Extended Cox model and cox.zph

This question deserves a more up-to-date answer on a few accounts. First, the cox.zph() function has substantially changed with recent versions of the survival package, so there might be confusion with outputs not looking the same. Second, there can be some hidden "gotchas" when you are dealing with time-dependent covariates, as in this question. Third, although much of another answer is fine, there is a serious error in one of the proposed ways to specify time-dependent coefficients. Finally, the proper way to deal with that last problem makes it impossible (currently, at least) to use cox.zph() to check proportional hazards (PH) in the final model.

1. For many years the cox.zph() function performed its tests of PH with an approximation, the correlation coefficient between scaled Schoenfeld residuals and (possibly transformed) time. That correlation coefficient was reported as "rho", as shown in another answer. Since Version 3.0-10 of the package, cox.zph() is now an exact score test. There is no longer a value of "rho" to report.

2. With time-dependent covariates there can be a problem with causality. For example, I recently helped analyze some data in which patients' use of a drug prescribed for chronic conditions was included as a covariate. As people get older they are increasingly likely to be using that drug. To include that drug as a time-dependent covariate would be problematic, as it might just be a marker of already having survived longer. A time-dependent covariate can too easily be a proxy for longer survival, which (in addition to the causality problem) might show up as a PH problem. I suspect that might have been part of the problem in the initial question here. To quote from the time-dependent vignette by Therneau, Crowson and Atkinson:

The key rule for time dependent covariates in a Cox model is simple and essentially the same as that for gambling: you cannot look into the future.

3. Time-dependent coefficients can help with PH problems whether or not there are time-dependent covariates. Modeling coefficients with step functions as a function of time, one of the approaches proposed in another answer, is valid. As @bandwagoner notes in a comment on that question, the other proposed approach, a covariate-time interaction, is not. Quoting again from the vignette:

This mistake has been made often enough th[at] the coxph routine has been updated to print an error message for such attempts. The issue is that the above code does not actually create a time dependent covariate, rather it creates a time-static value for each subject based on their value for the covariate time; no differently than if we had constructed the variable outside of a coxph call. This variable most definitely breaks the rule about not looking into the future, and one would quickly find the circularity: large values of time appear to predict long survival because long survival leads to large values for time.

4. The survival package provides a correct way to specify coefficients as arbitrary functions of time, through a user-defined tt() function. Unfortunately, as the NEWS file for the package says, from version 3.1-2 "The cox.zph command now refuses models with tt() terms, before it had an incorrect computation." So for now it seems that evaluation of time-dependent coefficients will depend on how well the user-defined tt() function matches the form of the time-dependency seen with the time-independent coefficients and on other graphical evaluations.

An extended Cox model is really technically the same as a regular Cox model. If your data set is properly constructed to accommodate time dependent covariates (multiple rows per subject, start and end times etc..), than cox.ph and cox.zph should handle your data just fine.

Having time dependent covariates doesn't change the fact that you should check for proportionality assumption, in this case using the Schoenfeld residuals against the transformed time using cox.zph. Having very small p values indicates that there are time dependent coefficients which you need to take care of.

Two main methods are (1) time interactions and (2) step functions. The former is easier to do and to read, but if it does not change the p values than use the later:

Note that it would be easier if you provided your own data, so the following is based on sample data I use

(1) Interaction with time

Here we use simple interaction with time on the problematic variable(s). Note that you don't need to add time itself to the model as it is the baseline.

> model.coxph0 <- coxph(Surv(t1, t2, event) ~ female + var2, data = data)
> summary(model.coxph0)
                      coef  exp(coef)   se(coef)      z Pr(>|z|)
female           0.1699562  1.1852530  0.1605322  1.059    0.290
var2            -0.0002503  0.9997497  0.0004652 -0.538    0.591

Checking for proportional assumption violations:

> (viol.cox0<- cox.zph(model.coxph0))
                   rho chisq      p
female          0.0501  1.16 0.2811
var2            0.1020  4.35 0.0370
GLOBAL              NA  5.31 0.0704

So var2 is problematic. lets try using interaction with time:

> model.coxph0 <- coxph(Surv(t1, t2, event) ~
            + female + var2 + var2:t2, data = data)
> summary(model.coxph0)
                      coef  exp(coef)   se(coef)      z Pr(>|z|)   
female           1.665e-01  1.181e+00  1.605e-01  1.038  0.29948   
var2            -1.358e-03  9.986e-01  6.852e-04 -1.982  0.04746 * 
var2:t2          5.803e-05  1.000e+00  2.106e-05  2.756  0.00586 **

Now lets check again with zph:

> (viol.cox0<- cox.zph(model.coxph0))
            rho chisq     p
female   0.0486 1.095 0.295
var2    -0.0250 0.258 0.611
var2:t2  0.0282 0.322 0.570
GLOBAL       NA 1.462 0.691

As you can see - that's the ticket.

(2) Step functions

Here we create a model devided by time segments according to how the residuals are plotted, and add a strata to the specific problematic variable(s).

> model.coxph1 <- coxph(Surv(t1, t2, event) ~ 
                      female + contributions, data = data)
> summary(model.coxph1)
                                  coef exp(coef)  se(coef)     z Pr(>|z|)    
    female               1.204e-01 1.128e+00 1.609e-01 0.748    0.454    
    contributions        2.138e-04 1.000e+00 3.584e-05 5.964 2.46e-09 ***

Now with zph:

> (viol.cox1<- cox.zph(model.coxph1))
                        rho chisq        p
female               0.0296  0.41 5.22e-01
contributions        0.2068 21.31 3.91e-06
GLOBAL                   NA 22.38 1.38e-05

> plot(viol.cox1)

So the contributions coefficient appears to be time dependent. I tried interaction with time that didn't work. So here is using step functions: You first need to view the graph (above) and visually check where the lines change angle. Here it seems to be around time spell 8 and 40. So we will create data using survSplit grouping at the aforementioned times:

sandbox_data <- survSplit(Surv(t1, t2, event) ~ 
                      female +contributions,
                      data = data, cut = c(8,40), episode = "tgroup", id = "id")

And then run the model with strata:

> model.coxph2 <- coxph(Surv(t1, t2, event) ~ 
                          female + contributions:strata(tgroup), data = sandbox_data)
> summary(model.coxph2)
                                          coef exp(coef)  se(coef)     z Pr(>|z|)    
female                               1.249e-01 1.133e+00 1.615e-01 0.774   0.4390    
contributions:strata(tgroup)tgroup=1 1.048e-04 1.000e+00 5.380e-05 1.948   0.0514 .  
contributions:strata(tgroup)tgroup=2 3.119e-04 1.000e+00 5.825e-05 5.355 8.54e-08 ***
contributions:strata(tgroup)tgroup=3 6.894e-04 1.001e+00 1.179e-04 5.845 5.06e-09 ***

And viola -

> (viol.cox1<- cox.zph(model.coxph1))
                                        rho chisq     p
female                               0.0410 0.781 0.377
contributions:strata(tgroup)tgroup=1 0.0363 0.826 0.364
contributions:strata(tgroup)tgroup=2 0.0479 0.958 0.328
contributions:strata(tgroup)tgroup=3 0.0172 0.140 0.708
GLOBAL                                   NA 2.956 0.565