R/markPH.aipw.R
In fei-heng/cmprskPH: Cause-specific proportional hazards model for competing risks analysis
Defines functions
print.markPH.aipw
markPH.aipw
Documented in
markPH.aipw
#' AIPW Estimation and Hypothesis Testing of Strain-Specific
#' Vaccine Efficacy Adjusted for Covariate Effects with Missing Causes
#'
#' Fit a stratified cause-specific proportional hazards regression model
#' postulates that the conditional cause-specific hazard function
#' for cause j for an individual in the k-th stratum with the covariate value
#' \eqn{z} equals
#' \deqn{ \lambda_{kj}(t|z)=\lambda_{0kj}(t)\exp(\beta_j^{T}z) }
#' for j=1,...,J and k=1, ..., K, where \eqn{\lambda_{0kj}(\cdot)}
#' is an unspecified cause-specific baseline hazard function
#' for the k-th stratum and K is the number of strata.
#'
#' @param cmprskPHformula a formula object with the response on the left of a '~'
#' operator, and the independent terms on the right as covariates in the
#' cause-specific proportional hazards regression model. The response
#' must be in the format of cbind(time, delta, cause), where time is the minimum
#' of the failure time and the censoring time, delta specifies the censoring status
#' (1: failure is observed; 0: right-censored), cause is the cause of failure.
#' If cause is missing, cause=NA.
#' @param trtpos the position of the treatment group indicator in the RHS of
#' the formula object cmprskPHformula. The default value is 1.
#' @param strata the column name of the strata variable in the data.
#' @param causelevels types of causes.
#' @param missformula a formula object started with a '~' operator, and the independent
#' terms on the right of '~' are variables used for predicting the probability
#' of being not missing under a logistic regression model.
#' @param markformula a formula object started with a '~' operator, and the independent
#' terms on the right of '~' are variables used for predicting the probability
#' of being cause j under a multinomial log-linear regression model (function
#' multinom() in nnet package).
#' @param data a data.frame with the variables.
#' @param VEnull the assumed VE value in the null hypothesis. The default value
#' is 0.3.
#' @param maxit Maximum number of iterations to attempt for convergence. The
#' default is 15.
#' @param ipw Whether to conduct the IPW estimation. The default value is TRUE.
#' @param cc Whether to conduct the complete-case estimation. The default value is TRUE.
#' @param lambda0 Whether to estimate the cumulative baseline hazard. The default value is TRUE.
#'
#' @return returns an object of type 'markPH.aipw'. With the following arguments:
#' \item{causes}{the types of causes of failure}
#' \item{coef.cc}{estimates of covariate coefficients using a complete-case analysis}
#' \item{se.cc}{estimates of standard errors of estimators for covariate coefficients
#' using a complete-case analysis}
#' \item{coef.ipw}{estimates of covariate coefficients using IPW method}
#' \item{se.ipw}{estimates of standard errors of estimators for covariate coefficients
#' using IPW method}
#' \item{coef}{estimates of covariate coefficients using AIPW method}
#' \item{se}{estimates of standard errors of estimators for covariate coefficients
#' using AIPW method}
#' \item{coef.VE}{estimates of the strain-specific vaccine efficacy to reduce
#' susceptibility to strain j using AIPW method, j=1,...,J}
#' \item{se.VE}{estimates of standard errors of estimators for the strain-specific
#' vaccine efficacy to reduce susceptibility to strain j using AIPW method, j=1,...,J}
#' \item{coef.VD}{estimates of VD using AIPW method}
#' \item{se.VD}{estimates of standard errors of estimators for VD using AIPW method}
#' \item{U1}{test statistic for testing against the alternative hypothesis
#' HA1: VE(j)>=VEnull with strict inequality for some 1<=j<=J}
#' \item{U2}{test statistic for testing against the alternative hypothesis
#' HA2: VE(j) is not equal to VEnull for some 1<=j<=J}
#' \item{U1j}{test statistic for testing against the alternative hypothesis
#' HAj1: VE(j)>VEnull}
#' \item{U2j}{test statistic for testing against the alternative hypothesis
#' HAj2: VE(j) is not equal to VEnull}
#' \item{T1}{test statistic for testing against the alternative hypothesis
#' HB1: VE(1)>=...>=VE(J) with at least one strict inequality}
#' \item{T2}{test statistic for testing against the alternative hypothesis
#' HB2: VE(i) is not equal to VE(j) for at least one pair of i and j, 1<=i<j<=J}
#' \item{pval.A1}{p value for testing against the alternative hypothesis
#' HA1: VE(j)>=VEnull with strict inequality for some 1<=j<=J}
#' \item{pval.A2}{p value for testing against the alternative hypothesis
#' HA2: VE(j) is not equal to VEnull for some 1<=j<=J}
#' \item{pval.A1j}{p value for testing against the alternative hypothesis
#' HAj1: VE(j)>VEnull}
#' \item{pval.A2j}{p value for testing against the alternative hypothesis
#' HAj2: VE(j) is not equal to VEnull}
#' \item{pval.B1}{p value for testing against the alternative hypothesis
#' HB1: VE(1)>=...>=VE(J) with at least one strict inequality}
#' \item{pval.B2}{p value for testing against the alternative hypothesis
#' HB2: VE(i) is not equal to VE(j) for at least one pair of i and j, 1<=i<j<=J}
#' \item{tgrid}{specifies the time grids for estimating \eqn{\lambda_{0kj}(\cdot)}
#' , the unspecified cause-specific baseline hazard function
#' for cause j and the k-th stratum.}
#' \item{lambda0}{estimates of \eqn{\lambda_{0kj}(\cdot)} using AIPW method}
#' \item{covariates}{covariates in the cause-specific proportional hazards model.}
#' \item{trtpos}{the position of the treatment group indicator in the RHS of
#' the formula object cmprskPHformula.}
#' \item{VEnull}{the assumed VE value in the null hypothesis.}
#' \item{diffsigma}{estimates of standard deviation of \eqn{\hat\alpha_j-\hat\alpha_{j-1}}, j=2,...,J.}
#' \item{cov.alpha}{estimated covariance matrix of \eqn{(\hat\alpha_1,\dots,\hat\alpha_J)}.}
#' \item{pval.C}{p values for testing pairwise multiple comparisons of VE.}
#'
#' @author Fei Heng
#'
#' @references (2021+)
#'
#' @examples
#'
#' ## Example 1: Simulated competing risks data with 2 causes subject to missingness
#' data(sim2cs)
#'
#' res.aipw <- markPH.aipw(cmprskPHformula=cbind(time,delta,cause)~z1+z2,
#' trtpos=1,
#' strata="strata",
#' causelevels=c(1,2),
#' missformula=~z1+A,
#' markformula=~time+z1+A,
#' data=sim2cs,
#' VEnull=0.3,
#' maxit=15)
#' res.aipw # print the result
#'
#' ## Example 2: Simulated competing risks data with 3 causes subject to missingness
#' data(sim3cs)
#'
#' res.aipw <- markPH.aipw(cmprskPHformula=cbind(time,delta,cause)~z1+z2,
#' trtpos=1,
#' strata="strata",
#' causelevels=c(1,2,3),
#' missformula=~z1+A,
#' markformula=~time+z1+A,
#' data=sim3cs,
#' VEnull=0.3,
#' maxit=15)
#' res.aipw # print the result
#'
#' @import MASS nnet caret
#'
