Extint_linear<-function(y, W_int, W_ext ,C=NULL, print.summary=TRUE, standardize = TRUE){ # w is matrix containing replicates, C=vector of cut points
####################################
## Lambda.par=(mux, s2x) and su2
#####################################
n<-dim(W_int)[1]
r<- dim(W_int)[2] # if W_int has replicates, this indicating the number of replicates
####################################
# first we standardize the raw data W
####################################
mux_hat<-mean(W_int)
if(standardize == TRUE){
a0 = 1/sd(W_int)
b0 = -mux_hat*a0}else{
a0 = 1
b0 = 0}
w = a0*W_int + b0
W_ext = a0*W_ext + b0
mux_hat<-mean(w)
qq = apply(W_ext,1,var)
su2_ext=qq
su2e= mean(qq)
mux_hat= wmean = mean(w)
s2x_hat=max((var(w)-su2e/r),0.2*var(w))
lambda_hat<-max(s2x_hat/(s2x_hat+su2e/r),0.2)
#############################################
#Cut points
if(is.null(C)){
C=rep(0,4)
C[1]=qnorm(0.2,mean =mux_hat , sd = sqrt(s2x_hat))
C[2]=qnorm(0.4,mean =mux_hat , sd = sqrt(s2x_hat))
C[3]=qnorm(0.6,mean =mux_hat , sd = sqrt(s2x_hat))
C[4]=qnorm(0.8,mean =mux_hat , sd = sqrt(s2x_hat))
}else{C = a0*C + b0}
sizeC<-length(C)
try(if(sizeC != 4) stop("the size of cutting points is different from 4!"))
J=length(C)+1 #Number of sets
##################
# Basic functions
##################
fMx<-function(x){
Mx<-vector()
Mx[1]=ifelse(x<C[1],1,0)
Mx[2]=ifelse((C[1]<=x)& (x<C[2]),1,0)
Mx[3]=ifelse((C[2]<=x)&(x<C[3]),1,0)
Mx[4]=ifelse((C[3]<=x)&(x<C[4]),1,0)
Mx[5]=ifelse(x>=C[4],1,0)
return (Mx)
}
fHM<-function(x,theta){
1/(1+exp(-(fMx(x)%*%theta)))
}
fH1<-function(x){1/(1+exp(-x))*(1-1/(1+exp(-x)))}
fH<-function(x){1/(1+exp(-x))}
K=c(-Inf,C,Inf)
###############################################
#Estimate \Theta
TmpIntegrate1 <- function(x){
n_W = length(w)
TMP = 0
for (i in 1:n_W){
muxw=mux_hat*(1-lambda_hat)+lambda_hat*w[i]
varxw=lambda_hat*su2e/r
f = dnorm(x,muxw,sqrt(varxw))
#Hmx=1/(1+exp(-(betac[1]+betac[2]*x)))
Hmx=betac[1]+betac[2]*x
TMP = TMP + Hmx* f
}
return(TMP)}
TmpIntegrate2 <- function(x){
n_W = length(w)
TMP = 0
for (i in 1:n_W){
muxw=mux_hat*(1-lambda_hat)+lambda_hat*w[i]
varxw=lambda_hat*su2e/r
f = dnorm(x,muxw,sqrt(varxw))
TMP = TMP + f
}
return(TMP)}
################################################
# Estimate function for Lambda(mux,sx2)
V_int<-function(Lambda.par){
c(mean(w-Lambda.par[1]), mean((w-Lambda.par[1])^2-Lambda.par[2]-su2e/r))
}
S.E.Sigma_thetac_Int_Ext<-function(beta.par, theta.par, Lambda.par,su2e){
mux_hat=Lambda.par[1]
s2x_hat=Lambda.par[2]
lambda_hat=max(s2x_hat/(s2x_hat+su2e/r),0.2)
npsi<-length(beta.par)+length(theta.par)+length(Lambda.par)
J<-length(theta.par)
########################### Find An
#### we need to find: E[derivative Phi with respect Lambda], E[derivative Phi with respect beta]
## E[derivative Q with respect Lambda], E[derivative Q with respect beta], E[derivative Q with respect theta]
#####
An<-matrix(0,nrow = npsi,ncol = npsi)
#### 1st E[derivative Phi with respect to beta]
An[3:4,3:4]<-matrix(c(-1,-mux_hat,-mux_hat,-s2x_hat-mux_hat^2),nrow = 2,ncol = 2)
##################
## 2nd E[derivative Q with respect beta]
fQ_i_su2e<-function(su2e ){
#mux_hat=Lambda.par[1]
#sx2_hat=Lambda.par[2]
lambda_hat=s2x_hat/(s2x_hat+su2e)
fQi<-matrix(0, ncol = 2, nrow = J)
for (j in 1:J){
muxw=mux_hat*(1-lambda_hat)+lambda_hat*w[i]
varxw=lambda_hat*su2e
fxw<-function(x){ dnorm(x,muxw,sqrt(varxw)) } # density of X/W
fxf<-function(x){
#Hmx=1/(1+exp(-(beta.par[1]+beta.par[2]*x)))
return(x*fxw(x))
