/(1+exp(cibootsse[2])) #NOW trans back, going to construct a 95% rectangle for boots
bspeclb=pnorm(cisp[1])#exp(cibootssp[1])/(1+exp(cibootssp[1])) #NOW trans back, going to construct a 95% rectangle for boots
bspecub=pnorm(cisp[2])#exp(cibootssp[2])/(1+exp(cibootssp[2])) #NOW trans back, going to construct a 95% rectangle for boots
#-----End of the rectangle------------
if (plots=="on"){
#-----Start--plot---Rectangle--------------------
lines(c(1-bspeclb,1-bspecub),c(bsenslb,bsenslb),col="black")
lines(c(1-bspecub,1-bspecub),c(bsenslb,bsensub),col="black")
lines(c(1-bspecub,1-bspeclb),c(bsensub,bsensub),col="black")
lines(c(1-bspeclb,1-bspeclb),c(bsenslb,bsensub),col="black")
#-----End--plot---Rectangle--------------------
}
brectx=c(1-bspeclb,1-bspecub, 1-bspecub,1-bspecub,1-bspecub,1-bspeclb,1-bspeclb,1-bspeclb)
brecty=c(bsenslb,bsenslb ,bsenslb,bsensub, bsensub,bsensub ,bsenslb,bsensub)
brectx=c(brectx, brectx[1]);
brecty=c(brecty, brecty[1]);
re <- cbind(brectx, brecty)
arearect=areapl(re)
#line for the Youden:
if (plots=="on"){
lines(c(1-Sp,1-Sp),c(rocuseless(1-Sp),Se),col="blue")
}
#================AUC================
invAUC= ((m2hat-m1hat)/s2hat)/(sqrt(1+(s1hat/s2hat)^2));
auc=pnorm(invAUC)
#====================AUC1==============================
dm1 =-1/(s2hat*sqrt(1+(s1hat/s2hat)^2));
ds1 = (s1hat*(m1hat-m2hat))/(s2hat^3*(1+(s1hat/s2hat)^2)^(3/2))
dm2 =1/(s2hat*sqrt(1+(s1hat/s2hat)^2));
ds2 =((m1hat-m2hat))/(s2hat^2*(1+(s1hat/s2hat)^2)^(3/2))
I=zeros(5,5);
sh=sqrt(1/(length(transx))*sum((transx-mean(transx))^2))
sd=sqrt(1/(length(transy))*sum((transy-mean(transy))^2))
mh=m1hat;
md=m2hat;
n=n1;
m=n2;
xlam=transx;
ylam=transy;
lam=cc$transformation.parameter
I[1,1]=n/sh^2;
I[2,2]=-(n/sh^2-3/sh^4*sum((xlam-mh)^2));
I[3,3]=m/sd^2;
I[4,4]=-(m/sd^2-3/sd^4*sum((ylam-md)^2));
kk= sum(((mh - (x^lam - 1)/lam)*((2*(x^lam - 1))/lam^3 - (2*x^lam*log(x))/lam^2 + (x^lam*log(x)^2)/lam))/sh^2) + - sum(((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)^2/sd^2) + - sum(((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)^2/sh^2) + + sum(((md - (y^lam - 1)/lam)*((2*(y^lam - 1))/lam^3 - (2*y^lam*log(y))/lam^2 + (y^lam*log(y)^2)/lam))/sd^2);
I[5,5]=-kk ;
I[1,5]=sum(((2*(x^lam - 1))/lam^2 - (2*x^lam*log(x))/lam)/(2*sh^2));
I[2,5]=-sum((2*((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)*(mh - (x^lam - 1)/lam))/sh^3);
I[3,5]=-sum(-((2*(y^lam - 1))/lam^2 - (2*y^lam*log(y))/lam)/(2*sd^2));
I[4,5]=-sum((2*((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)*(md - (y^lam - 1)/lam))/sd^3);
I[5,1]=I[1,5]
I[5,2]=I[2,5]
I[5,3]=I[3,5]
I[5,4]=I[4,5]
S=inv(I)
S=S[1:4,1:4]
## EVERYTHING ELSE ================================================================================
varauc=t(c(dm1,ds1,dm2,ds2))%*% S %*% t(t(c(dm1,ds1,dm2,ds2)))
CIauc=c(invAUC-Za*sqrt(varauc), invAUC+Za*sqrt(varauc))
CIauc=pnorm(CIauc)
CIauc
Zauc=invAUC/sqrt(varauc)
pvalauc=2*pnorm(-abs(Zauc))
#================= DeltaBClamJT================
mx=m1hat;
my=m2hat;
sx=s1hat;
sy=s2hat;
c=cutoff;
b=sy/sx;
a=my-mx;
Jhat=pnorm((m2hat-cutoff)/s2hat)+pnorm((cutoff-m1hat)/s1hat)-1;
radd=a^2+(b^2-1)*sx^2*log(b^2);
zy=(my-c)/sy;
zx=(c-mx)/sx;
dcdmx= (b^2 + a*b*radd^(-1/2)*(-1))/(b^2-1);
dcdsx= (-2*a*b^2)/((b^2-1)^2*sx) + (((b*(b^2+1))*(radd)^(1/2))/((b^2-1)^2*sx) - (sx*b*radd^(-1/2))/(b^2-1)*(log(b^2)+b^(2)-1) );
dcdmy= (-1 + a*b*radd^(-1/2))/(b^2-1);
dcdsy= (2*a*b)/((b^2-1)^2*sx) + (((-b^2-1)*(radd)^(1/2))/((b^2-1)^2*sx) + (sy*b*radd^(-1/2))/(b^2-1)*(log(b^2)+1-b^(-2)) );
d1=-sx^(-1)*dnorm(zx)+dcdmx*(sx^(-1)*dnorm(zx)-(sy^(-1)*dnorm(zy) ));
d2=-zx*sx^(-1)*dnorm(zx)+dcdsx*(sx^(-1)*dnorm(zx)-(sy^(-1)*dnorm(zy) )) ;
d3=sy^(-1)*dnorm(zy)+dcdmy*(sx^(-1)*dnorm(zx)-(sy^(-1)*dnorm(zy) )) ;
d4=-zy*sy^(-1)*dnorm(zy)+dcdsy*(sx^(-1)*dnorm(zx)-(sy^(-1)*dnorm(zy) ));
#varJ=t(c(d1, d2, d3, d4))%*%S%*%[d1 d2 d3 d4]';
varJ=t(c(d1,d2,d3,d4))%*% S %*% t(t(c(d1,d2,d3,d4)))
# Jstar=(Jhat+1)/2;
# Jstar=log(Jstar/(1-Jstar));
# VarJstar=(4/(Jhat^2 - 1)^2)*varJ
Jstar=qnorm(Jhat);
varJstar=(1/(dnorm(qnorm(Jhat))))^2*varJ
ZJstar=Jstar/sqrt(varJstar)
pvalJ=2*pnorm(-abs(ZJstar))
CIstar=c(Jstar-Za*sqrt(varJstar), Jstar+Za*sqrt(varJstar));
CI=c(0,0)
#CI[1]=exp(CIstar[1])/(1+exp(CIstar[1]))*2-1
#CI[2]=exp(CIstar[2])/(1+exp(CIstar[2]))*2-1
CI[1]=pnorm(CIstar[1])
CI[2]=pnorm(CIstar[2])
CIJ=CI;
Jhat
CIwidth=max(CI)-min(CI)
boots=0;c1hat_boots=0;c1hat_boots_or=0; c1hat=cutoff;
#===========CIs for the cutoff==================
for (boots in 1:1000){
#if (boots>=2){print((m1hat_boots*(b^2-1)-a+b*sqrt(a^2+(b^2-1)*s1hat_boots^2*log(b^2)))/(b^2-1))}
#tryCatch({
atx = sample(1:n1,n1,replace=T)
aty = sample(1:n2,n2,replace=T)
if (printProgress) {
if (boots==100){print(boots);print("bootstap samples out of 1000...")}
if (boots==200){print(boots);print("bootstap samples out of 1000...")}
if (boots==300){print(boots);print("bootstap samples out of 1000...")}
if (boots==400){print(boots);print("bootstap samples out of 1000...")}
if (boots==500){print(boots);print("bootstap samples out of 1000...")}
if (boots==600){print(boots);print("bootstap samples out of 1000...")}
if (boots==700){print(boots);print("bootstap samples out of 1000...")}
if (boots==800){print(boots);print("bootstap samples out of 1000...")}
if (boots==900){print(boots);print("bootstap samples out of 1000...")}
if (boots==1000){print(boots);print("Bootstrapping complete. You can request all results through the summary() function.")}
}
xboots=xor[atx]
yboots=yor[aty]
ccb=boxcoxleo2(xboots,yboots)
lam_boots=ccb$transformation.parameter
transx1_boots=ccb$transx;
transx2_boots=ccb$transy;
m1hat_boots=mean(transx1_boots)
m2hat_boots=mean(transx2_boots)
s1hat_boots=sqrt(var(transx1_boots)*(length(transx1_boots)-1)/length(transx1_boots))
s2hat_boots=sqrt(var(transx2_boots)*(length(transx2_boots)-1)/length(transx2_boots))
b=s2hat_boots/s1hat_boots
a=m2hat_boots-m1hat_boots
#print(b)
#print(a)
c1hat_boots[boots]=(m1hat_boots*(b^2-1)-a+b*sqrt(a^2+(b^2-1)*s1hat_boots^2*log(b^2)))/(b^2-1)
kr=(m1hat_boots*(b^2-1)-a+b*sqrt(a^2+(b^2-1)*s1hat_boots^2*log(b^2)))/(b^2-1)
c1hat_boots_or[boots] = ((kr*lam_boots+1))^(1/lam_boots)
#}, error=function(e){})
}
varboots=var(na.omit(c1hat_boots_or))
ccc=((c1hat*lam+1))^(1/lam)
CI_BCAN=c(ccc-Za*sqrt(var(na.omit(c1hat_boots_or))), ccc+Za*sqrt(var(na.omit(c1hat_boots_or))))
CIwidth=max(CI_BCAN)-min(CI_BCAN)
CIcutoff=CI_BCAN;
cutoff=ccc;
allc1=c1hat_boots_or
#CI_BCPB=c(quantile(na.omit(c1hat_boots_or),0.025,"type"=2), quantile(na.omit(c1hat_boots_or),0.975,"type"=2))
#CIwidth=max(CI_BCPB)-min(CI_BCPB)
#CI_BCPB
#legend("bottomright", legend=c("ROC estimate", "max of Youden index (J)", "rectangular 95% confidence region of the optimal operating point", "rectangular 95% confidence region of the optimal operating point"),
# col=c("red", "blue", "black", "green"), lty=c(1,1,1,1), cex=0.8)
if (plots=="on"){
legend("bottomright", legend=c(paste("ROC estimate with AUC = ",
formattable(auc, digits = 4, format = "f"),
", CI: (",
formattable(CIauc[1], digits = 4, format = "f"),
", ",
formattable(CIauc[2], digits = 4, format = "f") ,")", sep = ""),
paste("Maximized Youden index = ", formattable(Jhat, digits = 4, format = "f"),
", CI: (",
formattable((CIJ[1]), digits = 4, format = "f"),
", ",
formattable((CIJ[2]), digits = 4, format = "f"), ")", sep = ""),
paste("Optimal pair of (FPR,TPR): (",
formattable(1-Sp, digits = 4, format = "f"),
", ",
formattable(Se, digits = 4, format = "f"), ")", sep = ""),
paste("Area of the rect. conf. region =", formattable(arearect, digits = 4, format = "f")),
paste("Area of the egg conf. region =", formattable(areaegg, digits = 4, format = "f")),
paste("Youden based cutoff: ", formattable(cutoff, digits = 4, format = "f"),", CI: (",
formattable(CIcutoff[1], digits = 4, format = "f"),
", ",
formattable(CIcutoff[2], digits = 4, format = "f"), ")", sep = ""),
paste("Sp Marginal CI: (",
c(formattable(margcisp[1], digits = 4, format = "f")), ", ", formattable(margcisp[2], digits = 4, format = "f"),
")", sep = ""),
paste("Se Marginal CI: (",
c(formattable(margcise[1], digits = 4, format = "f")), ", ", formattable(margcise[2], digits = 4, format = "f"), ")",
sep = "")),
col=c("red", "blue", "red", "black", "green", "white", "white", "white"),
lty=c(1,1,1,1,1,1,NA, NA),
pch = c(NA, NA, 19, NA, NA, NA, NA, NA), cex=0.6)
}
#return(list(transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam , transformation.parameter=lam, AUC=auc, AUCCI=CIauc, J=Jhat, JCI=CIJ, Sens=Se, CImarginalSens=margcise, Spec=Sp, CImarginalSpec=margcisp, cutoff=cutoff, CIcutoff=CIcutoff, areaegg=areaegg, arearect=arearect, mxlam=mean(transx), sxlam=std(transx), mylam=mean(transy), sylam=std(transy)))
res <- matrix(c(auc,CIauc[1],CIauc[2],
Jhat,CIJ[1],CIJ[2],
cutoff,CIcutoff[1],CIcutoff[2],
Sp,margcisp[1],margcisp[2],
Se,margcise[1],margcise[2],
lam,NA,NA),ncol=3,byrow=TRUE)
colnames(res) <- c("Estimate","Confidence","Interval")
rownames(res) <- c("AUC","Jhat","cutoff","Sp","Se", "Lambda")
res <- data.frame(res)
res <- formattable(as.matrix(res), digits = 4, format = "f")
res
if (auc < 0.5) {
print("NOTE: AUC < 0.5; the ordering of the two groups may need to be reversed.")
}
return(list(transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam , transformation.parameter=lam, AUC=auc, AUCCI=CIauc, pvalueAUC=pvalauc, J=Jhat, JCI=CIJ, pvalueJ=pvalJ, Sens=Se, CImarginalSens=margcise, Spec=Sp, CImarginalSpec=margcisp, cutoff=cutoff, CIcutoff=CIcutoff, areaegg=areaegg, arearect=arearect, mxlam=mean(transx), sxlam=sqrt(var(transx)*(length(transx)-1)/length(transx)), mylam=mean(transy), sylam=sqrt(var(transy)*(length(transy)-1)/length(transy)), results=res , rocfun=rocfun))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
rocboxcoxCI<-function(marker, D, givenSP, givenSE, alpha, plots){
if (plots!="on"){plots="off"}
if (length(marker) != length(D)) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (sum(is.na(marker)) > 0 | sum(is.na(D)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if (alpha <= 0 | alpha >= 1) {
stop("ERROR: The level of significance, alpha, should be set between 0 and 1. A common choice is 0.05.")
} else if (!((sum(is.na(givenSP)) == 0 & sum(is.na(givenSE) > 0)) |
(sum(is.na(givenSE)) == 0 & sum(is.na(givenSP) > 0)))) {
stop("ERROR: Exactly one of 'givenSP' and 'givenSE' must be set to NA.")
} else if (sum(marker < 0) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
#graphics.off()
if (plots!="on"){plots="off"}
xor=marker[D==0]
yor=marker[D==1]
x = xor
y = yor
likbox<-function(x,y,h){
n=length(x);
m=length(y);
out=c();
for (i in 1:length(h)){
#print(i)
if (h[i]==0){
xh=log(x);
yh=log(y);
} else {
xh=((x^h[i])-1)/h[i];
yh=((y^h[i])-1)/h[i];
}
out[i]<-c( -n/2*log(sum((xh-sum(xh)/n)^2)/n) -m/2*log(sum((yh-sum(yh)/m)^2)/m) +(h[i]-1)*(sum(log(x))+sum(log(y))))
}
return(out)
}
boxcoxleo<-function(x,y){
init=1;
logL<-function(h){
-likbox(x,y,h)
}
#lam=fminsearch(logL,init)
#lam=c(lam$optbase$xopt)
lam<- optim(1,logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
transx=((x^lam)-1)/lam
transy=((y^lam)-1)/lam
#test1=print(shapiro.test(transx))
#test2=print(shapiro.test(transy))
return(list(transformation.parameter=lam,transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam ))
}
boxcoxleo2<-function(x,y){
init=1;
logL<-function(h){
-likbox(x,y,h)
}
lam<- optim(1,logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
#lam=fminsearch(logL,init)
#lam=c(lam$optbase$xopt)
transx=((x^lam)-1)/lam
transy=((y^lam)-1)/lam
return(list(transformation.parameter=lam,transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam ))
}
cc=boxcoxleo2(x,y)
transx=cc$transx
transy=cc$transy
#x11()
#qqnorm(x)
#title(main=" for X")
#x11()
#qqnorm(y)
#title(main=" for Y")
roc<-function(t,trans_x,trans_y){
na = length(trans_x)
nb = length(trans_y)
stransx=sqrt(var(trans_x)*(na-1)/na)
stransy=sqrt(var(trans_y)*(nb-1)/nb)
1-pnorm(qnorm(1-t,
mean=mean(trans_x),
sd=stransx),
mean=mean(trans_y),
sd=stransy)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
# x11()
#plot.new()
if (plots == "on") {
txt <- paste("Box-Cox Based ROC, alpha =", round(alpha,3) )
plot(linspace(0,1,1000),roc(linspace(0,1,1000),transx,transy),main=" ",xlab="FPR",ylab="TPR",type="l",col="red")
lines(linspace(0,1,10),linspace(0,1,10),type="l", lty=2)
title(main=txt)
}
m1hat=mean(transx)
m2hat=mean(transy)
# s1hat=std(transx)
# s2hat=std(transy)
n1=length(x)
n2=length(y)
I=zeros(5,5);
sh=sqrt(1/(length(transx))*sum((transx-mean(transx))^2))
sd=sqrt(1/(length(transy))*sum((transy-mean(transy))^2))
s1hat = sh
s2hat = sd
mh=m1hat;
md=m2hat;
n=n1;
m=n2;
xlam=transx;
ylam=transy;
lam=cc$transformation.parameter
I[1,1]=n/sh^2;
I[2,2]=-(n/sh^2-3/sh^4*sum((xlam-mh)^2));
I[3,3]=m/sd^2;
I[4,4]=-(m/sd^2-3/sd^4*sum((ylam-md)^2));
kk= sum(((mh - (x^lam - 1)/lam)*((2*(x^lam - 1))/lam^3 - (2*x^lam*log(x))/lam^2 + (x^lam*log(x)^2)/lam))/sh^2) + - sum(((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)^2/sd^2) + - sum(((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)^2/sh^2) + + sum(((md - (y^lam - 1)/lam)*((2*(y^lam - 1))/lam^3 - (2*y^lam*log(y))/lam^2 + (y^lam*log(y)^2)/lam))/sd^2);
I[5,5]=-kk ;
I[1,5]=sum(((2*(x^lam - 1))/lam^2 - (2*x^lam*log(x))/lam)/(2*sh^2));
I[2,5]=-sum((2*((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)*(mh - (x^lam - 1)/lam))/sh^3);
I[3,5]=-sum(-((2*(y^lam - 1))/lam^2 - (2*y^lam*log(y))/lam)/(2*sd^2));
I[4,5]=-sum((2*((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)*(md - (y^lam - 1)/lam))/sd^3);
I[5,1]=I[1,5]
I[5,2]=I[2,5]
I[5,3]=I[3,5]
I[5,4]=I[4,5]
S=inv(I)
S=S[1:4,1:4]
#=============================================
#GRID LINES:
kk=1;CIROC=c(1,1);CIROCvec1=0;CIROCvec2=0;Sehat=0;
tt=c(linspace(0,1,2000))
for (t in tt){
norminvROC= ((m2hat-m1hat)/s2hat+s1hat/s2hat*qnorm(t))
#====================ROC1==============================
dm1 =-1/s2hat;
ds1 =-(2^(1/2)*erfcinv(2*t))/s2hat;
dm2 =1/s2hat;
ds2 =(m1hat - m2hat)/s2hat^2 + (2^(1/2)*s1hat*erfcinv(2*t))/s2hat^2
vROC=t(c(dm1,ds1,dm2,ds2))%*% S %*% t(t(c(dm1,ds1,dm2,ds2)))
CIinvROC=c(norminvROC-qnorm(1-alpha/2)*sqrt(vROC), norminvROC+qnorm(1-alpha/2)*sqrt(vROC))
CIROC=c( pnorm(CIinvROC[1]), pnorm(CIinvROC[2]))
CIROCvec1[kk]=CIROC[1]
CIROCvec2[kk]=CIROC[2]
Sehat[kk]=pnorm(norminvROC)
#points(t,Sehat[length(Sehat)], col = "red")
#lines(c(t,t),c(CIROC[1],CIROC[1]),col="black",lty=2)
#lines(c(t,t),c(CIROC[2],CIROC[2]),col="black")
kk=kk+1
}
if (plots=="on"){
lines(c(tt),c(CIROCvec1),col="black",lty=2)
lines(c(tt),c(CIROCvec2),col="black",lty=2)
#====END GRID LINES=========================
#=============================================
# x11()
# plot.new()
txt <- paste("Box-Cox Based ROC, alpha =", round(alpha,3) )
plot(linspace(0,1,1000),roc(linspace(0,1,1000),transx,transy),main=" ",xlab="FPR",ylab="TPR",type="l",col="red")
lines(linspace(0,1,10),linspace(0,1,10),type="l", lty=2)
title(main=txt)
}
kk=1;CIROC=c(1,1);CIROCvec1=0;CIROCvec2=0;Sehat=0;
tt=c(givenSP);tt=1-givenSP;
#tt=c(0.05,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9)
for (t in tt){
norminvROC= ((m2hat-m1hat)/s2hat+s1hat/s2hat*qnorm(t))
#====================ROC1==============================
dm1 =-1/s2hat;
ds1 =-(2^(1/2)*erfcinv(2*t))/s2hat;
dm2 =1/s2hat;
ds2 =(m1hat - m2hat)/s2hat^2 + (2^(1/2)*s1hat*erfcinv(2*t))/s2hat^2
vROC=t(c(dm1,ds1,dm2,ds2))%*% S %*% t(t(c(dm1,ds1,dm2,ds2)))
CIinvROC=c(norminvROC-qnorm(1-alpha/2)*sqrt(vROC), norminvROC+qnorm(1-alpha/2)*sqrt(vROC))
CIROC=c( pnorm(CIinvROC[1]), pnorm(CIinvROC[2]))
CIROCvec1[kk]=CIROC[1]
CIROCvec2[kk]=CIROC[2]
Sehat[kk]=pnorm(norminvROC)
if (plots=="on"){
points(t,Sehat[length(Sehat)], col = "green")
lines(c(t,t),c(CIROC[1],CIROC[2]),col="green")
}
kk=kk+1
}
Spvalues=1-tt;FPRvalues=tt;
SEandCIs = matrix( c(Spvalues, Sehat, CIROCvec1,CIROCvec2), nrow=length(CIROCvec1), ncol=4)
colnames(SEandCIs) <- c("Given Sp", "Sehat","LL of 95%CI","UL of 95%CI")
SEandCIs
CIlowSe=CIROCvec1
CIuppSe=CIROCvec2
CIse=t(rbind(CIlowSe,CIuppSe))
#==================ROCinv1 (CIs for Sp)===================
mm1=m1hat;
mm2=m2hat;
ss1=s1hat;
ss2=s2hat;
m1hat=mm2;
m2hat=mm1;
s1hat=ss2;
s2hat=ss1;
kk=1;CIROC=c(1,1);CIROCvec1=0;CIROCvec2=0;Sphat=0;
mh=m1hat;
md=m2hat;
n=n1;
m=n2;
xlam=transx;
ylam=transy;
lam=cc$transformation.parameter
I[1,1]=n/sh^2;
I[2,2]=-(n/sh^2-3/sh^4*sum((xlam-mh)^2));
I[3,3]=m/sd^2;
I[4,4]=-(m/sd^2-3/sd^4*sum((ylam-md)^2));
kk= sum(((mh - (x^lam - 1)/lam)*((2*(x^lam - 1))/lam^3 - (2*x^lam*log(x))/lam^2 + (x^lam*log(x)^2)/lam))/sh^2) + - sum(((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)^2/sd^2) + - sum(((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)^2/sh^2) + + sum(((md - (y^lam - 1)/lam)*((2*(y^lam - 1))/lam^3 - (2*y^lam*log(y))/lam^2 + (y^lam*log(y)^2)/lam))/sd^2);
I[5,5]=-kk ;
I[1,5]=sum(((2*(x^lam - 1))/lam^2 - (2*x^lam*log(x))/lam)/(2*sh^2));
I[2,5]=-sum((2*((x^lam - 1)/lam^2 - (x^lam*log(x))/lam)*(mh - (x^lam - 1)/lam))/sh^3);
I[3,5]=-sum(-((2*(y^lam - 1))/lam^2 - (2*y^lam*log(y))/lam)/(2*sd^2));
I[4,5]=-sum((2*((y^lam - 1)/lam^2 - (y^lam*log(y))/lam)*(md - (y^lam - 1)/lam))/sd^3);
I[5,1]=I[1,5]
I[5,2]=I[2,5]
I[5,3]=I[3,5]
I[5,4]=I[4,5]
S=inv(I)
S=S[1:4,1:4]
pp=c(givenSE);
#pp=c(0.05,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9)
kk=1;CIROC=c(1,1);CIROCvec1=0;CIROCvec2=0;Sphat=0;
for (t in pp){
norminvROC= ((m2hat-m1hat)/s2hat+s1hat/s2hat*qnorm(t))
#====================ROC1==============================
dm1 =-1/s2hat;
ds1 =-(2^(1/2)*erfcinv(2*t))/s2hat;
dm2 =1/s2hat;
ds2 =(m1hat - m2hat)/s2hat^2 + (2^(1/2)*s1hat*erfcinv(2*t))/s2hat^2
vROC=t(c(dm1,ds1,dm2,ds2))%*% S %*% t(t(c(dm1,ds1,dm2,ds2)))
CIinvROC=c(norminvROC-qnorm(1-alpha/2)*sqrt(vROC), norminvROC+qnorm(1-alpha/2)*sqrt(vROC))
CIROC=c( pnorm(CIinvROC[1]), pnorm(CIinvROC[2]))
CIROCvec1[kk]=CIROC[1]
CIROCvec2[kk]=CIROC[2]
Sphat[kk]=pnorm(norminvROC)
if (plots=="on"){
points(Sphat[length(Sphat)],t, col = "purple")
lines(c(CIROC[1],CIROC[2]),c(t,t),col="purple")
}
kk=kk+1
}
#SPandCIs = matrix( c(Sphat, CIROCvec1,CIROCvec2), nrow=length(CIROCvec1), ncol=3)
#colnames(SPandCIs) <- c("Sphat","LL of 95%CI","UL of 95%CI")
#SPandCIs
Sevalues=givenSE;
SPandCIs = matrix( c(Sevalues, Sphat, CIROCvec1,CIROCvec2), nrow=length(CIROCvec1), ncol=4)
colnames(SPandCIs) <- c("Given Se", "Sphat","LL of 95%CI","UL of 95%CI")
SPandCIs
CIlowSp=CIROCvec1
CIuppSp=CIROCvec2
CIsp=t(rbind(CIlowSp,CIuppSp))
return(list(SPandCIs=formattable(as.matrix(SPandCIs), digits = 4, format = "f"),
SEandCIs=formattable(as.matrix(SEandCIs), digits = 4, format = "f"),
Sevalues=Sevalues,
Sphat=Sphat,
CIsp=formattable(as.matrix(CIsp), digits = 4, format = "f"),
Spvalues=Spvalues,
Sehat=Sehat,
CIse=formattable(as.matrix(CIse), digits = 4, format = "f")))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
checkboxcox<-function(marker, D, plots, printShapiro = FALSE){
oldpar <- par(no.readonly = TRUE)
on.exit(par(oldpar))
if (plots!="on"){plots="off"}
if (length(marker) != length(D)) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (sum(is.na(marker)) > 0 | sum(is.na(D)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if (sum(marker < 0) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
statusD=D;
xor=marker[D==0]
yor=marker[D==1]
x = xor
y = yor
scores = marker
# scores = c(x, y)
#====THIS IS THE LIKELIHOOD OF THE BOXCOX TRANSFORMATION=======================================
#INPUT ARGUMENTS: TWO GROUPS (x,y) and plots="on" or "off"====================================
likbox<-function(x,y,h){
n=length(x);
m=length(y);
out=c();
for (i in 1:length(h)){
#print(i)
if (h[i]==0){
xh=log(x);
yh=log(y);
} else {
xh=((x^h[i])-1)/h[i];
yh=((y^h[i])-1)/h[i];
}
out[i]<-c( -n/2*log(sum((xh-sum(xh)/n)^2)/n) -m/2*log(sum((yh-sum(yh)/m)^2)/m) +(h[i]-1)*(sum(log(x))+sum(log(y))))
}
return(out)
}
#====THIS THE FUNCTION NEEDED TO APPLY THE BOXCOX TRANSFORMATION INPUT ARGUMENTS:
#====TWO GROUPS (x,y) and plots="on" or "off"====================================
boxcoxleo<-function(x,y,plots){
if (plots!="on"){plots="off"}
init=1;
logL<-function(h){
-likbox(x,y,h)
}
#lam=fminsearch(logL,init)
# lam <- optimize(logL, c(-100, 100), tol = 0.0001)
# lam=c(lam$minimum) #c(lam$optbase$xopt)
lam<- optim(1,logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
transx=((x^lam)-1)/lam
transy=((y^lam)-1)/lam
if (!printShapiro) {
test1=shapiro.test(x)
test2=shapiro.test(y)
test3=shapiro.test(transx)
test4=shapiro.test(transy)
}
if (printShapiro) {
test1=print(shapiro.test(x))
test2=print(shapiro.test(y))
test3=print(shapiro.test(transx))
test4=print(shapiro.test(transy))
}
pval1=test1$p.value
pval2=test2$p.value
pval3=test3$p.value
pval4=test4$p.value
pvalues=c(pval1,pval2,pval3,pval4)
return(list(transformation.parameter=lam,transx=((x^lam)-1)/lam, transy=((y^lam)-1)/lam,pvalues=pvalues))
}
#====APPLY THE BOXCOX TRANSFORMATION WITH OR WITHOUT THE PLOTS================
cc=boxcoxleo(x,y,plots)
lam=cc$transformation.parameter
pvalues=cc$pvalues
transx=cc$transx #transformed scores for healthy
transy=cc$transy #transformed scores for diseased
if (plots!="on"){plots="off"}
#====IF PLOTS ARE REQUESTED PROVIDE THE HISTOGRAMS AND QQ-PLOTS FOR EACH GROUP
#====BEFORE AND AFTER THE TRANSFORMATION=======================================
if (plots=="on") {
# x11()
par(mfrow=c(2,2))
par(mar = c(4.1, 3.1, 3.1, 1.1))
hist(x, main = "Histogram of X", xlab = "X")
qqnorm(x, main = "Normal Q-Q Plot of X")
qqline(x, col = 2,lwd=2,lty=2)
hist(transx, main = "Histogram of Transformed X", xlab = "Transformed X")
qqnorm(transx, main = "Normal Q-Q Plot of Transformed X")
qqline(transx, col = 2,lwd=2,lty=2)
par(mfrow=c(2,2))
hist(y, main = "Histogram of Y", xlab = "Y")
qqnorm(y, main = "Normal Q-Q Plot of Y")
qqline(y, col = 2,lwd=2,lty=2)
hist(transy, main = "Histogram of Transformed Y", xlab = "Transformed Y")
qqnorm(transy, main = "Normal Q-Q Plot of Transformed Y")
qqline(transy, col = 2,lwd=2,lty=2)
par(mar = c(5.1, 4.1, 4.1, 2.1))
par(mfrow=c(1,1))
}
