R/sievePHipw.R

In sievePH: Sieve Analysis Methods for Proportional Hazards Models

# 've' returns vaccine efficacy values given parameters for the grid of mark values, v,
# and values for alpha, beta, and gamma
VE <- function(v, alpha, beta, gamma){ 1 - exp(alpha + beta*v + gamma) }

# 'covEstIPW' returns the estimated covariance matrix of 'phiHat' and 'lambdaHat'
# in the IPW scenario, using Theorem 1 in Juraska and Gilbert (2013, Biometrics)
# 'eventTime' is the observed time, defined as the minimum of failure, censoring, and study times
# 'eventType' is the failure indicator (0 if censored, 1 if failure)
# 'mark' is a data frame (with the same number of rows as the length of 'eventTime') specifying a multivariate mark (a numeric vector for a univariate mark is allowed), with NA for subjects with eventType=0.
# 'tx' is the treatment group indicator (1 if treatment, 0 if control)
# 'aux' is a data frame of auxiliary covariates
# 'formulaMiss' is a one-sided formula specifying the logistic regression model for estimating the probability of observing the mark; all variables in the formula except 'tx' must be included in 'aux'
# 'phiHat' is a vector of the alpha and beta estimates
# 'lambdaHat' is the estimate for lambda in the mark density ratio model
# 'gammaHat' is the estimate for gamma obtained in the marginal hazards model
covEstIPW <- function(eventTime, eventType, mark, tx, aux=NULL, formulaMiss, phiHat, lambdaHat, gammaHat){
  # convert either a numeric vector or a data frame into a matrix
  mark <- as.matrix(mark)

  n <- length(eventTime)
  isNA <- apply(mark, 1, function(row){ sum(is.na(row)) })
  m.complete <- sum(eventType==1 & isNA==0)
  m.f <- sum(eventType==1)
  eventTime.f <- eventTime[eventType==1]
  eventTime.fM <- matrix(eventTime.f, nrow=n, ncol=m.f, byrow=TRUE)
  eventTime.complete <- eventTime[eventType==1 & isNA==0]
  V.f <- cbind(1,mark[eventType==1,])
  V.complete <- na.omit(V.f)
  na.idx <- attr(V.complete,"na.action")
  tx.complete <- tx[eventType==1 & isNA==0]
  tx.f <- tx[eventType==1]
  if (!is.null(aux)){
    aux.f <- data.frame(aux[eventType==1, ])
    colnames(aux.f) <- colnames(aux)
  }
  eventTime.completeM <- matrix(eventTime.complete, nrow=n, ncol=m.complete, byrow=TRUE)
  VV.complete <- apply(V.complete,1,tcrossprod)
  nmark <- NCOL(V.complete)

  g <- function(phi){ exp(drop(V.complete %*% phi)) }
  dG <- function(phi){ t(g(phi) * V.complete) }
  d2G <- function(phi){ array(t(t(VV.complete)*g(phi)),dim=c(nmark,nmark,m.complete)) }
  dGdG <- function(phi){ array(apply(dG(phi),2,tcrossprod),dim=c(nmark,nmark,m.complete)) }

