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I'm able to obtain predicted survival probabilities of cox regression using either survfit.coxph or predictSurvProb from pec package. However, using these approaches I'm unable to predict the probability of new observation which has the time set after the study, or the end time of dataset used to construct the model, e.g. the period of study is from 0 to 1000, and the time for new observation is 1200.

Is this the limitation of cox regression? Otherwise, is there any alternative to calculate the aforesaid probability?

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It is a natural limitation of Cox regression, because it avoids the assumption of any particular parametric baseline hazard function so that there is no in-buildt assumption of what would happen beyond the time observed. Additionally, not that even at the end of your observation period (where few subjects are still at risk) predictions will be not great (i.e. variable and with large uncertainty). This is a consequence of wanting an analysis that is more assumption free than a parametric survival model and (besides the loss in efficiency that may not be all that bad in many cases) a price one pays for using a Cox model.

One very logical approach for prediction is to use a sufficiently flexible parametric survival model (e.g. Weibull), which makes it (mathematically) pretty easy to predict beyond the observation period. However, the further you predict beyond that, the more I would to start to worry about how justifiable the parametric model assumption are (e.g. is there a non-susceptible proportion or large between subject variability in frailty or does something fundamental happen that would change the hazard rate like a new process for events becoming relevant).

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