How can I compare parametric and semiparametric survival models?

On a given dataset, I am running a semiprametric Cox proportional hazards model, together with a series of parametric models (Weibul, gamma, lognormal, exponential, etc.). How can I know which is better? During my PhD, I was told by methodologists that you cannot use Likelihood-based statistics to compare parametric and semiparametric duration models. (However, some papers actually employed AIC and other measures to make this comparison.)

What can I use to compare parametric and semiparametric models? Any help is appreciated.

• Due to the way the AIC-criterion is defined, parametric and semi-parametric survival models are not comparable via AIC. I think the (easiest) sensible way of comparing parametric and semi-parametric survival models would be through some kind of measure based on predictions such as Brier score, AUC, etc. If you like to read some mathematics, you could also read link.springer.com/article/10.1007%2Fs10985-018-9450-7, where the authors develop an information criterion for these kinds of comparisons. – Simon Boge Brant Feb 4 at 18:39
• Thank you for the suggestions, I never heard of these measures. This raises two more questions: 1) What are the reasons of incompatibility of AICs? 2) Those papers that compare parametric and semiparametric models using AIC, BIC, etc... are they junk science? – Leevo Feb 4 at 21:03
• AIC is a comparison based on the likelihood. The "trick" in Cox regression is to write out the likelihood in terms of a hazard of the form $\alpha(t\vert x) = \alpha_0(t)e^{\eta(x)},$ so that $\alpha_0(t)$ can be factored out. Due to this, what you end up with maximising is not an actual likelihood, but a partial likelihood, and therefore the comparison between the Cox partial likelihood and the likelihood of a parametric model does not make sense, they are not on the same scale. – Simon Boge Brant Feb 5 at 16:23
• It seems to me that they do not justify the comparison between these models via the AIC, or discuss some method of making the AIC scores comparible (I do not know if this is possible), so that part of the paper certainly looks strange to me. The conclusions may be valid for all I know, but it seems like the authors of that paper, and the reviewers (if there were any) have insufficient knowledge of quite basic statistics. – Simon Boge Brant Feb 5 at 16:30