从扩展的Cox模型（带有外部时间相关的协变量）获得基线危险函数/生存函数

Question

I am applying an extended Cox model with external time-dependent covariates. Here is a small example (df) which I borrowed and modified from Themeau and Grambsch's book, Modeling survival data : extending the Cox model (2001):

id start stop  event trt bili albumin
1  0      188    0    1  1.8 2.54
1  188    372    0    1  1.6 2.88
1  372    729    0    1  1.7 2.80
1  729    1254   0    1  3.2 2.92
1  1254   1462   0    1  3.7 2.59
1  1462   1824   0    1  4.0 2.59
1  1824   1925   1    1  5.3 1.83
2  0      56     0    0  1.8 2.36
2  56     172    0    0  1.6 1.89
2  172    521    1    0  1.7 1.56
3  0      36     0    1  3.2 2.10
3  36     232    0    1  3.7 2.32
3  232    352    0    1  4.0 1.96
3  352    610    1    1  5.3 2.05

I would like to obtain the baseline hazard/survival function from the extended Cox model. In the classical Cox PH model which handles time-independent covariates, it seems that we can obtain the estimate of H(t) using the Nelson-Aalen estimator:

fit1<- coxph(Surv(time, event) ~ tidc's, data=df)
sfit<-survfit(fit1)
sfit$surv
H<- -log(sfit$surv)
H<- c(H, tail(H, 1))

I am wondering how to obtain the baseline hazard/survival function from the extended Cox model, when external time-dependent covariates are used instead? Could I use the similar method like this?

model_1<-coxph(Surv(start,stop,event) ~ treat+log(bili)+log(albumin),data=df)
mfit<-survfit(model_1)
mfit$surv
H1<- -log(mfit$surv)
H1<- c(H1, tail(H1, 1))

Thanks.

Answer 1

The survfit object also has an element named 'cumhaz'. That would seem to be the correct item to pull rather than recalculating it.

> all.equal( -log(mfit$surv), mfit$cumhaz )
[1] TRUE

It's unclear what you expect as a "baseline hazard" if you are using time-dependent covariates. At least in the survival package a "baseline hazard" is calculated on the basis of the estimated hazard for a hypothetical subject who had the mean value for each of the covariates. The packages author, Terry Therneau, devotes the entire first paragraph in the Details section of the help page for survfit.coxph discussing why he thinks this is often delivers a result of questionable value. I seriously doubt that he would consider a baseline hazard to be meaningful in a time-dependent covariate model where the baseline would be jumping around. He has specifically suggested that calculating survival curves in that setting is statistically unsupportable (despite the fact that they often appear in medical articles.)