# R/wtsc-all.r In weightedScores: Weighted Scores Method for Regression Models with Dependent Data

#### Documented in approxbvncdfbclcl1dmargmodelgodambeieemarglikpmargmodelscoreCovsolvewtscweightMatwtscwtsc.wrapper

```# bivariate standard normal log-likelihood for one observation
# input:
# low the vector of lower limits of length n.
# upp the vector of upper limits of length n.
# r the correlation parameter
# output:
# bivariate standard normal log-likelihood for one observation
bivlik<-function(low,upp,r)
{ rmat<-matrix(c(1,r,r,1),2,2)
prob<-pmvnorm(lower=low,upper=upp,mean=rep(0,2),corr=rmat)[1]
log(prob)
}

# the mean values of the univariate marginal distribution
# corresonding to the used link function
# input:
# x the matix of the covariates
# b the vector with the regression coefficients
# link has three options: 1. "log", 2. "logit". 3. "probit"
# output:
# the mean values of the univariate marginal distribution
{ mu<-exp(x %*% b)
}
else
{ expnu<-exp(x %*% b)
mu<-expnu/(1+expnu)
}
else
mu<-pnorm(x %*% b) }
}
mu
}

# Density  of the univariate marginal distribution
# input:
# y the vector of (non-negative integer) quantiles.
# mu the mean parameter of the univariate distribution.
# gam the parameter gamma of  the negative binomial distribution.
# invgam the inverse of parameter  gamma of negative binomial distribution.
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson, ?bernoulli? for
# Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2 parametrization of negative
# binomial in Cameron and Trivedi (1998).
# output:
# the density of the univariate marginal  distribution
dmargmodel<-function(y,mu,gam,invgam,margmodel)
{ if(margmodel=="poisson")
{ dpois(y,mu)
}
else
{ if(margmodel=="bernoulli")
{ dbinom(y,size=1,prob=mu) }
else
{ if(margmodel=="nb1")
{ dnbinom(y,prob=1/(1+gam),size=mu*invgam) }
else
{ # margmodel=="nb2"
dnbinom(y,size=invgam,mu=mu) }}}
}

# CDF  of the univariate marginal distribution
# input:
# y the vector of (non-negative integer) quantiles.
# mu the mean parameter of the univariate distribution.
# gam the parameter gamma of  the negative binomial distribution.
# invgam the inverse of parameter  gamma of negative binomial distribution.
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson, ?bernoulli? for
# Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2 parametrization of negative
# binomial in Cameron and Trivedi (1998).
# output:
# the cdf of the univariate marginal  distribution
pmargmodel<-function(y,mu,gam,invgam,margmodel)
{ if(margmodel=="poisson")
{ ppois(y,mu)
}
else
{ if(margmodel=="bernoulli")
{ pbinom(y,size=1,prob=mu) }
else
{ if(margmodel=="nb1")
{ pnbinom(y,prob=1/(1+gam),size=mu*invgam) }
else
{ # margmodel=="nb2"
pnbinom(y,size=invgam,mu=mu) }}}
}

# quantile  of the univariate marginal distribution
# input:
# y the vector of probabilities
# mu the mean parameter of the univariate distribution.
# gam the parameter gamma of  the negative binomial distribution.
# invgam the inverse of parameter  gamma of negative binomial distribution.
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson, ?bernoulli? for
# Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2 parametrization of negative
# binomial in Cameron and Trivedi (1998).
# output:
# the quantile of the univariate marginal  distribution
qmargmodel<-function(y,mu,gam,invgam,margmodel)
{ if(margmodel=="poisson")
{ qpois(y,mu)
}
else
{ if(margmodel=="bernoulli")
{ qbinom(y,size=1,prob=mu) }
else
{ if(margmodel=="nb1")
{ qnbinom(y,prob=1/(1+gam),size=mu*invgam) }
else
{ # margmodel=="nb2"
qnbinom(y,size=invgam,mu=mu) }}}
}

# negative univariate logikelihood assuming independence within clusters
# input:
# param the vector of regression and not regression parameters
# xdat the matrix of covariates (use the constant 1 for the first covariate)
# ydat the vector with the response
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson,
# ?bernoulli? for Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2
# parametrization of negative binomial in Cameron and Trivedi (1998)
# ?logit? for the logit link function, and ?probit? for the probit link function.
# output:
# negative univariate logikelihood assuming independence within clusters
{ p<-dim(xdat)[2]
b<-param[1:p]
if(margmodel=="nb1" | margmodel=="nb2")
{ gam<-param[p+1]
invgam<-1/gam
}
#else  gam<-invgam<-0
-sum(log(dmargmodel(ydat,mu,gam,invgam,margmodel)))
}