  score1.vect <- function(phi, lambda){
    vect <- matrix(0, nrow=nmark, ncol=m.f)
    vect[,-na.idx] <- t((-lambda/(pi*(1+lambda*(g(phi)-1))) + tx.complete/(pi*g(phi))) * t(dG(phi)))
    vect
  }
  xi <- function(gamma){ crossprod(eventTime>=eventTime.fM, tx*exp(gamma*tx)) }
  zeta <- function(gamma){ crossprod(eventTime>=eventTime.fM, exp(gamma*tx)) }
  eta <- drop(xi(gammaHat)/zeta(gammaHat))
  score3.vect <- function(gamma){ tx.f-eta }
  l.vect <- function(gamma){
    survprob.vect <- c(1, summary(survfit(Surv(eventTime,eventType)~1), times=sort(eventTime.f))$surv)
    surv.increm <- survprob.vect[-length(survprob.vect)] - survprob.vect[-1]
    eventTime.fMsq <- eventTime.fM[1:m.f,]
    crossprod(eventTime.f>=eventTime.fMsq, surv.increm*(tx.f*exp(gamma*tx.f) - eta*exp(gamma*tx.f))/zeta(gamma))
  }
  score1 <- function(phi, lambda){
    drop(-lambda * dG(phi) %*% (1/(pi*(1+lambda*(g(phi)-1)))) + dG(phi) %*% (tx.complete/(pi*g(phi))))
  }
  score2 <- function(phi, lambda){
    -sum((g(phi)-1)/(pi*(1+lambda*(g(phi)-1))))
  }
  score <- function(phi, lambda){ c(score1(phi,lambda),score2(phi,lambda)) }
  jack11 <- function(phi, lambda){
    d2Gperm <- aperm(d2G(phi), c(3,1,2))
    dGdGperm <- aperm(dGdG(phi), c(3,1,2))
    term1 <- apply(aperm(d2Gperm*(1/(pi*(1+lambda*(g(phi)-1)))), c(2,3,1)),c(1,2),sum)
    term2 <- apply(aperm(dGdGperm*(1/(pi*(1+lambda*(g(phi)-1))^2)), c(2,3,1)),c(1,2),sum)
    term3 <- apply(aperm(d2Gperm*(tx.complete/(pi*g(phi))), c(2,3,1)),c(1,2),sum)
    term4 <- apply(aperm(dGdGperm*(tx.complete/(pi*g(phi)^2)), c(2,3,1)),c(1,2),sum)
    -lambda*(term1 - lambda*term2) + term3 - term4
  }
  jack21 <- function(phi, lambda){
    drop(-dG(phi) %*% (1/(pi*(1+lambda*(g(phi)-1))^2)))
  }
  jack22 <- function(phi, lambda){
    sum(((g(phi)-1)^2)/(pi*(1+lambda*(g(phi)-1))^2))
  }
  jack <- function(phi, lambda){
    j21 <- jack21(phi,lambda)
    (cbind(rbind(jack11(phi,lambda),j21),c(j21,jack22(phi,lambda))))/n
  }
  jack33 <- sum(eta*(eta-1))/n

  r <- apply(V.f, 1, function(row){ ifelse(sum(is.na(row))>0,0,1) })

  mData <- data.frame(r, tx=tx.f)
  if (!is.null(aux)){ mData <- cbind(mData, aux.f) }

  formulaMissDecomp <- strsplit(strsplit(paste(deparse(formulaMiss), collapse = ""), " *[~] *")[[1]], " *[+] *")
  formulaMiss2sided <- as.formula(paste0("r ~ ", paste(formulaMissDecomp[[2]], collapse="+")))

  pi.all <- glm(formulaMiss2sided, family=binomial, data=mData)$fitted
  if (!is.null(na.idx)){
    pi <- pi.all[-na.idx]
  } else {
    pi <- pi.all
  }
  if (any(pi<0.005)){ stop("Selection probabilities not bounded away from 0.") }

  p <- mean(eventType==1)
  omega <- drop(score1.vect(phiHat,lambdaHat) %*% (score3.vect(gammaHat) + p*l.vect(gammaHat))/n -
                  sum(score3.vect(gammaHat) + p*l.vect(gammaHat))*apply(score1.vect(phiHat,lambdaHat),1,sum)/(n^2))
  return(drop(solve(jack(phiHat,lambdaHat))[1:nmark,1:nmark] %*% omega)/(n*jack33))
}