#' @export
markPH.aipw <- function(cmprskPHformula,
trtpos=1,
strata,
causelevels=NULL,
missformula,
markformula,
data=parent.frame(),
VEnull=0.3,
maxit=15,
ipw=T,
cc=T,
lambda0=T){
# next version:
# ...
data[data=="NaN"] <- NA
a <- model.frame(cmprskPHformula, data=data, na.action = na.pass)
time <- model.response(a)[,1]
delta <- model.response(a)[,2] # 0: censored, 1: observed failure
if (is.null(causelevels)){
cause.fa <- factor(model.response(a)[,3])
} else {
# define causes following the order of causelevels
cause.fa <- factor(model.response(a)[,3], labels=causelevels)
}
causelevels <- levels(cause.fa)
cause <- as.numeric(cause.fa) # cause = 1,2,...,J
cause[is.na(cause.fa)] <- NA
data$cause <- cause
covar2 <- as.matrix(model.matrix(a, data=a, na.action = na.pass)[,-1])
#A <- data[,Aux]
strata.fa <- factor(data[,strata])
strata.num <- as.numeric(strata.fa)
R <- (delta==0)|(!is.na(cause))
data$R <- R # cen.code=0
nstrt <- length(levels(strata.fa))# number of strata
ncs <- length(unique(cause[!is.na(cause)]))# number of causes
ncov <- ncol(covar2)# number of covariates
nsamp <- length(time)
# set grid points for lambda0
time0 <- min(time)
tau <- max(time)
ngrid <- 50 # number of grids (hard coding)
tgrid <- seq(time0,tau,length.out=ngrid)
b <- (tau-time0)/5 # bandwidth for lambda_0(t) (hard coding)
missformula <- formula(paste(c("R",missformula),collapse = " "))
markformula <- formula(paste(c("cause",markformula),collapse = " "))
## predict P(R=1) and P(cause=k) in jj-th stratum
# predict P(R=1) (in R, use subset=)
# logistic regression
wipw <- R
a <- model.frame(missformula, data=data, na.action = na.pass)
covar.miss <- as.matrix(model.matrix(a, data=a, na.action = na.pass)[,-1])
colnames(covar.miss) <- colnames(a)[-1]
npsi <- ncol(covar.miss)+1
dr <- matrix(0,nsamp,npsi)
Ipsi <- array(0, c(npsi,npsi,nstrt))
Spsi <- matrix(0,nsamp,npsi)
for (jj in 1:nstrt){
temp <- which((delta==1)&(strata.num==jj))
miss.res <- glm(missformula, data=data, family='binomial',subset=temp)
newdata <- as.data.frame(covar.miss[temp,])
colnames(newdata) <- colnames(a)[-1]
rhat <- predict(miss.res, newdata, type="response")
Rjj <- R[temp]
njj <- length(Rjj)
drjj <- matrix(0,njj,npsi)
Spsijj <- matrix(0,njj,npsi)
W <- cbind(rep(1,njj), covar.miss[temp,])
for (ipsi in 1:npsi){
Spsijj[,ipsi] <- (Rjj-rhat)*W[,ipsi]
drjj[,ipsi] <- rhat*(1-rhat)*W[,ipsi]
}
Ipsi[,,jj] <- crossprod(W,drjj)/njj
dr[temp,] <- drjj
Spsi[temp,] <- Spsijj
wipw[temp] <- Rjj/rhat
}
# predict cause/mark V for delta==1
# multinomial logistic regression
# library(nnet)
a <- model.frame(markformula, data=data, na.action = na.pass)
covar.mark <- as.matrix(model.matrix(a, data=a, na.action = na.pass)[,-1])
rhohat <- matrix(0,nsamp,ncs)
colnames(rhohat) <- causelevels
if (ncs==2){
for (jj in 1:nstrt){
# ## method 1
# # multinom() from package nnet
# # Fits multinomial log-linear models via neural networks
# # other options: mlogit() from package mlogit
# temp <- (delta==1)&(strata.num==jj)&R
# mark.res <- multinom(markformula, data=data, subset=temp, trace=F)
# # test for ncs>2!!!
# # newdata <- as.data.frame(covar.mark[(delta==1)&(strata.num==jj),])
# # colnames(newdata) <- colnames(a)[-1]
# newdata <- data[(delta==1)&(strata.num==jj), ]
# rhohat[(delta==1)&(strata.num==jj),2] <- predict(mark.res, newdata, type="probs")
# method 2 knn
proba <- predict(knn3(covar.mark[(delta==1)&(strata.num==jj)&R,],
factor(data$cause[(delta==1)&(strata.num==jj)&R]),
k = 5),
covar.mark[(delta==1)&(strata.num==jj),])
proba.class <- colnames(rhohat)[as.numeric(colnames(proba))]
