CG.Gumbel {compound.Cox}

Copula-graphic estimator under the Gumbel copula.

Description

This function computes the copula-graphic (CG) estimator (Rivest & Wells 2001) under the Gumbel copula.

Usage

CG.Gumbel(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+1)"

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) or Section 3.2 of Yeh et al.(2023). The output shows the survival probabilities at given time points of "t.vec". The input requires to specify an association parameter "alpha" of the Gumbel copula (alpha>=0), where alpha=0 corresponds to the independence copula. Emura and Chen (2016, 2018) and Yeh et al.(2023) applied the CG estimator to assess survival prognosis for lung cancer patients.

Value

tau

Kendall's tau (=alpha/(alpha+1))

time

sort(t.vec)

n.risk

the number of patients at-risk

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.

Yeh CT, Liao GY, Emura T (2023). Sensitivity analysis for survival prognostic prediction with gene selection: a copula method for dependent censoring, Biomedicines 11(3):797.

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.Gumbel(t.vec,d.vec,alpha=9,S.col="blue")
### CG.Gumbel gives identical results with the Kaplan-Meier estimator with alpha=0 ### 
CG.Gumbel(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.Gumbel(t.good,d.good,alpha=9,S.plot=TRUE,S.col="blue")

t.poor=t.vec[Poor]
d.poor=d.vec[Poor]
CG.Gumbel(t.poor,d.poor,alpha=9,S.plot=TRUE,S.col="red")