}
l2=integrate(fxf,lower=K[j], upper=K[j+1])$value
l1=integrate(fxw,lower=K[j], upper=K[j+1])$value
fQi[j, ]<-c(l1, l2)
}
return(fQi)
}
sumdphi<-0
for(i in 1:n){
sumdphi<-sumdphi+fQ_i_su2e(su2e)}
An[5:(J+4),3:4]<-sumdphi/(n)
##################
##### E[derivative Q with respect to theta]
An[5:(J+4),5:(J+4)]<-diag(-sumdphi[,1]/(n))
################ E[derivative of Phi with respect Lambda]
##########################################
# all components zero
### E[derivative of Q with respect Lambda]
##########################################
##################### derivative of Q with respect to Lambda
fQ_i_Lambda<-function(Lambda.par1 ){
#mux_hat=Lambda.par1[1]
#sx2_hat=Lambda.par1[2]
lambda_hat=Lambda.par1[2]/(Lambda.par1[2]+su2e)
fQi<-vector()
for (j in 1:J){
muxw=Lambda.par1[1]*(1-lambda_hat)+lambda_hat*w[i]
varxw=lambda_hat*su2e
fxw<-function(x){ dnorm(x,muxw,sqrt(varxw)) } # density of X/W
fxf<-function(x){
#Hmx=1/(1+exp(-(beta.par[1]+beta.par[2]*x)))
mx=beta.par[1]+beta.par[2]*x
return(mx*fxw(x))
}
l1=integrate(fxf,lower=K[j], upper=K[j+1])$value
l2=integrate(fxw,lower=K[j], upper=K[j+1])$value
fQi[j]<-l1-theta.par[j]*l2
}
return(fQi)
}
sumdphi<-0
for(i in 1:n){
sumdphi<-sumdphi+numDeriv::jacobian(fQ_i_Lambda,Lambda.par)}
EdQ_Lambda<- sumdphi/(n)
An[5:(J+4),1:2]<-EdQ_Lambda
##################### E[derivative of V with respect Lambda]
An[1:2,1:2]<--diag(2)
#An[2,3]<--1
############################ %%% %%%%%%%% Complete An %%%%%%%% ############################
############################ ############################ ############################ ############################
########################### Find Bn
#### we need to find: Sigma, cov(Psi), E[derivative Phi with respect Lambda], E[derivative Q with respect Lambda]
############################################################################
################### cov(Psi)
### Aproximate of cov(psi)
###################### cov(psi) %%%%%%%%% Term : E(Psi Psi^T)
phi_i_parameter<-function(beta.par,su2e){
phi_i<-c(y[i]-beta.par[1]-beta.par[2]*w[i], w[i]*y[i]-beta.par[1]*w[i]-beta.par[2]*w[i]^2+beta.par[2]*su2e)
return( phi_i)
}
fQ_i_parameter<-function(beta.par,theta.par,Lambda.par){
mux_hat=Lambda.par[1]
s2x_hat=Lambda.par[2]
lambda_hat=s2x_hat/(s2x_hat+su2e)
fQi<-vector()
for (j in 1:J){
muxw=mux_hat*(1-lambda_hat)+lambda_hat*w[i]
varxw=lambda_hat*su2e
fxw<-function(x){ dnorm(x,muxw,sqrt(varxw)) } # density of X/W
fxf<-function(x){
#Hmx=1/(1+exp(-(beta.par[1]+beta.par[2]*x)))
Hmx=beta.par[1]+beta.par[2]*x
return(Hmx*fxw(x))
}
l1=integrate(fxf,lower=K[j], upper=K[j+1])$value
l2=integrate(fxw,lower=K[j], upper=K[j+1])$value
fQi[j]<-l1-theta.par[j]*l2
}
return(fQi)
}
V_int_i<-function(Lambda.par){
c(w[i]-Lambda.par[1], (w[i]-Lambda.par[1])^2-Lambda.par[2]-su2e/r) # r=1
}
psi_i_parameter<- function(beta.par,theta.par,Lambda.par){
c(t( V_int_i(Lambda.par)),t(phi_i_parameter(beta.par,su2e)), t(fQ_i_parameter(beta.par,theta.par,Lambda.par)))
}
sumdpsi<-0
for(i in 1:n){
sumdpsi<-sumdpsi+ psi_i_parameter(beta.par,theta.par,Lambda.par)%*%t( psi_i_parameter(beta.par,theta.par,Lambda.par)) }#
Epsi_psi<-sumdpsi/(n) # Aproximate E(Psi Psi^T) ==> aprox cov(psi)
######################################## cov(psi) %%%%%%%%% Finally we calculate cov(Psi)
covPsi<- Epsi_psi
######################################## Sigma %%%%%%%%%
Sigma<-mean((su2_ext-su2e)^2) # su2_ext has to be defined out of the function S.E.Sigma_thetac_Int_Ext --- doesn't divide by r!!