rhohat[(delta==1)&(strata.num==jj),proba.class] <- proba
}
} else {
for (jj in 1:nstrt){
# method 1
# temp <- (delta==1)&(strata.num==jj)&R
# # multinom() from package nnet
# # Fits multinomial log-linear models via neural networks
# # other options: mlogit() from package mlogit
# mark.res <- multinom(markformula, data=data, subset=temp, trace=F)
# # test for ncs>2!!!
# newdata <- as.data.frame(covar.mark[(delta==1)&(strata.num==jj),])
# colnames(newdata) <- colnames(a)[-1]
# rhohat[(delta==1)&(strata.num==jj), mark.res$lab] <- predict(mark.res, newdata, type="probs")
# method 2 knn
proba <- predict(knn3(covar.mark[(delta==1)&(strata.num==jj)&R,],
factor(data$cause[(delta==1)&(strata.num==jj)&R]),
k = 5),
covar.mark[(delta==1)&(strata.num==jj),])
proba.class <- colnames(rhohat)[as.numeric(colnames(proba))]
rhohat[(delta==1)&(strata.num==jj),proba.class] <- proba
}
}
## estimation for each cause
# initialization
sbeta_c <- matrix(NA, ncov, ncs)
sstd_c <- matrix(NA, ncov, ncs)
sbeta_ic <- matrix(NA, ncov, ncs)
sstd_ic <- matrix(NA, ncov, ncs)
sbeta_acc <- matrix(NA, ncov, ncs)
sstd_acc <- matrix(NA, ncov, ncs)
sVE_acc <- rep(NA,ncs)
sVEstd_acc <- rep(NA,ncs)
sVD_acc <- rep(NA,ncs-1)
sVDstd_acc <- rep(NA,ncs-1)
for (ics in 1:ncs){
# deltacs: indicator of non-censoring for each cause
deltacs <- cause == ics# be carefule: NaN ->F
deltacs[is.na(cause)] <- F
# complete estimation
if (cc==T){
beta0 <- rep(0,ncov)
res.cc <- estf(time,covar2,deltacs,beta0,strata.num,maxit,subset=R)
# complete result
sbeta_c[,ics] <- res.cc$est
sstd_c[,ics] <- res.cc$se
}
# ipw estimation
if (ipw==T){
beta0 <- rep(0,ncov)
res.ipw <- esti(time,covar2,deltacs,beta0,wipw,strata.num,maxit,nstrt,dr,Ipsi,Spsi)
# ipw-c result
sbeta_ic[,ics] <- res.ipw$est
sstd_ic[,ics] <- res.ipw$se
}
# aipw estimation
beta0 <- rep(0,ncov)
res.aipw <- esta(time,covar2,deltacs,beta0,wipw,rhohat[,ics],delta,strata.num,maxit)
# aipw_cc result
sbeta_acc[,ics] <- res.aipw$est
sstd_acc[,ics] <- res.aipw$se
}
## calculate the cumulative baseline function
# aipw
# sLamtA0_c <- array(0, c(ngrid,nstrt,ncs))
slamtA0_c <- array(0, c(ngrid,nstrt,ncs))
# slamtA0_c_bd <- array(0, c(ngrid,nstrt,ncs)) boundary correction
if (lambda0==T){
for (ics in 1:ncs){
deltacs <- cause == ics# be carefule: NaN ->F
deltacs[is.na(cause)] <- F
ebz <- exp(covar2%*%sbeta_acc[,ics])
Sf_acc <- rep(0, nsamp)
for (i in 1:nsamp){
if (delta[i]){
Sf_acc[i] <- sum(ebz*(time >= time[i])*(strata.num==strata.num[i]))
}
}
for (jj in 1:nstrt){
for (it in 1:ngrid){
tvalue <- tgrid[it]
wt <- (wipw*deltacs+(1-wipw)*rhohat[,ics])*delta
# temp=which((wt!=0)&(Sf_acc!=0)&(time<tvalue)&(strata.num==jj))
# sLamtA0_c[it,jj,ics] <- sum(wt[temp]/Sf_acc[temp])
temp=which((wt!=0)&(Sf_acc!=0)&(strata.num==jj))
slamtA0_c[it,jj,ics] <- sum(epanker(tvalue, time[temp], b)*wt[temp]/Sf_acc[temp])
# ttstep <- 0.001
# ttgrid <- seq(time0,tau,by=ttstep)
# slamtA0_c_bd[it,jj,ics] <- sum(epanker(tvalue, time[temp], b)*wt[temp]/Sf_acc[temp])/sum(epanker(tvalue, ttgrid, b)*ttstep)
}
}
}
}
## hypothesis tests for causes with non-NA estimates
icov <- trtpos# test the treatment effect (trtpos)
U1 <- U2 <- T1 <- T2 <- pval.A1 <- pval.A2 <- pval.B1 <- pval.B2 <- NA
U1j <- U2j <- pval.A1j <- pval.A2j <- rep(NA, ncs)
diffsigma <- rep(NA, ncs)
# causes <- levels(cause.fa)
causes <- 1:ncs
nonna <- !is.na(sbeta_acc[icov,])
causes.na <- causes[!nonna]
id.test <- !(cause%in%causes.na)
time.test <- time[id.test]
covar2.test <- covar2[id.test,,drop=F]
cause.test <- cause[id.test]
sbeta_acc.test <- sbeta_acc[,nonna,drop=F]
wipw.test <- wipw[id.test]
rhohat.test <- rhohat[id.test,nonna]
delta.test <- delta[id.test]
strata.num.test <- strata.num[id.test]
ncs.test <- sum(nonna)
alphanull <- rep(log(1-VEnull),ncs.test)
cov_acc <- cova(time.test,covar2.test,cause.test,sbeta_acc.test,wipw.test,rhohat.test,delta.test,strata.num.test,maxit,causes[nonna])
alpha <- sbeta_acc[icov,nonna]
cov_alpha <- matrix(NA, ncs.test, ncs.test)
for (ics in 1:ncs.test){
for (jcs in 1:ncs.test){
cov_alpha[ics,jcs] <- cov_acc[(ics-1)*ncov+icov,(jcs-1)*ncov+icov]
}
}
sigma <- sstd_acc[icov,nonna]
# approximate the distribution of test statistics
D <- 1000
tuta <- mvrnorm(D,rep(0,ncs.test),cov_alpha)
U_tuta <- tuta%*%diag(1/sigma)
U1_tuta <- apply(U_tuta,1,FUN = min)
U2_tuta <- rowSums(U_tuta^2)
# H0: VE(j) <= VEnull
# HA1 \alpha_j <= c0
U1 <- min((alpha-alphanull)/sigma)
pval.A1 <- mean(U1_tuta<U1)
# H0: VE(j) = VEnull
# HA2 \alpha_j \ne c0
U2 <- sum(((alpha-alphanull)/sigma)^2)
pval.A2 <- mean(U2_tuta>U2)
# H0: VE(0) <= VEnull
# H0: VE(1) <= VEnull
# HA1 HA2 for each cause j
# HA1 \alpha_j <= c0
U1j <- (alpha-alphanull)/sigma
pval.A1j <- colMeans(U_tuta<matrix(rep(U1j,D),nrow=D,byrow=T))