# Independent estimating equations for binary, Poisson or
# negative binomial regression.
# xdat the matrix of covariates (use the constant 1 for the first covariate)
# ydat the vector with the response
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson,
# ?bernoulli? for Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2
# parametrization of negative binomial in Cameron and Trivedi (1998)
# ?logit? for the logit link function, and ?probit? for the probit link function.
# output: A list containing the following components:
# coef the vector with the ML estimated regression parameters
# gam the ML estimate of gamma parameter
{ #if(margmodel=="bernoulli")  family=binomial else family=poisson
if(margmodel=="nb1" | margmodel=="nb2" | margmodel=="poisson") link="log"
if(margmodel=="poisson")
res<-as.vector(uni\$coef)
list(reg=res)
} else {
if(margmodel=="bernoulli")
} else {
res<-as.vector(uni\$coef)
list(reg=res)
} else
{ p<-dim(xdat)[2]
uni<-nlm(marglik,c(rep(0,p),1),margmodel=margmodel,
res1<-uni\$e[1:p]
res2<-uni\$e[p+1]
list(reg=res1,gam=res2) }}
}

# corralation matrix
# input:
# d the dimension
# r a vector with correaltion parameters
# corstr indicates the latent correlation structure of normal copula.
# Choices are ?exch?, ?ar?, and ?unstr? for exchangeable, ar(1) and
# unstrucutred correlation structure, respectively.
# output:
# the correlation matrix
cormat<-function(d,r,pairs,corstr)
{ rmat<-matrix(1,d,d)
lr<-nrow(pairs)
if(length(r)==1) r<-rep(r,d^2)
if(corstr=="exch" | corstr=="unstr")
{
for(j in 1:lr)
{ rmat[pairs[j,1],pairs[j,2]]<-r[j]
rmat[pairs[j,2],pairs[j,1]]<-r[j]
}
} else {
for(j in 1:lr)
{ temp<-pairs[j,2]-pairs[j,1]
rmat[pairs[j,1],pairs[j,2]]<-r[j]^temp
rmat[pairs[j,2],pairs[j,1]]<-r[j]^temp
}
}
rmat
}

# Bivariate composite likelihood for multivariate normal copula with Poisson,
# binary, or negative binomial regression.
# input:
# r the vector of normal copula parameters
# b the regression coefficients
# gam the gamma parameter
# xdat the matrix of covariates (use the constant 1 for the first covariate)
# ydat the vector with the response
# id the the vector with the id
# tvec the time related vector
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson,
# ?bernoulli? for Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2
# parametrization of negative binomial in Cameron and Trivedi (1998)
# ?logit? for the logit link function, and ?probit? for the probit link function.
# corstr indicates the latent correlation structure of normal copula.
# Choices are ?exch?, ?ar?, and ?unstr? for exchangeable, ar(1) and
# unstrucutred correlation structure, respectively.
# output:
# negative bivariate composite likelihood for multivariate normal copula
# with Poisson, binary, or negative binomial regression.
{ s<-0
if(margmodel=="nb1" | margmodel=="nb2" | margmodel=="poisson") link="log"
if(margmodel=="nb1" | margmodel=="nb2") invgam<-1/gam else invgam<-NULL
uid<-unique(id)
d<-id.size(id)
maxd<-max(d)
tvec<-id.time(tvec,d)
n<-1:dim(xdat)[1]
pairs<-maxpairs(d)
rmat<-cormat(maxd,r,pairs,corstr)
c1<-(r[1] > 1 | r[1] < (-1/(maxd-1)))
c2<-(r[1] > 1 | r[1] < -1)
c3<-(det(rmat)<0 | min(rmat)< -1 | max(rmat)>1)
if((corstr=="exch" & c1) | (corstr=="ar" & c2 ) | (corstr=="unstr" & c3)) {s<-1e10}
else {
for(i in uid)
{ cases<-id==i
irow=n[cases]
yi<-ydat[irow]
ti<-tvec[irow]
newyi<-rep(NA,maxd)
newmui<-rep(NA,maxd)
newyi[ti]<-yi
x<-xdat[cases,]
newmui[ti]<-mui
vlow<-pmargmodel(newyi-1,newmui,gam,invgam,margmodel)
tem<-dmargmodel(newyi,newmui,gam,invgam,margmodel)
vupp<-vlow+tem
zlow=qnorm(vlow)
zupp=qnorm(vupp)
for(j in 1:dim(pairs)[1])
{ k1<-pairs[j,][1]
k2<-pairs[j,][2]
if((sum(k1==ti)==1) & (sum(k2==ti)==1))
{ if(corstr=="exch")
{ s<-s +bivlik(zlow[c(k1,k2)],zupp[c(k1,k2)],r) }
else
{ if(corstr=="ar")
{ s<-s +bivlik(zlow[c(k1,k2)],zupp[c(k1,k2)],r^(k2-k1)) }
else
{ # corstr=="unstr"
s<-s +bivlik(zlow[c(k1,k2)],zupp[c(k1,k2)],r[j])
}}}
else { s<-s }
}
}
-s
}
}