# 'densRatioIPW' applies the mark density ratio model with missing multivariate marks using
# the inferential procedure defined by inverse probability weighting (IPW) of complete cases.
# The function calculates the mark density ratio and returns a list containing:
#     'coef': estimates for alpha, beta, and lambda
#     'var': the corresponding covariance matrix
#     'jack': the first two rows and columns of the limit estimating function in matrix form
#     'conv': a logical value indicating convergence of the estimating functions
# 'mark' is a data frame representing a multivariate mark variable (a numeric vector for a univariate mark is allowed)
# 'tx' is the treatment group indicator (1 if treatment, 0 if control)
# 'aux' is a data frame of auxiliary covariates; the rows of 'aux' correspond to the rows of 'mark'
# 'formulaMiss' is a one-sided formula specifying the logistic regression model for estimating the probability of observing the mark; all variables in the formula except 'tx' must be included in 'aux'
densRatioIPW <- function(mark, tx, aux=NULL, formulaMiss){
  # convert either a numeric vector or a data frame into a matrix
  mark <- as.matrix(mark)

  V <- cbind(1,mark)
  V.complete <- na.omit(V)
  z <- tx
  na.idx <- attr(V.complete,"na.action")
  if (!is.null(na.idx)){
    z.complete <- z[-na.idx]
  } else {
    z.complete <- z
  }
  nmark <- NCOL(V.complete)
  ninf <- NROW(V.complete)
  VV.complete <- apply(V.complete,1,tcrossprod)

  g <- function(theta){ exp(drop(V.complete %*% theta)) }
  dG <- function(theta){ t(g(theta) * V.complete) }
  d2G <- function(theta){ array(t(t(VV.complete)*g(theta)),dim=c(nmark,nmark,ninf)) }
  dGdG <- function(theta){ array(apply(dG(theta),2,tcrossprod),dim=c(nmark,nmark,ninf)) }

  score1 <- function(theta, lambda){
    drop(-lambda * dG(theta) %*% (1/(pi*(1+lambda*(g(theta)-1)))) + dG(theta) %*% (z.complete/(pi*g(theta))))
  }
  score2 <- function(theta, lambda){
    -sum((g(theta)-1)/(pi*(1+lambda*(g(theta)-1))))
  }
  score <- function(theta, lambda){ c(score1(theta,lambda),score2(theta,lambda)) }
  jack11 <- function(theta, lambda){
    d2Gperm <- aperm(d2G(theta), c(3,1,2))
    dGdGperm <- aperm(dGdG(theta), c(3,1,2))
    term1 <- apply(aperm(d2Gperm*(1/(pi*(1+lambda*(g(theta)-1)))), c(2,3,1)),c(1,2),sum)
    term2 <- apply(aperm(dGdGperm*(1/(pi*(1+lambda*(g(theta)-1))^2)), c(2,3,1)),c(1,2),sum)
    term3 <- apply(aperm(d2Gperm*(z.complete/(pi*g(theta))), c(2,3,1)),c(1,2),sum)
    term4 <- apply(aperm(dGdGperm*(z.complete/(pi*g(theta)^2)), c(2,3,1)),c(1,2),sum)
    -lambda*(term1 - lambda*term2) + term3 - term4
  }
  jack21 <- function(theta, lambda){
    drop(-dG(theta) %*% (1/(pi*(1+lambda*(g(theta)-1))^2)))
  }
  jack22 <- function(theta, lambda){
    sum(((g(theta)-1)^2)/(pi*(1+lambda*(g(theta)-1))^2))
  }
  jack <- function(theta, lambda){
    j21 <- jack21(theta,lambda)
    cbind(rbind(jack11(theta,lambda),j21),c(j21,jack22(theta,lambda)))
  }

  r <- apply(V, 1, function(row){ ifelse(sum(is.na(row))>0,0,1) })

  mData <- data.frame(r, tx)
  if (!is.null(aux)){ mData <- cbind(mData, aux) }

  formulaMissDecomp <- strsplit(strsplit(paste(deparse(formulaMiss), collapse = ""), " *[~] *")[[1]], " *[+] *")
  formulaMiss2sided <- as.formula(paste0("r ~ ", paste(formulaMissDecomp[[2]], collapse="+")))