# H0: VE(0) = VEnull
# H0: VE(1) = VEnull
# HA2 \alpha_j \ne c0
U2j <- ((alpha-alphanull)/sigma)^2
pval.A2j <- colMeans(U_tuta^2>matrix(rep(U2j,D),nrow=D,byrow=T))
# H0: VE(0) <= VE(1)
# HB1 \alpha_1 <=...<= \alpha_J
# w1=wJ+1=0, wj=1 for j=2,...,J-1
w <- c(0,rep(1, ncs.test-1),0)
omega <- matrix(0, ncs.test, ncs.test-1)
for (ics in 2:ncs.test){
omega[ics-1,ics-1] <- -w[ics]
omega[ics,ics-1] <- w[ics]
}
diffsigma <- sqrt(diag(t(omega)%*%cov_alpha%*%omega))
T1 <- min(alpha%*%omega/diffsigma)
if (ncs.test>2){
T_tuta <- tuta%*%omega%*%diag(1/diffsigma)
} else {
T_tuta <- tuta%*%omega/diffsigma
}
T1_tuta <- apply(T_tuta,1,FUN = min)
pval.B1 <- mean(T1_tuta>T1)
# H0: VE(0) = VE(1)
# HB2 \alpha_i \ne \alpha_j, i<j
T2 <- sum((alpha%*%omega/diffsigma)^2)
T2_tuta <- rowSums(T_tuta^2)
pval.B2 <- mean(T2_tuta>T2)
## Point estimate and 95# CI for VE
sVE_acc <- 1-exp(alpha)
sVEstd_acc <- sigma*exp(alpha)
# VD=(1-VE(j)) / (1-VE(j-1))
sVD_acc <- as.vector(exp(alpha%*%omega))
sVDstd_acc <- as.vector(diffsigma*exp(alpha%*%omega))
## added by Fei - 5/22/2023
# H0: VE(i) = VE(j)
# HC: VE(i) \ne VE(j)
pval.C <- matrix(0, nrow=ncs.test, ncol=ncs.test)
for (ics in 1:(ncs.test-1)){
for(jcs in (ics+1):ncs.test){
T2ij <- (alpha[ics]-alpha[jcs])^2/(cov_alpha[ics,ics]-2*cov_alpha[ics,jcs]+cov_alpha[jcs,jcs])
T2ij_tuta <- (tuta[,ics]-tuta[,jcs])^2/(cov_alpha[ics,ics]-2*cov_alpha[ics,jcs]+cov_alpha[jcs,jcs])
pval.C[ics, jcs] <- mean(T2ij_tuta>T2ij)
}
}
res <- list(causes=causes[nonna],
causes.na=causes.na,
coef.cc=sbeta_c[,nonna,drop=F],
se.cc=sstd_c[,nonna,drop=F],
coef.ipw=sbeta_ic[,nonna,drop=F],
se.ipw=sstd_ic[,nonna,drop=F],
coef=sbeta_acc[,nonna,drop=F],
se=sstd_acc[,nonna,drop=F],
coef.VE=sVE_acc,
se.VE=sVEstd_acc,
coef.VD=sVD_acc,
se.VD=sVDstd_acc,
U1=U1,
U2=U2,
U1j=U1j,
U2j=U2j,
T1=T1,
T2=T2,
pval.A1=pval.A1,
pval.A2=pval.A2,
pval.A1j=pval.A1j,
pval.A2j=pval.A2j,
pval.B1=pval.B1,
pval.B2=pval.B2,
tgrid=tgrid,
lambda0=slamtA0_c[,,nonna,drop=F],
covariates=colnames(covar2),
trtpos=trtpos,
VEnull=VEnull,
diffsigma=diffsigma,
cov.alpha=cov_alpha,
pval.C=pval.C)
class(res) <- "markPH.aipw"
return(res)
}
##' @export
print.markPH.aipw <- function(x, digits=4,...){
ncs <- ncol(x$coef)
ncov <- nrow(x$coef)
cat("\n*********************************************************************\n")
cat("Causes {", x$causes.na, "} with NA estimates are ignored in the results\n")
cat("*********************************************************************\n\n")
cat("Table 1: estimates of covaraite coefficients\n")
for (ics in 1:ncs){
cat("Cause:", x$causes[ics], "\n")
table1 <- cbind(x$coef[,ics],
x$se[,ics],
pnorm(-abs(x$coef[,ics]/x$se[,ics]))*2,
x$coef.ipw[,ics],
x$se.ipw[,ics],
pnorm(-abs(x$coef.ipw[,ics]/x$se.ipw[,ics]))*2,
x$coef.cc[,ics],
x$se.cc[,ics],
pnorm(-abs(x$coef.cc[,ics]/x$se.cc[,ics]))*2)
colnames(table1) <- c("coef",
"se",
"pval",
"coef.ipw",
"se.ipw",
"pval.ipw",
"coef.cc",
"se.cc",
"pval.cc")
rownames(table1) <- x$covariates
print.default(table1, digits=digits)
cat("\n")
}
cat("Table 2: results for VE\n")
table2 <- cbind(x$coef.VE,x$se.VE,
1-(1-x$coef.VE)*exp(qnorm(0.975)*x$se[x$trtpos,]),
1-(1-x$coef.VE)*exp(-qnorm(0.975)*x$se[x$trtpos,]),
x$U1j,
x$pval.A1j,
x$U2j,
x$pval.A2j)
colnames(table2) <- c("est",
"se",
"95% LL",
"95% UL",
"U1j",
"p-value for HAj1",
"U2j",
"p-value for HAj2")
rownames(table2) <- paste0("VE(",x$causes,")")
print.default(table2, digits=digits)
cat(paste0("HAj1: VE(j)>",x$VEnull),"\n")
cat(paste0("HAj2: VE(j) not equal to ",x$VEnull),"\n")
cat("\n")
cat("Table 3: results for VD\n")
cat("VD(j,k)=(1-VE(j))/(1-VE(k))\n")
table3 <- rbind(cbind(x$coef.VD,x$se.VD,x$coef.VD*exp(-qnorm(0.975)*x$diffsigma),x$coef.VD*exp(qnorm(0.975)*x$diffsigma)),cbind(1/x$coef.VD,x$se.VD/x$coef.VD^2,1/x$coef.VD*exp(-qnorm(0.975)*x$diffsigma),1/x$coef.VD*exp(qnorm(0.975)*x$diffsigma)))
colnames(table3) <- c("est",
"se",
"95% LL",
"95% UL")
rnames <- rep(0, (ncs-1)*2)
for (ics in 2:ncs){
rnames[ics-1] <- paste0("VD(",x$causes[ics],",",x$causes[ics-1],")")
rnames[ncs-1+ics-1] <- paste0("VD(",x$causes[ics-1],",",x$causes[ics],")")
}
rownames(table3) <- rnames
print.default(table3, digits=digits)
cat("\n")
cat("Table 4: results for hypothesis tests\n")
cat("HA1: VE(j)>=", x$VEnull, "with strict inequality for some j\n")
cat("U1=",x$U1,"p-value=", x$pval.A1, "\n")
cat("HA2: VE(j) not equal to", x$VEnull, "for some j\n")
cat("U2=",x$U2,"p-value=", x$pval.A2, "\n")
cat("HB1: VE(1)>=...>=VE(J) with at least one strict inequality\n")
cat("T1=",x$T1,"p-value=", x$pval.B1, "\n")
cat("HB2: VE(i) not equal to VE(j) for at least one pair of i and j, 1<=i<j<=J\n")
cat("T2=",x$T2,"p-value=", x$pval.B2, "\n")
cat("\n")
}
fei-heng/cmprskPH documentation built on May 24, 2023, 3:54 p.m.