# optimization routine for composite likelihood for MVN copula
# b the regression coefficients
# gam the gamma parameter
# xdat the matrix of covariates (use the constant 1 for the first covariate)
# ydat the vector with the response
# id the the vector with the id
# tvec the time related vector
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson,
# ?bernoulli? for Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2
# parametrization of negative binomial in Cameron and Trivedi (1998)
# ?logit? for the logit link function, and ?probit? for the probit link function.
# corstr indicates the latent correlation structure of normal copula.
# Choices are ?exch?, ?ar?, and ?unstr? for exchangeable, ar(1) and
# unstrucutred correlation structure, respectively.
# output: A list containing the following components:
# minimum the value of the estimated minimum of CL1 for MVN copula
# estimate the CL1 estimates
# code an integer indicating why the optimization process terminated, see nlm.
{ if(margmodel=="nb1" | margmodel=="nb2" | margmodel=="poisson") link="log"
d<-id.size(id)
pairs<-maxpairs(d)
if(corstr=="unstr")
{ nom<-dim(pairs)[1]
}
else
}

# the dimension of each id
# input: the vector with the id
# id the vector with the id
# output:
# a vector with the dimension of each id
id.size<-function(id)
{ d<-NULL
uid<-unique(id)
for(i in uid)
{ di<-sum(id==i)
d<-c(d,di) }
d
}

# the transformed time points, i.e, 1,2,3... for each subject
# input:
# tvec: untrasformed time points
# d the dimension for all subjects
# output:
# the transformed time points, i.e, 1,2,3... for each subject
id.time<-function(tvec,d)
{ maxd<-max(d)
ut<-sort(unique(tvec))
newtvec<-rep(NA,length(tvec))
for(j in 1:maxd)
{ newtvec[tvec==ut[j]]<-j }
newtvec
}

# the maximum number of bivariate pairs
# input:
# d the dimension for all subjects
# output:
# the maximum number of bivariate pairs
maxpairs<-function(d)
{ pairs<-NULL
maxd<-max(d)
for(id1 in 1:(maxd-1))
{ for(id2 in (id1+1):maxd)
{ pairs<-rbind(pairs,c(id1,id2)) } }
pairs
}

# derivative of the marginal loglikelihood with respect to nu
# input:
# mu the mean parameter
# gam the gamma parameter
# invgam the inverse of gamma parameter
# ub the truncation value
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson,
# ?bernoulli? for Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2
# parametrization of negative binomial in Cameron and Trivedi (1998)
# ?logit? for the logit link function, and ?probit? for the probit link function.
# output:
# the vector with the derivatives of the margmodel loglikelihood with respect to nu
{ if(mu==1 & is.finite(mu)){mu<-0.9999}
nu<-qnorm(mu)
(0:1-mu)/mu/(1-mu)*dnorm(nu)
}
else
{ if(margmodel=="nb1")
{ j<-0:(ub-1)
s<-c(0,cumsum(1/(mu+gam*j)))
(s-invgam*log(1+gam))*mu
}
else
{ if(margmodel=="nb2")
{ pr<-1/(mu*gam+1)
(0:ub-mu)*pr
}
else { 0:ub-mu }}}
}

