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I've been learning about time-to-event analysis and playing with open datasets + fitting various Cox and parametric models for practice.

Other than by visually inspecting the estimated survival curves for each model against the Kaplan-Meier estimates, how can I compare goodness of fit across Cox and parametric models? I've been told I can't use deviance or information criteria because the models are fit by maximizing different likelihoods (partial likelihood for Cox, ordinary maximum likelihood for parametric). Are there other ways of comparing the goodness of fit across these different types of models?

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The importance of visually comparing predictions and observations shouldn't be underestimated. Comparisons against Kaplan-Meier curves, however, can be misleading as they either (a) don't take covariate values into account or, if they use covariate-defined subsets, (b) the numbers of cases for comparisons are severely limited unless you have a very large data set.

You can compare different model types on the same data set via bootstrapping. You fit each model type to multiple bootstrap samples of the data and then evaluate performance on the entire data set. Several numerical goodness-of-fit criteria can be evaluated. In particular, you can evaluate the calibration of model survival-probability predictions with respect to observations. That calibration is best assessed visually, to see how well the model works over the entire range of predictions.

See the following pages and their links for more details:

what is a good measure of goodness of fit for survival models that can be used for comparison between models?

How to do cross-validation with a Cox proportional hazards model?

Training, testing, validating in a survival analysis problem

Evaluate the performance of a model with bootstrap

Bootstrap with statistics for survival object

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