In order to verify the assumption of proportional hazards, I plotted the following: $$\log(-\log(S(t))) = \log(\Lambda(t))$$
If the hazards are indeed proportional for two groups, these curves should be parallel with the vertical distance being the log hazard ratio (which is constant if the proportional hazards assumption holds).
I used the following R code to make such a plot (using the
foo <- survfit(Surv(TIME, HADEVENT)~CASE, data=bar) survplot(foo, loglog=T, xlim=c(0,5*365.25), ylim=c(-10,-2), col=c("red","blue"))
I obtained the figure shown below for the groups in my study, and I am unsure what to make of it. For small values of $t$ ($t \leq 120$ days), the proportional hazards assumption does not hold at all based on visual inspection. For larger values ($ \gt 1$ year), it seems to be very reasonable.
I also fitted a Cox PH model and used
cox.zph to assess the proportionality assumption, with the following results:
foocox <- coxph(Surv(TIME, HADEVENT)~CASE, data=bar) print(cox.zph(foocox)) rho chisq p CASE -0.0372 46 1.18e-11
Based on the $p$-value, I conclude that the null hypothesis of proportionality is strongly rejected. Here is the
cox.zph plot, which also doesn't look too promising (e.g. it should be constant):
Here's a frequency table for my data (
CASE=0 is a 1:1 matched sample for case
CASE event no event n 0 10.464 93.832 104.296 1 22.903 81.393 104.296
Edit: after filtering out very large times -- which were inaccurate -- I obtain a highly significant $p$-value via
cox.zph, indicating the proportionality assumption is violated.
What are my options given this situation? Is it justified to continue working with proportional hazards models based on the graph of
cox.zph for larger values of $t$? Should I somehow refrain from using them for $t \lt 1$ year based on the visual inspection?