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A Cox Proportional Hazard model outputs both the coefficients $\beta$ and the baseline hazard $\lambda_0$ (eg, Breslow's method). From these two quantities one can estimate the distribution of time to event ($T$) for a specific subject with covariate vector $x_i$, ie, $\hat{S}(t|x_i)$ (or $\hat{S}_i(t)$).

However, this estimated distribution is truncated: it is not defined beyond the study period. That is, if a study lasts 10 years, then $\hat{S}_i(t)$ is not defined beyond 10 years, even though $T_i$ can have nonzero probability mass beyond 10 years (ie, $\hat{S}_i(10) \ne 0$).

This is fundamentally due to the nonparametric nature regarding the $\lambda_0$ part of the model. In order to estimate the full distribution of $T_i$, defined on $[0, \infty)$, one has assume a parametric form for $\lambda_0$ (amounting to assuming a specific distribution for $T_i$).

My questions are: What are the choices of the parametric form for $\lambda_0$, and what are their pros and cons? What are the most expressive parametric models? (My dataset is large so can cope with a model with many parameters.)

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  • $\begingroup$ If you fixed a model based on the data from 0-3 years old children, my guess that probability of 60 year old man living for other 100 year > 90% if exploration based on fitted model. $\endgroup$ – user158565 Oct 6 '18 at 5:11
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Indeed to be able to extrapolate outside the study period you need to use a parametric model. Some standard choices are the Weibull, the log-family (i.e., log-normal, log-logistic, log-Student's-t), and the Gamma distribution. All these have two parameters (actually the Student's-t has three because of the degrees of freedom). There are also the generalized Weibull and Gamma distributions that have three parameters, and several other proposals in the literature, see, e.g., here. It all boils down to which of these distributions provides the best fit to your data.

However, note that extrapolation can be dangerous. Even if you have a lot of data in your specific study period, say from 0 to 10 years, you don't have any data at later time points. Hence, you will be able to predict what happens at e.g., at 11 or 15 years by making the (strong) assumption that the shape of the distribution from the first 10 years can be extrapolated beyond this period. Typically, for a relatively short period (e.g., 1-2 years) it should be OK, but for if you want to extrapolate further than that can be far away from the truth.

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