# Cox Model and proportional hazards

I'm trying to fit a Cox model, but there is some problems. I have the following variables in the model.

• Group: 1, 2, ..., 9

• Sex: 1 female and 0 male

• Weight

• Age

The first thing that I did is split the variables Age and Weight in 4 different groups and check if the assumption of proportional hazards is met for each variable. I did the plot of $-log(-log S(t))\times t$.

The plot below is from Groups

For all the four variables the proportional hazard assumption is violated (crossed curves). Then I check it with hypothesis test and run the model

model<-coxph(Surv(Time,Event)~ Group + Sex + Weight + Age,data= dataset)
summary(model)
coef     exp(coef)  se(coef)   z     Pr(>|z|)
G2  0.1705602  1.1859691  0.1956226  0.872 0.383272
G3 -1.0036611  0.3665351  0.2386762 -4.205 2.61e-05 ***
G4 -0.8381683  0.4325020  0.2399613 -3.493 0.000478 ***
G5 -0.4544249  0.6348130  0.2092611 -2.172 0.029888 *
G6 -0.9123168  0.4015927  0.3471589 -2.628 0.008590 **
G7 -0.9977854  0.3686950  0.2413699 -4.134 3.57e-05 ***
G8 -1.7056585  0.1816527  0.3097035 -5.507 3.64e-08 ***
G9 -1.1614730  0.3130248  0.2488757 -4.667 3.06e-06 ***
Sex    -0.0307328  0.9697347  0.1331374 -0.231 0.817443
Weight 0.0004572  1.0004573  0.0004121  1.109 0.267330
Age    0.0044168  1.0044266  0.0036702  1.203 0.228815


From the summary of model, Sex, Weight, Age are not significant. Then the model just have groups as variables.

So I did

cox.zph(model,transform="rank",global=TRUE)

rho   chisq        p
G2 -0.1142  4.2426 0.039423
G3 -0.1732 10.6197 0.001119
G4 -0.0989  3.2302 0.072293
G5 -0.1588  8.7741 0.003055
G6 -0.1284  5.4636 0.019416
G7 -0.0508  0.9136 0.339165
G8  0.0984  3.3136 0.068709
G9 -0.1062  4.1598 0.041395
Sex    0.0085  0.0242 0.876276
Weight     0.1121  5.1191 0.023664
Age      -0.0109  0.0372 0.846986
GLOBAL         NA 36.2568 0.000153


I don't understand well this output, Group7 have prorportional hazard alone? How Sex, Age have proportional hazards if the curves of plot crossed?

If one level of categorical variable not hold the proportional hazard assumption, then the categorical variable not met the assumption right?

I made a several tests about proportionality, with graphs and tests with time dependent covariates, and in fact this assumption is not met, but I adjusted a stratified cox model by groups, and the output is below

                 coef  exp(coef)   se(coef)      z Pr(>|z|)
Sex -0.0295480  0.9708843  0.1331459 -0.222    0.824
Weight    0.0004545  1.0004546  0.0004111  1.105    0.269
Age     0.0043919  1.0044016  0.0036679  1.197    0.231
exp(coef) exp(-coef) lower .95 upper .95
Sex    0.9709     1.0300    0.7479     1.260
Weight     1.0005     0.9995    0.9996     1.001
Age       1.0044     0.9956    0.9972     1.012

Concordance= 0.532  (se = 0.045 )
Rsquare= 0.001   (max possible= 0.719 )
Likelihood ratio test= 3.11  on 3 df,   p=0.3745
Wald test            = 3.14  on 3 df,   p=0.3712
Score (logrank) test = 3.14  on 3 df,   p=0.3712


Here what I see is:

• The variables are not statistically significant

• The effects (hazards) of each variable are really closed to 1 for Weight and Age and for Sex a litle less. Then this variables have no effect on the survival time.

So I have no reason to keep them in the model, which would leave me with only the variable group that does not meet the proportionality hypothesis.

I begin to think that a parametric model is the best option for this case.

Also, be careful about how you interpret the p-values for the cox.zph tests. A low p-value for a coefficient is evidence that the proportional hazards assumption doesn't hold for its associated predictor, but a "non-significant" p-value is not proof that the proportional hazards assumption is met. As with any statistical test, a non-significant p-value might simply mean two few cases or too much variability to argue against the null hypothesis of a proportional hazard. It's hard to tell from your graph, but that might explain why crossing plots have p-values that do not rule out the PH assumption.
• @Roland : If PH is violated for Groups of interest then the Cox model wouldn't be appropriate. A completely non-parametric test, like that provided by survdiff(), could document that the Groups have different survival curves. A fully parametric model may have difficulties of its own, as there will be many parameters to fit and you might be at risk of overfitting. – EdM Nov 10 '16 at 15:01