# derivative of the marginal loglikelihood with respect to nu
# input:
# mu the mean parameter
# gam the gamma parameter
# invgam the inverse of gamma parameter
# y  the value of a non-negative integer quantile
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson,
# ?bernoulli? for Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2
# parametrization of negative binomial in Cameron and Trivedi (1998)
# ?logit? for the logit link function, and ?probit? for the probit link function.
# output:
# the derivative of the margmodel loglikelihood with respect to nu
{ if(mu==1 & is.finite(mu)){mu<-0.9999}
nu<-qnorm(mu)
(y-mu)/mu/(1-mu)*dnorm(nu)
}
else
{ if(margmodel=="nb1")
{ s<-0
if(y>0)
{ j<-0:(y-1)
s<-sum(1/(mu+gam*j))
}
(s-invgam*log(1+gam))*mu
}
else
{ if(margmodel=="nb2")
{ pr<-1/(mu*gam+1)
(y-mu)*pr
}
else { y-mu }}}
}

# derivative of the NB loglikelihood with respect to gamma
# input:
# gam the gamma parameter
# invgam the inverse of gamma parameter
# mu the mean parameter
# ub the truncation value
# margmodel indicates the marginal model. Choices are  ?nb1? , ?nb2? for
# the NB1 and NB2  parametrization of negative binomial in
# Cameron and Trivedi (1998)
# output:
# the vector with the derivatives of the NB loglikelihood with respect to gamma
derlik.gam<-function(mu,gam,invgam,ub,margmodel)
{ j<-0:(ub-1)
if(margmodel=="nb1")
{ s<-c(0,cumsum(j/(mu+gam*j)))
s+invgam*invgam*mu*log(1+gam)-(0:ub+invgam*mu)/(1+gam)
}
else
{ #if(margmodel=="nb2")
pr<-1/(mu*gam+1)
s<-c(0,cumsum(j/(1+j*gam)))
s-log(pr)/(gam*gam)-(0:ub+invgam)*mu*pr
}
}

# derivative of the NB loglikelihood with respect to gamma
# input:
# gam the gamma parameter
# invgam the inverse of gamma parameter
# mu the mean parameter
# y  the value of a non-negative integer quantile
# margmodel indicates the marginal model. Choices are  ?nb1? , ?nb2? for
# the NB1 and NB2  parametrization of negative binomial in
# Cameron and Trivedi (1998)
# output:
# the derivative of the NB loglikelihood with respect to gamma
iderlik.gam<-function(mu,gam,invgam,y,margmodel)
{ s<-0
if(margmodel=="nb1")
{ if(y>0)
{ j<-0:(y-1)
s<-sum(j/(mu+gam*j))
}
s+invgam*invgam*mu*log(1+gam)-(y+invgam*mu)/(1+gam)
}
else
{ #if(margmodel=="nb2")
if(y>0)
{ j<-0:(y-1)
s<-sum(j/(1+gam*j))
}
pr<-1/(mu*gam+1)
s-log(pr)/(gam*gam)-(y+invgam)*mu*pr
}
}

# minus expectation of the second derivative of the marginal loglikelihood
# with resect to nu
# input:
# mu the mean parameter
# gam the gamma parameter
# invgam the inverse of gamma parameter
# u the univariate cdfs
# ub the truncation value
# margmodel indicates the marginal model. Choices are  ?nb1? , ?nb2? for
# the NB1 and NB2  parametrization of negative binomial in
# Cameron and Trivedi (1998)
# output:
# the vector with the minus expectations of the margmodel loglikelihood
# with respect to nu
{ mu }
else
{ mu*(1-mu) }
else
{ if(margmodel=="nb1")
{ j<-0:ub
s1<-sum(1/(mu+j*gam)/(mu+j*gam)*(1-u))
s2<-sum(1/(mu+j*gam)*(1-u))
(mu*s1-s2+invgam*log(1+gam))*mu
}
else
{ if(margmodel=="nb2")
{ pr<-1/(mu*gam+1)
mu*pr
}
else
if(mu==1 & is.finite(mu)){mu<-0.9999}
nu<-qnorm(mu)
1/mu/(1-mu)*dnorm(nu)^2}
}}}
}

# minus expectation of the second derivative of the marginal NB loglikelihood
# with resect to gamma
# input:
# mu the mean parameter
# gam the gamma parameter
# invgam the inverse of gamma parameter
# u the univariate cdfs
# ub the truncation value
# margmodel indicates the marginal model. Choices are  ?nb1? , ?nb2? for
# the NB1 and NB2  parametrization of negative binomial in
# Cameron and Trivedi (1998)
# output:
# the vector with the minus expectations of the margmodel loglikelihood
# with respect to gamma
fisher.gam<-function(mu,gam,invgam,u,ub,margmodel)
{ j<-0:ub
if(margmodel=="nb1")
{ pr<-1/(gam+1)
s<-sum((j/(mu+j*gam))^2*(1-u))
s+2*invgam*invgam*invgam*log(1+gam)*mu-2*invgam*invgam*mu*pr-mu/gam*pr
}
else
{ #if(margmodel=="nb2")
s<-sum((invgam+j)^(-2)*(1-u))
invgam^4*(s-gam*mu/(mu+invgam))
}
}

# minus expectation of the second derivative of the marginal loglikelihood
# with resect to nu and gamma
# input:
# mu the mean parameter
# gam the gamma parameter
# invgam the inverse of gamma parameter
# u the univariate cdfs
# ub the truncation value
# margmodel indicates the marginal model. Choices are  ?nb1? , ?nb2? for
# the NB1 and NB2  parametrization of negative binomial in
# Cameron and Trivedi (1998)
# output:
# the vector with the minus expectations of the NB loglikelihood
# with respect to nu and gamma
fisher.nu.gam<-function(mu,gam,invgam,u,ub,margmodel)
{ if(margmodel=="nb1")
{ pr<-1/(gam+1)
j<-0:ub
s<-sum(j/(mu+j*gam)/(mu+j*gam)*(1-u))
(s-invgam*invgam*log(1+gam)+invgam*pr)*mu
}
else {0}
}