#====TWO ROC FUNCTIONS: ONE IS THE BOXCOX AND THE OTHER THE REFERENCE LINE================
roc<-function(t){
na = length(transx)
nb = length(transy)
stransx=sqrt( var(transx)*(na-1)/na )
stransy=sqrt( var(transy)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(transx),
sd=stransx),
mean=mean(transy),
sd=stransy)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
#================IF PLOTS ARE REQUESTED PLOT THE ROCS==
if (plots=="on") {
# x11()
par(mfrow = c(1, 1))
plot(linspace(0,1,1000),roc(linspace(0,1,1000)),main="Empirical and Box-Cox Based ROC",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="red")
lines(linspace(0,1,1000),linspace(0,1,1000),type="l", lty=2)
lines(roc.curve(scores, statusD), col=1)
#================LEGEND OF THE ROC PLOT================
legend("bottomright", legend=c(paste("Box-Cox ROC estimate "),
paste("Empirical ROC estimate ")),
col=c("red", "black"), lty=c(1, 1), pch = c(NA, NA), cex=0.8)
}
#================OUTPUT ARGUMENTS======================
return(list(transformation.parameter=lam,
transx=((x^lam)-1)/lam,
transy=((y^lam)-1)/lam,
pval_x=pvalues[1],
pval_y=pvalues[2],
pval_transx=pvalues[3],
pval_transy=pvalues[4],
rocfun=roc ))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
threerocs=function(marker, D, plots)
{
x=marker[D==0];
y=marker[D==1];
# Plot the empirical ROC curve
roc_obj <- roc(D, marker, quiet = TRUE)
auc_value <- auc(roc_obj)
tpr=roc_obj$sensitivities
fpr=1-roc_obj$specificities
if (plots == "on") {
plot(fpr, tpr, type = "l", col = "blue",
xlim = c(0, 1), ylim = c(0, 1), xlab = "FPR = 1 - Specificity",
ylab = "TPR = Sensitivity", main = "Three Estimates of the ROC Curve")
}
t=linspace(0,1,10000) #set a grid for the FPR
############## BOX-COX #############################
roctbc=checkboxcox(marker,D, plots="off", printShapiro = FALSE);
myroc=rocboxcox(marker,D,0.05,plots="off",FALSE)
rocboxc=myroc$rocbc
AUCbc=myroc$AUC
if (plots == "on") {lines(t,roctbc$roc(t),col="red")}
############## Metz #############################
curve2= roc_curves(D,marker,method="binormal")
rocmrm=points(curve2, metric="specificity", values=1-c(t))
rocmrm$TPR
AUCmrm=trapz(t,rocmrm$TPR)
if (plots == "on") {lines(t,rocmrm$TPR,col="forestgreen", lty = "dashed")}
#legend(x = "right", legend=c("one","two","three"))
if (plots == "on") {
legend("bottomright", legend = c(paste("Empirical (AUC = ", formattable(auc_value, digits = 4, format = "f"), ")", sep = ""),
paste("Box-Cox (AUC = ", formattable(AUCbc, digits = 4, format = "f"), ")", sep = ""),
paste("Metz (AUC = ", formattable(AUCmrm, digits = 4, format = "f"), ")", sep = "")),
col = c("blue", "red", "forestgreen"),
lty = c("solid", "dotted", "dashed"),
cex = 0.8)
}
return(list(AUC_Empirical = as.numeric(auc_value),
AUC_Metz = AUCmrm,
AUC_BoxCox = AUCbc))
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
checkboxcox2 <-function (marker1, marker2, D, plots, printShapiro = FALSE){
if ((length(marker1) != length(D)) | (length(marker2) != length(D))) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if ((sum(marker1 < 0) > 0) | (sum(marker2 < 2)) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
erf <- function (x) 2 * pnorm(x * sqrt(2)) - 1
erfc <- function (x) 2 * pnorm(x * sqrt(2), lower.tail = FALSE)
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower.tail = FALSE)/sqrt(2)
if (plots!="on"){plots="off"}
W1a=marker1[D==0]
W1b=marker1[D==1]
W2a=marker2[D==0]
W2b=marker2[D==1]
Wa=cbind(W1a,W2a);
Wb=cbind(W1b,W2b);
na=length(W1a)
nb=length(W1b)
likbox2D<-function(W1a,W1b,W2a,W2b,lam){
na=length(W1a);
nb=length(W1b);
out=c();
#for (i in 1:length(h)){
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
cova= cov(Walam)*(na-1)/na;
covb= cov(Wblam)*(nb-1)/nb;
mualam=c(mean(W1alam), mean(W2alam))
mublam=c(mean(W1blam), mean(W2blam))
out= -sum(log(dmvnorm(Walam,mualam,cova)))-sum(log(dmvnorm(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b));
return(out)
}
lam=c(2,2)
likbox2D(W1a,W1b,W2a,W2b,lam)
logL<-function(h){
likbox2D(W1a,W1b,W2a,W2b,h)
}
# init=c(0.8,0.8)
# lam=fminsearch(logL,init)
# lam=c(lam$optbase$xopt)
lam<- optim(c(1,1),logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
lam
out <- optim(c(1,1), logL, method = "Nelder-Mead")
lam = out$par
###################################
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
m1ahat=mean(W1alam)
m1bhat=mean(W1blam)
m2ahat=mean(W2alam)
m2bhat=mean(W2blam)
s1ahat=sqrt( var(W1alam)*(na-1)/na )
s1bhat=sqrt( var(W1blam)*(nb-1)/nb )
s2ahat=sqrt( var(W2alam)*(na-1)/na )
s2bhat=sqrt( var(W2blam)*(nb-1)/nb )
cova= cov(Walam)*(na-1)/na;cova=cova[1,2];
covb= cov(Wblam)*(nb-1)/nb;covb=covb[1,2];
lam1=lam[1]
lam2=lam[2]
# ==== PERFORM SHAPIRO-WILK TESTS =============================================================
x1 = W1a; y1 = W1b; transx1 = W1alam; transy1 = W1blam
x2 = W2a; y2 = W2b; transx2 = W2alam; transy2 = W2blam
if (!printShapiro) {
test_x1=shapiro.test(x1)
test_y1=shapiro.test(y1)
test_t_x1=shapiro.test(transx1)
test_t_y1=shapiro.test(transy1)
test_x2=shapiro.test(x2)
test_y2=shapiro.test(y2)
test_t_x2=shapiro.test(transx2)
test_t_y2=shapiro.test(transy2)
}
if (printShapiro) {
test_x1=print(shapiro.test(x1))
test_y1=print(shapiro.test(y1))
test_t_x1=print(shapiro.test(transx1))
test_t_y1=print(shapiro.test(transy1))
test_x2=print(shapiro.test(x2))
test_y2=print(shapiro.test(y2))
test_t_x2=print(shapiro.test(transx2))
test_t_y2=print(shapiro.test(transy2))
}
pval_x1=test_x1$p.value
pval_y1=test_y1$p.value
pval_t_x1=test_t_x1$p.value
pval_t_y1=test_t_y1$p.value
pval_x2=test_x2$p.value
pval_y2=test_y2$p.value
pval_t_x2=test_t_x2$p.value
pval_t_y2=test_t_y2$p.value
pvalues=c(pval_x1, pval_y1, pval_t_x1, pval_t_y1,
pval_x2, pval_y2, pval_t_x2, pval_t_y2)
res_shapiro = list(test_x1 = test_x1,
test_y1 = test_y1,
test_t_x1 = test_t_x1,
test_t_y1 = test_t_y1,
test_x2 = test_x2,
test_y2 = test_y2,
test_t_x2 = test_t_x2,
test_t_y2 = test_t_y2)
#====TWO ROC FUNCTIONS: ONE IS THE BOXCOX AND THE OTHER THE REFERENCE LINE================
roc1<-function(t){
na = length(W1alam)
nb = length(W1blam)
s1alam=sqrt( var(W1alam)*(na-1)/na )
s1blam=sqrt( var(W1blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W1alam),
sd=s1alam),
mean=mean(W1blam),
sd=s1blam)
}
roc2<-function(t){
na = length(W2alam)
nb = length(W2blam)
s2alam=sqrt( var(W2alam)*(na-1)/na )
s2blam=sqrt( var(W2blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W2alam),
sd=s2alam),
mean=mean(W2blam),
sd=s2blam)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
#================IF PLOTS ARE REQUESTED PLOT ALL OF THE THINGS====
if (plots=="on") {
par(mfrow=c(2,2))
par(mar = c(4.1, 3.1, 3.1, 1.1))
# HISTOGRAMS & QQ, X1
hist(W1a, main = "Histogram of X1", xlab = "X1")
qqnorm(W1a, main = "Normal Q-Q Plot of X1")
qqline(W1a, col = 2,lwd=2,lty=2)
hist(W1alam, main = "Histogram of Transformed X1", xlab = "Transformed X1")
qqnorm(W1alam, main = "Normal Q-Q Plot of Transformed X1")
qqline(W1alam, col = 2,lwd=2,lty=2)
# HISTOGRAMS & QQ, Y1
hist(W1b, main = "Histogram of Y1", xlab = "Y1")
qqnorm(W1b, main = "Normal Q-Q Plot of Y1")
qqline(W1b, col = 2,lwd=2,lty=2)
hist(W1blam, main = "Histogram of Transformed Y1", xlab = "Transformed Y1")
qqnorm(W1blam, main = "Normal Q-Q Plot of Transformed Y1")
qqline(W1blam, col = 2,lwd=2,lty=2)
# HISTOGRAMS & QQ, X2
hist(W2a, main = "Histogram of X2", xlab = "X2")
qqnorm(W2a, main = "Normal Q-Q Plot of X2")
qqline(W2a, col = 2,lwd=2,lty=2)
hist(W2alam, main = "Histogram of Transformed X2", xlab = "Transformed X2")
qqnorm(W2alam, main = "Normal Q-Q Plot of Transformed X2")
qqline(W2alam, col = 2,lwd=2,lty=2)
# HISTOGRAMS & QQ, Y2
hist(W2b, main = "Histogram of Y2", xlab = "Y2")
qqnorm(W2b, main = "Normal Q-Q Plot of Y2")
qqline(W2b, col = 2,lwd=2,lty=2)
hist(W2blam, main = "Histogram of Transformed Y2", xlab = "Transformed Y2")
qqnorm(W2blam, main = "Normal Q-Q Plot of Transformed Y2")
qqline(W2blam, col = 2,lwd=2,lty=2)
par(mar = c(5.1, 4.1, 4.1, 2.1))
# ROC PLOT 1
par(mfrow = c(1, 2))
plot(linspace(0,1,1000),
roc1(linspace(0,1,1000)),
main="Marker1",
xlab="FPR = 1 - Specificity",
ylab="TPR = Sensitivity",
type="l",
col="red")
lines(linspace(0,1,1000),
linspace(0,1,1000),
type="l",
lty=2)
lines(roc.curve(marker1, D), col="red", lty = 3, lwd = 1.5)
# LEGEND
legend("bottomright", legend=c(paste("Box-Cox ROC Estimate"),
paste("Empirical ROC Estimate")),
col=c("red", "red"), lty=c(1, 3), lwd = c(1, 1.5), pch = c(NA, NA), cex=0.8)
# ROC PLOT 2
plot(linspace(0,1,1000),
roc2(linspace(0,1,1000)),
main="Marker2",
xlab="FPR = 1 - Specificity",
ylab="TPR = Sensitivity",
type="l",
col="black")
lines(linspace(0,1,1000),
linspace(0,1,1000),
type="l",
lty=2)
lines(roc.curve(marker2, D), col="black", lty = 3, lwd = 1.5)
# LEGEND
legend("bottomright", legend=c(paste("Box-Cox ROC estimate"),
paste("Empirical ROC estimate")),
col=c("black", "black"), lty=c(1, 3), lwd = c(1, 1.5), pch = c(NA, NA), cex=0.8)
par(mfrow = c(1, 1))
}
return(list(res_shapiro = res_shapiro,
transformation.parameter.1 = lam[1],
transx1 = W1alam,
transy1 = W1blam,
transformation.parameter.2 = lam[2],
transx2 = W2alam,
transy2 = W2blam,
pval_x1 = pvalues[1],
pval_y1 = pvalues[2],
pval_transx1 = pvalues[3],
pval_transy1 = pvalues[4],
pval_x2 = pvalues[5],
pval_y2 = pvalues[6],
pval_transx2 = pvalues[7],
pval_transy2 = pvalues[8],
roc1 = roc1,
roc2 = roc2))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
comparebcAUC <-function (marker1, marker2, D, alpha, plots){
if ((length(marker1) != length(D)) | (length(marker2) != length(D))) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (alpha <= 0 | alpha >= 1) {
stop("ERROR: The level of significance, alpha, should be set between 0 and 1. A common choice is 0.05.")
} else if (sum(is.na(marker1)) > 0 | sum(is.na(marker2)) > 0 | sum(is.na(D)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if ((sum(marker1 < 0) > 0) | (sum(marker2 < 2)) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
erf <- function (x) 2 * pnorm(x * sqrt(2)) - 1
erfc <- function (x) 2 * pnorm(x * sqrt(2), lower.tail = FALSE)
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower.tail = FALSE)/sqrt(2)
pmalpha=qnorm(1-alpha/2);
if (plots!="on"){plots="off"}
W1a=marker1[D==0]
W1b=marker1[D==1]
W2a=marker2[D==0]
W2b=marker2[D==1]
Wa=cbind(W1a,W2a);
Wb=cbind(W1b,W2b);
na=length(W1a)
nb=length(W1b)
likbox2D<-function(W1a,W1b,W2a,W2b,lam){
na=length(W1a);
nb=length(W1b);
out=c();
#for (i in 1:length(h)){
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
#cova=1/na*sum((W1alam-mean(W1alam))*(W2alam-mean(W2alam)))
#covb=1/nb*sum((W1blam-mean(W1blam))*(W2blam-mean(W2blam)))
cova= cov(Walam)*(na-1)/na;
covb= cov(Wblam)*(nb-1)/nb;
mualam=c(mean(W1alam), mean(W2alam))
mublam=c(mean(W1blam), mean(W2blam))
#dmvn(c(0,0), mu, mcov, log = F)
# out= (-sum(log(dmvn(Walam,mualam,cova)))-sum(log(dmvn(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b)));
out= -sum(log(dmvnorm(Walam,mualam,cova)))-sum(log(dmvnorm(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b));
#out1=-sum(log(dmvnorm(Walam,mualam,cova)))
#out2=-sum(log(dmvnorm(Wblam,mublam,covb)))
#out3=-(lam[1]-1)*sum(log(W1a))
#out4=-(lam[2]-1)*sum(log(W2a))
#out5=-(lam[1]-1)*sum(log(W1b))
#out6=-(lam[2]-1)*sum(log(W2b))
#out=c(out,out1,out2,out3,out4,out5,out6, cova, covb)
#out= (-sum(log(mvnpdf2([W1alam(g(1)) W2alam(g(2))],[mean(W1alam(g(1))) mean(W2alam(g(2)))],[std(W1alam(g(1)),1).^2 cova;cova (std(W2alam(g(2)),1)).^2])))+...
# -sum(log(mvnpdf2([W1blam(g(1)) W2blam(g(2))],[mean(W1blam(g(1))) mean(W2blam(g(2)))],[std(W1blam(g(1)),1).^2 covb;covb (std(W2blam(g(2)),1)).^2])))+...
# -(g(1)-1).*sum(log(W1a))-(g(2)-1).*sum(log(W2a))-(g(1)-1).*sum(log(W1b))-(g(2)-1).*sum(log(W2b)));
return(out)
}
lam=c(2,2)
likbox2D(W1a,W1b,W2a,W2b,lam)
logL<-function(h){
likbox2D(W1a,W1b,W2a,W2b,h)
}
# init=c(0.8,0.8)
# lam=fminsearch(logL,init)
# lam=c(lam$optbase$xopt)
lam<- optim(c(1,1),logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
lam
#init=c(-10,10)
#lam=optimize(logL,init)
#lam
#f <- function (x, a) (x - a)^2
#xmin <- optimize(f, c(0, 1), tol = 0.0001, a = 1/3)
#xmin
out <- optim(c(1,1), logL, method = "Nelder-Mead")
lam = out$par
###################################
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
m1ahat=mean(W1alam)
m1bhat=mean(W1blam)
m2ahat=mean(W2alam)
m2bhat=mean(W2blam)
s1ahat=sqrt( var(W1alam)*(na-1)/na )
s1bhat=sqrt( var(W1blam)*(nb-1)/nb )
s2ahat=sqrt( var(W2alam)*(na-1)/na )
s2bhat=sqrt( var(W2blam)*(nb-1)/nb )
cova= cov(Walam)*(na-1)/na;cova=cova[1,2];
covb= cov(Wblam)*(nb-1)/nb;covb=covb[1,2];
lam1=lam[1]
lam2=lam[2]
##################################
h1_1=sum(-2/(2*s1ahat^2 - (2*cova^2)/s2ahat^2))
h1_2=sum(-(2*s1ahat*s2ahat^2*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_3=0
h1_4=0;
h1_5=sum(cova/(- cova^2 + s1ahat^2*s2ahat^2));
h1_6=sum((s2ahat*(2*cova^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat) - 2*cova*s1ahat^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_7=0
h1_8=0
h1_9=sum((cova^2*(lam2*m2ahat - W2a^lam2 + 1) + s2ahat^2*((lam2*m2ahat - W2a^lam2 + 1)*s1ahat^2 - 2*cova*lam2*m1ahat))/(lam2*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s2ahat^2*(2*lam2 - 2*W1a^lam1*lam2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_10=0
h1_11=sum((2*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h1_12=sum(-(cova*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h2_1=sum(-(2*s1ahat*s2ahat^2*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_2=sum(- (3*s1ahat^2*s2ahat^6 + cova^4*(lam1^2*m2ahat^2 + lam1^2*s2ahat^2) - cova^3*(2*lam1*m2ahat*s2ahat^2 - 2*W1a^lam1*lam1*m2ahat*s2ahat^2 + 2*lam1^2*m1ahat*m2ahat*s2ahat^2) - cova*(6*lam1*m2ahat*s1ahat^2*s2ahat^4 - 6*W1a^lam1*lam1*m2ahat*s1ahat^2*s2ahat^4 + 6*lam1^2*m1ahat*m2ahat*s1ahat^2*s2ahat^4) + cova^2*(W1a^(2*lam1)*s2ahat^4 - 2*W1a^lam1*s2ahat^4 + s2ahat^4 + 2*lam1*m1ahat*s2ahat^4 + lam1^2*m1ahat^2*s2ahat^4 - 2*W1a^lam1*lam1*m1ahat*s2ahat^4 + 3*lam1^2*m2ahat^2*s1ahat^2*s2ahat^2) - 6*W1a^lam1*s1ahat^2*s2ahat^6 + 3*W1a^(2*lam1)*s1ahat^2*s2ahat^6 - lam1^2*s1ahat^4*s2ahat^6 + 3*lam1^2*m1ahat^2*s1ahat^2*s2ahat^6 + 6*lam1*m1ahat*s1ahat^2*s2ahat^6 - 6*W1a^lam1*lam1*m1ahat*s1ahat^2*s2ahat^6)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^4*(W2a^(2*lam2)*lam1^2 - 2*W2a^lam2*lam1^2 + lam1^2) + cova^2*(3*lam1^2*s1ahat^2*s2ahat^2 + 3*W2a^(2*lam2)*lam1^2*s1ahat^2*s2ahat^2 - 6*W2a^lam2*lam1^2*s1ahat^2*s2ahat^2) - lam2*((2*W2a^lam2*lam1^2*m2ahat - 2*lam1^2*m2ahat)*cova^4 + (2*lam1*s2ahat^2 + 2*lam1^2*m1ahat*s2ahat^2 - 2*W1a^lam1*lam1*s2ahat^2 - 2*W2a^lam2*lam1*s2ahat^2 - 2*W2a^lam2*lam1^2*m1ahat*s2ahat^2 + 2*W1a^lam1*W2a^lam2*lam1*s2ahat^2)*cova^3 + (6*W2a^lam2*lam1^2*m2ahat*s1ahat^2*s2ahat^2 - 6*lam1^2*m2ahat*s1ahat^2*s2ahat^2)*cova^2 + (6*lam1*s1ahat^2*s2ahat^4 + 6*lam1^2*m1ahat*s1ahat^2*s2ahat^4 - 6*W1a^lam1*lam1*s1ahat^2*s2ahat^4 - 6*W2a^lam2*lam1*s1ahat^2*s2ahat^4 - 6*W2a^lam2*lam1^2*m1ahat*s1ahat^2*s2ahat^4 + 6*W1a^lam1*W2a^lam2*lam1*s1ahat^2*s2ahat^4)*cova))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_3=sum(0)
h2_4=sum(0)
h2_5=sum((s1ahat*(2*cova^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat) - 2*cova*s2ahat^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_6=sum((lam2*((2*cova*s2ahat^3*(2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat) - 2*cova*s2ahat*(4*cova*lam1^2*m2ahat - 4*W2a^lam2*cova*lam1^2*m2ahat))*s1ahat^3 + 2*cova*s2ahat*(2*cova^2*lam1 + 2*cova^2*lam1^2*m1ahat - 2*W1a^lam1*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1^2*m1ahat + 2*W1a^lam1*W2a^lam2*cova^2*lam1)*s1ahat) - 2*cova*s1ahat^3*s2ahat*(2*cova*lam1^2 + 2*W2a^(2*lam2)*cova*lam1^2 - 4*W2a^lam2*cova*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - ((4*cova^2*lam1^2*m2ahat^2*s2ahat - 2*cova*s2ahat^3*(2*lam1*m2ahat + cova*lam1^2 + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat))*s1ahat^3 + (2*cova*(2*cova + 2*W1a^(2*lam1)*cova - 4*W1a^lam1*cova + 2*cova*lam1^2*m1ahat^2 + 4*cova*lam1*m1ahat - 4*W1a^lam1*cova*lam1*m1ahat)*s2ahat^3 + 2*cova*(cova^3*lam1^2 - 2*cova^2*lam1*m2ahat + 2*W1a^lam1*cova^2*lam1*m2ahat - 2*cova^2*lam1^2*m1ahat*m2ahat)*s2ahat)*s1ahat)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_7=sum(0)
h2_8=sum(0)
h2_9=sum(- (cova*(s2ahat^4*(- 4*lam2^2*m1ahat^2*s1ahat + 2*lam2^2*s1ahat^3) - s1ahat^3*s2ahat^2*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2)) - cova^3*(s1ahat*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2) + 2*lam2^2*s1ahat*s2ahat^2) + s1ahat^3*s2ahat^4*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + cova^2*s1ahat*s2ahat^2*(6*lam2*m1ahat + 6*lam2^2*m1ahat*m2ahat - 6*W2a^lam2*lam2*m1ahat))/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2ahat - 6*W1a^lam1*lam2 - 6*W2a^lam2*lam2 + 6*W1a^lam1*W2a^lam2*lam2 - 6*W1a^lam1*lam2^2*m2ahat)*cova^2*s1ahat*s2ahat^2 + (8*W1a^lam1*lam2^2*m1ahat - 8*lam2^2*m1ahat)*cova*s1ahat*s2ahat^4 + (2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat)*s1ahat^3*s2ahat^4) - cova*s1ahat*s2ahat^4*(4*W1a^(2*lam1)*lam2^2 - 8*W1a^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_10=sum(0)
h2_11=sum((2*s1ahat*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_12=sum(-(2*cova*s1ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h3_1=sum(0)
h3_2=sum(0)
h3_3=sum(-2/(2*s1bhat^2 - (2*covb^2)/s2bhat^2))
h3_4=sum(-(2*s1bhat*s2bhat^2*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_5=sum(0)
h3_6=sum(0)
h3_7=sum(covb/(- covb^2 + s1bhat^2*s2bhat^2))
h3_8=sum((s2bhat*(2*covb^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat) - 2*covb*s1bhat^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_9=sum(0)
h3_10=sum((covb^2*(lam2*m2bhat - W2b^lam2 + 1) + s2bhat^2*((lam2*m2bhat - W2b^lam2 + 1)*s1bhat^2 - 2*covb*lam2*m1bhat))/(lam2*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s2bhat^2*(2*lam2 - 2*W1b^lam1*lam2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_11=sum((2*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h3_12=sum(-(covb*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h4_1=sum(0)
h4_2=sum(0)
h4_3=sum(-(2*s1bhat*s2bhat^2*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_4=sum(- (3*s1bhat^2*s2bhat^6 + covb^4*(lam1^2*m2bhat^2 + lam1^2*s2bhat^2) - covb^3*(2*lam1*m2bhat*s2bhat^2 - 2*W1b^lam1*lam1*m2bhat*s2bhat^2 + 2*lam1^2*m1bhat*m2bhat*s2bhat^2) - covb*(6*lam1*m2bhat*s1bhat^2*s2bhat^4 - 6*W1b^lam1*lam1*m2bhat*s1bhat^2*s2bhat^4 + 6*lam1^2*m1bhat*m2bhat*s1bhat^2*s2bhat^4) + covb^2*(W1b^(2*lam1)*s2bhat^4 - 2*W1b^lam1*s2bhat^4 + s2bhat^4 + 2*lam1*m1bhat*s2bhat^4 + lam1^2*m1bhat^2*s2bhat^4 - 2*W1b^lam1*lam1*m1bhat*s2bhat^4 + 3*lam1^2*m2bhat^2*s1bhat^2*s2bhat^2) - 6*W1b^lam1*s1bhat^2*s2bhat^6 + 3*W1b^(2*lam1)*s1bhat^2*s2bhat^6 - lam1^2*s1bhat^4*s2bhat^6 + 3*lam1^2*m1bhat^2*s1bhat^2*s2bhat^6 + 6*lam1*m1bhat*s1bhat^2*s2bhat^6 - 6*W1b^lam1*lam1*m1bhat*s1bhat^2*s2bhat^6)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^4*(W2b^(2*lam2)*lam1^2 - 2*W2b^lam2*lam1^2 + lam1^2) + covb^2*(3*lam1^2*s1bhat^2*s2bhat^2 + 3*W2b^(2*lam2)*lam1^2*s1bhat^2*s2bhat^2 - 6*W2b^lam2*lam1^2*s1bhat^2*s2bhat^2) - lam2*((2*W2b^lam2*lam1^2*m2bhat - 2*lam1^2*m2bhat)*covb^4 + (2*lam1*s2bhat^2 + 2*lam1^2*m1bhat*s2bhat^2 - 2*W1b^lam1*lam1*s2bhat^2 - 2*W2b^lam2*lam1*s2bhat^2 - 2*W2b^lam2*lam1^2*m1bhat*s2bhat^2 + 2*W1b^lam1*W2b^lam2*lam1*s2bhat^2)*covb^3 + (6*W2b^lam2*lam1^2*m2bhat*s1bhat^2*s2bhat^2 - 6*lam1^2*m2bhat*s1bhat^2*s2bhat^2)*covb^2 + (6*lam1*s1bhat^2*s2bhat^4 + 6*lam1^2*m1bhat*s1bhat^2*s2bhat^4 - 6*W1b^lam1*lam1*s1bhat^2*s2bhat^4 - 6*W2b^lam2*lam1*s1bhat^2*s2bhat^4 - 6*W2b^lam2*lam1^2*m1bhat*s1bhat^2*s2bhat^4 + 6*W1b^lam1*W2b^lam2*lam1*s1bhat^2*s2bhat^4)*covb))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_5=sum(0)
h4_6=sum(0)
h4_7=sum((s1bhat*(2*covb^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat) - 2*covb*s2bhat^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_8=sum((lam2*((2*covb*s2bhat^3*(2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat) - 2*covb*s2bhat*(4*covb*lam1^2*m2bhat - 4*W2b^lam2*covb*lam1^2*m2bhat))*s1bhat^3 + 2*covb*s2bhat*(2*covb^2*lam1 + 2*covb^2*lam1^2*m1bhat - 2*W1b^lam1*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1^2*m1bhat + 2*W1b^lam1*W2b^lam2*covb^2*lam1)*s1bhat) - 2*covb*s1bhat^3*s2bhat*(2*covb*lam1^2 + 2*W2b^(2*lam2)*covb*lam1^2 - 4*W2b^lam2*covb*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - ((4*covb^2*lam1^2*m2bhat^2*s2bhat - 2*covb*s2bhat^3*(2*lam1*m2bhat + covb*lam1^2 + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat))*s1bhat^3 + (2*covb*(2*covb + 2*W1b^(2*lam1)*covb - 4*W1b^lam1*covb + 2*covb*lam1^2*m1bhat^2 + 4*covb*lam1*m1bhat - 4*W1b^lam1*covb*lam1*m1bhat)*s2bhat^3 + 2*covb*(covb^3*lam1^2 - 2*covb^2*lam1*m2bhat + 2*W1b^lam1*covb^2*lam1*m2bhat - 2*covb^2*lam1^2*m1bhat*m2bhat)*s2bhat)*s1bhat)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_9=sum(0)
h4_10=sum(- (covb*(s2bhat^4*(- 4*lam2^2*m1bhat^2*s1bhat + 2*lam2^2*s1bhat^3) - s1bhat^3*s2bhat^2*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2)) - covb^3*(s1bhat*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2) + 2*lam2^2*s1bhat*s2bhat^2) + s1bhat^3*s2bhat^4*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + covb^2*s1bhat*s2bhat^2*(6*lam2*m1bhat + 6*lam2^2*m1bhat*m2bhat - 6*W2b^lam2*lam2*m1bhat))/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2bhat - 6*W1b^lam1*lam2 - 6*W2b^lam2*lam2 + 6*W1b^lam1*W2b^lam2*lam2 - 6*W1b^lam1*lam2^2*m2bhat)*covb^2*s1bhat*s2bhat^2 + (8*W1b^lam1*lam2^2*m1bhat - 8*lam2^2*m1bhat)*covb*s1bhat*s2bhat^4 + (2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat)*s1bhat^3*s2bhat^4) - covb*s1bhat*s2bhat^4*(4*W1b^(2*lam1)*lam2^2 - 8*W1b^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_11=sum((2*s1bhat*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_12=sum(-(2*covb*s1bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h5_1=sum(cova/(- cova^2 + s1ahat^2*s2ahat^2))
h5_2=sum((s1ahat*(2*cova^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat) - 2*cova*s2ahat^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_3=0
