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In a survival analysis, is it possible to use a likelihood based measure like AIC to compare the fit obtained from a Cox model and a parametric regression model? I have seen different views, from (1) no you can't to (2) finding a variety of published papers that do so!

While the more I think about it the more I wonder what is the AIC in the Cox model (given that you do not assume a model for the distribution, how would you calculate a likelihood to begin with?), I was hoping that someone could provide a definitive answer to whether this AIC comparison is possible or not and, if not, what are the suggested ways to choose between the two approaches.

Many thanks in advance for any feedback sent this way.

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  • $\begingroup$ This might help - mailman.columbia.edu/research/population-health-methods/…. Good site for investigation of your question $\endgroup$
    – David Hall
    Commented Oct 10, 2017 at 16:53
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    $\begingroup$ A specific quote from that site that is directly on topic: "AIC cannot be used to compare parametric and semi-parametric models, however, since parametric models are based on observed event times and semi-parametric models are based on the order of event times." $\endgroup$
    – EdM
    Commented Oct 10, 2017 at 17:38
  • $\begingroup$ @DavidHall Many thanks - very useful resource $\endgroup$
    – Tiago Marques
    Commented Oct 11, 2017 at 8:28

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The AIC can only be used to compare results derived from the same sample for the same outcome estimated from the same procedure with differing independent variables. The AIC is based on maximum likelihood estimators intrinsic to a given model. One cannot compare between models as they do not share MLEs. For example, one can generate AIC for linear regression and Poisson regression models, which are not comparable. That Cox regression and parametric regression measure time-to-event make them no more comparable.

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  • $\begingroup$ This answer is incorrect. One can compare AIC across models with different responses (e.g. Poisson vs Gaussian), with our without the same covariates. What you can't do is compare models which have used different data. $\endgroup$
    – Tiago Marques
    Commented Oct 11, 2017 at 8:27
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    $\begingroup$ I can not locate a citation which firmly suggests that AIC can be compared for two distinct procedures. The following CV post suggests comparison is not feasible in this case. stats.stackexchange.com/questions/139201/… $\endgroup$
    – Todd D
    Commented Nov 25, 2017 at 21:59
  • $\begingroup$ @Tiago Marques: I am with Todd here. For comparability, the likelihoods must be defined with respect to the same dominating measure, so discrete (Poisson) and continuous cannot be compared $\endgroup$
    – kjetil b halvorsen ♦
    Commented May 18, 2022 at 2:21

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