cox.zph p-value indicates proportional hazards but plot shows non-flat and non-straight line

Question

I created a Cox model using the following code:

final_model = coxph(Surv(time, death) ~ WDist + age + gender + bmi)
summary(final_model)

Call:
coxph(formula = Surv(time, death) ~ WDist + age + gender + bmi)

  n= 504, number of events= 153 

             coef  exp(coef)   se(coef)      z Pr(>|z|)    
WDist  -0.0074495  0.9925782  0.0008042 -9.263  < 2e-16 ***
age     0.0670862  1.0693877  0.0137782  4.869 1.12e-06 ***
gender -2.0294116  0.1314128  0.7269030 -2.792  0.00524 ** 
bmi    -0.0475984  0.9535167  0.0185177 -2.570  0.01016 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

       exp(coef) exp(-coef) lower .95 upper .95
WDist     0.9926     1.0075   0.99101    0.9941
age       1.0694     0.9351   1.04090    1.0987
gender    0.1314     7.6096   0.03162    0.5462
bmi       0.9535     1.0487   0.91953    0.9888

Concordance= 0.757  (se = 0.021 )
Likelihood ratio test= 138.3  on 4 df,   p=<2e-16
Wald test            = 140.6  on 4 df,   p=<2e-16
Score (logrank) test = 139.8  on 4 df,   p=<2e-16

After that, I decided to check the proportional hazards assumption using cox.zph, so I used the following code:

ph_test <- cox.zph(final_model, transform="rank")
ph_test 


       chisq df    p
WDist  0.289  1 0.59
age    1.748  1 0.19
gender 0.368  1 0.54
bmi    0.479  1 0.49
GLOBAL 3.163  4 0.53

Looking at the p-values we can claim that each of the values satisfy the proportional hazard assumption. However, when I plot that I get the following result for the "age" variable:

plot(ph_test)

The plot shows non-flat and non-straight line for the "age" variable, what contradicts the previous p-value result. Why is that? What am I missing? How to interpret this results?

Answer 1

Three thoughts in no particular order.

First, your choice of transform="rank" puts an awful lot of visual emphasis on late times when there are few events and thus little information. That can lead to the appearance of problems when there aren't really any if you take all the data into account. The default "km" choice is often more informative.

Second, although there is some waviness in the smoothed plot, it's not clear that the magnitude of waviness is very large compared to the confidence limits of the smooth. As with normality testing, you almost never get perfectly proportional hazards (PH) in practice. You have to apply your knowledge of the subject matter to determine if the deviation from proportionality is large enough to matter.

Third, an apparent lack of PH with a continuous predictor can arise from improper specification of the shape of its association with outcome. See this thread. Spline fits for age (with pspline() or natural/restricted cubic spline terms) might remove this apparent problem while allowing the data to tell you the shape of the association.