I'm aware of testing the proportional hazards assumption in the context of the Cox PH models, but I haven't encountered anything relating to parametric models? Is there a feasible way to test the PH assumption of certain parametric models?
It seems like there should be given that parametric models are only slightly different from the semi-parametric Cox models?
For example, if I wanted to fit a Gompertz mortality curve (as below), how would I test for the PH assumption?
$$\begin{align}
\mu_{x}&=abe^{ax+\beta Z}\\
H_{x}(t)&=\int_{0}^{t}\mu_{x+t}\,dt=b(e^{at}-1)e^{ax+\beta Z}\\
S_{x}(t)&=\text{exp}(-H_{x}(t))
\end{align}$$
I suppose in general what I'm asking is: for parametric survival models, what are some ways of evaluating goodness of fit of the model and also testing for assumptions (if any) of the model?
Do I need to check for PH assumptions in a parametric model or is that just for Cox models?
Best Answer
A complete answer depends on the nature of your parametric survival model.
If your parametric model incorporates covariates in a way that the relative hazards for any 2 sets of covariates are in a fixed proportion over time (as your Gompertz model seems to), then your parametric model is making an implicit proportional hazards assumption that must be validated in one way or another. As this answer by @CliffAB points out for the specific baseline hazard assumed by a parametric model:
This would suggest that you first try a Cox survival regression to test proportionality of hazards. If the assumption is violated with the empirical baseline hazard determined by the Cox regression, then there is little point to proceeding with any parametric model that implicitly assumes proportional hazards. If you can proceed with such a parametric model, the R
survivalpackage provides several types of residuals for evaluating parametric models with theresiduals()method forsurvregobjects, in addition to the suggestions made by @Theodor.If, alternatively, your model incorporates some covariates in a way that provides for non-proportional hazards as functions of covariate values (e.g., different baseline hazard shapes), then there is no need to test specifically for proportional hazards with respect to those covariates. Stratifying on those covariates would allow tests of proportional hazards for covariates that are assumed to involve proportional hazards. You will of course need to test how well the data fit the assumptions of your model, but insofar as proportional hazards aren't assumed (explicitly or implicitly) then they don't need to be tested.
For further background, Harrell's Regression Modeling Strategies devotes chapter 18 to building and evaluating parametric survival models; more cryptic but useful coverage of this topic can be found in examples worked through in his freely available course notes.