  pi.all <- glm(formulaMiss2sided, family=binomial, data=mData)$fitted

  if (!is.null(na.idx)){
    pi <- pi.all[-na.idx]
  } else {
    pi <- pi.all
  }
  if (any(pi<0.005)){ stop("Selection probabilities not bounded away from 0.") }

  param.old <- numeric(nmark+1)
  param.new <- c(numeric(nmark),0.5)
  while (sum((param.new - param.old)^2)>1e-8){
    param.old <- param.new
    jackInv <- try(solve(jack(param.old[-(nmark+1)],param.old[nmark+1])), silent=TRUE)
    if (!inherits(jackInv, "try-error")){
      param.new <- param.old - drop(jackInv %*% score(param.old[-(nmark+1)],param.old[nmark+1]))
    }
    if (sum(is.nan(param.new))>0) break
  }
  theta.new <- param.new[-(nmark+1)]
  lambda.new <- param.new[nmark+1]

  Resid <- function(theta, lambda){
    U <- matrix(0,nrow=length(z),ncol=nmark+1)
    U[-na.idx,1:nmark] <- -lambda * t(dG(theta)) * (1/(pi*(1+lambda*(g(theta)-1)))) + t(dG(theta)) * (z.complete/(pi*g(theta)))
    U[-na.idx,nmark+1] <- (g(theta)-1)/(pi*(1+lambda*(g(theta)-1)))

    formulaScore2sided <- as.formula(paste0("Ui ~ ", paste(formulaMissDecomp[[2]], collapse="+")))

    mf <- model.frame(formulaMiss2sided, mData)
    X <- model.matrix(terms(formulaMiss2sided), mf)
    S <- (r - pi.all) * X
    resids <- lapply(1:NCOL(U), function(i, U, S){ lm(formulaScore2sided, data=data.frame(Ui=U[, i], S))$resid }, U=U, S=S)

    Resids <- do.call("cbind", resids)
    return(crossprod(Resids) / ninf)
  }

  JackInv <- try(solve(jack(theta.new,lambda.new)), silent=TRUE)
  if (!inherits(JackInv, "try-error")){
    Var <- ninf * JackInv %*% Resid(theta.new,lambda.new) %*% JackInv
    names(param.new) <- rownames(Var) <- colnames(Var) <- c("alpha",paste("beta",1:(nmark-1),sep=""),"lambda")
  } else {
    Var <- NULL
  }

  return(list(coef=param.new, var=Var, jack=jack11(theta.new,lambda.new), probs=pi, conv=!(inherits(jackInv, "try-error") | inherits(JackInv, "try-error"))))
}