# Calculating the truncation value for the univariate distribution
# input:
# mu the mean parameter of the univariate distribution.
# gam the parameter gamma of  the negative binomial distribution.
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson, ?bernoulli? for
# Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2 parametrization of negative
# binomial in Cameron and Trivedi (1998).
# output:
# the truncation value--upper bound
truncation<-function(mu,gam,margmodel)
{ if(margmodel=="poisson")
{ ub<-round(max(10,mu+7*sqrt(mu),na.rm=T))
}
else
{ if(margmodel=="bernoulli") ub<-1
else
{ if(margmodel=="nb1")
{ pr<-1/(gam+1)
v<-mu/pr
ub<-round(max(10,mu+10*sqrt(v),na.rm=T))
}
else
{ pr<-1/(mu*gam+1)
v<-mu/pr
ub<-round(max(10,mu+7*sqrt(v),na.rm=T))
}}}
ub
}

# approximation of bivariate normal cdf (Johnson&Kotz, 1972)
# For rho<=0.4 the series truncated   at rho^3.
# For larger rho truncation at rho^5.
# input:
# r the normal copula parameter
approxbvncdf<-function(r,x1,x2,x1s,x2s,x1c,x2c,x1f,x2f,t1,t2)
{ r2=r*r; r3=r*r2; r4=r2*r2; r5=r4*r
tem3=r+r2*outer(x1,x2)/2+r3*outer(x1s-1,x2s-1)/6
if(r>0.4)
{ tem4=tem3+r4*outer(x1c-3*x1, x2c-3*x2)/24
tem5=tem4+r5*outer(x1f-6*x1s+3, x2f-6*x2s+3)/120
pr.a5=t1+t2*tem5
pr.a5
} else { pr.a3=t1+t2*tem3; pr.a3}
}

# covariance matrix of the scores Omega_i
# input:
# scnu the matrix of the score functions with respect to nu
# scgam the matrix of the score functions with respect to gam
# index the bivariate pair
# pmf the matrix of rectangle probabilities
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson, ?bernoulli? for
# Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2 parametrization of negative
# binomial in Cameron and Trivedi (1998).
scoreCov<-function(scnu,scgam,pmf,index,margmodel)
{ j1<-index[1]
j2<-index[2]
cov11<-t(scnu[,j1])%*%pmf%*%scnu[,j2]
if(margmodel=="bernoulli" | margmodel=="poisson")
{ cov11 }
else
{ cov12<-t(scnu[,j1])%*%pmf%*%scgam[,j2]
cov21<-t(scgam[,j1])%*%pmf%*%scnu[,j2]
cov22<-t(scgam[,j1])%*%pmf%*%scgam[,j2]
matrix(c(cov11,cov12,cov21,cov22),2,2)
}
}

# select the present column and lines in Omega, Delta and X matrices
# for unbalanced data
# input:
# tvec a vector of the time for an individual
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson, ?bernoulli? for
# Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2 parametrization of negative
# binomial in Cameron and Trivedi (1998).
subselect<-function(tvec,margmodel)
{ if(margmodel=="bernoulli" | margmodel=="poisson") sel<-tvec
else
{ sel<-NULL
for(i in 1:length(tvec))
{ tm<-2*tvec[i]
k<-c(tm-1,tm)
sel<-c(sel,k)
}}
sel
}