h5_4=0
h5_5=sum(-2/(2*s2ahat^2 - (2*cova^2)/s1ahat^2))
h5_6=sum(-(2*s1ahat^2*s2ahat*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_7=0
h5_8=0
h5_9=sum((cova^2*(lam1*m1ahat - W1a^lam1 + 1) + s1ahat^2*((lam1*m1ahat - W1a^lam1 + 1)*s2ahat^2 - 2*cova*lam1*m2ahat))/(lam1*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s1ahat^2*(2*lam1 - 2*W2a^lam2*lam1))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_10=0
h5_11=sum(-(cova*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h5_12=sum((2*s1ahat^2*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h6_1=sum((s2ahat*(2*cova^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat) - 2*cova*s1ahat^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_2=sum((lam2*((2*cova*s2ahat^3*(2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat) - 2*cova*s2ahat*(4*cova*lam1^2*m2ahat - 4*W2a^lam2*cova*lam1^2*m2ahat))*s1ahat^3 + 2*cova*s2ahat*(2*cova^2*lam1 + 2*cova^2*lam1^2*m1ahat - 2*W1a^lam1*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1^2*m1ahat + 2*W1a^lam1*W2a^lam2*cova^2*lam1)*s1ahat) - 2*cova*s1ahat^3*s2ahat*(2*cova*lam1^2 + 2*W2a^(2*lam2)*cova*lam1^2 - 4*W2a^lam2*cova*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - ((4*cova^2*lam1^2*m2ahat^2*s2ahat - 2*cova*s2ahat^3*(2*lam1*m2ahat + cova*lam1^2 + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat))*s1ahat^3 + (2*cova*(2*cova + 2*W1a^(2*lam1)*cova - 4*W1a^lam1*cova + 2*cova*lam1^2*m1ahat^2 + 4*cova*lam1*m1ahat - 4*W1a^lam1*cova*lam1*m1ahat)*s2ahat^3 + 2*cova*(cova^3*lam1^2 - 2*cova^2*lam1*m2ahat + 2*W1a^lam1*cova^2*lam1*m2ahat - 2*cova^2*lam1^2*m1ahat*m2ahat)*s2ahat)*s1ahat)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_3=0
h6_4=0
h6_5=sum(-(2*s1ahat^2*s2ahat*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_6=sum(- (3*s1ahat^6*s2ahat^2 + cova^4*(lam2^2*m1ahat^2 + lam2^2*s1ahat^2) - cova^3*(2*lam2*m1ahat*s1ahat^2 - 2*W2a^lam2*lam2*m1ahat*s1ahat^2 + 2*lam2^2*m1ahat*m2ahat*s1ahat^2) - cova*(6*lam2*m1ahat*s1ahat^4*s2ahat^2 - 6*W2a^lam2*lam2*m1ahat*s1ahat^4*s2ahat^2 + 6*lam2^2*m1ahat*m2ahat*s1ahat^4*s2ahat^2) + cova^2*(W2a^(2*lam2)*s1ahat^4 - 2*W2a^lam2*s1ahat^4 + s1ahat^4 + 2*lam2*m2ahat*s1ahat^4 + lam2^2*m2ahat^2*s1ahat^4 - 2*W2a^lam2*lam2*m2ahat*s1ahat^4 + 3*lam2^2*m1ahat^2*s1ahat^2*s2ahat^2) - 6*W2a^lam2*s1ahat^6*s2ahat^2 + 3*W2a^(2*lam2)*s1ahat^6*s2ahat^2 - lam2^2*s1ahat^6*s2ahat^4 + 3*lam2^2*m2ahat^2*s1ahat^6*s2ahat^2 + 6*lam2*m2ahat*s1ahat^6*s2ahat^2 - 6*W2a^lam2*lam2*m2ahat*s1ahat^6*s2ahat^2)/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^4*(W1a^(2*lam1)*lam2^2 - 2*W1a^lam1*lam2^2 + lam2^2) + cova^2*(3*lam2^2*s1ahat^2*s2ahat^2 + 3*W1a^(2*lam1)*lam2^2*s1ahat^2*s2ahat^2 - 6*W1a^lam1*lam2^2*s1ahat^2*s2ahat^2) - lam1*((2*W1a^lam1*lam2^2*m1ahat - 2*lam2^2*m1ahat)*cova^4 + (2*lam2*s1ahat^2 + 2*lam2^2*m2ahat*s1ahat^2 - 2*W1a^lam1*lam2*s1ahat^2 - 2*W2a^lam2*lam2*s1ahat^2 - 2*W1a^lam1*lam2^2*m2ahat*s1ahat^2 + 2*W1a^lam1*W2a^lam2*lam2*s1ahat^2)*cova^3 + (6*W1a^lam1*lam2^2*m1ahat*s1ahat^2*s2ahat^2 - 6*lam2^2*m1ahat*s1ahat^2*s2ahat^2)*cova^2 + (6*lam2*s1ahat^4*s2ahat^2 + 6*lam2^2*m2ahat*s1ahat^4*s2ahat^2 - 6*W1a^lam1*lam2*s1ahat^4*s2ahat^2 - 6*W2a^lam2*lam2*s1ahat^4*s2ahat^2 - 6*W1a^lam1*lam2^2*m2ahat*s1ahat^4*s2ahat^2 + 6*W1a^lam1*W2a^lam2*lam2*s1ahat^4*s2ahat^2)*cova))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_7=0
h6_8=0
h6_9=sum(- (cova*(s1ahat^4*(- 4*lam1^2*m2ahat^2*s2ahat + 2*lam1^2*s2ahat^3) - s1ahat^2*s2ahat^3*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2)) - cova^3*(s2ahat*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2) + 2*lam1^2*s1ahat^2*s2ahat) + s1ahat^4*s2ahat^3*(2*lam1*m2ahat + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat) + cova^2*s1ahat^2*s2ahat*(6*lam1*m2ahat + 6*lam1^2*m1ahat*m2ahat - 6*W1a^lam1*lam1*m2ahat))/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1ahat - 6*W1a^lam1*lam1 - 6*W2a^lam2*lam1 + 6*W1a^lam1*W2a^lam2*lam1 - 6*W2a^lam2*lam1^2*m1ahat)*cova^2*s1ahat^2*s2ahat + (8*W2a^lam2*lam1^2*m2ahat - 8*lam1^2*m2ahat)*cova*s1ahat^4*s2ahat + (2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat)*s1ahat^4*s2ahat^3) - cova*s1ahat^4*s2ahat*(4*W2a^(2*lam2)*lam1^2 - 8*W2a^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_10=0
h6_11=sum(-(2*cova*s2ahat*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_12=sum((2*s1ahat^2*s2ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h7_1=0
h7_2=0
h7_3=sum(covb/(- covb^2 + s1bhat^2*s2bhat^2))
h7_4=sum((s1bhat*(2*covb^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat) - 2*covb*s2bhat^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_5=0
h7_6=0
h7_7=sum(-2/(2*s2bhat^2 - (2*covb^2)/s1bhat^2))
h7_8=sum(-(2*s1bhat^2*s2bhat*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_9=0
h7_10=sum((covb^2*(lam1*m1bhat - W1b^lam1 + 1) + s1bhat^2*((lam1*m1bhat - W1b^lam1 + 1)*s2bhat^2 - 2*covb*lam1*m2bhat))/(lam1*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s1bhat^2*(2*lam1 - 2*W2b^lam2*lam1))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_11=sum(-(covb*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h7_12=sum((2*s1bhat^2*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h8_1=0
h8_2=0
h8_3=sum((s2bhat*(2*covb^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat) - 2*covb*s1bhat^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_4=sum((lam2*((2*covb*s2bhat^3*(2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat) - 2*covb*s2bhat*(4*covb*lam1^2*m2bhat - 4*W2b^lam2*covb*lam1^2*m2bhat))*s1bhat^3 + 2*covb*s2bhat*(2*covb^2*lam1 + 2*covb^2*lam1^2*m1bhat - 2*W1b^lam1*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1^2*m1bhat + 2*W1b^lam1*W2b^lam2*covb^2*lam1)*s1bhat) - 2*covb*s1bhat^3*s2bhat*(2*covb*lam1^2 + 2*W2b^(2*lam2)*covb*lam1^2 - 4*W2b^lam2*covb*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - ((4*covb^2*lam1^2*m2bhat^2*s2bhat - 2*covb*s2bhat^3*(2*lam1*m2bhat + covb*lam1^2 + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat))*s1bhat^3 + (2*covb*(2*covb + 2*W1b^(2*lam1)*covb - 4*W1b^lam1*covb + 2*covb*lam1^2*m1bhat^2 + 4*covb*lam1*m1bhat - 4*W1b^lam1*covb*lam1*m1bhat)*s2bhat^3 + 2*covb*(covb^3*lam1^2 - 2*covb^2*lam1*m2bhat + 2*W1b^lam1*covb^2*lam1*m2bhat - 2*covb^2*lam1^2*m1bhat*m2bhat)*s2bhat)*s1bhat)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_5=0
h8_6=0
h8_7=sum(-(2*s1bhat^2*s2bhat*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_8=sum(- (3*s1bhat^6*s2bhat^2 + covb^4*(lam2^2*m1bhat^2 + lam2^2*s1bhat^2) - covb^3*(2*lam2*m1bhat*s1bhat^2 - 2*W2b^lam2*lam2*m1bhat*s1bhat^2 + 2*lam2^2*m1bhat*m2bhat*s1bhat^2) - covb*(6*lam2*m1bhat*s1bhat^4*s2bhat^2 - 6*W2b^lam2*lam2*m1bhat*s1bhat^4*s2bhat^2 + 6*lam2^2*m1bhat*m2bhat*s1bhat^4*s2bhat^2) + covb^2*(W2b^(2*lam2)*s1bhat^4 - 2*W2b^lam2*s1bhat^4 + s1bhat^4 + 2*lam2*m2bhat*s1bhat^4 + lam2^2*m2bhat^2*s1bhat^4 - 2*W2b^lam2*lam2*m2bhat*s1bhat^4 + 3*lam2^2*m1bhat^2*s1bhat^2*s2bhat^2) - 6*W2b^lam2*s1bhat^6*s2bhat^2 + 3*W2b^(2*lam2)*s1bhat^6*s2bhat^2 - lam2^2*s1bhat^6*s2bhat^4 + 3*lam2^2*m2bhat^2*s1bhat^6*s2bhat^2 + 6*lam2*m2bhat*s1bhat^6*s2bhat^2 - 6*W2b^lam2*lam2*m2bhat*s1bhat^6*s2bhat^2)/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^4*(W1b^(2*lam1)*lam2^2 - 2*W1b^lam1*lam2^2 + lam2^2) + covb^2*(3*lam2^2*s1bhat^2*s2bhat^2 + 3*W1b^(2*lam1)*lam2^2*s1bhat^2*s2bhat^2 - 6*W1b^lam1*lam2^2*s1bhat^2*s2bhat^2) - lam1*((2*W1b^lam1*lam2^2*m1bhat - 2*lam2^2*m1bhat)*covb^4 + (2*lam2*s1bhat^2 + 2*lam2^2*m2bhat*s1bhat^2 - 2*W1b^lam1*lam2*s1bhat^2 - 2*W2b^lam2*lam2*s1bhat^2 - 2*W1b^lam1*lam2^2*m2bhat*s1bhat^2 + 2*W1b^lam1*W2b^lam2*lam2*s1bhat^2)*covb^3 + (6*W1b^lam1*lam2^2*m1bhat*s1bhat^2*s2bhat^2 - 6*lam2^2*m1bhat*s1bhat^2*s2bhat^2)*covb^2 + (6*lam2*s1bhat^4*s2bhat^2 + 6*lam2^2*m2bhat*s1bhat^4*s2bhat^2 - 6*W1b^lam1*lam2*s1bhat^4*s2bhat^2 - 6*W2b^lam2*lam2*s1bhat^4*s2bhat^2 - 6*W1b^lam1*lam2^2*m2bhat*s1bhat^4*s2bhat^2 + 6*W1b^lam1*W2b^lam2*lam2*s1bhat^4*s2bhat^2)*covb))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_9=0
h8_10=sum(- (covb*(s1bhat^4*(- 4*lam1^2*m2bhat^2*s2bhat + 2*lam1^2*s2bhat^3) - s1bhat^2*s2bhat^3*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2)) - covb^3*(s2bhat*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2) + 2*lam1^2*s1bhat^2*s2bhat) + s1bhat^4*s2bhat^3*(2*lam1*m2bhat + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat) + covb^2*s1bhat^2*s2bhat*(6*lam1*m2bhat + 6*lam1^2*m1bhat*m2bhat - 6*W1b^lam1*lam1*m2bhat))/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1bhat - 6*W1b^lam1*lam1 - 6*W2b^lam2*lam1 + 6*W1b^lam1*W2b^lam2*lam1 - 6*W2b^lam2*lam1^2*m1bhat)*covb^2*s1bhat^2*s2bhat + (8*W2b^lam2*lam1^2*m2bhat - 8*lam1^2*m2bhat)*covb*s1bhat^4*s2bhat + (2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat)*s1bhat^4*s2bhat^3) - covb*s1bhat^4*s2bhat*(4*W2b^(2*lam2)*lam1^2 - 8*W2b^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_11=sum(-(2*covb*s2bhat*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_12=sum((2*s1bhat^2*s2bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h9_1=sum((cova^2*(lam2*m2ahat - W2a^lam2 + 1) + s2ahat^2*((lam2*m2ahat - W2a^lam2 + 1)*s1ahat^2 - 2*cova*lam2*m1ahat))/(lam2*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s2ahat^2*(2*lam2 - 2*W1a^lam1*lam2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_2=sum(- (cova*(s2ahat^4*(- 4*lam2^2*m1ahat^2*s1ahat + 2*lam2^2*s1ahat^3) - s1ahat^3*s2ahat^2*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2)) - cova^3*(s1ahat*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2) + 2*lam2^2*s1ahat*s2ahat^2) + s1ahat^3*s2ahat^4*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + cova^2*s1ahat*s2ahat^2*(6*lam2*m1ahat + 6*lam2^2*m1ahat*m2ahat - 6*W2a^lam2*lam2*m1ahat))/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2ahat - 6*W1a^lam1*lam2 - 6*W2a^lam2*lam2 + 6*W1a^lam1*W2a^lam2*lam2 - 6*W1a^lam1*lam2^2*m2ahat)*cova^2*s1ahat*s2ahat^2 + (8*W1a^lam1*lam2^2*m1ahat - 8*lam2^2*m1ahat)*cova*s1ahat*s2ahat^4 + (2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat)*s1ahat^3*s2ahat^4) - cova*s1ahat*s2ahat^4*(4*W1a^(2*lam1)*lam2^2 - 8*W1a^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_3=0
h9_4=0
h9_5=sum((cova^2*(lam1*m1ahat - W1a^lam1 + 1) + s1ahat^2*((lam1*m1ahat - W1a^lam1 + 1)*s2ahat^2 - 2*cova*lam1*m2ahat))/(lam1*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s1ahat^2*(2*lam1 - 2*W2a^lam2*lam1))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_6=sum(- (cova*(s1ahat^4*(- 4*lam1^2*m2ahat^2*s2ahat + 2*lam1^2*s2ahat^3) - s1ahat^2*s2ahat^3*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2)) - cova^3*(s2ahat*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2) + 2*lam1^2*s1ahat^2*s2ahat) + s1ahat^4*s2ahat^3*(2*lam1*m2ahat + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat) + cova^2*s1ahat^2*s2ahat*(6*lam1*m2ahat + 6*lam1^2*m1ahat*m2ahat - 6*W1a^lam1*lam1*m2ahat))/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1ahat - 6*W1a^lam1*lam1 - 6*W2a^lam2*lam1 + 6*W1a^lam1*W2a^lam2*lam1 - 6*W2a^lam2*lam1^2*m1ahat)*cova^2*s1ahat^2*s2ahat + (8*W2a^lam2*lam1^2*m2ahat - 8*lam1^2*m2ahat)*cova*s1ahat^4*s2ahat + (2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat)*s1ahat^4*s2ahat^3) - cova*s1ahat^4*s2ahat*(4*W2a^(2*lam2)*lam1^2 - 8*W2a^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_7=0
h9_8=0
h9_9=sum(- (s1ahat^2*(cova^2*(6*lam2*m2ahat - 6*W2a^lam2 + 3*W2a^(2*lam2) + 3*lam2^2*m2ahat^2 - 6*W2a^lam2*lam2*m2ahat + 3) - cova*(6*lam2*m1ahat*s2ahat^2 - 6*W2a^lam2*lam2*m1ahat*s2ahat^2 + 6*lam2^2*m1ahat*m2ahat*s2ahat^2) + lam2^2*m1ahat^2*s2ahat^4) - cova^3*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + s1ahat^4*(W2a^(2*lam2)*s2ahat^2 - 2*W2a^lam2*s2ahat^2 + s2ahat^2 - lam2^2*s2ahat^4 + 2*lam2*m2ahat*s2ahat^2 + lam2^2*m2ahat^2*s2ahat^2 - 2*W2a^lam2*lam2*m2ahat*s2ahat^2) + cova^4*lam2^2 + 3*cova^2*lam2^2*m1ahat^2*s2ahat^2)/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^2*(3*lam2^2*s2ahat^2 - 6*W1a^lam1*lam2^2*s2ahat^2 + 3*W1a^(2*lam1)*lam2^2*s2ahat^2) + s1ahat^2*(lam2^2*s2ahat^4 - 2*W1a^lam1*lam2^2*s2ahat^4 + W1a^(2*lam1)*lam2^2*s2ahat^4) - lam1*(cova^3*(2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat) - cova^2*(6*lam2^2*m1ahat*s2ahat^2 - 6*W1a^lam1*lam2^2*m1ahat*s2ahat^2) + s1ahat^2*(cova*(6*lam2*s2ahat^2 + 6*lam2^2*m2ahat*s2ahat^2 - 6*W1a^lam1*lam2*s2ahat^2 - 6*W2a^lam2*lam2*s2ahat^2 - 6*W1a^lam1*lam2^2*m2ahat*s2ahat^2 + 6*W1a^lam1*W2a^lam2*lam2*s2ahat^2) - 2*lam2^2*m1ahat*s2ahat^4 + 2*W1a^lam1*lam2^2*m1ahat*s2ahat^4)))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_10=0
h9_11=sum(-((W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova^2*lam1 - 2*cova*lam2*s2ahat^2 + lam1*s1ahat^2*s2ahat^2 - W2a^lam2*cova^2*lam1 + 2*W1a^lam1*cova*lam2*s2ahat^2 + cova^2*lam1*lam2*m2ahat - W2a^lam2*lam1*s1ahat^2*s2ahat^2 + lam1*lam2*m2ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_12=sum(-((W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova^2*lam2 - 2*cova*lam1*s1ahat^2 + lam2*s1ahat^2*s2ahat^2 - W1a^lam1*cova^2*lam2 + 2*W2a^lam2*cova*lam1*s1ahat^2 + cova^2*lam1*lam2*m1ahat - W1a^lam1*lam2*s1ahat^2*s2ahat^2 + lam1*lam2*m1ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h10_1=0
h10_2=0
h10_3=sum((covb^2*(lam2*m2bhat - W2b^lam2 + 1) + s2bhat^2*((lam2*m2bhat - W2b^lam2 + 1)*s1bhat^2 - 2*covb*lam2*m1bhat))/(lam2*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s2bhat^2*(2*lam2 - 2*W1b^lam1*lam2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_4=sum(- (covb*(s2bhat^4*(- 4*lam2^2*m1bhat^2*s1bhat + 2*lam2^2*s1bhat^3) - s1bhat^3*s2bhat^2*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2)) - covb^3*(s1bhat*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2) + 2*lam2^2*s1bhat*s2bhat^2) + s1bhat^3*s2bhat^4*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + covb^2*s1bhat*s2bhat^2*(6*lam2*m1bhat + 6*lam2^2*m1bhat*m2bhat - 6*W2b^lam2*lam2*m1bhat))/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2bhat - 6*W1b^lam1*lam2 - 6*W2b^lam2*lam2 + 6*W1b^lam1*W2b^lam2*lam2 - 6*W1b^lam1*lam2^2*m2bhat)*covb^2*s1bhat*s2bhat^2 + (8*W1b^lam1*lam2^2*m1bhat - 8*lam2^2*m1bhat)*covb*s1bhat*s2bhat^4 + (2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat)*s1bhat^3*s2bhat^4) - covb*s1bhat*s2bhat^4*(4*W1b^(2*lam1)*lam2^2 - 8*W1b^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_5=0
h10_6=0
h10_7=sum((covb^2*(lam1*m1bhat - W1b^lam1 + 1) + s1bhat^2*((lam1*m1bhat - W1b^lam1 + 1)*s2bhat^2 - 2*covb*lam1*m2bhat))/(lam1*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s1bhat^2*(2*lam1 - 2*W2b^lam2*lam1))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_8=sum(- (covb*(s1bhat^4*(- 4*lam1^2*m2bhat^2*s2bhat + 2*lam1^2*s2bhat^3) - s1bhat^2*s2bhat^3*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2)) - covb^3*(s2bhat*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2) + 2*lam1^2*s1bhat^2*s2bhat) + s1bhat^4*s2bhat^3*(2*lam1*m2bhat + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat) + covb^2*s1bhat^2*s2bhat*(6*lam1*m2bhat + 6*lam1^2*m1bhat*m2bhat - 6*W1b^lam1*lam1*m2bhat))/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1bhat - 6*W1b^lam1*lam1 - 6*W2b^lam2*lam1 + 6*W1b^lam1*W2b^lam2*lam1 - 6*W2b^lam2*lam1^2*m1bhat)*covb^2*s1bhat^2*s2bhat + (8*W2b^lam2*lam1^2*m2bhat - 8*lam1^2*m2bhat)*covb*s1bhat^4*s2bhat + (2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat)*s1bhat^4*s2bhat^3) - covb*s1bhat^4*s2bhat*(4*W2b^(2*lam2)*lam1^2 - 8*W2b^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_9=0
h10_10=sum(- (s1bhat^2*(covb^2*(6*lam2*m2bhat - 6*W2b^lam2 + 3*W2b^(2*lam2) + 3*lam2^2*m2bhat^2 - 6*W2b^lam2*lam2*m2bhat + 3) - covb*(6*lam2*m1bhat*s2bhat^2 - 6*W2b^lam2*lam2*m1bhat*s2bhat^2 + 6*lam2^2*m1bhat*m2bhat*s2bhat^2) + lam2^2*m1bhat^2*s2bhat^4) - covb^3*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + s1bhat^4*(W2b^(2*lam2)*s2bhat^2 - 2*W2b^lam2*s2bhat^2 + s2bhat^2 - lam2^2*s2bhat^4 + 2*lam2*m2bhat*s2bhat^2 + lam2^2*m2bhat^2*s2bhat^2 - 2*W2b^lam2*lam2*m2bhat*s2bhat^2) + covb^4*lam2^2 + 3*covb^2*lam2^2*m1bhat^2*s2bhat^2)/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^2*(3*lam2^2*s2bhat^2 - 6*W1b^lam1*lam2^2*s2bhat^2 + 3*W1b^(2*lam1)*lam2^2*s2bhat^2) + s1bhat^2*(lam2^2*s2bhat^4 - 2*W1b^lam1*lam2^2*s2bhat^4 + W1b^(2*lam1)*lam2^2*s2bhat^4) - lam1*(covb^3*(2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat) - covb^2*(6*lam2^2*m1bhat*s2bhat^2 - 6*W1b^lam1*lam2^2*m1bhat*s2bhat^2) + s1bhat^2*(covb*(6*lam2*s2bhat^2 + 6*lam2^2*m2bhat*s2bhat^2 - 6*W1b^lam1*lam2*s2bhat^2 - 6*W2b^lam2*lam2*s2bhat^2 - 6*W1b^lam1*lam2^2*m2bhat*s2bhat^2 + 6*W1b^lam1*W2b^lam2*lam2*s2bhat^2) - 2*lam2^2*m1bhat*s2bhat^4 + 2*W1b^lam1*lam2^2*m1bhat*s2bhat^4)))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_11=sum(-((W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb^2*lam1 - 2*covb*lam2*s2bhat^2 + lam1*s1bhat^2*s2bhat^2 - W2b^lam2*covb^2*lam1 + 2*W1b^lam1*covb*lam2*s2bhat^2 + covb^2*lam1*lam2*m2bhat - W2b^lam2*lam1*s1bhat^2*s2bhat^2 + lam1*lam2*m2bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_12=sum(-((W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb^2*lam2 - 2*covb*lam1*s1bhat^2 + lam2*s1bhat^2*s2bhat^2 - W1b^lam1*covb^2*lam2 + 2*W2b^lam2*covb*lam1*s1bhat^2 + covb^2*lam1*lam2*m1bhat - W1b^lam1*lam2*s1bhat^2*s2bhat^2 + lam1*lam2*m1bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_1=sum((2*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h11_2=sum((2*s1ahat*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_3=sum((2*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h11_4=sum((2*s1bhat*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_5=sum(-(cova*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h11_6=sum(-(2*cova*s2ahat*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_7=sum(-(covb*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h11_8=sum(-(2*covb*s2bhat*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_9=sum(-((W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova^2*lam1 - 2*cova*lam2*s2ahat^2 + lam1*s1ahat^2*s2ahat^2 - W2a^lam2*cova^2*lam1 + 2*W1a^lam1*cova*lam2*s2ahat^2 + cova^2*lam1*lam2*m2ahat - W2a^lam2*lam1*s1ahat^2*s2ahat^2 + lam1*lam2*m2ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_10=sum(-((W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb^2*lam1 - 2*covb*lam2*s2bhat^2 + lam1*s1bhat^2*s2bhat^2 - W2b^lam2*covb^2*lam1 + 2*W1b^lam1*covb*lam2*s2bhat^2 + covb^2*lam1*lam2*m2bhat - W2b^lam2*lam1*s1bhat^2*s2bhat^2 + lam1*lam2*m2bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_11=sum(((2*((W1a^lam1-1)/lam1^2-(W1a^lam1*log(W1a))/lam1)^2)/s1ahat^2-(2*(m1ahat-(W1a^lam1-1)/lam1)*((2*W1a^lam1-2)/lam1^3-(2*W1a^lam1*log(W1a))/lam1^2+(W1a^lam1*log(W1a)^2)/lam1))/s1ahat^2+(2*cova*(m2ahat-(W2a^lam2-1)/lam2)*((2*W1a^lam1-2)/lam1^3-(2*W1a^lam1*log(W1a))/lam1^2+(W1a^lam1*log(W1a)^2)/lam1))/(s1ahat^2*s2ahat^2))/((2*cova^2)/(s1ahat^2*s2ahat^2)-2))+sum(((2*((W1b^lam1-1)/lam1^2-(W1b^lam1*log(W1b))/lam1)^2)/s1bhat^2-(2*(m1bhat-(W1b^lam1-1)/lam1)*((2*W1b^lam1-2)/lam1^3-(2*W1b^lam1*log(W1b))/lam1^2+(W1b^lam1*log(W1b)^2)/lam1))/s1bhat^2+(2*covb*(m2bhat-(W2b^lam2-1)/lam2)*((2*W1b^lam1-2)/lam1^3-(2*W1b^lam1*log(W1b))/lam1^2+(W1b^lam1*log(W1b)^2)/lam1))/(s1bhat^2*s2bhat^2))/((2*covb^2)/(s1bhat^2*s2bhat^2)-2))
h11_12=sum((2*cova*(W1a^lam1*lam1*log(W1a)-W1a^lam1+1)*(W2a^lam2*lam2*log(W2a)-W2a^lam2+1))/(lam1^2*lam2^2*(-2*cova^2+2*s1ahat^2*s2ahat^2)))+sum((2*covb*(W1b^lam1*lam1*log(W1b)-W1b^lam1+1)*(W2b^lam2*lam2*log(W2b)-W2b^lam2+1))/(lam1^2*lam2^2*(-2*covb^2+2*s1bhat^2*s2bhat^2)))
h12_1=sum(-(cova*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h12_2=sum(-(2*cova*s1ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_3=sum(-(covb*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h12_4=sum(-(2*covb*s1bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_5=sum((2*s1ahat^2*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h12_6=sum((2*s1ahat^2*s2ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_7=sum((2*s1bhat^2*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h12_8=sum((2*s1bhat^2*s2bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_9=sum(-((W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova^2*lam2 - 2*cova*lam1*s1ahat^2 + lam2*s1ahat^2*s2ahat^2 - W1a^lam1*cova^2*lam2 + 2*W2a^lam2*cova*lam1*s1ahat^2 + cova^2*lam1*lam2*m1ahat - W1a^lam1*lam2*s1ahat^2*s2ahat^2 + lam1*lam2*m1ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_10=sum(-((W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb^2*lam2 - 2*covb*lam1*s1bhat^2 + lam2*s1bhat^2*s2bhat^2 - W1b^lam1*covb^2*lam2 + 2*W2b^lam2*covb*lam1*s1bhat^2 + covb^2*lam1*lam2*m1bhat - W1b^lam1*lam2*s1bhat^2*s2bhat^2 + lam1*lam2*m1bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_11=sum((2*cova*(W1a^lam1*lam1*log(W1a)-W1a^lam1+1)*(W2a^lam2*lam2*log(W2a)-W2a^lam2+1))/(lam1^2*lam2^2*(-2*cova^2+2*s1ahat^2*s2ahat^2)))+sum((2*covb*(W1b^lam1*lam1*log(W1b)-W1b^lam1+1)*(W2b^lam2*lam2*log(W2b)-W2b^lam2+1))/(lam1^2*lam2^2*(-2*covb^2+2*s1bhat^2*s2bhat^2)))
h12_12=sum(((2*((W2a^lam2-1)/lam2^2-(W2a^lam2*log(W2a))/lam2)^2)/s2ahat^2-(2*(m2ahat-(W2a^lam2-1)/lam2)*((2*W2a^lam2-2)/lam2^3-(2*W2a^lam2*log(W2a))/lam2^2+(W2a^lam2*log(W2a)^2)/lam2))/s2ahat^2+(2*cova*(m1ahat-(W1a^lam1-1)/lam1)*((2*W2a^lam2-2)/lam2^3-(2*W2a^lam2*log(W2a))/lam2^2+(W2a^lam2*log(W2a)^2)/lam2))/(s1ahat^2*s2ahat^2))/((2*cova^2)/(s1ahat^2*s2ahat^2)-2))+sum(((2*((W2b^lam2-1)/lam2^2-(W2b^lam2*log(W2b))/lam2)^2)/s2bhat^2-(2*(m2bhat-(W2b^lam2-1)/lam2)*((2*W2b^lam2-2)/lam2^3-(2*W2b^lam2*log(W2b))/lam2^2+(W2b^lam2*log(W2b)^2)/lam2))/s2bhat^2+(2*covb*(m1bhat-(W1b^lam1-1)/lam1)*((2*W2b^lam2-2)/lam2^3-(2*W2b^lam2*log(W2b))/lam2^2+(W2b^lam2*log(W2b)^2)/lam2))/(s1bhat^2*s2bhat^2))/((2*covb^2)/(s1bhat^2*s2bhat^2)-2))
h1r=c(h1_1,h1_2,h1_3,h1_4,h1_5,h1_6,h1_7,h1_8,h1_9,h1_10,h1_11,h1_12)
h2r=c(h2_1,h2_2,h2_3,h2_4,h2_5,h2_6,h2_7,h2_8,h2_9,h2_10,h2_11,h2_12)
h3r=c(h3_1,h3_2,h3_3,h3_4,h3_5,h3_6,h3_7,h3_8,h3_9,h3_10,h3_11,h3_12)
h4r=c(h4_1,h4_2,h4_3,h4_4,h4_5,h4_6,h4_7,h4_8,h4_9,h4_10,h4_11,h4_12)
h5r=c(h5_1,h5_2,h5_3,h5_4,h5_5,h5_6,h5_7,h5_8,h5_9,h5_10,h5_11,h5_12)