#' Semiparametric Inverse Probability Weighted Complete-Case Estimation of Coefficients in a Mark-Specific Proportional Hazards Model
#' with a Multivariate Continuous Mark, Missing-at-Random in Some Failures
#'
#' \code{sievePHipw} implements the semiparametric inverse probability weighted (IPW) complete-case estimation method of Juraska and Gilbert (2015) for the multivariate mark-
#' specific hazard ratio, with the mark subject to missingness at random. It extends Juraska and Gilbert (2013) by weighting complete cases by the inverse of their estimated
#' probabilities given auxiliary covariates and/or treatment. The probabilities are estimated by fitting a logistic regression model with a user-specified linear predictor.
#' Coefficients in the treatment-to-placebo mark density ratio model (Qin, 1998) are estimated by solving the IPW estimating equations. The ordinary method of maximum partial likelihood
#' estimation is employed for estimating the overall log hazard ratio in the Cox model.
#'
#' @param eventTime a numeric vector specifying the observed right-censored event time
#' @param eventInd a numeric vector indicating the event of interest (1 if event, 0 if right-censored)
#' @param mark either a numeric vector specifying a univariate continuous mark or a data frame specifying a multivariate continuous mark subject to missingness at random. Missing mark values should be set to \code{NA}.
#' For subjects with \code{eventInd = 0}, the value(s) in \code{mark} should also be set to \code{NA}.
#' @param tx a numeric vector indicating the treatment group (1 if treatment, 0 if placebo)
#' @param aux a data frame specifying auxiliary covariates predictive of the probability of observing the mark. The mark missingness model only requires that the auxiliary covariates be observed in subjects who experienced the event of interest. For subjects with \code{eventInd = 0}, the value(s) in \code{aux} may be set to \code{NA}.
#' @param strata a numeric vector specifying baseline strata (\code{NULL} by default). If specified, a stratified Cox model is fit for estimating the marginal hazard ratio (i.e., a separate baseline hazard is assumed for each stratum). No stratification is used in estimation of the mark density ratio.
#' @param formulaMiss a one-sided formula object specifying (on the right side of the \code{~} operator) the linear predictor in the logistic regression model used for predicting the probability of observing
#' the mark. All terms in the formula except \code{tx} must be evaluable in the data frame \code{aux}.
#'
#' @details
#' \code{sievePHipw} considers data from a randomized placebo-controlled treatment efficacy trial with a time-to-event endpoint.
#' The parameter of interest, the mark-specific hazard ratio, is the ratio (treatment/placebo) of the conditional mark-specific hazard functions.
#' It factors as the product of the mark density ratio (treatment/placebo) and the ordinary marginal hazard function ignoring mark data.
#' The mark density ratio is estimated using the IPW complete-case estimation method, extending Qin (1998), and
#' the marginal hazard ratio is estimated using \code{coxph()} in the \code{survival} package.
#' The asymptotic properties of the IPW complete-case estimator are detailed in Juraska and Gilbert (2015).
#'
#' @return An object of class \code{sievePH} which can be processed by
#' \code{\link{summary.sievePH}} to obtain or print a summary of the results. An object of class
#' \code{sievePH} is a list containing the following components:
#' \itemize{
#' \item \code{DRcoef}: a numeric vector of estimates of coefficients \eqn{\phi} in the weight function \eqn{g(v, \phi)} in the density ratio model
#' \item \code{DRlambda}: an estimate of the Lagrange multiplier in the profile score functions for \eqn{\phi} (that arises by profiling out the nuisance parameter)
#' \item \code{DRconverged}: a logical value indicating whether the estimation procedure in the density ratio model converged
#' \item \code{logHR}: an estimate of the marginal log hazard ratio from \code{coxph()} in the \code{survival} package
#' \item \code{cov}: the estimated joint covariance matrix of \code{DRcoef} and \code{logHR}
#' \item \code{coxphFit}: an object returned by the call of \code{coxph()}
#' \item \code{nPlaEvents}: the number of events observed in the placebo group
#' \item \code{nTxEvents}: the number of events observed in the treatment group
#' \item \code{mark}: the input object
#' \item \code{tx}: the input object
#' }
#'
#' @references Juraska, M., and Gilbert, P. B. (2015), Mark-specific hazard ratio model with missing multivariate marks. \emph{Lifetime Data Analysis} 22(4): 606-25.
#'
#' Juraska, M. and Gilbert, P. B. (2013), Mark-specific hazard ratio model with multivariate continuous marks: an application to vaccine efficacy. \emph{Biometrics} 69(2):328-337.
#'
#' Qin, J. (1998), Inferences for case-control and semiparametric two-sample density ratio models. \emph{Biometrika} 85, 619-630.
#'
#' @examples
#' n <- 500
#' tx <- rep(0:1, each=n / 2)
#' tm <- c(rexp(n / 2, 0.2), rexp(n / 2, 0.2 * exp(-0.4)))
#' cens <- runif(n, 0, 15)
#' eventTime <- pmin(tm, cens, 3)
#' eventInd <- as.numeric(tm <= pmin(cens, 3))
#' mark1 <- ifelse(eventInd==1, c(rbeta(n / 2, 2, 5), rbeta(n / 2, 2, 2)), NA)
#' mark2 <- ifelse(eventInd==1, c(rbeta(n / 2, 1, 3), rbeta(n / 2, 5, 1)), NA)
#' # a continuous auxiliary covariate
#' A <- (mark1 + 0.4 * runif(n)) / 1.4
#' linPred <- -0.8 + 0.4 * tx + 0.8 * A
#' probs <- exp(linPred) / (1 + exp(linPred))
#' R <- rep(NA, length(probs))
#' while (sum(R, na.rm=TRUE) < 10){
#'   R[eventInd==1] <- sapply(probs[eventInd==1], function(p){ rbinom(1, 1, p) })
#' }
#' # produce missing-at-random marks
#' mark1[eventInd==1] <- ifelse(R[eventInd==1]==1, mark1[eventInd==1], NA)
#' mark2[eventInd==1] <- ifelse(R[eventInd==1]==1, mark2[eventInd==1], NA)
#'
#' # fit a model with a bivariate mark
#' fit <- sievePHipw(eventTime, eventInd, mark=data.frame(mark1, mark2), tx,
#'                   aux=data.frame(A), formulaMiss= ~ tx * A)
#'
#' @seealso \code{\link{summary.sievePH}}, \code{\link{plot.summary.sievePH}}, \code{\link{testIndepTimeMark}} and \code{\link{testDensRatioGOF}}
#'
#' @import survival
#'
#' @export
sievePHipw <- function(eventTime, eventInd, mark, tx, aux=NULL, strata=NULL, formulaMiss){
  if (is.numeric(mark)){ mark <- data.frame(mark) }
  if (!is.null(aux)){ if (!is.data.frame(aux)){ stop("'aux' must be a data frame.") } }