######################################## End Sigma
######################################## E(omega psi respect to su2) %%%%%%%%%
phi_i_parameter_su2e<-function(su2e){
phi_i<-c(y[i]-beta.par[1]-beta.par[2]*w[i], w[i]*y[i]-beta.par[1]*w[i]-beta.par[2]*w[i]^2+beta.par[2]*su2e)
return( phi_i)
}
fQ_i_parameter_su2e<-function(su2e){
mux_hat=Lambda.par[1]
s2x_hat=Lambda.par[2]
#cl=lambda/3
lambda_hat=max(s2x_hat/(s2x_hat+su2e/r),0.2) #max(s2x_hat/(s2x_hat+su2e),cl)
fQi<-vector()
for (j in 1:J){
muxw=mux_hat*(1-lambda_hat)+lambda_hat*w[i]
varxw=lambda_hat*su2e/r
fxw<-function(x){ dnorm(x,muxw,sqrt(varxw)) } # density of X/W
fHf<-function(x){
#Hmx=1/(1+exp(-(beta.par[1]+beta.par[2]*x)))
Hmx=beta.par[1]+beta.par[2]*x
return(Hmx*fxw(x))
}
l1=integrate(fHf,lower=K[j], upper=K[j+1])$value
l2=integrate(fxw,lower=K[j], upper=K[j+1])$value
fQi[j]<-l1-theta.par[j]*l2
}
return(fQi)
}
V_int_su2e_i<-function(su2e){
c(w[i]-Lambda.par[1], (w[i]-Lambda.par[1])^2-Lambda.par[2]-su2e/r)
}
psi_i_parameter_su2e<- function(su2e){
c(t( V_int_su2e_i(su2e)),t(phi_i_parameter_su2e(su2e)), t(fQ_i_parameter_su2e(su2e)))
}
sumdpsi<-0
for(i in 1:n){sumdpsi<-sumdpsi+numDeriv::jacobian(psi_i_parameter_su2e,su2e)}#
EOmegaPsi_su2e<-sumdpsi/n # Aproximate E(Psi Psi^T) ==> aprox cov(psi)
############### Then Bn is:
Bn<-matrix()
m<-length(W_ext[,1]) # size external dataset
Bn<-covPsi+(n/m)*EOmegaPsi_su2e%*%Sigma%*%t(EOmegaPsi_su2e)
############################ %%%%%%%% Complete Bn %%%%%%%% ############################
Sigma_whole = solve(An)%*%Bn%*%t(solve(An))
Sigma_theta<-Sigma_whole[5:9,5:9]
VarSigma_thetac<-Sigma_theta/n
s.e_thetac<-sqrt(diag(VarSigma_thetac))
s.e_thetac_J1<- sqrt((VarSigma_thetac[J,J]+VarSigma_thetac[1,1]-2*VarSigma_thetac[1,J]))
s.e_thetacfull<-c(s.e_thetac, s.e_thetac_J1) # SE of the estimators of theta_1,...,theta_J and (theta_J-theta_1)
#####################
Sigma_par<-Sigma_whole[1:4,1:4]
VarSigma_par<-Sigma_par/n
s.e_par<-sqrt(diag(VarSigma_par))
covab = VarSigma_par[3,4]
##################
return(list(s.e_par,s.e_thetacfull,covab,sqrt(Sigma/n) ))
}
### store estimates related to \theta
thetac_w_e=vector()
s.e_thetac_w_e=vector() # Store s.e of \theta_c by using MLE of beta
thetac.width.cover_w_e=vector()
##################
## beta.par
##################
### Log likelihood to find estimate of \beta
Lambda.par=c(mux_hat,s2x_hat)
### using estimating equation to find \beta
ymean<-mean(y)
swy<-(y%*%w)/n-ymean*wmean
sww<- sum(w*w) /n - wmean^2
betac= c(0,0)
betac[2] = swy/(sww-su2e)
betac[1] = ymean - betac[2]*wmean
beta.par = betac
##################
## theta.par
##################
### Log likelihood to find estimate of \theta
for (j in 1:J){