h6r=c(h6_1,h6_2,h6_3,h6_4,h6_5,h6_6,h6_7,h6_8,h6_9,h6_10,h6_11,h6_12)
h7r=c(h7_1,h7_2,h7_3,h7_4,h7_5,h7_6,h7_7,h7_8,h7_9,h7_10,h7_11,h7_12)
h8r=c(h8_1,h8_2,h8_3,h8_4,h8_5,h8_6,h8_7,h8_8,h8_9,h8_10,h8_11,h8_12)
h9r=c(h9_1,h9_2,h9_3,h9_4,h9_5,h9_6,h9_7,h9_8,h9_9,h9_10,h9_11,h9_12)
h10r=c(h10_1,h10_2,h10_3,h10_4,h10_5,h10_6,h10_7,h10_8,h10_9,h10_10,h10_11,h10_12)
h11r=c(h11_1,h11_2,h11_3,h11_4,h11_5,h11_6,h11_7,h11_8,h11_9,h11_10,h11_11,h11_12)
h12r=c(h12_1,h12_2,h12_3,h12_4,h12_5,h12_6,h12_7,h12_8,h12_9,h12_10,h12_11,h12_12)
HCF=rbind(h1r,h2r,h3r,h4r,h5r,h6r,h7r,h8r,h9r,h10r,h11r,h12r)
HCF[1,2]=0;HCF[1,6]=0;HCF[1,9]=0;
HCF[2,1]=0;HCF[2,5]=0;
HCF[3,4]=0;HCF[3,8]=0;HCF[3,10]=0;HCF[4,3]=0;
HCF[4,7]=0;
HCF[5,2]=0;HCF[5,6]=0;HCF[5,9]=0;
HCF[6,1]=0;HCF[6,5]=0;
HCF[7,4]=0;HCF[7,8]=0;HCF[7,10]=0;
HCF[8,3]=0;HCF[8,7]=0;
HCF[9,1]=0;HCF[9,5]=0;
HCF[10,3]=0;
HCF[10,7]=0;
HCF[1,1]=na*HCF[1,1];
HCF[1,5]=na*HCF[1,5];
HCF[5,1]=na*HCF[5,1];
HCF[3,3]=nb*HCF[3,3];
HCF[3,7]=nb*HCF[3,7];
HCF[7,3]=nb*HCF[7,3];
HCF[7,7]=nb*HCF[7,7];
HCF[5,5]=na*HCF[5,5];
VV=inv(-HCF)
#############################################################################
#DEALING WITH ORIGINAL AUCS
#############################################################################
AUC1 = erfc((2^(1/2)*(m1ahat - m1bhat))/(2*(s1ahat^2 + s1bhat^2)^(1/2)))/2
dm1a = -(2^(1/2)*exp(-(m1ahat - m1bhat)^2/(2*(s1ahat^2 + s1bhat^2))))/(2*pi^(1/2)*(s1ahat^2 + s1bhat^2)^(1/2))
dm1b = (2^(1/2)*exp(-(m1ahat - m1bhat)^2/(2*(s1ahat^2 + s1bhat^2))))/(2*pi^(1/2)*(s1ahat^2 + s1bhat^2)^(1/2))
ds1a = (2^(1/2)*s1ahat*exp(-(m1ahat - m1bhat)^2/(2*(s1ahat^2 + s1bhat^2)))*(m1ahat - m1bhat))/(2*pi^(1/2)*(s1ahat^2 + s1bhat^2)^(3/2))
ds1b = (2^(1/2)*s1bhat*exp(-(m1ahat - m1bhat)^2/(2*(s1ahat^2 + s1bhat^2)))*(m1ahat - m1bhat))/(2*pi^(1/2)*(s1ahat^2 + s1bhat^2)^(3/2))
AUC2 = erfc((2^(1/2)*(m2ahat - m2bhat))/(2*(s2ahat^2 + s2bhat^2)^(1/2)))/2
dm2a = -(2^(1/2)*exp(-(m2ahat - m2bhat)^2/(2*(s2ahat^2 + s2bhat^2))))/(2*pi^(1/2)*(s2ahat^2 + s2bhat^2)^(1/2))
dm2b = (2^(1/2)*exp(-(m2ahat - m2bhat)^2/(2*(s2ahat^2 + s2bhat^2))))/(2*pi^(1/2)*(s2ahat^2 + s2bhat^2)^(1/2))
ds2a = (2^(1/2)*s2ahat*exp(-(m2ahat - m2bhat)^2/(2*(s2ahat^2 + s2bhat^2)))*(m2ahat - m2bhat))/(2*pi^(1/2)*(s2ahat^2 + s2bhat^2)^(3/2))
ds2b = (2^(1/2)*s2bhat*exp(-(m2ahat - m2bhat)^2/(2*(s2ahat^2 + s2bhat^2)))*(m2ahat - m2bhat))/(2*pi^(1/2)*(s2ahat^2 + s2bhat^2)^(3/2))
varAUC1=c(dm1a, ds1a, dm1b, ds1b)%*%VV[1:4,1:4]%*%t(t(c(dm1a, ds1a, dm1b, ds1b)));
varAUC2=c(dm2a, ds2a, dm2b, ds2b)%*%VV[5:8,5:8]%*%t(t(c(dm2a, ds2a, dm2b, ds2b)));
varAUC1=c(varAUC1)
varAUC2=c(varAUC2)
V=zeros(8,8);
covAUC=c(dm1a, ds1a, dm1b, ds1b, 0, 0, 0, 0)%*%VV[1:8,1:8]%*%t(t(c(0, 0, 0, 0, dm2a, ds2a, dm2b, ds2b)))
covAUC=c(covAUC)
ZdAUC=(AUC2-AUC1)/(sqrt(varAUC1+varAUC2-2*covAUC))
SEdAUC=sqrt(varAUC1+varAUC2-2*covAUC)
CIdiffdAUC=c(AUC2-AUC1-pmalpha*SEdAUC, AUC2-AUC1+pmalpha*SEdAUC)
pval2tdAUC=2*pnorm(-abs(ZdAUC))
#############################################################################
#PROBIT VERSIONS
#############################################################################
#norminv versions
AUC1= ((m1bhat-m1ahat)/s1bhat)/(sqrt(1+(s1ahat/s1bhat)^2));
AUC2= ((m2bhat-m2ahat)/s2bhat)/(sqrt(1+(s2ahat/s2bhat)^2));
AUC1MC=AUC1
AUC2MC=AUC2
AUC1original=pnorm(AUC1)
AUC2original=pnorm(AUC2)
#====================norminv AUC1==============================
dm1a =-1/(s1bhat*sqrt(1+(s1ahat/s1bhat)^2));
ds1a = (s1ahat*(m1ahat-m1bhat))/(s1bhat^3*(1+(s1ahat/s1bhat)^2)^(3/2))
dm1b =1/(s1bhat*sqrt(1+(s1ahat/s1bhat)^2));
ds1b =((m1ahat-m1bhat))/(s1bhat^2*(1+(s1ahat/s1bhat)^2)^(3/2))
varAUC1=c(dm1a, ds1a, dm1b, ds1b)%*%VV[1:4,1:4]%*%t(t(c(dm1a, ds1a, dm1b, ds1b)))
#====================norminv AUC2==============================
dm2a =-1/(s2bhat*sqrt(1+(s2ahat/s2bhat)^2));
ds2a = (s2ahat*(m2ahat-m2bhat))/(s2bhat^3*(1+(s2ahat/s2bhat)^2)^(3/2))
dm2b =1/(s2bhat*sqrt(1+(s2ahat/s2bhat)^2));
ds2b =((m2ahat-m2bhat))/(s2bhat^2*(1+(s2ahat/s2bhat)^2)^(3/2))
varAUC2=c(dm2a, ds2a, dm2b, ds2b)%*%VV[5:8,5:8]%*%t(t(c(dm2a, ds2a, dm2b, ds2b)));
#====================norminv cov(AUC1,AUC2)==============================
V=zeros(8,8);
covAUC=c(dm1a, ds1a, dm1b, ds1b, 0, 0, 0, 0)%*%VV[1:8,1:8]%*%t(t(c(0, 0, 0, 0, dm2a, ds2a, dm2b, ds2b)))
covAUC=c(covAUC)
Z=(AUC2MC-AUC1MC)/(sqrt(varAUC1+varAUC2-2*covAUC))
SE=sqrt(varAUC1+varAUC2-2*covAUC)
CIdiff=c(AUC2MC-AUC1MC-pmalpha*SE, AUC2MC-AUC1MC+pmalpha*SE)
pval2t=2*pnorm(-abs(Z))
#################### IF PLOTS ARE REQUESTED #################################
#====TWO ROC FUNCTIONS: ONE IS THE BOXCOX AND THE OTHER THE REFERENCE LINE================
roc1<-function(t){
na = length(W1alam)
nb = length(W1blam)
s1alam=sqrt( var(W1alam)*(na-1)/na )
s1blam=sqrt( var(W1blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W1alam),
sd=s1alam),
mean=mean(W1blam),
sd=s1blam)
}
roc2<-function(t){
na = length(W2alam)
nb = length(W2blam)
s2alam=sqrt( var(W2alam)*(na-1)/na )
s2blam=sqrt( var(W2blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W2alam),
sd=s2alam),
mean=mean(W2blam),
sd=s2blam)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
#================IF PLOTS ARE REQUESTED PLOT THE ROCS==
if (plots=="on") {
# x11()
plot(linspace(0,1,1000),roc1(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="red")
lines(linspace(0,1,1000),roc2(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="black")
lines(linspace(0,1,1000),linspace(0,1,1000),type="l", lty=2)
legend("bottomright", legend=c(paste("ROC for Marker 1 with AUC =", formattable(AUC1original, digits = 4, format = "f")),
paste("ROC for Marker 2 with AUC =", formattable(AUC2original, digits = 4, format = "f")),
paste("P-value for the probit AUC difference:", formattable(pval2t, digits = 4, format = "f"), " ")),
col=c("red", "black", "white"), lty=c(1, 1, NA), pch = c(NA, NA, NA), cex=0.8)
}
###############################################################################
###############################################################################
#================OUTPUT ARGUMENTS======================
res <- data.frame(AUC1 = formattable(AUC1original, digits = 4, format = "f"),
AUC2 = formattable(AUC2original, digits = 4, format = "f"),
diff = formattable(AUC2original - AUC1original, digits = 4, format = "f"),
p_value_probit = formattable(pval2t, digits = 4, format = "f"),
p_value = formattable(pval2tdAUC, digits = 4, format = "f"),
ci_ll = formattable(CIdiffdAUC[1], digits = 4, format = "f"),
ci_ul = formattable(CIdiffdAUC[2], digits = 4, format = "f"))
rownames(res) <- c("Estimates:")
colnames(res) <- c("AUC 1", "AUC 2", "Diff", "P-Val (Probit)", "P-Val", "CI (LL)", "CI (UL)")
res <- formattable(as.matrix(res), digits = 4, format = "f")
res
#return(list(AUCmarker1=AUC1original,AUCmarker2=AUC2original, pvalue_difference= pval2t, CI_difference= pnorm(CIdiff), rocbc1=roc1, rocbc2=roc2))
return(list(resultstable=res,
AUCmarker1=AUC1original,
AUCmarker2=AUC2original,
pvalue_probit_difference= pval2t,
pvalue_difference= pval2tdAUC,
CI_difference= CIdiffdAUC,
roc1=roc1,
roc2=roc2,
transx1=W1alam,
transy1=W1blam,
transformation.parameter.1 = lam[1],
transx2=W2alam,
transy2=W2blam,
transformation.parameter.2 = lam[2]))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
comparebcJ <-function (marker1, marker2, D, alpha, plots){
if ((length(marker1) != length(D)) | (length(marker2) != length(D))) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (alpha <= 0 | alpha >= 1) {
stop("ERROR: The level of significance, alpha, should be set between 0 and 1. A common choice is 0.05.")
} else if (sum(is.na(marker1)) > 0 | sum(is.na(marker2)) > 0 | sum(is.na(D)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if ((sum(marker1 < 0) > 0) | (sum(marker2 < 2)) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
erf <- function (x) 2 * pnorm(x * sqrt(2)) - 1
erfc <- function (x) 2 * pnorm(x * sqrt(2), lower.tail = FALSE)
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower.tail = FALSE)/sqrt(2)
pmalpha=qnorm(1-alpha/2);
if (plots!="on"){plots="off"}
W1a=marker1[D==0]
W1b=marker1[D==1]
W2a=marker2[D==0]
W2b=marker2[D==1]
Wa=cbind(W1a,W2a);
Wb=cbind(W1b,W2b);
na=length(W1a)
nb=length(W1b)
likbox2D<-function(W1a,W1b,W2a,W2b,lam){
na=length(W1a);
nb=length(W1b);
out=c();
#for (i in 1:length(h)){
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
#cova=1/na*sum((W1alam-mean(W1alam))*(W2alam-mean(W2alam)))
#covb=1/nb*sum((W1blam-mean(W1blam))*(W2blam-mean(W2blam)))
cova= cov(Walam)*(na-1)/na;
covb= cov(Wblam)*(nb-1)/nb;
mualam=c(mean(W1alam), mean(W2alam))
mublam=c(mean(W1blam), mean(W2blam))
#dmvn(c(0,0), mu, mcov, log = F)
# out= (-sum(log(dmvn(Walam,mualam,cova)))-sum(log(dmvn(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b)));
out= -sum(log(dmvnorm(Walam,mualam,cova)))-sum(log(dmvnorm(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b));
#out1=-sum(log(dmvnorm(Walam,mualam,cova)))
#out2=-sum(log(dmvnorm(Wblam,mublam,covb)))
#out3=-(lam[1]-1)*sum(log(W1a))
#out4=-(lam[2]-1)*sum(log(W2a))
#out5=-(lam[1]-1)*sum(log(W1b))
#out6=-(lam[2]-1)*sum(log(W2b))
#out=c(out,out1,out2,out3,out4,out5,out6, cova, covb)
#out= (-sum(log(mvnpdf2([W1alam(g(1)) W2alam(g(2))],[mean(W1alam(g(1))) mean(W2alam(g(2)))],[std(W1alam(g(1)),1).^2 cova;cova (std(W2alam(g(2)),1)).^2])))+...
# -sum(log(mvnpdf2([W1blam(g(1)) W2blam(g(2))],[mean(W1blam(g(1))) mean(W2blam(g(2)))],[std(W1blam(g(1)),1).^2 covb;covb (std(W2blam(g(2)),1)).^2])))+...
# -(g(1)-1).*sum(log(W1a))-(g(2)-1).*sum(log(W2a))-(g(1)-1).*sum(log(W1b))-(g(2)-1).*sum(log(W2b)));
return(out)
}
lam=c(2,2)
likbox2D(W1a,W1b,W2a,W2b,lam)
logL<-function(h){
likbox2D(W1a,W1b,W2a,W2b,h)
}
# init=c(0.8,0.8)
# lam=fminsearch(logL,init)
# lam=c(lam$optbase$xopt)
lam<- optim(c(1,1),logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
lam
#init=c(-10,10)
#lam=optimize(logL,init)
#lam
#f <- function (x, a) (x - a)^2
#xmin <- optimize(f, c(0, 1), tol = 0.0001, a = 1/3)
#xmin
out <- optim(c(1,1), logL, method = "Nelder-Mead")
lam = out$par
###################################
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
m1ahat=mean(W1alam)
m1bhat=mean(W1blam)
m2ahat=mean(W2alam)
m2bhat=mean(W2blam)
s1ahat=sqrt( var(W1alam)*(na-1)/na )
s1bhat=sqrt( var(W1blam)*(nb-1)/nb )
s2ahat=sqrt( var(W2alam)*(na-1)/na )
s2bhat=sqrt( var(W2blam)*(nb-1)/nb )
cova= cov(Walam)*(na-1)/na;cova=cova[1,2];
covb= cov(Wblam)*(nb-1)/nb;covb=covb[1,2];
lam1=lam[1]
lam2=lam[2]
##################################
h1_1=sum(-2/(2*s1ahat^2 - (2*cova^2)/s2ahat^2))
h1_2=sum(-(2*s1ahat*s2ahat^2*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_3=0
h1_4=0;
h1_5=sum(cova/(- cova^2 + s1ahat^2*s2ahat^2));
h1_6=sum((s2ahat*(2*cova^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat) - 2*cova*s1ahat^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_7=0
h1_8=0
h1_9=sum((cova^2*(lam2*m2ahat - W2a^lam2 + 1) + s2ahat^2*((lam2*m2ahat - W2a^lam2 + 1)*s1ahat^2 - 2*cova*lam2*m1ahat))/(lam2*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s2ahat^2*(2*lam2 - 2*W1a^lam1*lam2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_10=0
h1_11=sum((2*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h1_12=sum(-(cova*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h2_1=sum(-(2*s1ahat*s2ahat^2*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_2=sum(- (3*s1ahat^2*s2ahat^6 + cova^4*(lam1^2*m2ahat^2 + lam1^2*s2ahat^2) - cova^3*(2*lam1*m2ahat*s2ahat^2 - 2*W1a^lam1*lam1*m2ahat*s2ahat^2 + 2*lam1^2*m1ahat*m2ahat*s2ahat^2) - cova*(6*lam1*m2ahat*s1ahat^2*s2ahat^4 - 6*W1a^lam1*lam1*m2ahat*s1ahat^2*s2ahat^4 + 6*lam1^2*m1ahat*m2ahat*s1ahat^2*s2ahat^4) + cova^2*(W1a^(2*lam1)*s2ahat^4 - 2*W1a^lam1*s2ahat^4 + s2ahat^4 + 2*lam1*m1ahat*s2ahat^4 + lam1^2*m1ahat^2*s2ahat^4 - 2*W1a^lam1*lam1*m1ahat*s2ahat^4 + 3*lam1^2*m2ahat^2*s1ahat^2*s2ahat^2) - 6*W1a^lam1*s1ahat^2*s2ahat^6 + 3*W1a^(2*lam1)*s1ahat^2*s2ahat^6 - lam1^2*s1ahat^4*s2ahat^6 + 3*lam1^2*m1ahat^2*s1ahat^2*s2ahat^6 + 6*lam1*m1ahat*s1ahat^2*s2ahat^6 - 6*W1a^lam1*lam1*m1ahat*s1ahat^2*s2ahat^6)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^4*(W2a^(2*lam2)*lam1^2 - 2*W2a^lam2*lam1^2 + lam1^2) + cova^2*(3*lam1^2*s1ahat^2*s2ahat^2 + 3*W2a^(2*lam2)*lam1^2*s1ahat^2*s2ahat^2 - 6*W2a^lam2*lam1^2*s1ahat^2*s2ahat^2) - lam2*((2*W2a^lam2*lam1^2*m2ahat - 2*lam1^2*m2ahat)*cova^4 + (2*lam1*s2ahat^2 + 2*lam1^2*m1ahat*s2ahat^2 - 2*W1a^lam1*lam1*s2ahat^2 - 2*W2a^lam2*lam1*s2ahat^2 - 2*W2a^lam2*lam1^2*m1ahat*s2ahat^2 + 2*W1a^lam1*W2a^lam2*lam1*s2ahat^2)*cova^3 + (6*W2a^lam2*lam1^2*m2ahat*s1ahat^2*s2ahat^2 - 6*lam1^2*m2ahat*s1ahat^2*s2ahat^2)*cova^2 + (6*lam1*s1ahat^2*s2ahat^4 + 6*lam1^2*m1ahat*s1ahat^2*s2ahat^4 - 6*W1a^lam1*lam1*s1ahat^2*s2ahat^4 - 6*W2a^lam2*lam1*s1ahat^2*s2ahat^4 - 6*W2a^lam2*lam1^2*m1ahat*s1ahat^2*s2ahat^4 + 6*W1a^lam1*W2a^lam2*lam1*s1ahat^2*s2ahat^4)*cova))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_3=sum(0)
h2_4=sum(0)
h2_5=sum((s1ahat*(2*cova^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat) - 2*cova*s2ahat^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_6=sum((lam2*((2*cova*s2ahat^3*(2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat) - 2*cova*s2ahat*(4*cova*lam1^2*m2ahat - 4*W2a^lam2*cova*lam1^2*m2ahat))*s1ahat^3 + 2*cova*s2ahat*(2*cova^2*lam1 + 2*cova^2*lam1^2*m1ahat - 2*W1a^lam1*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1^2*m1ahat + 2*W1a^lam1*W2a^lam2*cova^2*lam1)*s1ahat) - 2*cova*s1ahat^3*s2ahat*(2*cova*lam1^2 + 2*W2a^(2*lam2)*cova*lam1^2 - 4*W2a^lam2*cova*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - ((4*cova^2*lam1^2*m2ahat^2*s2ahat - 2*cova*s2ahat^3*(2*lam1*m2ahat + cova*lam1^2 + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat))*s1ahat^3 + (2*cova*(2*cova + 2*W1a^(2*lam1)*cova - 4*W1a^lam1*cova + 2*cova*lam1^2*m1ahat^2 + 4*cova*lam1*m1ahat - 4*W1a^lam1*cova*lam1*m1ahat)*s2ahat^3 + 2*cova*(cova^3*lam1^2 - 2*cova^2*lam1*m2ahat + 2*W1a^lam1*cova^2*lam1*m2ahat - 2*cova^2*lam1^2*m1ahat*m2ahat)*s2ahat)*s1ahat)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_7=sum(0)
h2_8=sum(0)
h2_9=sum(- (cova*(s2ahat^4*(- 4*lam2^2*m1ahat^2*s1ahat + 2*lam2^2*s1ahat^3) - s1ahat^3*s2ahat^2*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2)) - cova^3*(s1ahat*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2) + 2*lam2^2*s1ahat*s2ahat^2) + s1ahat^3*s2ahat^4*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + cova^2*s1ahat*s2ahat^2*(6*lam2*m1ahat + 6*lam2^2*m1ahat*m2ahat - 6*W2a^lam2*lam2*m1ahat))/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2ahat - 6*W1a^lam1*lam2 - 6*W2a^lam2*lam2 + 6*W1a^lam1*W2a^lam2*lam2 - 6*W1a^lam1*lam2^2*m2ahat)*cova^2*s1ahat*s2ahat^2 + (8*W1a^lam1*lam2^2*m1ahat - 8*lam2^2*m1ahat)*cova*s1ahat*s2ahat^4 + (2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat)*s1ahat^3*s2ahat^4) - cova*s1ahat*s2ahat^4*(4*W1a^(2*lam1)*lam2^2 - 8*W1a^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_10=sum(0)
h2_11=sum((2*s1ahat*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_12=sum(-(2*cova*s1ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h3_1=sum(0)
h3_2=sum(0)
h3_3=sum(-2/(2*s1bhat^2 - (2*covb^2)/s2bhat^2))
h3_4=sum(-(2*s1bhat*s2bhat^2*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_5=sum(0)
h3_6=sum(0)
h3_7=sum(covb/(- covb^2 + s1bhat^2*s2bhat^2))
h3_8=sum((s2bhat*(2*covb^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat) - 2*covb*s1bhat^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_9=sum(0)
h3_10=sum((covb^2*(lam2*m2bhat - W2b^lam2 + 1) + s2bhat^2*((lam2*m2bhat - W2b^lam2 + 1)*s1bhat^2 - 2*covb*lam2*m1bhat))/(lam2*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s2bhat^2*(2*lam2 - 2*W1b^lam1*lam2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_11=sum((2*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h3_12=sum(-(covb*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h4_1=sum(0)
h4_2=sum(0)
h4_3=sum(-(2*s1bhat*s2bhat^2*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_4=sum(- (3*s1bhat^2*s2bhat^6 + covb^4*(lam1^2*m2bhat^2 + lam1^2*s2bhat^2) - covb^3*(2*lam1*m2bhat*s2bhat^2 - 2*W1b^lam1*lam1*m2bhat*s2bhat^2 + 2*lam1^2*m1bhat*m2bhat*s2bhat^2) - covb*(6*lam1*m2bhat*s1bhat^2*s2bhat^4 - 6*W1b^lam1*lam1*m2bhat*s1bhat^2*s2bhat^4 + 6*lam1^2*m1bhat*m2bhat*s1bhat^2*s2bhat^4) + covb^2*(W1b^(2*lam1)*s2bhat^4 - 2*W1b^lam1*s2bhat^4 + s2bhat^4 + 2*lam1*m1bhat*s2bhat^4 + lam1^2*m1bhat^2*s2bhat^4 - 2*W1b^lam1*lam1*m1bhat*s2bhat^4 + 3*lam1^2*m2bhat^2*s1bhat^2*s2bhat^2) - 6*W1b^lam1*s1bhat^2*s2bhat^6 + 3*W1b^(2*lam1)*s1bhat^2*s2bhat^6 - lam1^2*s1bhat^4*s2bhat^6 + 3*lam1^2*m1bhat^2*s1bhat^2*s2bhat^6 + 6*lam1*m1bhat*s1bhat^2*s2bhat^6 - 6*W1b^lam1*lam1*m1bhat*s1bhat^2*s2bhat^6)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^4*(W2b^(2*lam2)*lam1^2 - 2*W2b^lam2*lam1^2 + lam1^2) + covb^2*(3*lam1^2*s1bhat^2*s2bhat^2 + 3*W2b^(2*lam2)*lam1^2*s1bhat^2*s2bhat^2 - 6*W2b^lam2*lam1^2*s1bhat^2*s2bhat^2) - lam2*((2*W2b^lam2*lam1^2*m2bhat - 2*lam1^2*m2bhat)*covb^4 + (2*lam1*s2bhat^2 + 2*lam1^2*m1bhat*s2bhat^2 - 2*W1b^lam1*lam1*s2bhat^2 - 2*W2b^lam2*lam1*s2bhat^2 - 2*W2b^lam2*lam1^2*m1bhat*s2bhat^2 + 2*W1b^lam1*W2b^lam2*lam1*s2bhat^2)*covb^3 + (6*W2b^lam2*lam1^2*m2bhat*s1bhat^2*s2bhat^2 - 6*lam1^2*m2bhat*s1bhat^2*s2bhat^2)*covb^2 + (6*lam1*s1bhat^2*s2bhat^4 + 6*lam1^2*m1bhat*s1bhat^2*s2bhat^4 - 6*W1b^lam1*lam1*s1bhat^2*s2bhat^4 - 6*W2b^lam2*lam1*s1bhat^2*s2bhat^4 - 6*W2b^lam2*lam1^2*m1bhat*s1bhat^2*s2bhat^4 + 6*W1b^lam1*W2b^lam2*lam1*s1bhat^2*s2bhat^4)*covb))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_5=sum(0)
h4_6=sum(0)
h4_7=sum((s1bhat*(2*covb^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat) - 2*covb*s2bhat^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_8=sum((lam2*((2*covb*s2bhat^3*(2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat) - 2*covb*s2bhat*(4*covb*lam1^2*m2bhat - 4*W2b^lam2*covb*lam1^2*m2bhat))*s1bhat^3 + 2*covb*s2bhat*(2*covb^2*lam1 + 2*covb^2*lam1^2*m1bhat - 2*W1b^lam1*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1^2*m1bhat + 2*W1b^lam1*W2b^lam2*covb^2*lam1)*s1bhat) - 2*covb*s1bhat^3*s2bhat*(2*covb*lam1^2 + 2*W2b^(2*lam2)*covb*lam1^2 - 4*W2b^lam2*covb*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - ((4*covb^2*lam1^2*m2bhat^2*s2bhat - 2*covb*s2bhat^3*(2*lam1*m2bhat + covb*lam1^2 + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat))*s1bhat^3 + (2*covb*(2*covb + 2*W1b^(2*lam1)*covb - 4*W1b^lam1*covb + 2*covb*lam1^2*m1bhat^2 + 4*covb*lam1*m1bhat - 4*W1b^lam1*covb*lam1*m1bhat)*s2bhat^3 + 2*covb*(covb^3*lam1^2 - 2*covb^2*lam1*m2bhat + 2*W1b^lam1*covb^2*lam1*m2bhat - 2*covb^2*lam1^2*m1bhat*m2bhat)*s2bhat)*s1bhat)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_9=sum(0)
h4_10=sum(- (covb*(s2bhat^4*(- 4*lam2^2*m1bhat^2*s1bhat + 2*lam2^2*s1bhat^3) - s1bhat^3*s2bhat^2*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2)) - covb^3*(s1bhat*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2) + 2*lam2^2*s1bhat*s2bhat^2) + s1bhat^3*s2bhat^4*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + covb^2*s1bhat*s2bhat^2*(6*lam2*m1bhat + 6*lam2^2*m1bhat*m2bhat - 6*W2b^lam2*lam2*m1bhat))/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2bhat - 6*W1b^lam1*lam2 - 6*W2b^lam2*lam2 + 6*W1b^lam1*W2b^lam2*lam2 - 6*W1b^lam1*lam2^2*m2bhat)*covb^2*s1bhat*s2bhat^2 + (8*W1b^lam1*lam2^2*m1bhat - 8*lam2^2*m1bhat)*covb*s1bhat*s2bhat^4 + (2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat)*s1bhat^3*s2bhat^4) - covb*s1bhat*s2bhat^4*(4*W1b^(2*lam1)*lam2^2 - 8*W1b^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_11=sum((2*s1bhat*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_12=sum(-(2*covb*s1bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h5_1=sum(cova/(- cova^2 + s1ahat^2*s2ahat^2))
h5_2=sum((s1ahat*(2*cova^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat) - 2*cova*s2ahat^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_3=0
h5_4=0
h5_5=sum(-2/(2*s2ahat^2 - (2*cova^2)/s1ahat^2))
h5_6=sum(-(2*s1ahat^2*s2ahat*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_7=0
h5_8=0
h5_9=sum((cova^2*(lam1*m1ahat - W1a^lam1 + 1) + s1ahat^2*((lam1*m1ahat - W1a^lam1 + 1)*s2ahat^2 - 2*cova*lam1*m2ahat))/(lam1*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s1ahat^2*(2*lam1 - 2*W2a^lam2*lam1))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_10=0
h5_11=sum(-(cova*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h5_12=sum((2*s1ahat^2*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h6_1=sum((s2ahat*(2*cova^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat) - 2*cova*s1ahat^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_2=sum((lam2*((2*cova*s2ahat^3*(2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat) - 2*cova*s2ahat*(4*cova*lam1^2*m2ahat - 4*W2a^lam2*cova*lam1^2*m2ahat))*s1ahat^3 + 2*cova*s2ahat*(2*cova^2*lam1 + 2*cova^2*lam1^2*m1ahat - 2*W1a^lam1*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1^2*m1ahat + 2*W1a^lam1*W2a^lam2*cova^2*lam1)*s1ahat) - 2*cova*s1ahat^3*s2ahat*(2*cova*lam1^2 + 2*W2a^(2*lam2)*cova*lam1^2 - 4*W2a^lam2*cova*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - ((4*cova^2*lam1^2*m2ahat^2*s2ahat - 2*cova*s2ahat^3*(2*lam1*m2ahat + cova*lam1^2 + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat))*s1ahat^3 + (2*cova*(2*cova + 2*W1a^(2*lam1)*cova - 4*W1a^lam1*cova + 2*cova*lam1^2*m1ahat^2 + 4*cova*lam1*m1ahat - 4*W1a^lam1*cova*lam1*m1ahat)*s2ahat^3 + 2*cova*(cova^3*lam1^2 - 2*cova^2*lam1*m2ahat + 2*W1a^lam1*cova^2*lam1*m2ahat - 2*cova^2*lam1^2*m1ahat*m2ahat)*s2ahat)*s1ahat)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_3=0
h6_4=0