  nPlaEvents <- sum(eventInd * (1-tx))
  nTxEvents <- sum(eventInd * tx)

  auxForEvents <- aux
  if (!is.null(aux)){
    auxForEvents <- data.frame(aux[eventInd==1, ])
    colnames(auxForEvents) <- colnames(aux)
  }

  dRatio <- densRatioIPW(mark[eventInd==1, ], tx[eventInd==1], aux=auxForEvents, formulaMiss=formulaMiss)

  # fit the Cox proportional hazards model to estimate the marginal hazard ratio
  fm.coxph <- as.formula(paste0("Surv(eventTime, eventInd) ~ tx", ifelse(is.null(strata), "", " + strata(strata)")))
  phReg <- survival::coxph(fm.coxph)

  # the estimate of the marginal log hazard ratio
  gammaHat <- phReg$coef

  # the output list
  out <- list(DRcoef=NA, DRlambda=NA, DRconverged=dRatio$conv, logHR=gammaHat, cov=NA, coxphFit=phReg, nPlaEvents=nPlaEvents, nTxEvents=nTxEvents, mark=mark, tx=tx)

  if (dRatio$conv){

    # a vector of estimates of the density ratio coefficients (alpha, beta1, beta2,..., betak) and the Lagrange multiplier
    thetaHat <- dRatio$coef

    # variance and covariance estimates
    # order of columns in 'dRatio$var': alpha, beta1, beta2,...betak, lambda, where k is the dimension of the mark
    lastComp <- length(thetaHat)
    vthetaHat <- dRatio$var[-lastComp, -lastComp]
    vgammaHat <- drop(phReg$var)
    covThG <- covEstIPW(eventTime, eventInd, mark, tx, aux=aux, formulaMiss=formulaMiss, thetaHat[-lastComp], thetaHat[lastComp], gammaHat)

    # covariance matrix for alpha, beta1, beta2,..., betak, gamma
    Sigma <- cbind(rbind(vthetaHat, covThG), c(covThG, vgammaHat))
    colnames(Sigma) <- rownames(Sigma) <- c("alpha", paste0("beta", 1:NCOL(mark)), "gamma")

    out$DRcoef <- thetaHat[-lastComp]
    out$DRlambda <- thetaHat[lastComp]
    out$cov <- Sigma
  } else {
    warning("The estimation method in the density ratio model did not converge.")
  }

  class(out) <- "sievePH"
  return(out)
}