# weight matrix fixed at values from the CL1 estimator
# input:
# b the regression coefficients
# gam the gamma parameter
# xdat the matrix of covariates (use the constant 1 for the first covariate)
# ydat the vector with the response
# id the the vector with the id
# tvec the time related vector
# rh the vector with CL1 estimates
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson,
# ?bernoulli? for Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2
# parametrization of negative binomial in Cameron and Trivedi (1998)
# ?logit? for the logit link function, and ?probit? for the probit link function.
# corstr indicates the latent correlation structure of normal copula.
# Choices are ?exch?, ?ar?, and ?unstr? for exchangeable, ar(1) and
# unstrucutred correlation structure, respectively.
# output: A list containing the following components:
# omega the array with the Omega matrices
# delta the array with the Delta matrices
# X the array with the X matrices
{ if(margmodel=="nb1" | margmodel=="nb2" | margmodel=="poisson") link="log"
if(margmodel=="nb1" | margmodel=="nb2") invgam<-1/gam else invgam<-NULL
uid<-unique(id)
d<-id.size(id)
maxd<-max(d)
q<-length(gam)
lid<-length(uid)
dim<-dim(xdat)
n<-1:dim[1]
p<-dim[2]
pairs<-maxpairs(d)
tvec<-id.time(tvec,d)
omega<-array(NA,c(maxd*(1+q),maxd*(1+q),lid))
X<-array(NA,c(p+q,maxd*(1+q),lid))
delta<-array(NA,c(maxd*(1+q),maxd*(1+q),lid))
dom<-maxd*(1+q)
pos<-seq(1,dom-1,by=2)    #not used for binary
m<-0
for(i in uid)
{ m<-m+1
cases<-id==i
irow=n[cases]
newx<-matrix(NA,maxd,p)
newyi<-rep(NA,maxd)
newmui<-rep(NA,maxd)
ti<-tvec[irow]
x<-xdat[cases,]
newx[ti,]<-x
newmui[ti]<-mui
ub<-truncation(newmui,gam,margmodel)
du<-scnu<-scgam<-matrix(NA,1+ub,maxd)
for(j in ti)
{ du[,j]<-dmargmodel(0:ub,newmui[j],gam,invgam,margmodel)
if(margmodel=="nb1" | margmodel=="nb2")
{ scgam[,j]<-derlik.gam(newmui[j],gam,invgam,ub,margmodel) }
}
u<-apply(du,2,cumsum)
z<-qnorm(u)
z[is.nan(z)]<-7
z[z>4&margmodel=="bernoulli"]<-7
z<-rbind(-7,z)
pz<-pnorm(z)
dz<-dnorm(z)
zs<-z*z
zc<-z*zs
zf<-zs*zs
xi<-NULL
if(margmodel=="bernoulli" | margmodel=="poisson")
{ diagonali<-rep(NA,maxd)
} else {
diagonali<-array(NA,c(2,2,maxd)) }
for(j in 1:maxd)
if(margmodel=="nb1" | margmodel=="nb2")
{ temp<-cbind(newx[j,],0)
xi<-cbind(xi,rbind(temp,c(0,1)))
f2<-fisher.gam(newmui[j],gam,invgam,u[,j],ub,margmodel)
f3<-fisher.nu.gam(newmui[j],gam,invgam,u[,j],ub,margmodel)
diagonali[,,j]<-matrix(c(f1,f3,f3,f2),2,2)
}
else
{ temp<-newx[j,]
xi<-cbind(xi,temp)
diagonali[j]<-f1
}
}
if(margmodel=="bernoulli" | margmodel=="poisson")
{ deltai<-diag(diagonali)
offi<-rep(NA,dim(pairs)[1])
} else {
deltai<-matrix(0,dom,dom)
minus<-0
for(j in pos)
{ deltai[j:(j+1),j:(j+1)]<-diagonali[,,j-minus]
minus<-minus+1
}
offi<-array(NA,c(2,2,dim(pairs)[1]))
}
for(k in 1:dim(pairs)[1])
{ k1<-pairs[k,][1]
k2<-pairs[k,][2]
if((sum(k1==ti)==1) & (sum(k2==ti)==1))
{ x1=z[,k1]; x2=z[,k2]
x1s=zs[,k1]; x2s=zs[,k2]
x1c=zc[,k1]; x1f=zf[,k1]
x2c=zc[,k2]; x2f=zf[,k2]
t1=outer(pz[,k1],pz[,k2])
t2=outer(dz[,k1],dz[,k2])
if(corstr=="exch")
{ cdf<-approxbvncdf(rh,x1,x2,x1s,x2s,x1c,x2c,x1f,x2f,t1,t2) }
else
{ if(corstr=="ar")
{ cdf<-approxbvncdf(rh^(k2-k1),x1,x2,x1s,x2s,x1c,x2c,x1f,x2f,t1,t2) }
else
{ # corstr=="unstr"
cdf<-approxbvncdf(rh[k],x1,x2,x1s,x2s,x1c,x2c,x1f,x2f,t1,t2)
}}
cdf1=apply(cdf,2,diff)
pmf=apply(t(cdf1),2,diff)
pmf=t(pmf)
if(margmodel=="bernoulli" | margmodel=="poisson")
{offi[k]<-scoreCov(scnu,scgam,pmf,pairs[k,],margmodel)}
else {offi[,,k]<-scoreCov(scnu,scgam,pmf,pairs[k,],margmodel)}
}
}
omegai<-deltai
if(margmodel=="bernoulli" | margmodel=="poisson")
{ for(j in 1:dim(pairs)[1])
{ omegai[pairs[j,1],pairs[j,2]]<-offi[j]
omegai[pairs[j,2],pairs[j,1]]<-offi[j]
}}
else
{ ch1<-0
ch2<-0
for(j in 1:(maxd-1))
{ for(r in pos[-(1:j)])
{ omegai[(j+ch1):(j+1+ch1),r:(r+1)]<-offi[,,(j+ch2-ch1)]
omegai[r:(r+1),(j+ch1):(j+1+ch1)]<-t(offi[,,(j+ch2-ch1)])
ch2<-ch2+1
}
ch1<-ch1+1
}}
X[,,m]<-xi
delta[,,m]<-deltai
omega[,,m]<-omegai
}
list(omega=omega,X=X,delta=delta)
}