h6_5=sum(-(2*s1ahat^2*s2ahat*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_6=sum(- (3*s1ahat^6*s2ahat^2 + cova^4*(lam2^2*m1ahat^2 + lam2^2*s1ahat^2) - cova^3*(2*lam2*m1ahat*s1ahat^2 - 2*W2a^lam2*lam2*m1ahat*s1ahat^2 + 2*lam2^2*m1ahat*m2ahat*s1ahat^2) - cova*(6*lam2*m1ahat*s1ahat^4*s2ahat^2 - 6*W2a^lam2*lam2*m1ahat*s1ahat^4*s2ahat^2 + 6*lam2^2*m1ahat*m2ahat*s1ahat^4*s2ahat^2) + cova^2*(W2a^(2*lam2)*s1ahat^4 - 2*W2a^lam2*s1ahat^4 + s1ahat^4 + 2*lam2*m2ahat*s1ahat^4 + lam2^2*m2ahat^2*s1ahat^4 - 2*W2a^lam2*lam2*m2ahat*s1ahat^4 + 3*lam2^2*m1ahat^2*s1ahat^2*s2ahat^2) - 6*W2a^lam2*s1ahat^6*s2ahat^2 + 3*W2a^(2*lam2)*s1ahat^6*s2ahat^2 - lam2^2*s1ahat^6*s2ahat^4 + 3*lam2^2*m2ahat^2*s1ahat^6*s2ahat^2 + 6*lam2*m2ahat*s1ahat^6*s2ahat^2 - 6*W2a^lam2*lam2*m2ahat*s1ahat^6*s2ahat^2)/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^4*(W1a^(2*lam1)*lam2^2 - 2*W1a^lam1*lam2^2 + lam2^2) + cova^2*(3*lam2^2*s1ahat^2*s2ahat^2 + 3*W1a^(2*lam1)*lam2^2*s1ahat^2*s2ahat^2 - 6*W1a^lam1*lam2^2*s1ahat^2*s2ahat^2) - lam1*((2*W1a^lam1*lam2^2*m1ahat - 2*lam2^2*m1ahat)*cova^4 + (2*lam2*s1ahat^2 + 2*lam2^2*m2ahat*s1ahat^2 - 2*W1a^lam1*lam2*s1ahat^2 - 2*W2a^lam2*lam2*s1ahat^2 - 2*W1a^lam1*lam2^2*m2ahat*s1ahat^2 + 2*W1a^lam1*W2a^lam2*lam2*s1ahat^2)*cova^3 + (6*W1a^lam1*lam2^2*m1ahat*s1ahat^2*s2ahat^2 - 6*lam2^2*m1ahat*s1ahat^2*s2ahat^2)*cova^2 + (6*lam2*s1ahat^4*s2ahat^2 + 6*lam2^2*m2ahat*s1ahat^4*s2ahat^2 - 6*W1a^lam1*lam2*s1ahat^4*s2ahat^2 - 6*W2a^lam2*lam2*s1ahat^4*s2ahat^2 - 6*W1a^lam1*lam2^2*m2ahat*s1ahat^4*s2ahat^2 + 6*W1a^lam1*W2a^lam2*lam2*s1ahat^4*s2ahat^2)*cova))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_7=0
h6_8=0
h6_9=sum(- (cova*(s1ahat^4*(- 4*lam1^2*m2ahat^2*s2ahat + 2*lam1^2*s2ahat^3) - s1ahat^2*s2ahat^3*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2)) - cova^3*(s2ahat*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2) + 2*lam1^2*s1ahat^2*s2ahat) + s1ahat^4*s2ahat^3*(2*lam1*m2ahat + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat) + cova^2*s1ahat^2*s2ahat*(6*lam1*m2ahat + 6*lam1^2*m1ahat*m2ahat - 6*W1a^lam1*lam1*m2ahat))/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1ahat - 6*W1a^lam1*lam1 - 6*W2a^lam2*lam1 + 6*W1a^lam1*W2a^lam2*lam1 - 6*W2a^lam2*lam1^2*m1ahat)*cova^2*s1ahat^2*s2ahat + (8*W2a^lam2*lam1^2*m2ahat - 8*lam1^2*m2ahat)*cova*s1ahat^4*s2ahat + (2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat)*s1ahat^4*s2ahat^3) - cova*s1ahat^4*s2ahat*(4*W2a^(2*lam2)*lam1^2 - 8*W2a^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_10=0
h6_11=sum(-(2*cova*s2ahat*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_12=sum((2*s1ahat^2*s2ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h7_1=0
h7_2=0
h7_3=sum(covb/(- covb^2 + s1bhat^2*s2bhat^2))
h7_4=sum((s1bhat*(2*covb^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat) - 2*covb*s2bhat^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_5=0
h7_6=0
h7_7=sum(-2/(2*s2bhat^2 - (2*covb^2)/s1bhat^2))
h7_8=sum(-(2*s1bhat^2*s2bhat*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_9=0
h7_10=sum((covb^2*(lam1*m1bhat - W1b^lam1 + 1) + s1bhat^2*((lam1*m1bhat - W1b^lam1 + 1)*s2bhat^2 - 2*covb*lam1*m2bhat))/(lam1*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s1bhat^2*(2*lam1 - 2*W2b^lam2*lam1))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_11=sum(-(covb*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h7_12=sum((2*s1bhat^2*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h8_1=0
h8_2=0
h8_3=sum((s2bhat*(2*covb^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat) - 2*covb*s1bhat^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_4=sum((lam2*((2*covb*s2bhat^3*(2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat) - 2*covb*s2bhat*(4*covb*lam1^2*m2bhat - 4*W2b^lam2*covb*lam1^2*m2bhat))*s1bhat^3 + 2*covb*s2bhat*(2*covb^2*lam1 + 2*covb^2*lam1^2*m1bhat - 2*W1b^lam1*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1^2*m1bhat + 2*W1b^lam1*W2b^lam2*covb^2*lam1)*s1bhat) - 2*covb*s1bhat^3*s2bhat*(2*covb*lam1^2 + 2*W2b^(2*lam2)*covb*lam1^2 - 4*W2b^lam2*covb*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - ((4*covb^2*lam1^2*m2bhat^2*s2bhat - 2*covb*s2bhat^3*(2*lam1*m2bhat + covb*lam1^2 + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat))*s1bhat^3 + (2*covb*(2*covb + 2*W1b^(2*lam1)*covb - 4*W1b^lam1*covb + 2*covb*lam1^2*m1bhat^2 + 4*covb*lam1*m1bhat - 4*W1b^lam1*covb*lam1*m1bhat)*s2bhat^3 + 2*covb*(covb^3*lam1^2 - 2*covb^2*lam1*m2bhat + 2*W1b^lam1*covb^2*lam1*m2bhat - 2*covb^2*lam1^2*m1bhat*m2bhat)*s2bhat)*s1bhat)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_5=0
h8_6=0
h8_7=sum(-(2*s1bhat^2*s2bhat*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_8=sum(- (3*s1bhat^6*s2bhat^2 + covb^4*(lam2^2*m1bhat^2 + lam2^2*s1bhat^2) - covb^3*(2*lam2*m1bhat*s1bhat^2 - 2*W2b^lam2*lam2*m1bhat*s1bhat^2 + 2*lam2^2*m1bhat*m2bhat*s1bhat^2) - covb*(6*lam2*m1bhat*s1bhat^4*s2bhat^2 - 6*W2b^lam2*lam2*m1bhat*s1bhat^4*s2bhat^2 + 6*lam2^2*m1bhat*m2bhat*s1bhat^4*s2bhat^2) + covb^2*(W2b^(2*lam2)*s1bhat^4 - 2*W2b^lam2*s1bhat^4 + s1bhat^4 + 2*lam2*m2bhat*s1bhat^4 + lam2^2*m2bhat^2*s1bhat^4 - 2*W2b^lam2*lam2*m2bhat*s1bhat^4 + 3*lam2^2*m1bhat^2*s1bhat^2*s2bhat^2) - 6*W2b^lam2*s1bhat^6*s2bhat^2 + 3*W2b^(2*lam2)*s1bhat^6*s2bhat^2 - lam2^2*s1bhat^6*s2bhat^4 + 3*lam2^2*m2bhat^2*s1bhat^6*s2bhat^2 + 6*lam2*m2bhat*s1bhat^6*s2bhat^2 - 6*W2b^lam2*lam2*m2bhat*s1bhat^6*s2bhat^2)/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^4*(W1b^(2*lam1)*lam2^2 - 2*W1b^lam1*lam2^2 + lam2^2) + covb^2*(3*lam2^2*s1bhat^2*s2bhat^2 + 3*W1b^(2*lam1)*lam2^2*s1bhat^2*s2bhat^2 - 6*W1b^lam1*lam2^2*s1bhat^2*s2bhat^2) - lam1*((2*W1b^lam1*lam2^2*m1bhat - 2*lam2^2*m1bhat)*covb^4 + (2*lam2*s1bhat^2 + 2*lam2^2*m2bhat*s1bhat^2 - 2*W1b^lam1*lam2*s1bhat^2 - 2*W2b^lam2*lam2*s1bhat^2 - 2*W1b^lam1*lam2^2*m2bhat*s1bhat^2 + 2*W1b^lam1*W2b^lam2*lam2*s1bhat^2)*covb^3 + (6*W1b^lam1*lam2^2*m1bhat*s1bhat^2*s2bhat^2 - 6*lam2^2*m1bhat*s1bhat^2*s2bhat^2)*covb^2 + (6*lam2*s1bhat^4*s2bhat^2 + 6*lam2^2*m2bhat*s1bhat^4*s2bhat^2 - 6*W1b^lam1*lam2*s1bhat^4*s2bhat^2 - 6*W2b^lam2*lam2*s1bhat^4*s2bhat^2 - 6*W1b^lam1*lam2^2*m2bhat*s1bhat^4*s2bhat^2 + 6*W1b^lam1*W2b^lam2*lam2*s1bhat^4*s2bhat^2)*covb))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_9=0
h8_10=sum(- (covb*(s1bhat^4*(- 4*lam1^2*m2bhat^2*s2bhat + 2*lam1^2*s2bhat^3) - s1bhat^2*s2bhat^3*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2)) - covb^3*(s2bhat*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2) + 2*lam1^2*s1bhat^2*s2bhat) + s1bhat^4*s2bhat^3*(2*lam1*m2bhat + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat) + covb^2*s1bhat^2*s2bhat*(6*lam1*m2bhat + 6*lam1^2*m1bhat*m2bhat - 6*W1b^lam1*lam1*m2bhat))/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1bhat - 6*W1b^lam1*lam1 - 6*W2b^lam2*lam1 + 6*W1b^lam1*W2b^lam2*lam1 - 6*W2b^lam2*lam1^2*m1bhat)*covb^2*s1bhat^2*s2bhat + (8*W2b^lam2*lam1^2*m2bhat - 8*lam1^2*m2bhat)*covb*s1bhat^4*s2bhat + (2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat)*s1bhat^4*s2bhat^3) - covb*s1bhat^4*s2bhat*(4*W2b^(2*lam2)*lam1^2 - 8*W2b^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_11=sum(-(2*covb*s2bhat*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_12=sum((2*s1bhat^2*s2bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h9_1=sum((cova^2*(lam2*m2ahat - W2a^lam2 + 1) + s2ahat^2*((lam2*m2ahat - W2a^lam2 + 1)*s1ahat^2 - 2*cova*lam2*m1ahat))/(lam2*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s2ahat^2*(2*lam2 - 2*W1a^lam1*lam2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_2=sum(- (cova*(s2ahat^4*(- 4*lam2^2*m1ahat^2*s1ahat + 2*lam2^2*s1ahat^3) - s1ahat^3*s2ahat^2*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2)) - cova^3*(s1ahat*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2) + 2*lam2^2*s1ahat*s2ahat^2) + s1ahat^3*s2ahat^4*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + cova^2*s1ahat*s2ahat^2*(6*lam2*m1ahat + 6*lam2^2*m1ahat*m2ahat - 6*W2a^lam2*lam2*m1ahat))/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2ahat - 6*W1a^lam1*lam2 - 6*W2a^lam2*lam2 + 6*W1a^lam1*W2a^lam2*lam2 - 6*W1a^lam1*lam2^2*m2ahat)*cova^2*s1ahat*s2ahat^2 + (8*W1a^lam1*lam2^2*m1ahat - 8*lam2^2*m1ahat)*cova*s1ahat*s2ahat^4 + (2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat)*s1ahat^3*s2ahat^4) - cova*s1ahat*s2ahat^4*(4*W1a^(2*lam1)*lam2^2 - 8*W1a^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_3=0
h9_4=0
h9_5=sum((cova^2*(lam1*m1ahat - W1a^lam1 + 1) + s1ahat^2*((lam1*m1ahat - W1a^lam1 + 1)*s2ahat^2 - 2*cova*lam1*m2ahat))/(lam1*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s1ahat^2*(2*lam1 - 2*W2a^lam2*lam1))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_6=sum(- (cova*(s1ahat^4*(- 4*lam1^2*m2ahat^2*s2ahat + 2*lam1^2*s2ahat^3) - s1ahat^2*s2ahat^3*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2)) - cova^3*(s2ahat*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2) + 2*lam1^2*s1ahat^2*s2ahat) + s1ahat^4*s2ahat^3*(2*lam1*m2ahat + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat) + cova^2*s1ahat^2*s2ahat*(6*lam1*m2ahat + 6*lam1^2*m1ahat*m2ahat - 6*W1a^lam1*lam1*m2ahat))/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1ahat - 6*W1a^lam1*lam1 - 6*W2a^lam2*lam1 + 6*W1a^lam1*W2a^lam2*lam1 - 6*W2a^lam2*lam1^2*m1ahat)*cova^2*s1ahat^2*s2ahat + (8*W2a^lam2*lam1^2*m2ahat - 8*lam1^2*m2ahat)*cova*s1ahat^4*s2ahat + (2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat)*s1ahat^4*s2ahat^3) - cova*s1ahat^4*s2ahat*(4*W2a^(2*lam2)*lam1^2 - 8*W2a^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_7=0
h9_8=0
h9_9=sum(- (s1ahat^2*(cova^2*(6*lam2*m2ahat - 6*W2a^lam2 + 3*W2a^(2*lam2) + 3*lam2^2*m2ahat^2 - 6*W2a^lam2*lam2*m2ahat + 3) - cova*(6*lam2*m1ahat*s2ahat^2 - 6*W2a^lam2*lam2*m1ahat*s2ahat^2 + 6*lam2^2*m1ahat*m2ahat*s2ahat^2) + lam2^2*m1ahat^2*s2ahat^4) - cova^3*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + s1ahat^4*(W2a^(2*lam2)*s2ahat^2 - 2*W2a^lam2*s2ahat^2 + s2ahat^2 - lam2^2*s2ahat^4 + 2*lam2*m2ahat*s2ahat^2 + lam2^2*m2ahat^2*s2ahat^2 - 2*W2a^lam2*lam2*m2ahat*s2ahat^2) + cova^4*lam2^2 + 3*cova^2*lam2^2*m1ahat^2*s2ahat^2)/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^2*(3*lam2^2*s2ahat^2 - 6*W1a^lam1*lam2^2*s2ahat^2 + 3*W1a^(2*lam1)*lam2^2*s2ahat^2) + s1ahat^2*(lam2^2*s2ahat^4 - 2*W1a^lam1*lam2^2*s2ahat^4 + W1a^(2*lam1)*lam2^2*s2ahat^4) - lam1*(cova^3*(2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat) - cova^2*(6*lam2^2*m1ahat*s2ahat^2 - 6*W1a^lam1*lam2^2*m1ahat*s2ahat^2) + s1ahat^2*(cova*(6*lam2*s2ahat^2 + 6*lam2^2*m2ahat*s2ahat^2 - 6*W1a^lam1*lam2*s2ahat^2 - 6*W2a^lam2*lam2*s2ahat^2 - 6*W1a^lam1*lam2^2*m2ahat*s2ahat^2 + 6*W1a^lam1*W2a^lam2*lam2*s2ahat^2) - 2*lam2^2*m1ahat*s2ahat^4 + 2*W1a^lam1*lam2^2*m1ahat*s2ahat^4)))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_10=0
h9_11=sum(-((W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova^2*lam1 - 2*cova*lam2*s2ahat^2 + lam1*s1ahat^2*s2ahat^2 - W2a^lam2*cova^2*lam1 + 2*W1a^lam1*cova*lam2*s2ahat^2 + cova^2*lam1*lam2*m2ahat - W2a^lam2*lam1*s1ahat^2*s2ahat^2 + lam1*lam2*m2ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_12=sum(-((W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova^2*lam2 - 2*cova*lam1*s1ahat^2 + lam2*s1ahat^2*s2ahat^2 - W1a^lam1*cova^2*lam2 + 2*W2a^lam2*cova*lam1*s1ahat^2 + cova^2*lam1*lam2*m1ahat - W1a^lam1*lam2*s1ahat^2*s2ahat^2 + lam1*lam2*m1ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h10_1=0
h10_2=0
h10_3=sum((covb^2*(lam2*m2bhat - W2b^lam2 + 1) + s2bhat^2*((lam2*m2bhat - W2b^lam2 + 1)*s1bhat^2 - 2*covb*lam2*m1bhat))/(lam2*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s2bhat^2*(2*lam2 - 2*W1b^lam1*lam2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_4=sum(- (covb*(s2bhat^4*(- 4*lam2^2*m1bhat^2*s1bhat + 2*lam2^2*s1bhat^3) - s1bhat^3*s2bhat^2*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2)) - covb^3*(s1bhat*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2) + 2*lam2^2*s1bhat*s2bhat^2) + s1bhat^3*s2bhat^4*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + covb^2*s1bhat*s2bhat^2*(6*lam2*m1bhat + 6*lam2^2*m1bhat*m2bhat - 6*W2b^lam2*lam2*m1bhat))/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2bhat - 6*W1b^lam1*lam2 - 6*W2b^lam2*lam2 + 6*W1b^lam1*W2b^lam2*lam2 - 6*W1b^lam1*lam2^2*m2bhat)*covb^2*s1bhat*s2bhat^2 + (8*W1b^lam1*lam2^2*m1bhat - 8*lam2^2*m1bhat)*covb*s1bhat*s2bhat^4 + (2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat)*s1bhat^3*s2bhat^4) - covb*s1bhat*s2bhat^4*(4*W1b^(2*lam1)*lam2^2 - 8*W1b^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_5=0
h10_6=0
h10_7=sum((covb^2*(lam1*m1bhat - W1b^lam1 + 1) + s1bhat^2*((lam1*m1bhat - W1b^lam1 + 1)*s2bhat^2 - 2*covb*lam1*m2bhat))/(lam1*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s1bhat^2*(2*lam1 - 2*W2b^lam2*lam1))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_8=sum(- (covb*(s1bhat^4*(- 4*lam1^2*m2bhat^2*s2bhat + 2*lam1^2*s2bhat^3) - s1bhat^2*s2bhat^3*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2)) - covb^3*(s2bhat*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2) + 2*lam1^2*s1bhat^2*s2bhat) + s1bhat^4*s2bhat^3*(2*lam1*m2bhat + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat) + covb^2*s1bhat^2*s2bhat*(6*lam1*m2bhat + 6*lam1^2*m1bhat*m2bhat - 6*W1b^lam1*lam1*m2bhat))/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1bhat - 6*W1b^lam1*lam1 - 6*W2b^lam2*lam1 + 6*W1b^lam1*W2b^lam2*lam1 - 6*W2b^lam2*lam1^2*m1bhat)*covb^2*s1bhat^2*s2bhat + (8*W2b^lam2*lam1^2*m2bhat - 8*lam1^2*m2bhat)*covb*s1bhat^4*s2bhat + (2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat)*s1bhat^4*s2bhat^3) - covb*s1bhat^4*s2bhat*(4*W2b^(2*lam2)*lam1^2 - 8*W2b^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_9=0
h10_10=sum(- (s1bhat^2*(covb^2*(6*lam2*m2bhat - 6*W2b^lam2 + 3*W2b^(2*lam2) + 3*lam2^2*m2bhat^2 - 6*W2b^lam2*lam2*m2bhat + 3) - covb*(6*lam2*m1bhat*s2bhat^2 - 6*W2b^lam2*lam2*m1bhat*s2bhat^2 + 6*lam2^2*m1bhat*m2bhat*s2bhat^2) + lam2^2*m1bhat^2*s2bhat^4) - covb^3*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + s1bhat^4*(W2b^(2*lam2)*s2bhat^2 - 2*W2b^lam2*s2bhat^2 + s2bhat^2 - lam2^2*s2bhat^4 + 2*lam2*m2bhat*s2bhat^2 + lam2^2*m2bhat^2*s2bhat^2 - 2*W2b^lam2*lam2*m2bhat*s2bhat^2) + covb^4*lam2^2 + 3*covb^2*lam2^2*m1bhat^2*s2bhat^2)/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^2*(3*lam2^2*s2bhat^2 - 6*W1b^lam1*lam2^2*s2bhat^2 + 3*W1b^(2*lam1)*lam2^2*s2bhat^2) + s1bhat^2*(lam2^2*s2bhat^4 - 2*W1b^lam1*lam2^2*s2bhat^4 + W1b^(2*lam1)*lam2^2*s2bhat^4) - lam1*(covb^3*(2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat) - covb^2*(6*lam2^2*m1bhat*s2bhat^2 - 6*W1b^lam1*lam2^2*m1bhat*s2bhat^2) + s1bhat^2*(covb*(6*lam2*s2bhat^2 + 6*lam2^2*m2bhat*s2bhat^2 - 6*W1b^lam1*lam2*s2bhat^2 - 6*W2b^lam2*lam2*s2bhat^2 - 6*W1b^lam1*lam2^2*m2bhat*s2bhat^2 + 6*W1b^lam1*W2b^lam2*lam2*s2bhat^2) - 2*lam2^2*m1bhat*s2bhat^4 + 2*W1b^lam1*lam2^2*m1bhat*s2bhat^4)))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_11=sum(-((W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb^2*lam1 - 2*covb*lam2*s2bhat^2 + lam1*s1bhat^2*s2bhat^2 - W2b^lam2*covb^2*lam1 + 2*W1b^lam1*covb*lam2*s2bhat^2 + covb^2*lam1*lam2*m2bhat - W2b^lam2*lam1*s1bhat^2*s2bhat^2 + lam1*lam2*m2bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_12=sum(-((W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb^2*lam2 - 2*covb*lam1*s1bhat^2 + lam2*s1bhat^2*s2bhat^2 - W1b^lam1*covb^2*lam2 + 2*W2b^lam2*covb*lam1*s1bhat^2 + covb^2*lam1*lam2*m1bhat - W1b^lam1*lam2*s1bhat^2*s2bhat^2 + lam1*lam2*m1bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_1=sum((2*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h11_2=sum((2*s1ahat*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_3=sum((2*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h11_4=sum((2*s1bhat*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_5=sum(-(cova*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h11_6=sum(-(2*cova*s2ahat*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_7=sum(-(covb*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h11_8=sum(-(2*covb*s2bhat*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_9=sum(-((W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova^2*lam1 - 2*cova*lam2*s2ahat^2 + lam1*s1ahat^2*s2ahat^2 - W2a^lam2*cova^2*lam1 + 2*W1a^lam1*cova*lam2*s2ahat^2 + cova^2*lam1*lam2*m2ahat - W2a^lam2*lam1*s1ahat^2*s2ahat^2 + lam1*lam2*m2ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_10=sum(-((W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb^2*lam1 - 2*covb*lam2*s2bhat^2 + lam1*s1bhat^2*s2bhat^2 - W2b^lam2*covb^2*lam1 + 2*W1b^lam1*covb*lam2*s2bhat^2 + covb^2*lam1*lam2*m2bhat - W2b^lam2*lam1*s1bhat^2*s2bhat^2 + lam1*lam2*m2bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_11=sum(((2*((W1a^lam1-1)/lam1^2-(W1a^lam1*log(W1a))/lam1)^2)/s1ahat^2-(2*(m1ahat-(W1a^lam1-1)/lam1)*((2*W1a^lam1-2)/lam1^3-(2*W1a^lam1*log(W1a))/lam1^2+(W1a^lam1*log(W1a)^2)/lam1))/s1ahat^2+(2*cova*(m2ahat-(W2a^lam2-1)/lam2)*((2*W1a^lam1-2)/lam1^3-(2*W1a^lam1*log(W1a))/lam1^2+(W1a^lam1*log(W1a)^2)/lam1))/(s1ahat^2*s2ahat^2))/((2*cova^2)/(s1ahat^2*s2ahat^2)-2))+sum(((2*((W1b^lam1-1)/lam1^2-(W1b^lam1*log(W1b))/lam1)^2)/s1bhat^2-(2*(m1bhat-(W1b^lam1-1)/lam1)*((2*W1b^lam1-2)/lam1^3-(2*W1b^lam1*log(W1b))/lam1^2+(W1b^lam1*log(W1b)^2)/lam1))/s1bhat^2+(2*covb*(m2bhat-(W2b^lam2-1)/lam2)*((2*W1b^lam1-2)/lam1^3-(2*W1b^lam1*log(W1b))/lam1^2+(W1b^lam1*log(W1b)^2)/lam1))/(s1bhat^2*s2bhat^2))/((2*covb^2)/(s1bhat^2*s2bhat^2)-2))
h11_12=sum((2*cova*(W1a^lam1*lam1*log(W1a)-W1a^lam1+1)*(W2a^lam2*lam2*log(W2a)-W2a^lam2+1))/(lam1^2*lam2^2*(-2*cova^2+2*s1ahat^2*s2ahat^2)))+sum((2*covb*(W1b^lam1*lam1*log(W1b)-W1b^lam1+1)*(W2b^lam2*lam2*log(W2b)-W2b^lam2+1))/(lam1^2*lam2^2*(-2*covb^2+2*s1bhat^2*s2bhat^2)))
h12_1=sum(-(cova*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h12_2=sum(-(2*cova*s1ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_3=sum(-(covb*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h12_4=sum(-(2*covb*s1bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_5=sum((2*s1ahat^2*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h12_6=sum((2*s1ahat^2*s2ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_7=sum((2*s1bhat^2*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h12_8=sum((2*s1bhat^2*s2bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_9=sum(-((W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova^2*lam2 - 2*cova*lam1*s1ahat^2 + lam2*s1ahat^2*s2ahat^2 - W1a^lam1*cova^2*lam2 + 2*W2a^lam2*cova*lam1*s1ahat^2 + cova^2*lam1*lam2*m1ahat - W1a^lam1*lam2*s1ahat^2*s2ahat^2 + lam1*lam2*m1ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_10=sum(-((W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb^2*lam2 - 2*covb*lam1*s1bhat^2 + lam2*s1bhat^2*s2bhat^2 - W1b^lam1*covb^2*lam2 + 2*W2b^lam2*covb*lam1*s1bhat^2 + covb^2*lam1*lam2*m1bhat - W1b^lam1*lam2*s1bhat^2*s2bhat^2 + lam1*lam2*m1bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_11=sum((2*cova*(W1a^lam1*lam1*log(W1a)-W1a^lam1+1)*(W2a^lam2*lam2*log(W2a)-W2a^lam2+1))/(lam1^2*lam2^2*(-2*cova^2+2*s1ahat^2*s2ahat^2)))+sum((2*covb*(W1b^lam1*lam1*log(W1b)-W1b^lam1+1)*(W2b^lam2*lam2*log(W2b)-W2b^lam2+1))/(lam1^2*lam2^2*(-2*covb^2+2*s1bhat^2*s2bhat^2)))
h12_12=sum(((2*((W2a^lam2-1)/lam2^2-(W2a^lam2*log(W2a))/lam2)^2)/s2ahat^2-(2*(m2ahat-(W2a^lam2-1)/lam2)*((2*W2a^lam2-2)/lam2^3-(2*W2a^lam2*log(W2a))/lam2^2+(W2a^lam2*log(W2a)^2)/lam2))/s2ahat^2+(2*cova*(m1ahat-(W1a^lam1-1)/lam1)*((2*W2a^lam2-2)/lam2^3-(2*W2a^lam2*log(W2a))/lam2^2+(W2a^lam2*log(W2a)^2)/lam2))/(s1ahat^2*s2ahat^2))/((2*cova^2)/(s1ahat^2*s2ahat^2)-2))+sum(((2*((W2b^lam2-1)/lam2^2-(W2b^lam2*log(W2b))/lam2)^2)/s2bhat^2-(2*(m2bhat-(W2b^lam2-1)/lam2)*((2*W2b^lam2-2)/lam2^3-(2*W2b^lam2*log(W2b))/lam2^2+(W2b^lam2*log(W2b)^2)/lam2))/s2bhat^2+(2*covb*(m1bhat-(W1b^lam1-1)/lam1)*((2*W2b^lam2-2)/lam2^3-(2*W2b^lam2*log(W2b))/lam2^2+(W2b^lam2*log(W2b)^2)/lam2))/(s1bhat^2*s2bhat^2))/((2*covb^2)/(s1bhat^2*s2bhat^2)-2))
h1r=c(h1_1,h1_2,h1_3,h1_4,h1_5,h1_6,h1_7,h1_8,h1_9,h1_10,h1_11,h1_12)
h2r=c(h2_1,h2_2,h2_3,h2_4,h2_5,h2_6,h2_7,h2_8,h2_9,h2_10,h2_11,h2_12)
h3r=c(h3_1,h3_2,h3_3,h3_4,h3_5,h3_6,h3_7,h3_8,h3_9,h3_10,h3_11,h3_12)
h4r=c(h4_1,h4_2,h4_3,h4_4,h4_5,h4_6,h4_7,h4_8,h4_9,h4_10,h4_11,h4_12)
h5r=c(h5_1,h5_2,h5_3,h5_4,h5_5,h5_6,h5_7,h5_8,h5_9,h5_10,h5_11,h5_12)
h6r=c(h6_1,h6_2,h6_3,h6_4,h6_5,h6_6,h6_7,h6_8,h6_9,h6_10,h6_11,h6_12)
h7r=c(h7_1,h7_2,h7_3,h7_4,h7_5,h7_6,h7_7,h7_8,h7_9,h7_10,h7_11,h7_12)
h8r=c(h8_1,h8_2,h8_3,h8_4,h8_5,h8_6,h8_7,h8_8,h8_9,h8_10,h8_11,h8_12)
h9r=c(h9_1,h9_2,h9_3,h9_4,h9_5,h9_6,h9_7,h9_8,h9_9,h9_10,h9_11,h9_12)
h10r=c(h10_1,h10_2,h10_3,h10_4,h10_5,h10_6,h10_7,h10_8,h10_9,h10_10,h10_11,h10_12)
h11r=c(h11_1,h11_2,h11_3,h11_4,h11_5,h11_6,h11_7,h11_8,h11_9,h11_10,h11_11,h11_12)
h12r=c(h12_1,h12_2,h12_3,h12_4,h12_5,h12_6,h12_7,h12_8,h12_9,h12_10,h12_11,h12_12)
HCF=rbind(h1r,h2r,h3r,h4r,h5r,h6r,h7r,h8r,h9r,h10r,h11r,h12r)
HCF[1,2]=0;HCF[1,6]=0;HCF[1,9]=0;
HCF[2,1]=0;HCF[2,5]=0;
HCF[3,4]=0;HCF[3,8]=0;HCF[3,10]=0;HCF[4,3]=0;
HCF[4,7]=0;
HCF[5,2]=0;HCF[5,6]=0;HCF[5,9]=0;
HCF[6,1]=0;HCF[6,5]=0;
HCF[7,4]=0;HCF[7,8]=0;HCF[7,10]=0;
HCF[8,3]=0;HCF[8,7]=0;
HCF[9,1]=0;HCF[9,5]=0;
HCF[10,3]=0;
HCF[10,7]=0;
HCF[1,1]=na*HCF[1,1];
HCF[1,5]=na*HCF[1,5];
HCF[5,1]=na*HCF[5,1];
HCF[3,3]=nb*HCF[3,3];
HCF[3,7]=nb*HCF[3,7];
HCF[7,3]=nb*HCF[7,3];
HCF[7,7]=nb*HCF[7,7];
HCF[5,5]=na*HCF[5,5];
VV=inv(-HCF)
#############################################################################
#############################################################################
#================ DIFFERENCES OF THE YOUDEN INDICES=======================
m1a=m1ahat;
s1a=s1ahat;
m1b=m1bhat;
s1b=s1bhat;
m2a=m2ahat;
s2a=s2ahat;
m2b=m2bhat;
s2b=s2bhat;
a1=m1b-m1a;
b1=s1b/s1a;
c1=(m1a*(b1^2-1)-a1+b1*sqrt(a1^2+(b1^2-1)*s1a^2*log(b1^2)))/(b1^2-1);
J1=pnorm((m1b-c1)/s1b)+pnorm((c1-m1a)/s1a)-1
J1BCCC=J1
a2=m2b-m2a;
b2=s2b/s2a;
c2=(m2a*(b2^2-1)-a2+b2*sqrt(a2^2+(b2^2-1)*s2a^2*log(b2^2)))/(b2^2-1);
J2=pnorm((m2b-c2)/s2b)+pnorm((c2-m2a)/s2a)-1
J2BCCC=J2
dJ1_dm1a = (2^(1/2)*exp(-(m1a - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1a^2))*((s1b^2/s1a^2 + (s1b*(2*m1a - 2*m1b))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)))/(s1b^2/s1a^2 - 1) - 1))/(2*pi^(1/2)*s1a) - (2^(1/2)*exp(-(m1b - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1b^2))*(s1b^2/s1a^2 + (s1b*(2*m1a - 2*m1b))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))))/(2*pi^(1/2)*s1b*(s1b^2/s1a^2 - 1));
