# Extended Cox model and cox.zph

I have previously had experience only with Cox PH model and its assumptions checking. Now for the first time I have my clients data with most of the covariates varying in time, only a few are fixed valued. I did set it up in start/stop format and I have also fitted some models, but cox.zph is showing very small p-values. My understanding is, that extended Cox model is not a PH model. Do I even need the cox.zph test for time-dependent variables? Or when including fixed covariates, then would I need to see large p-values for those or not?

• What does .zph mean? – Andy Apr 5 '15 at 16:32
• In R, the cox.zph function will test proportionality of all the predictors in the model by creating interactions with time. I have been using it for assessing Cox PH assumption (+plots), but I am confused about Extended Cox model with time-dependent covariates. – Finance Apr 5 '15 at 17:24

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