dependCox.reg {compound.Cox}

Univariate Cox regression under dependent censoring.

Description

This function performs univariate Cox regression under dependent censoring, where dependence between survival time and censoring time is modeled via the Clayton copula (Emura and Chen 2016).

Usage

dependCox.reg(t.vec, d.vec, X.vec, alpha, var = TRUE, censor.reg=FALSE, baseline=FALSE)

Arguments

t.vec

A vector of survival times (time-to-death or censoring)

d.vec

A vector of censoring indicators, 1=death, 0=censoring

X.vec

A vector of covariates (multiple covariates are not allowed)

alpha

An copula parameter (Kendall's tau = alpha/(alpha+2)

var

If TRUE, the standard deviations are given (use FALSE to reduce the computational cost)

censor.reg

If TRUE, show the fitted results for both survival and censoring models

baseline

If TRUE, show the cumulative baseline hazards at the values of "t.vec"

Details

The Clayton model yields positive association between survival time and censoring time with Kendall's tau being equal to alpha/(alpha+2), where alpha > 0 is a copula parameter. The independence copula corresponds to alpha = 0.

Value

beta

The estimated regression coefficient

SE

The standard error for the estimated regression coefficient

Z

The Z-value for testing the null hypothesis of "beta=0" (the Wald test)

P

The P-value for testing the null hypothesis of "beta=0" (the Wald test)

Author(s)

Takeshi Emura

References

Emura T, Chen YH (2016). Gene Selection for Survival Data Under Dependent Censoring: a Copula-based Approach, Stat Methods Med Res 25(No.6): 2840-57.

Examples

### Joint Cox regression of survival and censoring ### 
data(Lung)
t.vec=Lung[,"t.vec"]# death or censoring times #
d.vec=Lung[,"d.vec"]# censoring indicators #
# 16-gene prognostic index (Emura and Chen 2016; 2018) #
X.vec=0.51*Lung[,"ZNF264"]+0.50*Lung[,"MMP16"]+
  0.50*Lung[,"HGF"]-0.49*Lung[,"HCK"]+0.47*Lung[,"NF1"]+
  0.46*Lung[,"ERBB3"]+0.57*Lung[,"NR2F6"]+0.77*Lung[,"AXL"]+
  0.51*Lung[,"CDC23"]+0.92*Lung[,"DLG2"]-0.34*Lung[,"IGF2"]+
  0.54*Lung[,"RBBP6"]+0.51*Lung[,"COX11"]+
  0.40*Lung[,"DUSP6"]-0.37*Lung[,"ENG"]-0.41*Lung[,"IHPK1"]
dependCox.reg(t.vec,d.vec,X.vec,alpha=18,censor.reg=TRUE)

temp=c(Lung[,"train"]==TRUE)
t.vec=Lung[temp,"t.vec"]
d.vec=Lung[temp,"d.vec"]
dependCox.reg(t.vec,d.vec,Lung[temp,"ZNF264"],alpha=18) 
# this reproduces Table 3 of Emura and Chen (2016) #

#### A simulation study under dependent censoring ####
beta_true=1.5 # true regression coefficient
alpha_true=2 # true copula parameter corresponding to Kendall's tau=0.5
n=150
t.vec=d.vec=X.vec=numeric(n)
set.seed(1)
for(i in 1:n){
  X.vec[i]=runif(1)
  eta=X.vec[i]*beta_true
  U=runif(1)
  V=runif(1)
  T=-1/exp(eta)*log(1-U) # Exp(eta) distribution
  W=(1-U)^(-alpha_true) # dependence produced by the Clayton copula
  C=1/alpha_true/exp(eta)*log(  1-W+W*(1-V)^(-alpha_true/(alpha_true+1))  ) # Exp(eta) distribution
  t.vec[i]=min(T,C)
  d.vec[i]=(T<=C) 
}

dependCox.reg(t.vec,d.vec,X.vec,alpha=alpha_true,var=FALSE) # faster computation by "var=FALSE"
beta_true# the above estimate is close to the true value
coxph(Surv(t.vec,d.vec)~X.vec)$coef
# this estimate is biased for the true value due to dependent censoring