dJ1_ds1a = (2^(1/2)*exp(-(m1b - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1b^2))*(((s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a^2 + (2*m1a*s1b^2)/s1a^3 + (s1b*(2*s1a*(s1b^2/s1a^2 - 1) + (2*s1b^2*log(s1b^2/s1a^2))/s1a - 2*s1a*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1)))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)))/(s1b^2/s1a^2 - 1) - (2*s1b^2*(m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a))/(s1a^3*(s1b^2/s1a^2 - 1)^2)))/(2*pi^(1/2)*s1b) - (exp(-(m1a - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1a^2))*((2^(1/2)*(((s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a^2 + (2*m1a*s1b^2)/s1a^3 + (s1b*(2*s1a*(s1b^2/s1a^2 - 1) + (2*s1b^2*log(s1b^2/s1a^2))/s1a - 2*s1a*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1)))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)))/(s1b^2/s1a^2 - 1) - (2*s1b^2*(m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a))/(s1a^3*(s1b^2/s1a^2 - 1)^2)))/(2*s1a) - (2^(1/2)*(m1a - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1)))/(2*s1a^2)))/pi^(1/2);
dJ1_dm1b = (2^(1/2)*exp(-(m1b - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1b^2))*(((s1b*(2*m1a - 2*m1b))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)) + 1)/(s1b^2/s1a^2 - 1) + 1))/(2*pi^(1/2)*s1b) - (2^(1/2)*exp(-(m1a - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1a^2))*((s1b*(2*m1a - 2*m1b))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)) + 1))/(2*pi^(1/2)*s1a*(s1b^2/s1a^2 - 1));
dJ1_ds1b = (2^(1/2)*exp(-(m1a - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1a^2))*((((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)/s1a + (2*m1a*s1b)/s1a^2 + (s1b*(2*s1b*log(s1b^2/s1a^2) + (2*s1a^2*(s1b^2/s1a^2 - 1))/s1b))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)))/(s1b^2/s1a^2 - 1) - (2*s1b*(m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a))/(s1a^2*(s1b^2/s1a^2 - 1)^2)))/(2*pi^(1/2)*s1a) - (exp(-(m1b - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1))^2/(2*s1b^2))*((2^(1/2)*((((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)/s1a + (2*m1a*s1b)/s1a^2 + (s1b*(2*s1b*log(s1b^2/s1a^2) + (2*s1a^2*(s1b^2/s1a^2 - 1))/s1b))/(2*s1a*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2)))/(s1b^2/s1a^2 - 1) - (2*s1b*(m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a))/(s1a^2*(s1b^2/s1a^2 - 1)^2)))/(2*s1b) + (2^(1/2)*(m1b - (m1a - m1b + m1a*(s1b^2/s1a^2 - 1) + (s1b*((m1a - m1b)^2 + s1a^2*log(s1b^2/s1a^2)*(s1b^2/s1a^2 - 1))^(1/2))/s1a)/(s1b^2/s1a^2 - 1)))/(2*s1b^2)))/pi^(1/2);
dJ1_dm2a =0;
dJ1_ds2a =0;
dJ1_dm2b =0;
dJ1_ds2b =0;
dJ2_dm1a =0;
dJ2_ds1a =0;
dJ2_dm1b =0;
dJ2_ds1b =0;
dJ2_dm2a = (2^(1/2)*exp(-(m2a - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2a^2))*((s2b^2/s2a^2 + (s2b*(2*m2a - 2*m2b))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)))/(s2b^2/s2a^2 - 1) - 1))/(2*pi^(1/2)*s2a) - (2^(1/2)*exp(-(m2b - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2b^2))*(s2b^2/s2a^2 + (s2b*(2*m2a - 2*m2b))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))))/(2*pi^(1/2)*s2b*(s2b^2/s2a^2 - 1));
dJ2_ds2a = (2^(1/2)*exp(-(m2b - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2b^2))*(((s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a^2 + (2*m2a*s2b^2)/s2a^3 + (s2b*(2*s2a*(s2b^2/s2a^2 - 1) + (2*s2b^2*log(s2b^2/s2a^2))/s2a - 2*s2a*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1)))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)))/(s2b^2/s2a^2 - 1) - (2*s2b^2*(m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a))/(s2a^3*(s2b^2/s2a^2 - 1)^2)))/(2*pi^(1/2)*s2b) - (exp(-(m2a - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2a^2))*((2^(1/2)*(((s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a^2 + (2*m2a*s2b^2)/s2a^3 + (s2b*(2*s2a*(s2b^2/s2a^2 - 1) + (2*s2b^2*log(s2b^2/s2a^2))/s2a - 2*s2a*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1)))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)))/(s2b^2/s2a^2 - 1) - (2*s2b^2*(m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a))/(s2a^3*(s2b^2/s2a^2 - 1)^2)))/(2*s2a) - (2^(1/2)*(m2a - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1)))/(2*s2a^2)))/pi^(1/2);
dJ2_dm2b = (2^(1/2)*exp(-(m2b - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2b^2))*(((s2b*(2*m2a - 2*m2b))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)) + 1)/(s2b^2/s2a^2 - 1) + 1))/(2*pi^(1/2)*s2b) - (2^(1/2)*exp(-(m2a - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2a^2))*((s2b*(2*m2a - 2*m2b))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)) + 1))/(2*pi^(1/2)*s2a*(s2b^2/s2a^2 - 1));
dJ2_ds2b = (2^(1/2)*exp(-(m2a - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2a^2))*((((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)/s2a + (2*m2a*s2b)/s2a^2 + (s2b*(2*s2b*log(s2b^2/s2a^2) + (2*s2a^2*(s2b^2/s2a^2 - 1))/s2b))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)))/(s2b^2/s2a^2 - 1) - (2*s2b*(m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a))/(s2a^2*(s2b^2/s2a^2 - 1)^2)))/(2*pi^(1/2)*s2a) - (exp(-(m2b - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1))^2/(2*s2b^2))*((2^(1/2)*((((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)/s2a + (2*m2a*s2b)/s2a^2 + (s2b*(2*s2b*log(s2b^2/s2a^2) + (2*s2a^2*(s2b^2/s2a^2 - 1))/s2b))/(2*s2a*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2)))/(s2b^2/s2a^2 - 1) - (2*s2b*(m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a))/(s2a^2*(s2b^2/s2a^2 - 1)^2)))/(2*s2b) + (2^(1/2)*(m2b - (m2a - m2b + m2a*(s2b^2/s2a^2 - 1) + (s2b*((m2a - m2b)^2 + s2a^2*log(s2b^2/s2a^2)*(s2b^2/s2a^2 - 1))^(1/2))/s2a)/(s2b^2/s2a^2 - 1)))/(2*s2b^2)))/pi^(1/2);
VVpars=VV[1:8,1:8];
VJ1=c(dJ1_dm1a, dJ1_ds1a, dJ1_dm1b, dJ1_ds1b, dJ1_dm2a, dJ1_ds2a, dJ1_dm2b, dJ1_ds2b)%*%VVpars%*%t(t(c(dJ1_dm1a, dJ1_ds1a, dJ1_dm1b, dJ1_ds1b, dJ1_dm2a, dJ1_ds2a, dJ1_dm2b, dJ1_ds2b)));
VJ2=c(dJ2_dm1a, dJ2_ds1a, dJ2_dm1b, dJ2_ds1b, dJ2_dm2a, dJ2_ds2a, dJ2_dm2b, dJ2_ds2b)%*%VVpars%*%t(t(c(dJ2_dm1a, dJ2_ds1a, dJ2_dm1b, dJ2_ds1b, dJ2_dm2a, dJ2_ds2a, dJ2_dm2b, dJ2_ds2b)));
COVJ12=c(dJ1_dm1a, dJ1_ds1a, dJ1_dm1b, dJ1_ds1b, dJ1_dm2a, dJ1_ds2a, dJ1_dm2b, dJ1_ds2b)%*%VVpars%*%t(t(c(dJ2_dm1a, dJ2_ds1a, dJ2_dm1b, dJ2_ds1b, dJ2_dm2a, dJ2_ds2a, dJ2_dm2b, dJ2_ds2b)));
kk1=c(VJ1, COVJ12)
kk2=c(COVJ12, VJ2)
V12=rbind(kk1,kk2)
Z=(J2-J1)/(sqrt(VJ1+VJ2-2*COVJ12));
J1original=J1;
J2original=J2;
SE=sqrt(VJ1+VJ2-2*COVJ12);
CIoriginal=c((J2-J1)-pmalpha*SE, (J2-J1)+pmalpha*SE);
pval2tJ=2*pnorm(-abs(Z))
#JT2=log(0.5*(J2+1)/(1-0.5*(J2+1)));
#JT1=log(0.5*(J1+1)/(1-0.5*(J1+1)));
#VJT1=4/(J1^2-1)^2*VJ1;
#VJT2=4/(J2^2-1)^2*VJ2;
#COVJT12=c(-2/(J1^2-1), 0)%*%V12%*%t(t(c(0, -2/(J2^2-1))))
#Zstar=(JT2-JT1)/(sqrt(VJT1+VJT2-2*COVJT12))
#SE=(sqrt(VJT1+VJT2-2*COVJT12))
#CIZstar=c((JT2-JT1)-pmalpha*SE, (JT2-JT1)+pmalpha*SE)
#CIZstar
JT2=qnorm(J2);
JT1=qnorm(J1);
VJT1=(1/(dnorm(qnorm(J1))))^2*VJ1;
VJT2=(1/(dnorm(qnorm(J2))))^2*VJ2;
COVJT12=c((1/(dnorm(qnorm(J1))))*VJ1, 0)%*%V12%*%t(t(c(0, (1/(dnorm(qnorm(J2))))*VJ2)))
Zstar=(JT2-JT1)/(sqrt(VJT1+VJT2-2*COVJT12))
SE=(sqrt(VJT1+VJT2-2*COVJT12))
CIZstar=c((JT2-JT1)-pmalpha*SE, (JT2-JT1)+pmalpha*SE)
CIZstar
pval2t=2*pnorm(-abs(Zstar))
#CIoriginal=c(2*(exp(CIZstar[1])/(1+exp(CIZstar[1])))-1 , 2*(exp(CIZstar[2])/(1+exp(CIZstar[2])))-1)
#====TWO ROC FUNCTIONS: ONE IS THE BOXCOX AND THE OTHER THE REFERENCE LINE================
roc1<-function(t){
na = length(W1alam)
nb = length(W1blam)
s1alam=sqrt( var(W1alam)*(na-1)/na )
s1blam=sqrt( var(W1blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W1alam),
sd=s1alam),
mean=mean(W1blam),
sd=s1blam)
}
roc2<-function(t){
na = length(W2alam)
nb = length(W2blam)
s2alam=sqrt( var(W2alam)*(na-1)/na )
s2blam=sqrt( var(W2blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W2alam),
sd=s2alam),
mean=mean(W2blam),
sd=s2blam)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
Sens1=1-pnorm(c1, mean=mean(W1blam), sd=sqrt(var(W1blam)*(nb-1)/nb))
Spec1= pnorm(c1, mean=mean(W1alam), sd=sqrt(var(W1alam)*(na-1)/na))
Sens2=1-pnorm(c2, mean=mean(W2blam), sd=sqrt(var(W2blam)*(nb-1)/nb))
Spec2= pnorm(c2, mean=mean(W2alam), sd=sqrt(var(W2alam)*(na-1)/na))
#================IF PLOTS ARE REQUESTED PLOT THE ROCS==
if (plots=="on") {
# x11()
plot(linspace(0,1,1000),roc1(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="red")
lines(linspace(0,1,1000),roc2(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="black")
lines(linspace(0,1,1000),linspace(0,1,1000),type="l", lty=2)
points(1-Spec1,Sens1, col = "red")
points(1-Spec2,Sens2, col = "black")
#line for the Youden:
lines(c(1-Spec1,1-Spec1),c(rocuseless(1-Spec1),Sens1),col="blue")
lines(c(1-Spec2,1-Spec2),c(rocuseless(1-Spec2),Sens2),col="green")
legend("bottomright", legend=c(paste("ROC for Marker 1 with J =", formattable(J1original, digits = 4, format = "f"), " "),
paste("ROC for Marker 2 with J =", formattable(J2original, digits = 4, format = "f"), " "),
paste("P-value for the difference:", formattable(pval2t, digits = 4, format = "f"), " ")),
col=c("red", "black", "white"), lty=c(1, 1, NA), pch = c(NA, NA, NA), cex=0.8)
}
# if (plots=="on"){
# points(1-Spec1,Sens1, col = "red")
# points(1-Spec2,Sens2, col = "black")
#
# #line for the Youden:
# lines(c(1-Spec1,1-Spec1),c(rocuseless(1-Spec1),Sens1),col="blue")
# lines(c(1-Spec2,1-Spec2),c(rocuseless(1-Spec2),Sens2),col="green")
#
# }
#======================================================
res <- data.frame(J1 = formattable(J1original, digits = 4, format = "f"),
J2 = formattable(J2original, digits = 4, format = "f"),
diff = formattable(J2original - J1original, digits = 4, format = "f"),
p_value_probit = formattable(pval2t, digits = 4, format = "f"),
p_value = formattable(pval2tJ, digits = 4, format = "f"),
ci_ll = formattable(CIoriginal[1], digits = 4, format = "f"),
ci_ul = formattable(CIoriginal[2], digits = 4, format = "f"))
rownames(res) <- c("Estimates:")
colnames(res) <- c("J 1", "J 2", "Diff", "P-Val (Probit)", "P-Val", "CI (LL)", "CI (UL)")
res <- formattable(as.matrix(res), digits = 4, format = "f")
res
#return(list(AUCmarker1=AUC1original,AUCmarker2=AUC2original, pvalue_difference= pval2t, CI_difference= pnorm(CIdiff), rocbc1=roc1, rocbc2=roc2))
#return(list(resultstable=res,J1=J1original,J2=J2original, pvalue_difference= pval2t, CI_difference= CIoriginal, rocbc1=roc1, rocbc2=roc2))
return(list(resultstable=res,
J1=J1original,
J2=J2original,
pvalue_probit_difference= pval2t,
pvalue_difference= pval2tJ,
CI_difference= CIoriginal,
roc1=roc1, roc2=roc2,
transx1=W1alam,
transy1=W1blam,
transformation.parameter.1 = lam[1],
transx2=W2alam,
transy2=W2blam,
transformation.parameter.2 = lam[2]))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
comparebcSens <-function (marker1, marker2, D, atSpec, alpha, plots){
if ((length(marker1) != length(D)) | (length(marker2) != length(D))) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (alpha <= 0 | alpha >= 1) {
stop("ERROR: The level of significance, alpha, should be set between 0 and 1. A common choice is 0.05.")
} else if (sum(is.na(marker1)) > 0 | sum(is.na(marker2)) > 0 | sum(is.na(D)) > 0 | sum(is.na(atSpec)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if ((!is.numeric(atSpec)) | (length(atSpec) != 1)) {
stop("ERROR: 'atSpec' must be a single numeric value.")
} else if ((sum(marker1 < 0) > 0) | (sum(marker2 < 2)) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
erf <- function (x) 2 * pnorm(x * sqrt(2)) - 1
erfc <- function (x) 2 * pnorm(x * sqrt(2), lower.tail = FALSE)
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower.tail = FALSE)/sqrt(2)
pmalpha=qnorm(1-alpha/2);
if (plots!="on"){plots="off"}
W1a=marker1[D==0]
W1b=marker1[D==1]
W2a=marker2[D==0]
W2b=marker2[D==1]
Wa=cbind(W1a,W2a);
Wb=cbind(W1b,W2b);
na=length(W1a)
nb=length(W1b)
likbox2D<-function(W1a,W1b,W2a,W2b,lam){
na=length(W1a);
nb=length(W1b);
out=c();
#for (i in 1:length(h)){
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
cova= cov(Walam)*(na-1)/na;
covb= cov(Wblam)*(nb-1)/nb;
mualam=c(mean(W1alam), mean(W2alam))
mublam=c(mean(W1blam), mean(W2blam))
out= -sum(log(dmvnorm(Walam,mualam,cova)))-sum(log(dmvnorm(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b));
return(out)
}
lam=c(2,2)
likbox2D(W1a,W1b,W2a,W2b,lam)
logL<-function(h){
likbox2D(W1a,W1b,W2a,W2b,h)
}
# init=c(0.8,0.8)
# lam=fminsearch(logL,init)
# lam=c(lam$optbase$xopt)
lam<- optim(c(1,1),logL,gr=NULL,method="BFGS", control=list(maxit=10000))
lam=c(lam$par)
lam
out <- optim(c(1,1), logL, method = "Nelder-Mead")
lam = out$par
###################################
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
m1ahat=mean(W1alam)
m1bhat=mean(W1blam)
m2ahat=mean(W2alam)
m2bhat=mean(W2blam)
s1ahat=sqrt( var(W1alam)*(na-1)/na )
s1bhat=sqrt( var(W1blam)*(nb-1)/nb )
s2ahat=sqrt( var(W2alam)*(na-1)/na )
s2bhat=sqrt( var(W2blam)*(nb-1)/nb )
cova= cov(Walam)*(na-1)/na;cova=cova[1,2];
covb= cov(Wblam)*(nb-1)/nb;covb=covb[1,2];
lam1=lam[1]
lam2=lam[2]
##################################
h1_1=sum(-2/(2*s1ahat^2 - (2*cova^2)/s2ahat^2))
h1_2=sum(-(2*s1ahat*s2ahat^2*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_3=0
h1_4=0;
h1_5=sum(cova/(- cova^2 + s1ahat^2*s2ahat^2));
h1_6=sum((s2ahat*(2*cova^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat) - 2*cova*s1ahat^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_7=0
h1_8=0
h1_9=sum((cova^2*(lam2*m2ahat - W2a^lam2 + 1) + s2ahat^2*((lam2*m2ahat - W2a^lam2 + 1)*s1ahat^2 - 2*cova*lam2*m1ahat))/(lam2*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s2ahat^2*(2*lam2 - 2*W1a^lam1*lam2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h1_10=0
h1_11=sum((2*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h1_12=sum(-(cova*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h2_1=sum(-(2*s1ahat*s2ahat^2*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_2=sum(- (3*s1ahat^2*s2ahat^6 + cova^4*(lam1^2*m2ahat^2 + lam1^2*s2ahat^2) - cova^3*(2*lam1*m2ahat*s2ahat^2 - 2*W1a^lam1*lam1*m2ahat*s2ahat^2 + 2*lam1^2*m1ahat*m2ahat*s2ahat^2) - cova*(6*lam1*m2ahat*s1ahat^2*s2ahat^4 - 6*W1a^lam1*lam1*m2ahat*s1ahat^2*s2ahat^4 + 6*lam1^2*m1ahat*m2ahat*s1ahat^2*s2ahat^4) + cova^2*(W1a^(2*lam1)*s2ahat^4 - 2*W1a^lam1*s2ahat^4 + s2ahat^4 + 2*lam1*m1ahat*s2ahat^4 + lam1^2*m1ahat^2*s2ahat^4 - 2*W1a^lam1*lam1*m1ahat*s2ahat^4 + 3*lam1^2*m2ahat^2*s1ahat^2*s2ahat^2) - 6*W1a^lam1*s1ahat^2*s2ahat^6 + 3*W1a^(2*lam1)*s1ahat^2*s2ahat^6 - lam1^2*s1ahat^4*s2ahat^6 + 3*lam1^2*m1ahat^2*s1ahat^2*s2ahat^6 + 6*lam1*m1ahat*s1ahat^2*s2ahat^6 - 6*W1a^lam1*lam1*m1ahat*s1ahat^2*s2ahat^6)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^4*(W2a^(2*lam2)*lam1^2 - 2*W2a^lam2*lam1^2 + lam1^2) + cova^2*(3*lam1^2*s1ahat^2*s2ahat^2 + 3*W2a^(2*lam2)*lam1^2*s1ahat^2*s2ahat^2 - 6*W2a^lam2*lam1^2*s1ahat^2*s2ahat^2) - lam2*((2*W2a^lam2*lam1^2*m2ahat - 2*lam1^2*m2ahat)*cova^4 + (2*lam1*s2ahat^2 + 2*lam1^2*m1ahat*s2ahat^2 - 2*W1a^lam1*lam1*s2ahat^2 - 2*W2a^lam2*lam1*s2ahat^2 - 2*W2a^lam2*lam1^2*m1ahat*s2ahat^2 + 2*W1a^lam1*W2a^lam2*lam1*s2ahat^2)*cova^3 + (6*W2a^lam2*lam1^2*m2ahat*s1ahat^2*s2ahat^2 - 6*lam1^2*m2ahat*s1ahat^2*s2ahat^2)*cova^2 + (6*lam1*s1ahat^2*s2ahat^4 + 6*lam1^2*m1ahat*s1ahat^2*s2ahat^4 - 6*W1a^lam1*lam1*s1ahat^2*s2ahat^4 - 6*W2a^lam2*lam1*s1ahat^2*s2ahat^4 - 6*W2a^lam2*lam1^2*m1ahat*s1ahat^2*s2ahat^4 + 6*W1a^lam1*W2a^lam2*lam1*s1ahat^2*s2ahat^4)*cova))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_3=sum(0)
h2_4=sum(0)
h2_5=sum((s1ahat*(2*cova^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat) - 2*cova*s2ahat^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_6=sum((lam2*((2*cova*s2ahat^3*(2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat) - 2*cova*s2ahat*(4*cova*lam1^2*m2ahat - 4*W2a^lam2*cova*lam1^2*m2ahat))*s1ahat^3 + 2*cova*s2ahat*(2*cova^2*lam1 + 2*cova^2*lam1^2*m1ahat - 2*W1a^lam1*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1^2*m1ahat + 2*W1a^lam1*W2a^lam2*cova^2*lam1)*s1ahat) - 2*cova*s1ahat^3*s2ahat*(2*cova*lam1^2 + 2*W2a^(2*lam2)*cova*lam1^2 - 4*W2a^lam2*cova*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - ((4*cova^2*lam1^2*m2ahat^2*s2ahat - 2*cova*s2ahat^3*(2*lam1*m2ahat + cova*lam1^2 + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat))*s1ahat^3 + (2*cova*(2*cova + 2*W1a^(2*lam1)*cova - 4*W1a^lam1*cova + 2*cova*lam1^2*m1ahat^2 + 4*cova*lam1*m1ahat - 4*W1a^lam1*cova*lam1*m1ahat)*s2ahat^3 + 2*cova*(cova^3*lam1^2 - 2*cova^2*lam1*m2ahat + 2*W1a^lam1*cova^2*lam1*m2ahat - 2*cova^2*lam1^2*m1ahat*m2ahat)*s2ahat)*s1ahat)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_7=sum(0)
h2_8=sum(0)
h2_9=sum(- (cova*(s2ahat^4*(- 4*lam2^2*m1ahat^2*s1ahat + 2*lam2^2*s1ahat^3) - s1ahat^3*s2ahat^2*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2)) - cova^3*(s1ahat*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2) + 2*lam2^2*s1ahat*s2ahat^2) + s1ahat^3*s2ahat^4*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + cova^2*s1ahat*s2ahat^2*(6*lam2*m1ahat + 6*lam2^2*m1ahat*m2ahat - 6*W2a^lam2*lam2*m1ahat))/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2ahat - 6*W1a^lam1*lam2 - 6*W2a^lam2*lam2 + 6*W1a^lam1*W2a^lam2*lam2 - 6*W1a^lam1*lam2^2*m2ahat)*cova^2*s1ahat*s2ahat^2 + (8*W1a^lam1*lam2^2*m1ahat - 8*lam2^2*m1ahat)*cova*s1ahat*s2ahat^4 + (2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat)*s1ahat^3*s2ahat^4) - cova*s1ahat*s2ahat^4*(4*W1a^(2*lam1)*lam2^2 - 8*W1a^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h2_10=sum(0)
h2_11=sum((2*s1ahat*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h2_12=sum(-(2*cova*s1ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h3_1=sum(0)
h3_2=sum(0)
h3_3=sum(-2/(2*s1bhat^2 - (2*covb^2)/s2bhat^2))
h3_4=sum(-(2*s1bhat*s2bhat^2*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_5=sum(0)
h3_6=sum(0)
h3_7=sum(covb/(- covb^2 + s1bhat^2*s2bhat^2))
h3_8=sum((s2bhat*(2*covb^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat) - 2*covb*s1bhat^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_9=sum(0)
h3_10=sum((covb^2*(lam2*m2bhat - W2b^lam2 + 1) + s2bhat^2*((lam2*m2bhat - W2b^lam2 + 1)*s1bhat^2 - 2*covb*lam2*m1bhat))/(lam2*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s2bhat^2*(2*lam2 - 2*W1b^lam1*lam2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h3_11=sum((2*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h3_12=sum(-(covb*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h4_1=sum(0)
h4_2=sum(0)
h4_3=sum(-(2*s1bhat*s2bhat^2*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_4=sum(- (3*s1bhat^2*s2bhat^6 + covb^4*(lam1^2*m2bhat^2 + lam1^2*s2bhat^2) - covb^3*(2*lam1*m2bhat*s2bhat^2 - 2*W1b^lam1*lam1*m2bhat*s2bhat^2 + 2*lam1^2*m1bhat*m2bhat*s2bhat^2) - covb*(6*lam1*m2bhat*s1bhat^2*s2bhat^4 - 6*W1b^lam1*lam1*m2bhat*s1bhat^2*s2bhat^4 + 6*lam1^2*m1bhat*m2bhat*s1bhat^2*s2bhat^4) + covb^2*(W1b^(2*lam1)*s2bhat^4 - 2*W1b^lam1*s2bhat^4 + s2bhat^4 + 2*lam1*m1bhat*s2bhat^4 + lam1^2*m1bhat^2*s2bhat^4 - 2*W1b^lam1*lam1*m1bhat*s2bhat^4 + 3*lam1^2*m2bhat^2*s1bhat^2*s2bhat^2) - 6*W1b^lam1*s1bhat^2*s2bhat^6 + 3*W1b^(2*lam1)*s1bhat^2*s2bhat^6 - lam1^2*s1bhat^4*s2bhat^6 + 3*lam1^2*m1bhat^2*s1bhat^2*s2bhat^6 + 6*lam1*m1bhat*s1bhat^2*s2bhat^6 - 6*W1b^lam1*lam1*m1bhat*s1bhat^2*s2bhat^6)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^4*(W2b^(2*lam2)*lam1^2 - 2*W2b^lam2*lam1^2 + lam1^2) + covb^2*(3*lam1^2*s1bhat^2*s2bhat^2 + 3*W2b^(2*lam2)*lam1^2*s1bhat^2*s2bhat^2 - 6*W2b^lam2*lam1^2*s1bhat^2*s2bhat^2) - lam2*((2*W2b^lam2*lam1^2*m2bhat - 2*lam1^2*m2bhat)*covb^4 + (2*lam1*s2bhat^2 + 2*lam1^2*m1bhat*s2bhat^2 - 2*W1b^lam1*lam1*s2bhat^2 - 2*W2b^lam2*lam1*s2bhat^2 - 2*W2b^lam2*lam1^2*m1bhat*s2bhat^2 + 2*W1b^lam1*W2b^lam2*lam1*s2bhat^2)*covb^3 + (6*W2b^lam2*lam1^2*m2bhat*s1bhat^2*s2bhat^2 - 6*lam1^2*m2bhat*s1bhat^2*s2bhat^2)*covb^2 + (6*lam1*s1bhat^2*s2bhat^4 + 6*lam1^2*m1bhat*s1bhat^2*s2bhat^4 - 6*W1b^lam1*lam1*s1bhat^2*s2bhat^4 - 6*W2b^lam2*lam1*s1bhat^2*s2bhat^4 - 6*W2b^lam2*lam1^2*m1bhat*s1bhat^2*s2bhat^4 + 6*W1b^lam1*W2b^lam2*lam1*s1bhat^2*s2bhat^4)*covb))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_5=sum(0)
h4_6=sum(0)
h4_7=sum((s1bhat*(2*covb^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat) - 2*covb*s2bhat^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_8=sum((lam2*((2*covb*s2bhat^3*(2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat) - 2*covb*s2bhat*(4*covb*lam1^2*m2bhat - 4*W2b^lam2*covb*lam1^2*m2bhat))*s1bhat^3 + 2*covb*s2bhat*(2*covb^2*lam1 + 2*covb^2*lam1^2*m1bhat - 2*W1b^lam1*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1^2*m1bhat + 2*W1b^lam1*W2b^lam2*covb^2*lam1)*s1bhat) - 2*covb*s1bhat^3*s2bhat*(2*covb*lam1^2 + 2*W2b^(2*lam2)*covb*lam1^2 - 4*W2b^lam2*covb*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - ((4*covb^2*lam1^2*m2bhat^2*s2bhat - 2*covb*s2bhat^3*(2*lam1*m2bhat + covb*lam1^2 + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat))*s1bhat^3 + (2*covb*(2*covb + 2*W1b^(2*lam1)*covb - 4*W1b^lam1*covb + 2*covb*lam1^2*m1bhat^2 + 4*covb*lam1*m1bhat - 4*W1b^lam1*covb*lam1*m1bhat)*s2bhat^3 + 2*covb*(covb^3*lam1^2 - 2*covb^2*lam1*m2bhat + 2*W1b^lam1*covb^2*lam1*m2bhat - 2*covb^2*lam1^2*m1bhat*m2bhat)*s2bhat)*s1bhat)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_9=sum(0)
h4_10=sum(- (covb*(s2bhat^4*(- 4*lam2^2*m1bhat^2*s1bhat + 2*lam2^2*s1bhat^3) - s1bhat^3*s2bhat^2*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2)) - covb^3*(s1bhat*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2) + 2*lam2^2*s1bhat*s2bhat^2) + s1bhat^3*s2bhat^4*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + covb^2*s1bhat*s2bhat^2*(6*lam2*m1bhat + 6*lam2^2*m1bhat*m2bhat - 6*W2b^lam2*lam2*m1bhat))/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2bhat - 6*W1b^lam1*lam2 - 6*W2b^lam2*lam2 + 6*W1b^lam1*W2b^lam2*lam2 - 6*W1b^lam1*lam2^2*m2bhat)*covb^2*s1bhat*s2bhat^2 + (8*W1b^lam1*lam2^2*m1bhat - 8*lam2^2*m1bhat)*covb*s1bhat*s2bhat^4 + (2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat)*s1bhat^3*s2bhat^4) - covb*s1bhat*s2bhat^4*(4*W1b^(2*lam1)*lam2^2 - 8*W1b^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h4_11=sum((2*s1bhat*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h4_12=sum(-(2*covb*s1bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h5_1=sum(cova/(- cova^2 + s1ahat^2*s2ahat^2))