# the weigted scores equations
# input:
# param the vector of regression and not regression parameters
# xdat the matrix of covariates (use the constant 1 for the first covariate)
# ydat the vector with the response
# id the the vector with the id
# tvec the time related vector
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson,
# ?bernoulli? for Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2
# parametrization of negative binomial in Cameron and Trivedi (1998)
# ?logit? for the logit link function, and ?probit? for the probit link function.
# corstr indicates the latent correlation structure of normal copula.
# Choices are ?exch?, ?ar?, and ?unstr? for exchangeable, ar(1) and
# unstrucutred correlation structure, respectively.
# WtScMat is a list containing the following components:
# omega the array with the Omega matrices
# delta the array with the Delta matrices
# X the array with the X matrices
# output
# the weigted scores equations
{ if(margmodel=="nb1" | margmodel=="nb2" | margmodel=="poisson") link="log"
uid<-unique(id)
d<-id.size(id)
maxd<-max(d)
dim<-dim(xdat)
n<-1:dim[1]
p<-dim[2]
tvec<-id.time(tvec,d)
b<-param[1:p]
if(p<length(param)) {gam<-param[p+1]; invgam<-1/gam }
g<-0
m<-0
for(i in uid)
{ cases<-id==i
irow=n[cases]
m<-m+1
x<-xdat[cases,]
ub<-truncation(mu,gam,margmodel)
scnu<-scgam<-matrix(NA,ub+1,d[m])
for(j in 1:d[m])
if(margmodel=="nb1" | margmodel=="nb2")
{ scgam[,j]<-derlik.gam(mu[j],gam,invgam,ub,margmodel) }
}
y<-ydat[irow]
sci<-NULL
for(j in 1:d[m])
{ if(y[j]>ub)
if(margmodel=="nb1" | margmodel=="nb2")
{ scgami<-iderlik.gam(mu[j],gam,invgam,y[j],margmodel)
} else {
scgami<-NULL}
}
else {
scnui<-scnu[y[j]+1,j]
if(margmodel=="nb1" | margmodel=="nb2")
{ scgami<-scgam[y[j]+1,j]
} else {
scgami<-NULL}}
sci<-c(sci,c(scnui,scgami))
}
ti<-tvec[irow]
seli<-subselect(ti,margmodel)
Xi<-WtScMat\$X[,,m]
Xi<-Xi[,seli]
deltai<-WtScMat\$delta[,,m]
deltai<-deltai[seli,seli]
omegai<-WtScMat\$omega[,,m]
omegai<-omegai[seli,seli]
gi<-Xi%*%t(deltai)%*%solve(omegai,sci)
g<-g+gi
}
g
}

# solving the weigted scores equations
# input:
# start the starting values (IEE estimates) for the vector of
# regression and not regression parameters
# xdat the matrix of covariates (use the constant 1 for the first covariate)
# ydat the vector with the response
# id the the vector with the id
# tvec the time related vector
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson,
# ?bernoulli? for Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2
# parametrization of negative binomial in Cameron and Trivedi (1998)
# ?logit? for the logit link function, and ?probit? for the probit link function.
# corstr indicates the latent correlation structure of normal copula.
# Choices are ?exch?, ?ar?, and ?unstr? for exchangeable, ar(1) and
# unstrucutred correlation structure, respectively.
# WtScMat is a list containing the following components:
# omega the array with the Omega matrices
# delta the array with the Delta matrices
# X the array with the X matrices
# output:
# the weighted scores estimates
{ multiroot(f=wtsc,start,atol=1e-4,rtol=1e-4,ctol=1e-4,