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Defines functions print.markPH.aipw markPH.aipw
Documented in markPH.aipw
#' AIPW Estimation and Hypothesis Testing of Strain-Specific
#' Vaccine Efficacy Adjusted for Covariate Effects with Missing Causes
#'
#' Fit a stratified cause-specific proportional hazards regression model
#' postulates that the conditional cause-specific hazard function
#' for cause j for an individual in the k-th stratum with the covariate value
#' \eqn{z} equals
#' \deqn{ \lambda_{kj}(t|z)=\lambda_{0kj}(t)\exp(\beta_j^{T}z) }
#' for j=1,...,J and k=1, ..., K, where \eqn{\lambda_{0kj}(\cdot)}
#' is an unspecified cause-specific baseline hazard function
#' for the k-th stratum and K is the number of strata.
#'
#' @param cmprskPHformula a formula object with the response on the left of a '~'
#' operator, and the independent terms on the right as covariates in the
#' cause-specific proportional hazards regression model. The response
#' must be in the format of cbind(time, delta, cause), where time is the minimum
#' of the failure time and the censoring time, delta specifies the censoring status
#' (1: failure is observed; 0: right-censored), cause is the cause of failure.
#' If cause is missing, cause=NA.
#' @param trtpos the position of the treatment group indicator in the RHS of
#' the formula object cmprskPHformula. The default value is 1.
#' @param strata the column name of the strata variable in the data.
#' @param causelevels types of causes.
#' @param missformula a formula object started with a '~' operator, and the independent
#' terms on the right of '~' are variables used for predicting the probability
#' of being not missing under a logistic regression model.
#' @param markformula a formula object started with a '~' operator, and the independent
#' terms on the right of '~' are variables used for predicting the probability
#' of being cause j under a multinomial log-linear regression model (function
#' multinom() in nnet package).
#' @param data a data.frame with the variables.
#' @param VEnull the assumed VE value in the null hypothesis. The default value
#' is 0.3.
#' @param maxit Maximum number of iterations to attempt for convergence. The
#' default is 15.
#' @param ipw Whether to conduct the IPW estimation. The default value is TRUE.
#' @param cc Whether to conduct the complete-case estimation. The default value is TRUE.
#' @param lambda0 Whether to estimate the cumulative baseline hazard. The default value is TRUE.
#'
#' @return returns an object of type 'markPH.aipw'. With the following arguments:
#' \item{causes}{the types of causes of failure}
#' \item{coef.cc}{estimates of covariate coefficients using a complete-case analysis}
#' \item{se.cc}{estimates of standard errors of estimators for covariate coefficients
#' using a complete-case analysis}
#' \item{coef.ipw}{estimates of covariate coefficients using IPW method}
#' \item{se.ipw}{estimates of standard errors of estimators for covariate coefficients
#' using IPW method}
#' \item{coef}{estimates of covariate coefficients using AIPW method}
#' \item{se}{estimates of standard errors of estimators for covariate coefficients
#' using AIPW method}
#' \item{coef.VE}{estimates of the strain-specific vaccine efficacy to reduce
#' susceptibility to strain j using AIPW method, j=1,...,J}
#' \item{se.VE}{estimates of standard errors of estimators for the strain-specific
#' vaccine efficacy to reduce susceptibility to strain j using AIPW method, j=1,...,J}
#' \item{coef.VD}{estimates of VD using AIPW method}
#' \item{se.VD}{estimates of standard errors of estimators for VD using AIPW method}
#' \item{U1}{test statistic for testing against the alternative hypothesis
#' HA1: VE(j)>=VEnull with strict inequality for some 1<=j<=J}
#' \item{U2}{test statistic for testing against the alternative hypothesis
#' HA2: VE(j) is not equal to VEnull for some 1<=j<=J}
#' \item{U1j}{test statistic for testing against the alternative hypothesis
#' HAj1: VE(j)>VEnull}
#' \item{U2j}{test statistic for testing against the alternative hypothesis
#' HAj2: VE(j) is not equal to VEnull}
#' \item{T1}{test statistic for testing against the alternative hypothesis
#' HB1: VE(1)>=...>=VE(J) with at least one strict inequality}
#' \item{T2}{test statistic for testing against the alternative hypothesis
#' HB2: VE(i) is not equal to VE(j) for at least one pair of i and j, 1<=i<j<=J}
#' \item{pval.A1}{p value for testing against the alternative hypothesis
#' HA1: VE(j)>=VEnull with strict inequality for some 1<=j<=J}
#' \item{pval.A2}{p value for testing against the alternative hypothesis
#' HA2: VE(j) is not equal to VEnull for some 1<=j<=J}
#' \item{pval.A1j}{p value for testing against the alternative hypothesis
#' HAj1: VE(j)>VEnull}
#' \item{pval.A2j}{p value for testing against the alternative hypothesis
#' HAj2: VE(j) is not equal to VEnull}
#' \item{pval.B1}{p value for testing against the alternative hypothesis
#' HB1: VE(1)>=...>=VE(J) with at least one strict inequality}
#' \item{pval.B2}{p value for testing against the alternative hypothesis
#' HB2: VE(i) is not equal to VE(j) for at least one pair of i and j, 1<=i<j<=J}
#' \item{tgrid}{specifies the time grids for estimating \eqn{\lambda_{0kj}(\cdot)}
#' , the unspecified cause-specific baseline hazard function
#' for cause j and the k-th stratum.}
#' \item{lambda0}{estimates of \eqn{\lambda_{0kj}(\cdot)} using AIPW method}
#' \item{covariates}{covariates in the cause-specific proportional hazards model.}
#' \item{trtpos}{the position of the treatment group indicator in the RHS of
#' the formula object cmprskPHformula.}
#' \item{VEnull}{the assumed VE value in the null hypothesis.}
#' \item{diffsigma}{estimates of standard deviation of \eqn{\hat\alpha_j-\hat\alpha_{j-1}}, j=2,...,J.}
#' \item{cov.alpha}{estimated covariance matrix of \eqn{(\hat\alpha_1,\dots,\hat\alpha_J)}.}
#' \item{pval.C}{p values for testing pairwise multiple comparisons of VE.}
#'
#' @author Fei Heng
#'
#' @references (2021+)
#'
#' @examples
#'
#' ## Example 1: Simulated competing risks data with 2 causes subject to missingness
#' data(sim2cs)
#'
#' res.aipw <- markPH.aipw(cmprskPHformula=cbind(time,delta,cause)~z1+z2,
#' trtpos=1,
#' strata="strata",
#' causelevels=c(1,2),
#' missformula=~z1+A,
#' markformula=~time+z1+A,
#' data=sim2cs,
#' VEnull=0.3,
#' maxit=15)
#' res.aipw # print the result
#'
#' ## Example 2: Simulated competing risks data with 3 causes subject to missingness
#' data(sim3cs)
#'
#' res.aipw <- markPH.aipw(cmprskPHformula=cbind(time,delta,cause)~z1+z2,
#' trtpos=1,
#' strata="strata",
#' causelevels=c(1,2,3),
#' missformula=~z1+A,
#' markformula=~time+z1+A,
#' data=sim3cs,
#' VEnull=0.3,
#' maxit=15)
#' res.aipw # print the result
#'
#' @import MASS nnet caret
#'
