Censored Quantile Regression & Monotonicity-Respecting Restoring

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Yijian Huang (yhuang5@emory.edu)

Package cequre performs censored quantile regression of Huang (2010), and restores monotonicity respecting via adaptive interpolation for dynamic regression of Huang (2017). The monotonicity-respecting restoration applies to general dynamic regression models including (uncensored or censored) quantile regression model, additive hazards model, and dynamic survival models of Peng and Huang (2007), among others.

Installation

cequre is available on CRAN:

install.packages("cequre")

Censored quantile regression of Huang (2010)

This procedure is illustrated with the Mayo primary biliary cholangitis dataset as given in package survival.

## Mayo PBC data
library(survival)
pbc_analy <- as.matrix(na.omit(pbc[,c("time","status","age","edema","bili","albumin","protime")]))
# log transformation for time, bili, albumin, and protime
pbc_analy[,c(1,5:7)] <- log(pbc_analy[,c(1,5:7)])
colnames(pbc_analy)[c(1,5:7)] <- paste("log",colnames(pbc_analy)[c(1,5:7)])
# convert status to censoring indicator
pbc_analy[,2] <- pbc_analy[,2]>1

## Censored quantile regression
library(cequre)
taus <- .1*(1:9)
fit <- cequre(pbc_analy[,1],pbc_analy[,2],pbc_analy[,-c(1,2)],
taus=taus,res=200)

Plot the estimated regression coefficient processes and associated pointwise 95% CI’s:

var.names <- c("baseline",colnames(pbc_analy)[-(1:2)])
par(mfrow=c(3,2),mai=c(.5,.5,.1,.1))
for(k in 1:6) {
  plot(fit$curve[7,],fit$curve[k,],type="s",lwd=2,xlab="",ylab="",
  ylim=c(-1,1)*max(abs(fit$curve[k,])))
  lines(c(0,1),c(0,0),lty=2)
  text(1,max(abs(fit$curve[k,])),var.names[k],adj=c(1,1))
# pointwise 95% CI
  for(i in 1:length(taus)) lines(c(1,1)*taus[i],fit$bt[k,i]+
    c(-1,1)*qnorm(.975)*sqrt(fit$va[k,k,i]))
  }
mtext("Estimated regression coefficient",side=2,line=-1,outer=T,cex=.8)
mtext("Probability",side=1,line=-1,outer=T,cex=.8)

Restoration of monotonicity respecting using adaptive interpolation of Huang (2017)

The monotonicity-respecting restoration is illustrated with the preceding censored quantile regression using the Mayo primary biliary cholangitis dataset:

# zch needs to include constant 1 as being the covariate for the intercept
mfit <- monodr(fit$curve,
               zch = cbind(rep(1,dim(pbc_analy)[1]),pbc_analy[,-(1:2)]),
               initau=fit$tau.bnd/2)
# plot the original regression coefficient (in black) and the one after the
# monotonicity-respecting restoration (in orange)
par(mfrow=c(3,2),mai=c(.5,.5,.1,.1))
for(k in 1:6) {
  plot(fit$curve[7,],fit$curve[k,],type="s",xlab="",ylab="",
  ylim=c(-1,1)*max(abs(fit$curve[k,])))
  lines(c(0,1),c(0,0),lty=2)
  text(1,max(abs(fit$curve[k,])),var.names[k],adj=c(1,1))
  lines(mfit$airc[7,],mfit$airc[k,],lwd=2,col="dark orange")
  }
mtext("Estimated regression coefficient",side=2,line=-1,outer=T,cex=.8)
mtext("Probability",side=1,line=-1,outer=T,cex=.8)

References

Huang, Y. (2010) Quantile calculus and censored regression, The Annals of Statistics 38, 1607–1637.

Huang, Y. (2017) Restoration of monotonicity respecting in dynamic regression. Journal of the American Statistical Association 112, 613–622.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.