h5_2=sum((s1ahat*(2*cova^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat) - 2*cova*s2ahat^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_3=0
h5_4=0
h5_5=sum(-2/(2*s2ahat^2 - (2*cova^2)/s1ahat^2))
h5_6=sum(-(2*s1ahat^2*s2ahat*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_7=0
h5_8=0
h5_9=sum((cova^2*(lam1*m1ahat - W1a^lam1 + 1) + s1ahat^2*((lam1*m1ahat - W1a^lam1 + 1)*s2ahat^2 - 2*cova*lam1*m2ahat))/(lam1*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s1ahat^2*(2*lam1 - 2*W2a^lam2*lam1))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h5_10=0
h5_11=sum(-(cova*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h5_12=sum((2*s1ahat^2*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h6_1=sum((s2ahat*(2*cova^2*(lam2 - W1a^lam1*lam2 + lam1*lam2*m1ahat) - 2*cova*s1ahat^2*(lam1 - W2a^lam2*lam1 + lam1*lam2*m2ahat)))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_2=sum((lam2*((2*cova*s2ahat^3*(2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat) - 2*cova*s2ahat*(4*cova*lam1^2*m2ahat - 4*W2a^lam2*cova*lam1^2*m2ahat))*s1ahat^3 + 2*cova*s2ahat*(2*cova^2*lam1 + 2*cova^2*lam1^2*m1ahat - 2*W1a^lam1*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1 - 2*W2a^lam2*cova^2*lam1^2*m1ahat + 2*W1a^lam1*W2a^lam2*cova^2*lam1)*s1ahat) - 2*cova*s1ahat^3*s2ahat*(2*cova*lam1^2 + 2*W2a^(2*lam2)*cova*lam1^2 - 4*W2a^lam2*cova*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - ((4*cova^2*lam1^2*m2ahat^2*s2ahat - 2*cova*s2ahat^3*(2*lam1*m2ahat + cova*lam1^2 + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat))*s1ahat^3 + (2*cova*(2*cova + 2*W1a^(2*lam1)*cova - 4*W1a^lam1*cova + 2*cova*lam1^2*m1ahat^2 + 4*cova*lam1*m1ahat - 4*W1a^lam1*cova*lam1*m1ahat)*s2ahat^3 + 2*cova*(cova^3*lam1^2 - 2*cova^2*lam1*m2ahat + 2*W1a^lam1*cova^2*lam1*m2ahat - 2*cova^2*lam1^2*m1ahat*m2ahat)*s2ahat)*s1ahat)/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_3=0
h6_4=0
h6_5=sum(-(2*s1ahat^2*s2ahat*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_6=sum(- (3*s1ahat^6*s2ahat^2 + cova^4*(lam2^2*m1ahat^2 + lam2^2*s1ahat^2) - cova^3*(2*lam2*m1ahat*s1ahat^2 - 2*W2a^lam2*lam2*m1ahat*s1ahat^2 + 2*lam2^2*m1ahat*m2ahat*s1ahat^2) - cova*(6*lam2*m1ahat*s1ahat^4*s2ahat^2 - 6*W2a^lam2*lam2*m1ahat*s1ahat^4*s2ahat^2 + 6*lam2^2*m1ahat*m2ahat*s1ahat^4*s2ahat^2) + cova^2*(W2a^(2*lam2)*s1ahat^4 - 2*W2a^lam2*s1ahat^4 + s1ahat^4 + 2*lam2*m2ahat*s1ahat^4 + lam2^2*m2ahat^2*s1ahat^4 - 2*W2a^lam2*lam2*m2ahat*s1ahat^4 + 3*lam2^2*m1ahat^2*s1ahat^2*s2ahat^2) - 6*W2a^lam2*s1ahat^6*s2ahat^2 + 3*W2a^(2*lam2)*s1ahat^6*s2ahat^2 - lam2^2*s1ahat^6*s2ahat^4 + 3*lam2^2*m2ahat^2*s1ahat^6*s2ahat^2 + 6*lam2*m2ahat*s1ahat^6*s2ahat^2 - 6*W2a^lam2*lam2*m2ahat*s1ahat^6*s2ahat^2)/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^4*(W1a^(2*lam1)*lam2^2 - 2*W1a^lam1*lam2^2 + lam2^2) + cova^2*(3*lam2^2*s1ahat^2*s2ahat^2 + 3*W1a^(2*lam1)*lam2^2*s1ahat^2*s2ahat^2 - 6*W1a^lam1*lam2^2*s1ahat^2*s2ahat^2) - lam1*((2*W1a^lam1*lam2^2*m1ahat - 2*lam2^2*m1ahat)*cova^4 + (2*lam2*s1ahat^2 + 2*lam2^2*m2ahat*s1ahat^2 - 2*W1a^lam1*lam2*s1ahat^2 - 2*W2a^lam2*lam2*s1ahat^2 - 2*W1a^lam1*lam2^2*m2ahat*s1ahat^2 + 2*W1a^lam1*W2a^lam2*lam2*s1ahat^2)*cova^3 + (6*W1a^lam1*lam2^2*m1ahat*s1ahat^2*s2ahat^2 - 6*lam2^2*m1ahat*s1ahat^2*s2ahat^2)*cova^2 + (6*lam2*s1ahat^4*s2ahat^2 + 6*lam2^2*m2ahat*s1ahat^4*s2ahat^2 - 6*W1a^lam1*lam2*s1ahat^4*s2ahat^2 - 6*W2a^lam2*lam2*s1ahat^4*s2ahat^2 - 6*W1a^lam1*lam2^2*m2ahat*s1ahat^4*s2ahat^2 + 6*W1a^lam1*W2a^lam2*lam2*s1ahat^4*s2ahat^2)*cova))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_7=0
h6_8=0
h6_9=sum(- (cova*(s1ahat^4*(- 4*lam1^2*m2ahat^2*s2ahat + 2*lam1^2*s2ahat^3) - s1ahat^2*s2ahat^3*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2)) - cova^3*(s2ahat*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2) + 2*lam1^2*s1ahat^2*s2ahat) + s1ahat^4*s2ahat^3*(2*lam1*m2ahat + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat) + cova^2*s1ahat^2*s2ahat*(6*lam1*m2ahat + 6*lam1^2*m1ahat*m2ahat - 6*W1a^lam1*lam1*m2ahat))/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1ahat - 6*W1a^lam1*lam1 - 6*W2a^lam2*lam1 + 6*W1a^lam1*W2a^lam2*lam1 - 6*W2a^lam2*lam1^2*m1ahat)*cova^2*s1ahat^2*s2ahat + (8*W2a^lam2*lam1^2*m2ahat - 8*lam1^2*m2ahat)*cova*s1ahat^4*s2ahat + (2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat)*s1ahat^4*s2ahat^3) - cova*s1ahat^4*s2ahat*(4*W2a^(2*lam2)*lam1^2 - 8*W2a^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h6_10=0
h6_11=sum(-(2*cova*s2ahat*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h6_12=sum((2*s1ahat^2*s2ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h7_1=0
h7_2=0
h7_3=sum(covb/(- covb^2 + s1bhat^2*s2bhat^2))
h7_4=sum((s1bhat*(2*covb^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat) - 2*covb*s2bhat^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_5=0
h7_6=0
h7_7=sum(-2/(2*s2bhat^2 - (2*covb^2)/s1bhat^2))
h7_8=sum(-(2*s1bhat^2*s2bhat*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_9=0
h7_10=sum((covb^2*(lam1*m1bhat - W1b^lam1 + 1) + s1bhat^2*((lam1*m1bhat - W1b^lam1 + 1)*s2bhat^2 - 2*covb*lam1*m2bhat))/(lam1*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s1bhat^2*(2*lam1 - 2*W2b^lam2*lam1))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h7_11=sum(-(covb*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h7_12=sum((2*s1bhat^2*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h8_1=0
h8_2=0
h8_3=sum((s2bhat*(2*covb^2*(lam2 - W1b^lam1*lam2 + lam1*lam2*m1bhat) - 2*covb*s1bhat^2*(lam1 - W2b^lam2*lam1 + lam1*lam2*m2bhat)))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_4=sum((lam2*((2*covb*s2bhat^3*(2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat) - 2*covb*s2bhat*(4*covb*lam1^2*m2bhat - 4*W2b^lam2*covb*lam1^2*m2bhat))*s1bhat^3 + 2*covb*s2bhat*(2*covb^2*lam1 + 2*covb^2*lam1^2*m1bhat - 2*W1b^lam1*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1 - 2*W2b^lam2*covb^2*lam1^2*m1bhat + 2*W1b^lam1*W2b^lam2*covb^2*lam1)*s1bhat) - 2*covb*s1bhat^3*s2bhat*(2*covb*lam1^2 + 2*W2b^(2*lam2)*covb*lam1^2 - 4*W2b^lam2*covb*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - ((4*covb^2*lam1^2*m2bhat^2*s2bhat - 2*covb*s2bhat^3*(2*lam1*m2bhat + covb*lam1^2 + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat))*s1bhat^3 + (2*covb*(2*covb + 2*W1b^(2*lam1)*covb - 4*W1b^lam1*covb + 2*covb*lam1^2*m1bhat^2 + 4*covb*lam1*m1bhat - 4*W1b^lam1*covb*lam1*m1bhat)*s2bhat^3 + 2*covb*(covb^3*lam1^2 - 2*covb^2*lam1*m2bhat + 2*W1b^lam1*covb^2*lam1*m2bhat - 2*covb^2*lam1^2*m1bhat*m2bhat)*s2bhat)*s1bhat)/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_5=0
h8_6=0
h8_7=sum(-(2*s1bhat^2*s2bhat*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_8=sum(- (3*s1bhat^6*s2bhat^2 + covb^4*(lam2^2*m1bhat^2 + lam2^2*s1bhat^2) - covb^3*(2*lam2*m1bhat*s1bhat^2 - 2*W2b^lam2*lam2*m1bhat*s1bhat^2 + 2*lam2^2*m1bhat*m2bhat*s1bhat^2) - covb*(6*lam2*m1bhat*s1bhat^4*s2bhat^2 - 6*W2b^lam2*lam2*m1bhat*s1bhat^4*s2bhat^2 + 6*lam2^2*m1bhat*m2bhat*s1bhat^4*s2bhat^2) + covb^2*(W2b^(2*lam2)*s1bhat^4 - 2*W2b^lam2*s1bhat^4 + s1bhat^4 + 2*lam2*m2bhat*s1bhat^4 + lam2^2*m2bhat^2*s1bhat^4 - 2*W2b^lam2*lam2*m2bhat*s1bhat^4 + 3*lam2^2*m1bhat^2*s1bhat^2*s2bhat^2) - 6*W2b^lam2*s1bhat^6*s2bhat^2 + 3*W2b^(2*lam2)*s1bhat^6*s2bhat^2 - lam2^2*s1bhat^6*s2bhat^4 + 3*lam2^2*m2bhat^2*s1bhat^6*s2bhat^2 + 6*lam2*m2bhat*s1bhat^6*s2bhat^2 - 6*W2b^lam2*lam2*m2bhat*s1bhat^6*s2bhat^2)/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^4*(W1b^(2*lam1)*lam2^2 - 2*W1b^lam1*lam2^2 + lam2^2) + covb^2*(3*lam2^2*s1bhat^2*s2bhat^2 + 3*W1b^(2*lam1)*lam2^2*s1bhat^2*s2bhat^2 - 6*W1b^lam1*lam2^2*s1bhat^2*s2bhat^2) - lam1*((2*W1b^lam1*lam2^2*m1bhat - 2*lam2^2*m1bhat)*covb^4 + (2*lam2*s1bhat^2 + 2*lam2^2*m2bhat*s1bhat^2 - 2*W1b^lam1*lam2*s1bhat^2 - 2*W2b^lam2*lam2*s1bhat^2 - 2*W1b^lam1*lam2^2*m2bhat*s1bhat^2 + 2*W1b^lam1*W2b^lam2*lam2*s1bhat^2)*covb^3 + (6*W1b^lam1*lam2^2*m1bhat*s1bhat^2*s2bhat^2 - 6*lam2^2*m1bhat*s1bhat^2*s2bhat^2)*covb^2 + (6*lam2*s1bhat^4*s2bhat^2 + 6*lam2^2*m2bhat*s1bhat^4*s2bhat^2 - 6*W1b^lam1*lam2*s1bhat^4*s2bhat^2 - 6*W2b^lam2*lam2*s1bhat^4*s2bhat^2 - 6*W1b^lam1*lam2^2*m2bhat*s1bhat^4*s2bhat^2 + 6*W1b^lam1*W2b^lam2*lam2*s1bhat^4*s2bhat^2)*covb))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_9=0
h8_10=sum(- (covb*(s1bhat^4*(- 4*lam1^2*m2bhat^2*s2bhat + 2*lam1^2*s2bhat^3) - s1bhat^2*s2bhat^3*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2)) - covb^3*(s2bhat*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2) + 2*lam1^2*s1bhat^2*s2bhat) + s1bhat^4*s2bhat^3*(2*lam1*m2bhat + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat) + covb^2*s1bhat^2*s2bhat*(6*lam1*m2bhat + 6*lam1^2*m1bhat*m2bhat - 6*W1b^lam1*lam1*m2bhat))/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1bhat - 6*W1b^lam1*lam1 - 6*W2b^lam2*lam1 + 6*W1b^lam1*W2b^lam2*lam1 - 6*W2b^lam2*lam1^2*m1bhat)*covb^2*s1bhat^2*s2bhat + (8*W2b^lam2*lam1^2*m2bhat - 8*lam1^2*m2bhat)*covb*s1bhat^4*s2bhat + (2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat)*s1bhat^4*s2bhat^3) - covb*s1bhat^4*s2bhat*(4*W2b^(2*lam2)*lam1^2 - 8*W2b^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h8_11=sum(-(2*covb*s2bhat*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h8_12=sum((2*s1bhat^2*s2bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h9_1=sum((cova^2*(lam2*m2ahat - W2a^lam2 + 1) + s2ahat^2*((lam2*m2ahat - W2a^lam2 + 1)*s1ahat^2 - 2*cova*lam2*m1ahat))/(lam2*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s2ahat^2*(2*lam2 - 2*W1a^lam1*lam2))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_2=sum(- (cova*(s2ahat^4*(- 4*lam2^2*m1ahat^2*s1ahat + 2*lam2^2*s1ahat^3) - s1ahat^3*s2ahat^2*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2)) - cova^3*(s1ahat*(4*lam2*m2ahat - 4*W2a^lam2 + 2*W2a^(2*lam2) + 2*lam2^2*m2ahat^2 - 4*W2a^lam2*lam2*m2ahat + 2) + 2*lam2^2*s1ahat*s2ahat^2) + s1ahat^3*s2ahat^4*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + cova^2*s1ahat*s2ahat^2*(6*lam2*m1ahat + 6*lam2^2*m1ahat*m2ahat - 6*W2a^lam2*lam2*m1ahat))/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2ahat - 6*W1a^lam1*lam2 - 6*W2a^lam2*lam2 + 6*W1a^lam1*W2a^lam2*lam2 - 6*W1a^lam1*lam2^2*m2ahat)*cova^2*s1ahat*s2ahat^2 + (8*W1a^lam1*lam2^2*m1ahat - 8*lam2^2*m1ahat)*cova*s1ahat*s2ahat^4 + (2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat)*s1ahat^3*s2ahat^4) - cova*s1ahat*s2ahat^4*(4*W1a^(2*lam1)*lam2^2 - 8*W1a^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_3=0
h9_4=0
h9_5=sum((cova^2*(lam1*m1ahat - W1a^lam1 + 1) + s1ahat^2*((lam1*m1ahat - W1a^lam1 + 1)*s2ahat^2 - 2*cova*lam1*m2ahat))/(lam1*(s1ahat^2*s2ahat^2 - cova^2)^2) - (cova*s1ahat^2*(2*lam1 - 2*W2a^lam2*lam1))/(lam1*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_6=sum(- (cova*(s1ahat^4*(- 4*lam1^2*m2ahat^2*s2ahat + 2*lam1^2*s2ahat^3) - s1ahat^2*s2ahat^3*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2)) - cova^3*(s2ahat*(4*lam1*m1ahat - 4*W1a^lam1 + 2*W1a^(2*lam1) + 2*lam1^2*m1ahat^2 - 4*W1a^lam1*lam1*m1ahat + 2) + 2*lam1^2*s1ahat^2*s2ahat) + s1ahat^4*s2ahat^3*(2*lam1*m2ahat + 2*lam1^2*m1ahat*m2ahat - 2*W1a^lam1*lam1*m2ahat) + cova^2*s1ahat^2*s2ahat*(6*lam1*m2ahat + 6*lam1^2*m1ahat*m2ahat - 6*W1a^lam1*lam1*m2ahat))/(lam1^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1ahat - 6*W1a^lam1*lam1 - 6*W2a^lam2*lam1 + 6*W1a^lam1*W2a^lam2*lam1 - 6*W2a^lam2*lam1^2*m1ahat)*cova^2*s1ahat^2*s2ahat + (8*W2a^lam2*lam1^2*m2ahat - 8*lam1^2*m2ahat)*cova*s1ahat^4*s2ahat + (2*lam1 + 2*lam1^2*m1ahat - 2*W1a^lam1*lam1 - 2*W2a^lam2*lam1 + 2*W1a^lam1*W2a^lam2*lam1 - 2*W2a^lam2*lam1^2*m1ahat)*s1ahat^4*s2ahat^3) - cova*s1ahat^4*s2ahat*(4*W2a^(2*lam2)*lam1^2 - 8*W2a^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_7=0
h9_8=0
h9_9=sum(- (s1ahat^2*(cova^2*(6*lam2*m2ahat - 6*W2a^lam2 + 3*W2a^(2*lam2) + 3*lam2^2*m2ahat^2 - 6*W2a^lam2*lam2*m2ahat + 3) - cova*(6*lam2*m1ahat*s2ahat^2 - 6*W2a^lam2*lam2*m1ahat*s2ahat^2 + 6*lam2^2*m1ahat*m2ahat*s2ahat^2) + lam2^2*m1ahat^2*s2ahat^4) - cova^3*(2*lam2*m1ahat + 2*lam2^2*m1ahat*m2ahat - 2*W2a^lam2*lam2*m1ahat) + s1ahat^4*(W2a^(2*lam2)*s2ahat^2 - 2*W2a^lam2*s2ahat^2 + s2ahat^2 - lam2^2*s2ahat^4 + 2*lam2*m2ahat*s2ahat^2 + lam2^2*m2ahat^2*s2ahat^2 - 2*W2a^lam2*lam2*m2ahat*s2ahat^2) + cova^4*lam2^2 + 3*cova^2*lam2^2*m1ahat^2*s2ahat^2)/(lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3) - (cova^2*(3*lam2^2*s2ahat^2 - 6*W1a^lam1*lam2^2*s2ahat^2 + 3*W1a^(2*lam1)*lam2^2*s2ahat^2) + s1ahat^2*(lam2^2*s2ahat^4 - 2*W1a^lam1*lam2^2*s2ahat^4 + W1a^(2*lam1)*lam2^2*s2ahat^4) - lam1*(cova^3*(2*lam2 + 2*lam2^2*m2ahat - 2*W1a^lam1*lam2 - 2*W2a^lam2*lam2 + 2*W1a^lam1*W2a^lam2*lam2 - 2*W1a^lam1*lam2^2*m2ahat) - cova^2*(6*lam2^2*m1ahat*s2ahat^2 - 6*W1a^lam1*lam2^2*m1ahat*s2ahat^2) + s1ahat^2*(cova*(6*lam2*s2ahat^2 + 6*lam2^2*m2ahat*s2ahat^2 - 6*W1a^lam1*lam2*s2ahat^2 - 6*W2a^lam2*lam2*s2ahat^2 - 6*W1a^lam1*lam2^2*m2ahat*s2ahat^2 + 6*W1a^lam1*W2a^lam2*lam2*s2ahat^2) - 2*lam2^2*m1ahat*s2ahat^4 + 2*W1a^lam1*lam2^2*m1ahat*s2ahat^4)))/(lam1^2*lam2^2*(s1ahat^2*s2ahat^2 - cova^2)^3))
h9_10=0
h9_11=sum(-((W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova^2*lam1 - 2*cova*lam2*s2ahat^2 + lam1*s1ahat^2*s2ahat^2 - W2a^lam2*cova^2*lam1 + 2*W1a^lam1*cova*lam2*s2ahat^2 + cova^2*lam1*lam2*m2ahat - W2a^lam2*lam1*s1ahat^2*s2ahat^2 + lam1*lam2*m2ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h9_12=sum(-((W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova^2*lam2 - 2*cova*lam1*s1ahat^2 + lam2*s1ahat^2*s2ahat^2 - W1a^lam1*cova^2*lam2 + 2*W2a^lam2*cova*lam1*s1ahat^2 + cova^2*lam1*lam2*m1ahat - W1a^lam1*lam2*s1ahat^2*s2ahat^2 + lam1*lam2*m1ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h10_1=0
h10_2=0
h10_3=sum((covb^2*(lam2*m2bhat - W2b^lam2 + 1) + s2bhat^2*((lam2*m2bhat - W2b^lam2 + 1)*s1bhat^2 - 2*covb*lam2*m1bhat))/(lam2*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s2bhat^2*(2*lam2 - 2*W1b^lam1*lam2))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_4=sum(- (covb*(s2bhat^4*(- 4*lam2^2*m1bhat^2*s1bhat + 2*lam2^2*s1bhat^3) - s1bhat^3*s2bhat^2*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2)) - covb^3*(s1bhat*(4*lam2*m2bhat - 4*W2b^lam2 + 2*W2b^(2*lam2) + 2*lam2^2*m2bhat^2 - 4*W2b^lam2*lam2*m2bhat + 2) + 2*lam2^2*s1bhat*s2bhat^2) + s1bhat^3*s2bhat^4*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + covb^2*s1bhat*s2bhat^2*(6*lam2*m1bhat + 6*lam2^2*m1bhat*m2bhat - 6*W2b^lam2*lam2*m1bhat))/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam1*((6*lam2 + 6*lam2^2*m2bhat - 6*W1b^lam1*lam2 - 6*W2b^lam2*lam2 + 6*W1b^lam1*W2b^lam2*lam2 - 6*W1b^lam1*lam2^2*m2bhat)*covb^2*s1bhat*s2bhat^2 + (8*W1b^lam1*lam2^2*m1bhat - 8*lam2^2*m1bhat)*covb*s1bhat*s2bhat^4 + (2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat)*s1bhat^3*s2bhat^4) - covb*s1bhat*s2bhat^4*(4*W1b^(2*lam1)*lam2^2 - 8*W1b^lam1*lam2^2 + 4*lam2^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_5=0
h10_6=0
h10_7=sum((covb^2*(lam1*m1bhat - W1b^lam1 + 1) + s1bhat^2*((lam1*m1bhat - W1b^lam1 + 1)*s2bhat^2 - 2*covb*lam1*m2bhat))/(lam1*(s1bhat^2*s2bhat^2 - covb^2)^2) - (covb*s1bhat^2*(2*lam1 - 2*W2b^lam2*lam1))/(lam1*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_8=sum(- (covb*(s1bhat^4*(- 4*lam1^2*m2bhat^2*s2bhat + 2*lam1^2*s2bhat^3) - s1bhat^2*s2bhat^3*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2)) - covb^3*(s2bhat*(4*lam1*m1bhat - 4*W1b^lam1 + 2*W1b^(2*lam1) + 2*lam1^2*m1bhat^2 - 4*W1b^lam1*lam1*m1bhat + 2) + 2*lam1^2*s1bhat^2*s2bhat) + s1bhat^4*s2bhat^3*(2*lam1*m2bhat + 2*lam1^2*m1bhat*m2bhat - 2*W1b^lam1*lam1*m2bhat) + covb^2*s1bhat^2*s2bhat*(6*lam1*m2bhat + 6*lam1^2*m1bhat*m2bhat - 6*W1b^lam1*lam1*m2bhat))/(lam1^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (lam2*((6*lam1 + 6*lam1^2*m1bhat - 6*W1b^lam1*lam1 - 6*W2b^lam2*lam1 + 6*W1b^lam1*W2b^lam2*lam1 - 6*W2b^lam2*lam1^2*m1bhat)*covb^2*s1bhat^2*s2bhat + (8*W2b^lam2*lam1^2*m2bhat - 8*lam1^2*m2bhat)*covb*s1bhat^4*s2bhat + (2*lam1 + 2*lam1^2*m1bhat - 2*W1b^lam1*lam1 - 2*W2b^lam2*lam1 + 2*W1b^lam1*W2b^lam2*lam1 - 2*W2b^lam2*lam1^2*m1bhat)*s1bhat^4*s2bhat^3) - covb*s1bhat^4*s2bhat*(4*W2b^(2*lam2)*lam1^2 - 8*W2b^lam2*lam1^2 + 4*lam1^2))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_9=0
h10_10=sum(- (s1bhat^2*(covb^2*(6*lam2*m2bhat - 6*W2b^lam2 + 3*W2b^(2*lam2) + 3*lam2^2*m2bhat^2 - 6*W2b^lam2*lam2*m2bhat + 3) - covb*(6*lam2*m1bhat*s2bhat^2 - 6*W2b^lam2*lam2*m1bhat*s2bhat^2 + 6*lam2^2*m1bhat*m2bhat*s2bhat^2) + lam2^2*m1bhat^2*s2bhat^4) - covb^3*(2*lam2*m1bhat + 2*lam2^2*m1bhat*m2bhat - 2*W2b^lam2*lam2*m1bhat) + s1bhat^4*(W2b^(2*lam2)*s2bhat^2 - 2*W2b^lam2*s2bhat^2 + s2bhat^2 - lam2^2*s2bhat^4 + 2*lam2*m2bhat*s2bhat^2 + lam2^2*m2bhat^2*s2bhat^2 - 2*W2b^lam2*lam2*m2bhat*s2bhat^2) + covb^4*lam2^2 + 3*covb^2*lam2^2*m1bhat^2*s2bhat^2)/(lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3) - (covb^2*(3*lam2^2*s2bhat^2 - 6*W1b^lam1*lam2^2*s2bhat^2 + 3*W1b^(2*lam1)*lam2^2*s2bhat^2) + s1bhat^2*(lam2^2*s2bhat^4 - 2*W1b^lam1*lam2^2*s2bhat^4 + W1b^(2*lam1)*lam2^2*s2bhat^4) - lam1*(covb^3*(2*lam2 + 2*lam2^2*m2bhat - 2*W1b^lam1*lam2 - 2*W2b^lam2*lam2 + 2*W1b^lam1*W2b^lam2*lam2 - 2*W1b^lam1*lam2^2*m2bhat) - covb^2*(6*lam2^2*m1bhat*s2bhat^2 - 6*W1b^lam1*lam2^2*m1bhat*s2bhat^2) + s1bhat^2*(covb*(6*lam2*s2bhat^2 + 6*lam2^2*m2bhat*s2bhat^2 - 6*W1b^lam1*lam2*s2bhat^2 - 6*W2b^lam2*lam2*s2bhat^2 - 6*W1b^lam1*lam2^2*m2bhat*s2bhat^2 + 6*W1b^lam1*W2b^lam2*lam2*s2bhat^2) - 2*lam2^2*m1bhat*s2bhat^4 + 2*W1b^lam1*lam2^2*m1bhat*s2bhat^4)))/(lam1^2*lam2^2*(s1bhat^2*s2bhat^2 - covb^2)^3))
h10_11=sum(-((W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb^2*lam1 - 2*covb*lam2*s2bhat^2 + lam1*s1bhat^2*s2bhat^2 - W2b^lam2*covb^2*lam1 + 2*W1b^lam1*covb*lam2*s2bhat^2 + covb^2*lam1*lam2*m2bhat - W2b^lam2*lam1*s1bhat^2*s2bhat^2 + lam1*lam2*m2bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h10_12=sum(-((W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb^2*lam2 - 2*covb*lam1*s1bhat^2 + lam2*s1bhat^2*s2bhat^2 - W1b^lam1*covb^2*lam2 + 2*W2b^lam2*covb*lam1*s1bhat^2 + covb^2*lam1*lam2*m1bhat - W1b^lam1*lam2*s1bhat^2*s2bhat^2 + lam1*lam2*m1bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_1=sum((2*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h11_2=sum((2*s1ahat*s2ahat^2*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_3=sum((2*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h11_4=sum((2*s1bhat*s2bhat^2*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_5=sum(-(cova*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1))/(lam1^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h11_6=sum(-(2*cova*s2ahat*(W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_7=sum(-(covb*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1))/(lam1^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h11_8=sum(-(2*covb*s2bhat*(W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_9=sum(-((W1a^lam1*lam1*log(W1a) - W1a^lam1 + 1)*(cova^2*lam1 - 2*cova*lam2*s2ahat^2 + lam1*s1ahat^2*s2ahat^2 - W2a^lam2*cova^2*lam1 + 2*W1a^lam1*cova*lam2*s2ahat^2 + cova^2*lam1*lam2*m2ahat - W2a^lam2*lam1*s1ahat^2*s2ahat^2 + lam1*lam2*m2ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m1ahat*s2ahat^2))/(lam1^3*lam2*(s1ahat^2*s2ahat^2 - cova^2)^2))
h11_10=sum(-((W1b^lam1*lam1*log(W1b) - W1b^lam1 + 1)*(covb^2*lam1 - 2*covb*lam2*s2bhat^2 + lam1*s1bhat^2*s2bhat^2 - W2b^lam2*covb^2*lam1 + 2*W1b^lam1*covb*lam2*s2bhat^2 + covb^2*lam1*lam2*m2bhat - W2b^lam2*lam1*s1bhat^2*s2bhat^2 + lam1*lam2*m2bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m1bhat*s2bhat^2))/(lam1^3*lam2*(s1bhat^2*s2bhat^2 - covb^2)^2))
h11_11=sum(((2*((W1a^lam1-1)/lam1^2-(W1a^lam1*log(W1a))/lam1)^2)/s1ahat^2-(2*(m1ahat-(W1a^lam1-1)/lam1)*((2*W1a^lam1-2)/lam1^3-(2*W1a^lam1*log(W1a))/lam1^2+(W1a^lam1*log(W1a)^2)/lam1))/s1ahat^2+(2*cova*(m2ahat-(W2a^lam2-1)/lam2)*((2*W1a^lam1-2)/lam1^3-(2*W1a^lam1*log(W1a))/lam1^2+(W1a^lam1*log(W1a)^2)/lam1))/(s1ahat^2*s2ahat^2))/((2*cova^2)/(s1ahat^2*s2ahat^2)-2))+sum(((2*((W1b^lam1-1)/lam1^2-(W1b^lam1*log(W1b))/lam1)^2)/s1bhat^2-(2*(m1bhat-(W1b^lam1-1)/lam1)*((2*W1b^lam1-2)/lam1^3-(2*W1b^lam1*log(W1b))/lam1^2+(W1b^lam1*log(W1b)^2)/lam1))/s1bhat^2+(2*covb*(m2bhat-(W2b^lam2-1)/lam2)*((2*W1b^lam1-2)/lam1^3-(2*W1b^lam1*log(W1b))/lam1^2+(W1b^lam1*log(W1b)^2)/lam1))/(s1bhat^2*s2bhat^2))/((2*covb^2)/(s1bhat^2*s2bhat^2)-2))
h11_12=sum((2*cova*(W1a^lam1*lam1*log(W1a)-W1a^lam1+1)*(W2a^lam2*lam2*log(W2a)-W2a^lam2+1))/(lam1^2*lam2^2*(-2*cova^2+2*s1ahat^2*s2ahat^2)))+sum((2*covb*(W1b^lam1*lam1*log(W1b)-W1b^lam1+1)*(W2b^lam2*lam2*log(W2b)-W2b^lam2+1))/(lam1^2*lam2^2*(-2*covb^2+2*s1bhat^2*s2bhat^2)))
h12_1=sum(-(cova*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- cova^2 + s1ahat^2*s2ahat^2)))
h12_2=sum(-(2*cova*s1ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam1 - lam2*s2ahat^2 - W2a^lam2*cova*lam1 + W1a^lam1*lam2*s2ahat^2 + cova*lam1*lam2*m2ahat - lam1*lam2*m1ahat*s2ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_3=sum(-(covb*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- covb^2 + s1bhat^2*s2bhat^2)))
h12_4=sum(-(2*covb*s1bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam1 - lam2*s2bhat^2 - W2b^lam2*covb*lam1 + W1b^lam1*lam2*s2bhat^2 + covb*lam1*lam2*m2bhat - lam1*lam2*m1bhat*s2bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_5=sum((2*s1ahat^2*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1))/(lam2^2*(- 2*cova^2 + 2*s1ahat^2*s2ahat^2)))
h12_6=sum((2*s1ahat^2*s2ahat*(W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova*lam2 - lam1*s1ahat^2 - W1a^lam1*cova*lam2 + W2a^lam2*lam1*s1ahat^2 + cova*lam1*lam2*m1ahat - lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_7=sum((2*s1bhat^2*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1))/(lam2^2*(- 2*covb^2 + 2*s1bhat^2*s2bhat^2)))