#' @export
markPH.aipw <- function(cmprskPHformula,
trtpos=1,
strata,
causelevels=NULL,
missformula,
markformula,
data=parent.frame(),
VEnull=0.3,
maxit=15,
ipw=T,
cc=T,
lambda0=T){
# next version:
# ...
data[data=="NaN"] <- NA
a <- model.frame(cmprskPHformula, data=data, na.action = na.pass)
time <- model.response(a)[,1]
delta <- model.response(a)[,2] # 0: censored, 1: observed failure
if (is.null(causelevels)){
cause.fa <- factor(model.response(a)[,3])
} else {
# define causes following the order of causelevels
cause.fa <- factor(model.response(a)[,3], labels=causelevels)
}
causelevels <- levels(cause.fa)
cause <- as.numeric(cause.fa) # cause = 1,2,...,J
cause[is.na(cause.fa)] <- NA
data$cause <- cause
covar2 <- as.matrix(model.matrix(a, data=a, na.action = na.pass)[,-1])
#A <- data[,Aux]
strata.fa <- factor(data[,strata])
strata.num <- as.numeric(strata.fa)
R <- (delta==0)|(!is.na(cause))
data$R <- R # cen.code=0
nstrt <- length(levels(strata.fa))# number of strata
ncs <- length(unique(cause[!is.na(cause)]))# number of causes
ncov <- ncol(covar2)# number of covariates
nsamp <- length(time)
# set grid points for lambda0
time0 <- min(time)
tau <- max(time)
ngrid <- 50 # number of grids (hard coding)
tgrid <- seq(time0,tau,length.out=ngrid)
b <- (tau-time0)/5 # bandwidth for lambda_0(t) (hard coding)
missformula <- formula(paste(c("R",missformula),collapse = " "))
markformula <- formula(paste(c("cause",markformula),collapse = " "))
## predict P(R=1) and P(cause=k) in jj-th stratum
# predict P(R=1) (in R, use subset=)
# logistic regression
wipw <- R
a <- model.frame(missformula, data=data, na.action = na.pass)
covar.miss <- as.matrix(model.matrix(a, data=a, na.action = na.pass)[,-1])
colnames(covar.miss) <- colnames(a)[-1]
npsi <- ncol(covar.miss)+1
dr <- matrix(0,nsamp,npsi)
Ipsi <- array(0, c(npsi,npsi,nstrt))
Spsi <- matrix(0,nsamp,npsi)
for (jj in 1:nstrt){
temp <- which((delta==1)&(strata.num==jj))
miss.res <- glm(missformula, data=data, family='binomial',subset=temp)
newdata <- as.data.frame(covar.miss[temp,])
colnames(newdata) <- colnames(a)[-1]
rhat <- predict(miss.res, newdata, type="response")
Rjj <- R[temp]
njj <- length(Rjj)
drjj <- matrix(0,njj,npsi)
Spsijj <- matrix(0,njj,npsi)
W <- cbind(rep(1,njj), covar.miss[temp,])
for (ipsi in 1:npsi){
Spsijj[,ipsi] <- (Rjj-rhat)*W[,ipsi]
drjj[,ipsi] <- rhat*(1-rhat)*W[,ipsi]
}
Ipsi[,,jj] <- crossprod(W,drjj)/njj
dr[temp,] <- drjj
Spsi[temp,] <- Spsijj
wipw[temp] <- Rjj/rhat
}
# predict cause/mark V for delta==1
# multinomial logistic regression
# library(nnet)
a <- model.frame(markformula, data=data, na.action = na.pass)
covar.mark <- as.matrix(model.matrix(a, data=a, na.action = na.pass)[,-1])
rhohat <- matrix(0,nsamp,ncs)
colnames(rhohat) <- causelevels
if (ncs==2){
for (jj in 1:nstrt){
# ## method 1
# # multinom() from package nnet
# # Fits multinomial log-linear models via neural networks
# # other options: mlogit() from package mlogit
# temp <- (delta==1)&(strata.num==jj)&R
# mark.res <- multinom(markformula, data=data, subset=temp, trace=F)
# # test for ncs>2!!!
# # newdata <- as.data.frame(covar.mark[(delta==1)&(strata.num==jj),])
# # colnames(newdata) <- colnames(a)[-1]
# newdata <- data[(delta==1)&(strata.num==jj), ]
# rhohat[(delta==1)&(strata.num==jj),2] <- predict(mark.res, newdata, type="probs")
# method 2 knn
proba <- predict(knn3(covar.mark[(delta==1)&(strata.num==jj)&R,],
factor(data$cause[(delta==1)&(strata.num==jj)&R]),
k = 5),
covar.mark[(delta==1)&(strata.num==jj),])
proba.class <- colnames(rhohat)[as.numeric(colnames(proba))]