# inverse Godambe matrix with Delta and Omega evaluated at IEE estimator
# input:
# b the regression coefficients
# gam the gamma parameter
# xdat the matrix of covariates (use the constant 1 for the first covariate)
# ydat the vector with the response
# id the the vector with the id
# tvec the time related vector
# rh the vector with CL1 estimates
# WtScMat a list containing the following components:
# omega the array with the Omega matrices
# delta the array with the Delta matrices
# X the array with the X matrices
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson,
# ?bernoulli? for Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2
# parametrization of negative binomial in Cameron and Trivedi (1998)
# ?logit? for the logit link function, and ?probit? for the probit link function.
# output:
# the inverse Godambe matrix
{ if(margmodel=="nb1" | margmodel=="nb2" | margmodel=="poisson") link="log"
uid<-unique(id)
d<-id.size(id)
maxd<-max(d)
dim<-dim(xdat)
n<-1:dim[1]
p<-dim[2]
tvec<-id.time(tvec,d)
b<-param[1:p]
if(margmodel=="nb1" | margmodel=="nb2")
{ gam<-param[p+1]
invgam<-1/gam
} else {
gam<-invgam<-NULL}
v<-v1<-v2<-v3<-0
fv<-fv1<-fv2<-fv3<-0
m<-0
for(i in uid)
{ m<-m+1
cases<-id==i
irow=n[cases]
newx<-matrix(NA,max(d),length(b))
newy<-rep(NA,maxd)
newmui<-rep(NA,maxd)
ti<-tvec[irow]
x<-xdat[cases,]
newx[ti,]<-x
newmui[ti]<-mui
ub<-truncation(newmui,gam,margmodel)
scnu<-scgam<-matrix(NA,1+ub,maxd)
for(j in ti)
if(margmodel=="nb1" | margmodel=="nb2")
{ scgam[,j]<-derlik.gam(newmui[j],gam,invgam,ub,margmodel) }
}
y<-ydat[irow]
newy[ti]<-y
sci<-NULL
for(j in 1:maxd)
{ if(sum(newy[j]>ub,na.rm=T)==1)
if(margmodel=="nb1" | margmodel=="nb2")
{ scgami<-iderlik.gam(newmui[j],gam,invgam,newy[j],margmodel)
} else {
scgami<-NULL}
}
else {
scnui<-scnu[newy[j]+1,j]
if(margmodel=="nb1" | margmodel=="nb2")
{ scgami<-scgam[newy[j]+1,j]
} else {
scgami<-NULL}}
sci<-c(sci,c(scnui,scgami))
}
sci<-sci[!is.na(sci)]
seli<-subselect(ti,margmodel)
Xi<-WtScMat\$X[,,m]
Xi<-Xi[,seli]
fdeltai<-WtScMat\$delta[,,m]
fdeltai<-fdeltai[seli,seli]
fomegai<-WtScMat\$omega[,,m]
fomegai<-fomegai[seli,seli]
fci<-fdeltai%*%t(Xi)
nrhs1=ncol(fdeltai)
nrhs2=ncol(fci)
tem.lin=solve(fomegai,cbind(fdeltai,fci))
finvai=t(tem.lin[,1:nrhs1])
xi.invai=Xi %*% finvai
fai<-solve(finvai)
tem=solve(t(fai),t(Xi))
fv1i<-xi.invai %*% fci
fv2i<-xi.invai %*% (sci%*% t(sci))  %*% tem
fv3i<-t(fci) %*% tem
fv1<-fv1+fv1i
fv2<-fv2+fv2i
fv3<-fv3+fv3i
}
solve(fv1,fv2)%*%solve(fv3)
}

# the weighted scores wrapper function: handles all the steps in the weighted scores
# xdat the matrix of covariates (use the constant 1 for the first covariate)
# ydat the vector with the response
# id the the vector with the id
# tvec the time related vector
# rh the vector with CL1 estimates
# WtScMat a list containing the following components:
# omega the array with the Omega matrices
# delta the array with the Delta matrices
# X the array with the X matrices
# margmodel indicates the marginal model. Choices are ?poisson? for Poisson,
# ?bernoulli? for Bernoulli, and ?nb1? , ?nb2? for the NB1 and NB2
# parametrization of negative binomial in Cameron and Trivedi (1998)
# corstr indicates the latent correlation structure of normal copula.
# Choices are ?exch?, ?ar?, and ?unstr? for exchangeable, ar(1) and
# unstrucutred correlation structure, respectively.
# ?logit? for the logit link function, and ?probit? for the probit link function.
# iprint indicates printing of some intermediate results, default FALSE
{ if(margmodel=="nb1" | margmodel=="nb2" | margmodel=="poisson") link="log"
if(iprint)
{ cat("\niest: IEE estimates\n")
print(c(i.est\$reg,i.est\$gam))
}
if(iprint)
{ cat("\nest.rho: CL1 estimates\n")
print(est.rho\$e)
cat("\nest.rho: CL1 likelihood\n")
print(-est.rho\$m)
}
WtScMat<-weightMat(b=i.est\$reg,gam=i.est\$gam,rh=est.rho\$e,
ws<-solvewtsc(start=c(i.est\$reg,i.est\$gam),WtScMat,xdat,ydat,id,
if(iprint)
{ cat("ws=parameter estimates\n")
print(ws\$r)
}
se<-sqrt(diag(acov))
if(iprint)
{ cat("\nacov: inverse Godambe matrix with W based on first-stage wt matrices\n")
print(acov)
cat("\nse: robust standard errors\n")
print(se)
res<-round(cbind(ws\$r,se),3)
cat("\nres: Weighted scores estimates and standard errors\n")
print(res)
}
list(IEEest=c(i.est\$reg,i.est\$gam),CL1est=est.rho\$e,CL1lik=-est.rho\$m,WSest=ws\$r, asympcov=acov)
}
```

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weightedScores documentation built on March 24, 2020, 1:07 a.m.