CG.Clayton {compound.Cox}R Documentation

Copula-graphic estimator under the Clayton copula.

Description

This function computes the copula-graphic (CG) estimator (Rivest & Wells 2001) of a survival function under the Clayton copula.

Usage

CG.Clayton(t.vec, d.vec, alpha, S.plot = TRUE, S.col = "black")

Arguments

t.vec

Vector of survival times (time to either death or censoring)

d.vec

Vector of censoring indicators, 1=death, 0=censoring

alpha

Association parameter that is related to Kendall's tau through "tau= alpha/(alpha+2)"

S.plot

If TRUE, the survival curve is displayed

S.col

Color of the survival curve in the plot

Details

The CG estimator is a variant of the Kaplan-Meier estimator for a survival function. The CG estimator relaxes the independent censoring assumption of the KM estimator through a copula-based dependent censoring model. The computational formula of the CG estimator is given in Appendix D of Emura et al. (2019). The output shows the survival probabilities at given time points of "t.vec". The input requires to specify an association parameter "alpha" of the Clayton copula (alpha>0), where alpha=0 corresponds to the independence copula. Emura and Chen (2016, 2018) applied the CG estimator to assess survival prognosis for lung cancer patients.

Value

tau

Kendall's tau (=alpha/(alpha+2))

time

sort(t.vec)

surv

survival probability at "time"

Author(s)

Takeshi Emura

References

Emura T, Matsui S, Chen HY (2019). compound.Cox: Univariate Feature Selection and Compound Covariate for Predicting Survival, Computer Methods and Programs in Biomedicine 168: 21-37.

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.

Emura T, Chen YH (2018). Analysis of Survival Data with Dependent Censoring, Copula-Based Approaches, JSS Research Series in Statistics, Springer, Singapore.

Rivest LP, Wells MT (2001). A Martingale Approach to the Copula-graphic Estimator for the Survival Function under Dependent Censoring, J Multivar Anal; 79: 138-55.

Examples

## Example 1 (a toy example of n=8) ##
t.vec=c(1,3,5,4,7,8,10,13)
d.vec=c(1,0,0,1,1,0,1,0)
CG.Clayton(t.vec,d.vec,alpha=18,S.col="blue")
### CG.Clayton gives identical results with the Kaplan-Meier estimator with alpha=0 ### 
CG.Clayton(t.vec,d.vec,alpha=0,S.plot=FALSE)$surv
survfit(Surv(t.vec,d.vec)~1)$surv

## Example 2 (Analysis of the lung cancer data) ##
data(Lung) # read the data
t.vec=Lung[,"t.vec"]
d.vec=Lung[,"d.vec"]
x.vec=Lung[,"MMP16"] # the gene associated with survival (Emura and Chen 2016, 2018) #
Poor=x.vec>median(x.vec) ## Indicator of poor survival
Good=x.vec<=median(x.vec) ## Indicator of good survival

par(mfrow=c(1,2))
###### Predicted survival curves via the CG estimator #####
t.good=t.vec[Good]
d.good=d.vec[Good]
CG.Clayton(t.good,d.good,alpha=18,S.plot=TRUE,S.col="blue")

t.poor=t.vec[Poor]
d.poor=d.vec[Poor]
CG.Clayton(t.poor,d.poor,alpha=18,S.plot=TRUE,S.col="red")

[Package compound.Cox version 3.20 Index]