rhohat[(delta==1)&(strata.num==jj),proba.class] <- proba
}
} else {
for (jj in 1:nstrt){
# method 1
# temp <- (delta==1)&(strata.num==jj)&R
# # multinom() from package nnet
# # Fits multinomial log-linear models via neural networks
# # other options: mlogit() from package mlogit
# mark.res <- multinom(markformula, data=data, subset=temp, trace=F)
# # test for ncs>2!!!
# newdata <- as.data.frame(covar.mark[(delta==1)&(strata.num==jj),])
# colnames(newdata) <- colnames(a)[-1]
# rhohat[(delta==1)&(strata.num==jj), mark.res$lab] <- predict(mark.res, newdata, type="probs")
# method 2 knn
proba <- predict(knn3(covar.mark[(delta==1)&(strata.num==jj)&R,],
factor(data$cause[(delta==1)&(strata.num==jj)&R]),
k = 5),
covar.mark[(delta==1)&(strata.num==jj),])
proba.class <- colnames(rhohat)[as.numeric(colnames(proba))]
rhohat[(delta==1)&(strata.num==jj),proba.class] <- proba
}
}
## estimation for each cause
# initialization
sbeta_c <- matrix(NA, ncov, ncs)
sstd_c <- matrix(NA, ncov, ncs)
sbeta_ic <- matrix(NA, ncov, ncs)
sstd_ic <- matrix(NA, ncov, ncs)
sbeta_acc <- matrix(NA, ncov, ncs)
sstd_acc <- matrix(NA, ncov, ncs)
sVE_acc <- rep(NA,ncs)
sVEstd_acc <- rep(NA,ncs)
sVD_acc <- rep(NA,ncs-1)
sVDstd_acc <- rep(NA,ncs-1)
for (ics in 1:ncs){
# deltacs: indicator of non-censoring for each cause
deltacs <- cause == ics# be carefule: NaN ->F
deltacs[is.na(cause)] <- F
# complete estimation
if (cc==T){
beta0 <- rep(0,ncov)
res.cc <- estf(time,covar2,deltacs,beta0,strata.num,maxit,subset=R)
# complete result
sbeta_c[,ics] <- res.cc$est
sstd_c[,ics] <- res.cc$se
}
# ipw estimation
if (ipw==T){
beta0 <- rep(0,ncov)
res.ipw <- esti(time,covar2,deltacs,beta0,wipw,strata.num,maxit,nstrt,dr,Ipsi,Spsi)
# ipw-c result
sbeta_ic[,ics] <- res.ipw$est
sstd_ic[,ics] <- res.ipw$se
}
# aipw estimation
beta0 <- rep(0,ncov)
res.aipw <- esta(time,covar2,deltacs,beta0,wipw,rhohat[,ics],delta,strata.num,maxit)
# aipw_cc result
sbeta_acc[,ics] <- res.aipw$est
sstd_acc[,ics] <- res.aipw$se
}
## calculate the cumulative baseline function
# aipw
# sLamtA0_c <- array(0, c(ngrid,nstrt,ncs))
slamtA0_c <- array(0, c(ngrid,nstrt,ncs))
# slamtA0_c_bd <- array(0, c(ngrid,nstrt,ncs)) boundary correction
if (lambda0==T){
for (ics in 1:ncs){
deltacs <- cause == ics# be carefule: NaN ->F
deltacs[is.na(cause)] <- F
ebz <- exp(covar2%*%sbeta_acc[,ics])
Sf_acc <- rep(0, nsamp)
for (i in 1:nsamp){
if (delta[i]){
Sf_acc[i] <- sum(ebz*(time >= time[i])*(strata.num==strata.num[i]))
}
}
for (jj in 1:nstrt){
for (it in 1:ngrid){
tvalue <- tgrid[it]
wt <- (wipw*deltacs+(1-wipw)*rhohat[,ics])*delta
# temp=which((wt!=0)&(Sf_acc!=0)&(time<tvalue)&(strata.num==jj))
# sLamtA0_c[it,jj,ics] <- sum(wt[temp]/Sf_acc[temp])
temp=which((wt!=0)&(Sf_acc!=0)&(strata.num==jj))
slamtA0_c[it,jj,ics] <- sum(epanker(tvalue, time[temp], b)*wt[temp]/Sf_acc[temp])
# ttstep <- 0.001
# ttgrid <- seq(time0,tau,by=ttstep)
# slamtA0_c_bd[it,jj,ics] <- sum(epanker(tvalue, time[temp], b)*wt[temp]/Sf_acc[temp])/sum(epanker(tvalue, ttgrid, b)*ttstep)
}
}
}
}
## hypothesis tests for causes with non-NA estimates
icov <- trtpos# test the treatment effect (trtpos)
U1 <- U2 <- T1 <- T2 <- pval.A1 <- pval.A2 <- pval.B1 <- pval.B2 <- NA
U1j <- U2j <- pval.A1j <- pval.A2j <- rep(NA, ncs)
diffsigma <- rep(NA, ncs)
# causes <- levels(cause.fa)
causes <- 1:ncs
nonna <- !is.na(sbeta_acc[icov,])
causes.na <- causes[!nonna]
id.test <- !(cause%in%causes.na)
time.test <- time[id.test]
covar2.test <- covar2[id.test,,drop=F]
cause.test <- cause[id.test]
sbeta_acc.test <- sbeta_acc[,nonna,drop=F]
wipw.test <- wipw[id.test]
rhohat.test <- rhohat[id.test,nonna]
delta.test <- delta[id.test]
strata.num.test <- strata.num[id.test]
ncs.test <- sum(nonna)
alphanull <- rep(log(1-VEnull),ncs.test)
cov_acc <- cova(time.test,covar2.test,cause.test,sbeta_acc.test,wipw.test,rhohat.test,delta.test,strata.num.test,maxit,causes[nonna])
alpha <- sbeta_acc[icov,nonna]
cov_alpha <- matrix(NA, ncs.test, ncs.test)
for (ics in 1:ncs.test){
for (jcs in 1:ncs.test){
cov_alpha[ics,jcs] <- cov_acc[(ics-1)*ncov+icov,(jcs-1)*ncov+icov]
}
}
sigma <- sstd_acc[icov,nonna]
# approximate the distribution of test statistics
D <- 1000
tuta <- mvrnorm(D,rep(0,ncs.test),cov_alpha)
U_tuta <- tuta%*%diag(1/sigma)
U1_tuta <- apply(U_tuta,1,FUN = min)
U2_tuta <- rowSums(U_tuta^2)
# H0: VE(j) <= VEnull
# HA1 \alpha_j <= c0
U1 <- min((alpha-alphanull)/sigma)
pval.A1 <- mean(U1_tuta<U1)
# H0: VE(j) = VEnull
# HA2 \alpha_j \ne c0
U2 <- sum(((alpha-alphanull)/sigma)^2)
pval.A2 <- mean(U2_tuta>U2)
# H0: VE(0) <= VEnull
# H0: VE(1) <= VEnull
# HA1 HA2 for each cause j
# HA1 \alpha_j <= c0
U1j <- (alpha-alphanull)/sigma
pval.A1j <- colMeans(U_tuta<matrix(rep(U1j,D),nrow=D,byrow=T))