l1=integrate(TmpIntegrate1, lower=K[j], upper=K[j+1])$value
l2=integrate(TmpIntegrate2, lower=K[j], upper=K[j+1])$value
thetac_w_e[j]=l1/l2
# thetac_w_e[j]=log(L/(1-L))
}
###############################################################
## Assymptotic variance and CI
###############################################################
theta.par=thetac_w_e
s.e_thetac_w_e<-S.E.Sigma_thetac_Int_Ext(beta.par, theta.par, Lambda.par,su2e)
UL.thetac<-c(theta.par,theta.par[5]-theta.par[1])+1.96*s.e_thetac_w_e[[2]]
LL.thetac<-c(theta.par,theta.par[5]-theta.par[1])-1.96*s.e_thetac_w_e[[2]]
I.C<-cbind( LL.thetac,UL.thetac)
## Estimate
thetac_output_w_e<-cbind(thetac_w_e, thetac_w_e[J]-thetac_w_e[1] )
##############################################################
##here we transfer all estimates back
##############################################################
mux_hat = (mux_hat - b0)/a0
s2x_hat = s2x_hat/(a0^2)
su2_hat = su2e/(a0^2)
beta.par[2] = a0*betac[2]
beta.par[1] = b0*betac[2] + betac[1]
#########################################
#transfer the variance
#########################################
se_Lambda = c((s.e_thetac_w_e[[1]])[1:2], s.e_thetac_w_e[[4]])
se_Lambda[1] = se_Lambda[1]/(a0)
se_Lambda[2] = se_Lambda[2]/(a0^2)
se_Lambda[3] = se_Lambda[3]/(a0^2)
se_beta = (s.e_thetac_w_e[[1]])[3:4]
se_beta[1] = sqrt((se_beta[1])^2+(b0^2)*(se_beta[2])^2+2*(b0)*s.e_thetac_w_e[[3]])
se_beta[2] = (a0)*se_beta[2]
########################## Output
#
# SUMMARY
cname <- c("Estimate", "Std. Error", "z-value", "Pr(>|z|)")
if (print.summary == TRUE) {
all_par<-c(mux_hat, s2x_hat,su2e,beta.par, theta.par)
s.e_all_par<- c(se_Lambda,se_beta, s.e_thetac_w_e[[2]][1:J])
tval <-all_par/s.e_all_par
pval <- 2 * (1 - pnorm(abs(tval)))
outmat <- cbind(all_par, s.e_all_par, tval, pval)
name_Lam_beta<-c("mu.x", "sigma^2.x","sigma^2.u", "alpha","beta")
name_theta<-vector()
for(i in 1:J){name_theta[i]<-paste("theta",i)}
rownames(outmat) <- c(name_Lam_beta, name_theta)
colnames(outmat) <- cname
cat("Summary", "\n")
cat(" ", "\n")
print(round(outmat, 5))
cat(" ", "\n")
dmat<-theta.par[J]-theta.par[1]
smat<-s.e_thetac_w_e[[2]][J+1]
tmat<-dmat/smat
outmat1 <- cbind(dmat, smat, tmat, 2 * (1 - pnorm(abs(tmat))))
rownames(outmat1) <- paste("theta",J,"- theta 1:")
colnames(outmat1) <- cname
print(round(outmat1, 5))
cat(" ", "\n")
}
lambda_beta<-c(mux_hat, s2x_hat,su2e,beta.par)
names(lambda_beta)<-name_Lam_beta
out1 <- list(outmat1, theta.par, lambda_beta, s.e_thetac_w_e[[2]], c(se_Lambda, s.e_thetac_w_e[[4]], se_beta))
names(out1) <- c( "theta5-theta1", "theta","nuisance", "se.theta","se.nuisance")
return(out1) }
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