h12_8=sum((2*s1bhat^2*s2bhat*(W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb*lam2 - lam1*s1bhat^2 - W1b^lam1*covb*lam2 + W2b^lam2*lam1*s1bhat^2 + covb*lam1*lam2*m1bhat - lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_9=sum(-((W2a^lam2*lam2*log(W2a) - W2a^lam2 + 1)*(cova^2*lam2 - 2*cova*lam1*s1ahat^2 + lam2*s1ahat^2*s2ahat^2 - W1a^lam1*cova^2*lam2 + 2*W2a^lam2*cova*lam1*s1ahat^2 + cova^2*lam1*lam2*m1ahat - W1a^lam1*lam2*s1ahat^2*s2ahat^2 + lam1*lam2*m1ahat*s1ahat^2*s2ahat^2 - 2*cova*lam1*lam2*m2ahat*s1ahat^2))/(lam1*lam2^3*(s1ahat^2*s2ahat^2 - cova^2)^2))
h12_10=sum(-((W2b^lam2*lam2*log(W2b) - W2b^lam2 + 1)*(covb^2*lam2 - 2*covb*lam1*s1bhat^2 + lam2*s1bhat^2*s2bhat^2 - W1b^lam1*covb^2*lam2 + 2*W2b^lam2*covb*lam1*s1bhat^2 + covb^2*lam1*lam2*m1bhat - W1b^lam1*lam2*s1bhat^2*s2bhat^2 + lam1*lam2*m1bhat*s1bhat^2*s2bhat^2 - 2*covb*lam1*lam2*m2bhat*s1bhat^2))/(lam1*lam2^3*(s1bhat^2*s2bhat^2 - covb^2)^2))
h12_11=sum((2*cova*(W1a^lam1*lam1*log(W1a)-W1a^lam1+1)*(W2a^lam2*lam2*log(W2a)-W2a^lam2+1))/(lam1^2*lam2^2*(-2*cova^2+2*s1ahat^2*s2ahat^2)))+sum((2*covb*(W1b^lam1*lam1*log(W1b)-W1b^lam1+1)*(W2b^lam2*lam2*log(W2b)-W2b^lam2+1))/(lam1^2*lam2^2*(-2*covb^2+2*s1bhat^2*s2bhat^2)))
h12_12=sum(((2*((W2a^lam2-1)/lam2^2-(W2a^lam2*log(W2a))/lam2)^2)/s2ahat^2-(2*(m2ahat-(W2a^lam2-1)/lam2)*((2*W2a^lam2-2)/lam2^3-(2*W2a^lam2*log(W2a))/lam2^2+(W2a^lam2*log(W2a)^2)/lam2))/s2ahat^2+(2*cova*(m1ahat-(W1a^lam1-1)/lam1)*((2*W2a^lam2-2)/lam2^3-(2*W2a^lam2*log(W2a))/lam2^2+(W2a^lam2*log(W2a)^2)/lam2))/(s1ahat^2*s2ahat^2))/((2*cova^2)/(s1ahat^2*s2ahat^2)-2))+sum(((2*((W2b^lam2-1)/lam2^2-(W2b^lam2*log(W2b))/lam2)^2)/s2bhat^2-(2*(m2bhat-(W2b^lam2-1)/lam2)*((2*W2b^lam2-2)/lam2^3-(2*W2b^lam2*log(W2b))/lam2^2+(W2b^lam2*log(W2b)^2)/lam2))/s2bhat^2+(2*covb*(m1bhat-(W1b^lam1-1)/lam1)*((2*W2b^lam2-2)/lam2^3-(2*W2b^lam2*log(W2b))/lam2^2+(W2b^lam2*log(W2b)^2)/lam2))/(s1bhat^2*s2bhat^2))/((2*covb^2)/(s1bhat^2*s2bhat^2)-2))
h1r=c(h1_1,h1_2,h1_3,h1_4,h1_5,h1_6,h1_7,h1_8,h1_9,h1_10,h1_11,h1_12)
h2r=c(h2_1,h2_2,h2_3,h2_4,h2_5,h2_6,h2_7,h2_8,h2_9,h2_10,h2_11,h2_12)
h3r=c(h3_1,h3_2,h3_3,h3_4,h3_5,h3_6,h3_7,h3_8,h3_9,h3_10,h3_11,h3_12)
h4r=c(h4_1,h4_2,h4_3,h4_4,h4_5,h4_6,h4_7,h4_8,h4_9,h4_10,h4_11,h4_12)
h5r=c(h5_1,h5_2,h5_3,h5_4,h5_5,h5_6,h5_7,h5_8,h5_9,h5_10,h5_11,h5_12)
h6r=c(h6_1,h6_2,h6_3,h6_4,h6_5,h6_6,h6_7,h6_8,h6_9,h6_10,h6_11,h6_12)
h7r=c(h7_1,h7_2,h7_3,h7_4,h7_5,h7_6,h7_7,h7_8,h7_9,h7_10,h7_11,h7_12)
h8r=c(h8_1,h8_2,h8_3,h8_4,h8_5,h8_6,h8_7,h8_8,h8_9,h8_10,h8_11,h8_12)
h9r=c(h9_1,h9_2,h9_3,h9_4,h9_5,h9_6,h9_7,h9_8,h9_9,h9_10,h9_11,h9_12)
h10r=c(h10_1,h10_2,h10_3,h10_4,h10_5,h10_6,h10_7,h10_8,h10_9,h10_10,h10_11,h10_12)
h11r=c(h11_1,h11_2,h11_3,h11_4,h11_5,h11_6,h11_7,h11_8,h11_9,h11_10,h11_11,h11_12)
h12r=c(h12_1,h12_2,h12_3,h12_4,h12_5,h12_6,h12_7,h12_8,h12_9,h12_10,h12_11,h12_12)
HCF=rbind(h1r,h2r,h3r,h4r,h5r,h6r,h7r,h8r,h9r,h10r,h11r,h12r)
HCF[1,2]=0;HCF[1,6]=0;HCF[1,9]=0;
HCF[2,1]=0;HCF[2,5]=0;
HCF[3,4]=0;HCF[3,8]=0;HCF[3,10]=0;HCF[4,3]=0;
HCF[4,7]=0;
HCF[5,2]=0;HCF[5,6]=0;HCF[5,9]=0;
HCF[6,1]=0;HCF[6,5]=0;
HCF[7,4]=0;HCF[7,8]=0;HCF[7,10]=0;
HCF[8,3]=0;HCF[8,7]=0;
HCF[9,1]=0;HCF[9,5]=0;
HCF[10,3]=0;
HCF[10,7]=0;
HCF[1,1]=na*HCF[1,1];
HCF[1,5]=na*HCF[1,5];
HCF[5,1]=na*HCF[5,1];
HCF[3,3]=nb*HCF[3,3];
HCF[3,7]=nb*HCF[3,7];
HCF[7,3]=nb*HCF[7,3];
HCF[7,7]=nb*HCF[7,7];
HCF[5,5]=na*HCF[5,5];
VV=inv(-HCF)
att=1-atSpec;
t=att;
s1alam=sqrt( var(W1alam)*(na-1)/na )
s1blam=sqrt( var(W1blam)*(nb-1)/nb )
s2alam=sqrt( var(W2alam)*(na-1)/na )
s2blam=sqrt( var(W2blam)*(nb-1)/nb )
m1alam=mean(W1alam)
m1blam=mean(W1blam)
m2alam=mean(W2alam)
m2blam=mean(W2blam)
#############################################################################
#############################################################################
#==================AT A GIVEN T FOR THE ORIGINAL===================
ROC1or= (1-pnorm(qnorm(1-t,m1alam,s1alam),m1blam,s1blam));
ROC2or= (1-pnorm(qnorm(1-t,m2alam,s2alam),m2blam,s2blam));
dm1a = -(2^(1/2)*exp(-((m1ahat - m1bhat)/s1bhat + (2^(1/2)*s1ahat*erfcinv(2*t))/s1bhat)^2/2))/(2*s1bhat*pi^(1/2))
dm1b = (2^(1/2)*exp(-((m1ahat - m1bhat)/s1bhat + (2^(1/2)*s1ahat*erfcinv(2*t))/s1bhat)^2/2))/(2*s1bhat*pi^(1/2))
ds1a = -(erfcinv(2*t)*exp(-((m1ahat - m1bhat)/s1bhat + (2^(1/2)*s1ahat*erfcinv(2*t))/s1bhat)^2/2))/(s1bhat*pi^(1/2))
ds1b = (2^(1/2)*exp(-((m1ahat - m1bhat)/s1bhat + (2^(1/2)*s1ahat*erfcinv(2*t))/s1bhat)^2/2)*((m1ahat - m1bhat)/s1bhat^2 + (2^(1/2)*s1ahat*erfcinv(2*t))/s1bhat^2))/(2*pi^(1/2))
der1=c(dm1a, ds1a, dm1b, ds1b);
vROC1=c(der1)%*%VV[1:4,1:4]%*%t(t(c(der1)))
vROC1=c(vROC1)
dm2a = -(2^(1/2)*exp(-((m2ahat - m2bhat)/s2bhat + (2^(1/2)*s2ahat*erfcinv(2*t))/s2bhat)^2/2))/(2*s2bhat*pi^(1/2))
dm2b = (2^(1/2)*exp(-((m2ahat - m2bhat)/s2bhat + (2^(1/2)*s2ahat*erfcinv(2*t))/s2bhat)^2/2))/(2*s2bhat*pi^(1/2))
ds2a = -(erfcinv(2*t)*exp(-((m2ahat - m2bhat)/s2bhat + (2^(1/2)*s2ahat*erfcinv(2*t))/s2bhat)^2/2))/(s2bhat*pi^(1/2))
ds2b = (2^(1/2)*exp(-((m2ahat - m2bhat)/s2bhat + (2^(1/2)*s2ahat*erfcinv(2*t))/s2bhat)^2/2)*((m2ahat - m2bhat)/s2bhat^2 + (2^(1/2)*s2ahat*erfcinv(2*t))/s2bhat^2))/(2*pi^(1/2))
der2=c(dm2a, ds2a, dm2b, ds2b);
vROC2=c(der2)%*%VV[5:8,5:8]%*%t(t(c(der2)))
covROC=c(der1, 0, 0, 0, 0)%*%VV[1:8,1:8]%*%t(t(c(0, 0, 0, 0, der2)))
Z=(ROC2or-ROC1or)/(sqrt(vROC2+vROC1-2*covROC))
SEZ=sqrt(vROC1+vROC2-2*covROC)
pval2tZ=2*pnorm(-abs(Z));
CIoriginal=c((ROC2or-ROC1or)-pmalpha*SEZ, (ROC2or-ROC1or)+pmalpha*SEZ)
CIoriginal
#############################################################################
#############################################################################
#==================AT A GIVEN T FOR THE TRANSORMED===================
ROC1MC= qnorm(1-pnorm(qnorm(1-t,m1alam,s1alam),m1blam,s1blam));
ROC2MC= qnorm(1-pnorm(qnorm(1-t,m2alam,s2alam),m2blam,s2blam));
dm1a =-1/s1blam;
ds1a =-(2^(1/2)*erfcinv(2*t))/s1blam;
dm1b =1/s1blam;
ds1b =(m1alam - m1blam)/s1blam^2 + (2^(1/2)*s1alam*erfcinv(2*t))/s1blam^2;
der1=c(dm1a, ds1a, dm1b, ds1b);
vROC1=c(der1)%*%VV[1:4,1:4]%*%t(t(c(der1)))
vROC1=c(vROC1)
dm2a =-1/s2blam;
ds2a =-(2^(1/2)*erfcinv(2*t))/s2blam;
dm2b =1/s2blam;
ds2b =(m2alam - m2blam)/s2blam^2 + (2^(1/2)*s2alam*erfcinv(2*t))/s2blam^2;
der2=c(dm2a, ds2a, dm2b, ds2b);
vROC2=c(der2)%*%VV[5:8,5:8]%*%t(t(c(der2)))
vROC2=c(vROC2)
covROC=c(der1, 0, 0, 0, 0)%*%VV[1:8,1:8]%*%t(t(c(0, 0, 0, 0, der2)))
covROC=c(covROC)
Zroct=(ROC2MC-ROC1MC)/(sqrt(vROC2+vROC1-2*covROC))
SEroct=sqrt(vROC1+vROC2-2*covROC)
pval2t_roct=2*pnorm(-abs(Zroct));pval2t=pval2t_roct;
CIZstar=c((ROC2MC-ROC1MC)-pmalpha*SEroct, (ROC2MC-ROC1MC)+pmalpha*SEroct)
CIZstar
SE1=pnorm(ROC1MC)
SE2=pnorm(ROC2MC)
res <- data.frame(Sens1 = formattable(SE1, digits = 4, format = "f"),
Sens2 = formattable(SE2, digits = 4, format = "f"),
diff = formattable(SE2 - SE1, digits = 4, format = "f"),
p_value_probit = formattable(pval2t, digits = 4, format = "f"),
p_value = formattable(pval2tZ, digits = 4, format = "f"),
ci_ll = formattable(CIoriginal[1], digits = 4, format = "f"),
ci_ul = formattable(CIoriginal[2], digits = 4, format = "f"))
rownames(res) <- c("Estimates:")
colnames(res) <- c("Sens 1", "Sens 2", "Diff", "P-Val (Probit)", "P-Val", "CI (LL)", "CI (UL)")
res <- formattable(as.matrix(res), digits = 4, format = "f")
res
#====TWO ROC FUNCTIONS: ONE IS THE BOXCOX AND THE OTHER THE REFERENCE LINE================
roc1<-function(t){
na = length(W1alam)
nb = length(W1blam)
s1alam=sqrt( var(W1alam)*(na-1)/na )
s1blam=sqrt( var(W1blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W1alam),
sd=s1alam),
mean=mean(W1blam),
sd=s1blam)
}
roc2<-function(t){
na = length(W2alam)
nb = length(W2blam)
s2alam=sqrt( var(W2alam)*(na-1)/na )
s2blam=sqrt( var(W2blam)*(nb-1)/nb )
1-pnorm(qnorm(1-t,
mean=mean(W2alam),
sd=s2alam),
mean=mean(W2blam),
sd=s2blam)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
#================IF PLOTS ARE REQUESTED PLOT THE ROCS==
if (plots=="on") {
# x11()
plot(linspace(0,1,1000),roc1(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="red")
lines(linspace(0,1,1000),roc2(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="black")
lines(linspace(0,1,1000),linspace(0,1,1000),type="l", lty=2)
points(1-atSpec,SE1, col = "red")
points(1-atSpec,SE2, col = "black")
legend("bottomright", legend=c(paste("ROC for Marker 1 with Sens =", formattable(SE1, digits = 4, format = "f"), "at Spec =", formattable(atSpec, digits = 4, format = "f")),
paste("ROC for Marker 2 with Sens =", formattable(SE2, digits = 4, format = "f"), "at Spec =", formattable(atSpec, digits = 4, format = "f")),
paste("P-value for the Sens difference:", formattable(pval2t, digits = 4, format = "f"))),
col=c("red", "black", "white"), lty=c(1, 1, NA), pch = c(NA, NA, NA), cex=0.8)
}
return(list(resultstable=res,
Sens1=SE1,
Sens2=SE2,
pvalue_probit_difference= pval2t,
pvalue_difference= pval2tZ,
CI_difference= CIoriginal,
roc1=roc1,
roc2=roc2,
transx1=W1alam,
transy1=W1blam,
transformation.parameter.1 = lam[1],
transx2=W2alam,
transy2=W2blam,
transformation.parameter.2 = lam[2]))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
comparebcSpec <-function (marker1, marker2, D, atSens, alpha, plots){
if ((length(marker1) != length(D)) | (length(marker2) != length(D))) {
stop("ERROR: The length of the 'marker' and 'D' inputs must be equal.")
} else if (min(D) != 0 | max(D) != 1) {
stop("ERROR: Controls must be assigned a value of 0; cases must be assigned a value of 1. Both controls and cases should be included in the dataset.")
} else if (alpha <= 0 | alpha >= 1) {
stop("ERROR: The level of significance, alpha, should be set between 0 and 1. A common choice is 0.05.")
} else if (sum(is.na(marker1)) > 0 | sum(is.na(marker2)) > 0 | sum(is.na(D)) > 0 | sum(is.na(atSens)) > 0) {
stop("ERROR: Please remove all missing data before running this function.")
} else if ((!is.numeric(atSens)) | (length(atSens) != 1)) {
stop("ERROR: 'atSens' must be a single numeric value.")
} else if ((sum(marker1 < 0) > 0) | (sum(marker2 < 2)) > 0) {
stop("ERROR: To use the Box-Cox transformation, all marker values must be positive.")
} else {
erf <- function (x) 2 * pnorm(x * sqrt(2)) - 1
erfc <- function (x) 2 * pnorm(x * sqrt(2), lower.tail = FALSE)
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower.tail = FALSE)/sqrt(2)
pmalpha=qnorm(1-alpha/2);
if (plots!="on"){plots="off"}
W1a=marker1[D==0]
W1b=marker1[D==1]
W2a=marker2[D==0]
W2b=marker2[D==1]
Dflip=abs(D-1);
atSpec=1-atSens;
#out=comparebcSens(marker1group1=W1b, marker1group2=W1a ,marker2group1=W2b, marker2group2=W2a, atSpec=atSens, plots="off")
out=comparebcSens(marker1, marker2, Dflip, atSpec=1-atSens, alpha, plots="off")
#return(list(resultstable=res,Sens1=SE1,Sens2=SE2, pvalue_probit_difference= pval2t, CI_probit_difference= CIZstar, pvalue_difference= pval2tZ, CI_difference= CIoriginal, roc1=roc1, roc2=roc2, transx1=W1alam, transy1=W1blam, transx2=W2alam, transy2=W2blam))
FPR1=out$Sens1
FPR2=out$Sens2
CIZstar=out$CI_probit_difference
CIoriginal=out$CI_difference
pval2t=out$pvalue_probit_difference
pval2tZ=out$pvalue_difference
W1alam=out$transx1
W1blam=out$transy1
W2alam=out$transx2
W2blam=out$transy2
#====TWO ROC FUNCTIONS: ONE IS THE BOXCOX AND THE OTHER THE REFERENCE LINE================
roct1<-function(t){
na = length(W1alam)
nb = length(W1blam)
s1alam=sqrt(var(W1alam)*(na-1)/na)
s1blam=sqrt(var(W1blam)*(nb-1)/nb)
1-pnorm(qnorm(1-t,
mean=mean(W1blam),
sd=s1blam),
mean=mean(W1alam),
sd=s1alam)
}
roct2<-function(t){
na = length(W2alam)
nb = length(W2blam)
s2alam=sqrt(var(W2alam)*(na-1)/na)
s2blam=sqrt(var(W2blam)*(nb-1)/nb)
1-pnorm(qnorm(1-t,
mean=mean(W2blam),
sd=s2blam),
mean=mean(W2alam),
sd=s2alam)
}
rocuseless<-function(t){
1-pnorm(qnorm(1-t,mean=1,sd=1),mean=1,sd=1)
}
#================IF PLOTS ARE REQUESTED PLOT THE ROCS==
if (plots=="on") {
# x11()
plot(linspace(0,1,1000),roct1(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="red")
lines(linspace(0,1,1000),roct2(linspace(0,1,1000)),main="Box-Cox Based ROCs",xlab="FPR = 1 - Specificity",ylab="TPR = Sensitivity",type="l",col="black")
lines(linspace(0,1,1000),linspace(0,1,1000),type="l", lty=2)
points(FPR1, 1-atSpec, col = "red")
points(FPR2, 1-atSpec, col = "black")
#line for the Youden:
#lines(c(1-atSpec,1-SE1),c(1-atSpec,1-SE2),col="green")
legend("bottomright", legend=c(paste("ROC for Marker 1 with FPR =", formattable(FPR1, digits = 4, format = "f"), "at Sens =", formattable(atSens, digits = 4, format = "f")),
paste("ROC for Marker 2 with FPR =", formattable(FPR2, digits = 4, format = "f"), "at Sens =", formattable(atSens, digits = 4, format = "f")),
paste("P-value for the Spec difference:", formattable(pval2t, digits = 4, format = "f"))),
col=c("red", "black", "white"), lty=c(1, 1, NA), pch = c(NA, NA, NA), cex=0.8)
}
res <- data.frame(FPR1 = formattable(FPR1, digits = 4, format = "f"),
FPR2 = formattable(FPR2, digits = 4, format = "f"),
formattable(FPR2 - FPR1, digits = 4, format = "f"),
p_value_probit = formattable(pval2t, digits = 4, format = "f"),
p_value = formattable(pval2tZ, digits = 4, format = "f"),
ci_ll = formattable(CIoriginal[1], digits = 4, format = "f"),
ci_ul = formattable(CIoriginal[2], digits = 4, format = "f"))
rownames(res) <- c("Estimates:")
colnames(res) <- c("FPR 1", "FPR 2", "Diff", "P-Val (Probit)", "P-Val", "CI (LL)", "CI (UL)")
res <- formattable(as.matrix(res), digits = 4, format = "f")
res
#return(list(resultstable=res,Sens1=SE1,Sens2=SE2, pvalue_probit_difference= pval2t, CI_probit_difference= CIZstar, pvalue_difference= pval2tZ, CI_difference= CIoriginal, roc1=roc1, roc2=roc2, transx1=W1alam, transy1=W1blam, transx2=W2alam, transy2=W2blam))
return(list(resultstable=res,
FPR1=FPR1,
FPR2=FPR2,
pvalue_probit_difference= pval2t,
pvalue_difference= pval2tZ,
CI_difference= CIoriginal,
roc1=roct1,
roc2=roct2,
transx1=W1alam,
transy1=W1blam,
transformation.parameter.1 = out$transformation.parameter.1,
transx2=W2alam,
transy2=W2blam,
transformation.parameter.2 = out$transformation.parameter.2))
#return(list(FPR1=FPR1))
}
}
## ----echo=FALSE, eval=TRUE----------------------------------------------------
threerocs2 <- function (marker1, marker2, D, plots) {
erf <- function (x) 2 * pnorm(x * sqrt(2)) - 1
erfc <- function (x) 2 * pnorm(x * sqrt(2), lower = FALSE)
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower = FALSE)/sqrt(2)
W1a=marker1[D==0]
W1b=marker1[D==1]
W2a=marker2[D==0]
W2b=marker2[D==1]
Wa=cbind(W1a,W2a);
Wb=cbind(W1b,W2b);
na=length(W1a)
nb=length(W1b)
likbox2D<-function(W1a,W1b,W2a,W2b,lam){
na=length(W1a);
nb=length(W1b);
out=c();
#for (i in 1:length(h)){
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
cova= cov(Walam)*(na-1)/na;
covb= cov(Wblam)*(nb-1)/nb;
mualam=c(mean(W1alam), mean(W2alam))
mublam=c(mean(W1blam), mean(W2blam))
out= -sum(log(dmvnorm(Walam,mualam,cova)))-sum(log(dmvnorm(Wblam,mublam,covb))) -(lam[1]-1)*sum(log(W1a))-(lam[2]-1)*sum(log(W2a))-(lam[1]-1)*sum(log(W1b))-(lam[2]-1)*sum(log(W2b));
return(out)
}
lam=c(2,2)
likbox2D(W1a,W1b,W2a,W2b,lam)
logL<-function(h){
likbox2D(W1a,W1b,W2a,W2b,h)
}
init=c(0.8,0.8)
lam=fminsearch(logL,init)
lam=c(lam$optbase$xopt)
lam
out <- optim(c(1,1), logL, method = "Nelder-Mead")
lam = out$par
###################################
W1alam= (W1a^lam[1]-1)/lam[1];
W2alam= (W2a^lam[2]-1)/lam[2];
W1blam= (W1b^lam[1]-1)/lam[1];
W2blam= (W2b^lam[2]-1)/lam[2];
Walam=cbind(W1alam,W2alam)
Wblam=cbind(W1blam,W2blam)
lam1=lam[1]
lam2=lam[2]
cova= cov(Walam)*(na-1)/na;cova=cova[1,2];
covb= cov(Wblam)*(nb-1)/nb;covb=covb[1,2];
s1alam=sqrt( var(W1alam)*(na-1)/na )
s1blam=sqrt( var(W1blam)*(nb-1)/nb )
s2alam=sqrt( var(W2alam)*(na-1)/na )
s2blam=sqrt( var(W2blam)*(nb-1)/nb )
m1alam=mean(W1alam)
m1blam=mean(W1blam)
m2alam=mean(W2alam)
m2blam=mean(W2blam)
roc1bc<-function(t){
1-pnorm(qnorm(1-t,mean=mean(W1alam),sd=s1alam),mean=mean(W1blam),sd=s1blam)
}
roc2bc<-function(t){
1-pnorm(qnorm(1-t,mean=mean(W2alam),sd=s2alam),mean=mean(W2blam),sd=s2blam)
}
####################### MRM ###################
rating=c(marker1,marker2)
truth=c(D,D)
reader=c(linspace(1,1,length(marker1)),linspace(2,2,length(marker2)))
curves_binorm <- roc_curves(truth, rating,
groups = list(Reader = reader),
method = "binormal")
params_binorm <- parameters(curves_binorm)
t=linspace(0,1,1000);
roct_mrm_1=pnorm(c(params_binorm$a[1])+c(params_binorm$b[1])*qnorm(t))
roct_mrm_2=pnorm(c(params_binorm$a[2])+c(params_binorm$b[2])*qnorm(t))
# Plot the empirical ROC curve
roc_obj1 <- roc(D, marker1, quiet = TRUE)
auc_emp1 <- auc(roc_obj1)
tpr1=roc_obj1$sensitivities
fpr1=1-roc_obj1$specificities
roc_obj2 <- roc(D, marker2, quiet = TRUE)
auc_emp2 <- auc(roc_obj2)
tpr2=roc_obj2$sensitivities
fpr2=1-roc_obj2$specificities
AUC_empirical_marker1=auc_emp1
AUC_empirical_marker2=auc_emp2
if (plots == "on") {
plot(t, roc1bc(t), type="l", lty = "dotted", col="indianred2", lwd=1.5, ylab="TPR = Sensitivity", xlab="FPR = 1 - Sensitivity")
lines(t, roc2bc(t), type="l", lty = "dotted", col="red4", lwd=1.5)
lines(t, roct_mrm_1, type="l", lty = "dashed", lwd=1.5, col="green2")
lines(t, roct_mrm_2, type="l", lty = "dashed", lwd=1.5, col="darkgreen")
lines(fpr1, tpr1, type="l", lty = "solid", lwd=1, col="deepskyblue")
lines(fpr2, tpr2, type="l",lty = "solid", lwd=1, col="blue4")
}
#plot(fpr1, tpr1, type = "l", col = "magenta", lwd = 2,
# xlim = c(0, 1), ylim = c(0, 1), xlab = "False Positive Rate (FPR)",
# ylab = "True Positive Rate (TPR)", main = "Three Estimates of the ROC Curve")
AUC1bc=trapz(t,roc1bc(t))
AUC2bc=trapz(t,roc2bc(t))
AUCmetz1=trapz(t,roct_mrm_1)
AUCmetz2=trapz(t,roct_mrm_2)
AUCemp1=abs(trapz(fpr1,tpr1))
AUCemp2=abs(trapz(fpr2,tpr2))
#legend("bottomright", legend = c("Box-Cox marker 1", "Box-Cox marker 2", "Metz marker 1",, "Metz marker 2","Empirical marker 1","Empirical marker 2"),
# col = c("black","red","blue","green", "magenta", "cyan"), lty = c("solid", "solid", "dashed","dashed","dashed","dashed"),
# lwd = 2, bty = "n")
#CORRECT ONE:
if (plots == "on") {
legend("bottomright",
legend = c(paste("Box-Cox Marker 1 (AUC = ", formattable(AUC1bc, digits = 4, format = "f"), ")", sep = ""),
paste("Box-Cox Marker 2 (AUC = ", formattable(AUC2bc, digits = 4, format = "f"), ")", sep = ""),
paste("Metz Marker 1 (AUC = ", formattable(AUCmetz1, digits = 4, format = "f"), ")", sep = ""),
paste("Metz Marker 2 (AUC = ", formattable(AUCmetz2, digits = 4, format = "f"), ")", sep = ""),
paste("Empirical Marker 1 (AUC = ", formattable(AUCemp1, digits = 4, format = "f"), ")", sep = ""),
paste("Empirical Marker 2 (AUC = ", formattable(AUCemp2, digits = 4, format = "f"), ")", sep = "")),
col = c("indianred2", "red4", "green2", "darkgreen", "deepskyblue", "blue4"),
lty = c("dotted", "dotted", "dashed", "dashed", "solid", "solid"),
cex = 0.8, lwd = 2)
title(main = "Three Estimates of the ROC Curve (Two Markers)")
}
return(list(AUC_BoxCox1 = AUC1bc,
AUC_BoxCox2 = AUC2bc,
AUC_Metz1 = AUCmetz1,
AUC_Metz2 = AUCmetz2,
AUC_Empirical1 = AUCemp1,
AUC_Empirical2 = AUCemp2))
}
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
#DATA GENERATION
set.seed(123)
x=rgamma(100, shape=2, rate = 8) # Generates biomarker data from a gamma distribution
# for the healthy group.
y=rgamma(100, shape=2, rate = 4) # Generates biomarker data from a gamma distribution
# for the diseased group.
scores=c(x,y)
D=c(zeros(1,100), ones(1,100))
out=checkboxcox(marker=scores, D, plots="on", printShapiro = TRUE)
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
#DATA GENERATION
set.seed(123)
x <- rgamma(100, shape=2, rate = 8) # generates biomarker data from a gamma
# distribution for the healthy group.
y <- rgamma(100, shape=2, rate = 4) # generates biomarker data from a gamma
# distribution for the diseased group.
scores <- c(x,y)
D=c(pracma::zeros(1,100), pracma::ones(1,100))
out=threerocs(marker=scores, D, plots="on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
set.seed(123)
x=rgamma(100, shape=2, rate = 8)
y=rgamma(100, shape=2, rate = 4)
scores=c(x,y)
D=c(zeros(1,100), ones(1,100))
out=rocboxcox(marker=scores,D, 0.05, plots="on", printProgress=FALSE)
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
givenSP=c(0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9)
givenSE=NA
out=rocboxcoxCI(marker=scores ,D, givenSP=givenSP, givenSE=NA, alpha=0.05, plots="on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
#DATA GENERATION
set.seed(123)
nx <- 100
Sx <- matrix(c(1, 0.5, 0.5, 1), nrow = 2, ncol = 2)
mux <- c(X = 10, Y = 12)
X = mvtnorm::rmvnorm(nx, mean = mux, sigma = Sx)
ny <- 100
Sy <- matrix(c(1.1, 0.6, 0.6, 1.1), nrow = 2, ncol = 2)
muy <- c(X = 11, Y = 13.7)
Y = mvtnorm::rmvnorm(ny, mean = muy, sigma = Sy)
dx = pracma::zeros(nx,1)
dy = pracma::ones(ny,1)
markers = rbind(X,Y);
marker1 = markers[,1]
marker2 = markers[,2]
D = c(rbind(dx,dy))
out=checkboxcox2(marker1, marker2, D, plots = "on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
#GENERATE SOME BIVARIATE DATA===
set.seed(123)
nx <- 100
Sx <- matrix(c(1, 0.5,
0.5, 1),
nrow = 2, ncol = 2)
mux <- c(X = 10, Y = 12)
X=rmvnorm(nx, mean = mux, sigma = Sx)
ny <- 100
Sy <- matrix(c(1.1, 0.6,
0.6, 1.1),
nrow = 2, ncol = 2)
muy <- c(X = 11, Y = 13.7)
Y=rmvnorm(ny, mean = muy, sigma = Sy)
dx=zeros(nx,1)
dy=ones(ny,1)
markers=rbind(X,Y);
marker1=markers[,1]
marker2=markers[,2]
D=c(rbind(dx,dy))
#==DATA HAVE BEEN GENERATED====
#===COMPARE THE AUCs of Marker 1 vs Maker 2
out=comparebcAUC(marker1, marker2, D, alpha=0.05, plots="on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
#==DATA HAVE BEEN GENERATED====
#===COMPARE THE Js of Marker 1 vs Maker 2
out=comparebcJ(marker1, marker2, D, alpha=0.05, plots="on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
out=comparebcSens(marker1=marker1, marker2=marker2, D=D, alpha =0.05, atSpec=0.8, plots="on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
out=comparebcSpec(marker1=marker1, marker2=marker2, D=D, alpha =0.05, atSens=0.8, plots="on")
summary(out)
## ----echo=TRUE, eval=TRUE-----------------------------------------------------
out=threerocs2(marker1, marker2, D, plots = "on")
summary(out)
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