# H0: VE(0) = VEnull
# H0: VE(1) = VEnull
# HA2 \alpha_j \ne c0
U2j <- ((alpha-alphanull)/sigma)^2
pval.A2j <- colMeans(U_tuta^2>matrix(rep(U2j,D),nrow=D,byrow=T))
# H0: VE(0) <= VE(1)
# HB1 \alpha_1 <=...<= \alpha_J
# w1=wJ+1=0, wj=1 for j=2,...,J-1
w <- c(0,rep(1, ncs.test-1),0)
omega <- matrix(0, ncs.test, ncs.test-1)
for (ics in 2:ncs.test){
omega[ics-1,ics-1] <- -w[ics]
omega[ics,ics-1] <- w[ics]
}
diffsigma <- sqrt(diag(t(omega)%*%cov_alpha%*%omega))
T1 <- min(alpha%*%omega/diffsigma)
if (ncs.test>2){
T_tuta <- tuta%*%omega%*%diag(1/diffsigma)
} else {
T_tuta <- tuta%*%omega/diffsigma
}
T1_tuta <- apply(T_tuta,1,FUN = min)
pval.B1 <- mean(T1_tuta>T1)
# H0: VE(0) = VE(1)
# HB2 \alpha_i \ne \alpha_j, i<j
T2 <- sum((alpha%*%omega/diffsigma)^2)
T2_tuta <- rowSums(T_tuta^2)
pval.B2 <- mean(T2_tuta>T2)
## Point estimate and 95# CI for VE
sVE_acc <- 1-exp(alpha)
sVEstd_acc <- sigma*exp(alpha)
# VD=(1-VE(j)) / (1-VE(j-1))
sVD_acc <- as.vector(exp(alpha%*%omega))
sVDstd_acc <- as.vector(diffsigma*exp(alpha%*%omega))
## added by Fei - 5/22/2023
# H0: VE(i) = VE(j)
# HC: VE(i) \ne VE(j)
pval.C <- matrix(0, nrow=ncs.test, ncol=ncs.test)
for (ics in 1:(ncs.test-1)){
for(jcs in (ics+1):ncs.test){
T2ij <- (alpha[ics]-alpha[jcs])^2/(cov_alpha[ics,ics]-2*cov_alpha[ics,jcs]+cov_alpha[jcs,jcs])
T2ij_tuta <- (tuta[,ics]-tuta[,jcs])^2/(cov_alpha[ics,ics]-2*cov_alpha[ics,jcs]+cov_alpha[jcs,jcs])
pval.C[ics, jcs] <- mean(T2ij_tuta>T2ij)
}
}
res <- list(causes=causes[nonna],
causes.na=causes.na,
coef.cc=sbeta_c[,nonna,drop=F],
se.cc=sstd_c[,nonna,drop=F],
coef.ipw=sbeta_ic[,nonna,drop=F],
se.ipw=sstd_ic[,nonna,drop=F],
coef=sbeta_acc[,nonna,drop=F],
se=sstd_acc[,nonna,drop=F],
coef.VE=sVE_acc,
se.VE=sVEstd_acc,
coef.VD=sVD_acc,
se.VD=sVDstd_acc,
U1=U1,
U2=U2,
U1j=U1j,
U2j=U2j,
T1=T1,
T2=T2,
pval.A1=pval.A1,
pval.A2=pval.A2,
pval.A1j=pval.A1j,
pval.A2j=pval.A2j,
pval.B1=pval.B1,
pval.B2=pval.B2,
tgrid=tgrid,
lambda0=slamtA0_c[,,nonna,drop=F],
covariates=colnames(covar2),
trtpos=trtpos,
VEnull=VEnull,
diffsigma=diffsigma,
cov.alpha=cov_alpha,
pval.C=pval.C)
class(res) <- "markPH.aipw"
return(res)
}
##' @export
print.markPH.aipw <- function(x, digits=4,...){
ncs <- ncol(x$coef)
ncov <- nrow(x$coef)
cat("\n*********************************************************************\n")
cat("Causes {", x$causes.na, "} with NA estimates are ignored in the results\n")
cat("*********************************************************************\n\n")
cat("Table 1: estimates of covaraite coefficients\n")
for (ics in 1:ncs){
cat("Cause:", x$causes[ics], "\n")
table1 <- cbind(x$coef[,ics],
x$se[,ics],
pnorm(-abs(x$coef[,ics]/x$se[,ics]))*2,
x$coef.ipw[,ics],
x$se.ipw[,ics],
pnorm(-abs(x$coef.ipw[,ics]/x$se.ipw[,ics]))*2,
x$coef.cc[,ics],
x$se.cc[,ics],
pnorm(-abs(x$coef.cc[,ics]/x$se.cc[,ics]))*2)
colnames(table1) <- c("coef",
"se",
"pval",
"coef.ipw",
"se.ipw",
"pval.ipw",
"coef.cc",
"se.cc",
"pval.cc")
rownames(table1) <- x$covariates
print.default(table1, digits=digits)
cat("\n")
}
cat("Table 2: results for VE\n")
table2 <- cbind(x$coef.VE,x$se.VE,
1-(1-x$coef.VE)*exp(qnorm(0.975)*x$se[x$trtpos,]),
1-(1-x$coef.VE)*exp(-qnorm(0.975)*x$se[x$trtpos,]),
x$U1j,
x$pval.A1j,
x$U2j,
x$pval.A2j)
colnames(table2) <- c("est",
"se",
"95% LL",
"95% UL",
"U1j",
"p-value for HAj1",
"U2j",
"p-value for HAj2")
rownames(table2) <- paste0("VE(",x$causes,")")
print.default(table2, digits=digits)
cat(paste0("HAj1: VE(j)>",x$VEnull),"\n")
cat(paste0("HAj2: VE(j) not equal to ",x$VEnull),"\n")
cat("\n")
cat("Table 3: results for VD\n")
cat("VD(j,k)=(1-VE(j))/(1-VE(k))\n")
table3 <- rbind(cbind(x$coef.VD,x$se.VD,x$coef.VD*exp(-qnorm(0.975)*x$diffsigma),x$coef.VD*exp(qnorm(0.975)*x$diffsigma)),cbind(1/x$coef.VD,x$se.VD/x$coef.VD^2,1/x$coef.VD*exp(-qnorm(0.975)*x$diffsigma),1/x$coef.VD*exp(qnorm(0.975)*x$diffsigma)))
colnames(table3) <- c("est",
"se",
"95% LL",
"95% UL")
rnames <- rep(0, (ncs-1)*2)
for (ics in 2:ncs){
rnames[ics-1] <- paste0("VD(",x$causes[ics],",",x$causes[ics-1],")")
rnames[ncs-1+ics-1] <- paste0("VD(",x$causes[ics-1],",",x$causes[ics],")")
}
rownames(table3) <- rnames
print.default(table3, digits=digits)
cat("\n")
cat("Table 4: results for hypothesis tests\n")
cat("HA1: VE(j)>=", x$VEnull, "with strict inequality for some j\n")
cat("U1=",x$U1,"p-value=", x$pval.A1, "\n")
cat("HA2: VE(j) not equal to", x$VEnull, "for some j\n")
cat("U2=",x$U2,"p-value=", x$pval.A2, "\n")
cat("HB1: VE(1)>=...>=VE(J) with at least one strict inequality\n")
cat("T1=",x$T1,"p-value=", x$pval.B1, "\n")
cat("HB2: VE(i) not equal to VE(j) for at least one pair of i and j, 1<=i<j<=J\n")
cat("T2=",x$T2,"p-value=", x$pval.B2, "\n")
cat("\n")
}
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