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I want to predict an event based on transactional data. I'm only interested in predicting either future hazard or time-to-event. My understanding is that to predict in the future I have to use a fully-parameterized proportional hazard model with time-varying covariates. By "fully parameterized" I mean that I assume the underlying distribution of the event in time.

So the PH model has two parts: the baseline hazard (whose exact formula I assumed, so this is not a Cox model) and the effet of the covariates (e^(linear combination of the covariates)). The value of the second part (the effect of the covariates) is varying in time until now; after that, I assume the value of the covariates stays constant. As for the first part, the baseline hazard, I know its function, so I can extrapolate the value of the baseline hazard in the future.

My first question: does it make sense to make predictions in the future in that manner with a PH model?

Also, I'm interested in predicting not only the hazard but the time-to-event. For that I need to create an AFT model. My understanding is that even though theoretically possible it is computationally too intensive to create an AFT model with time-varying covariates. So I want to build an AFT model with the last slice in time of my transactional data set. But I want to take into account my whole history to get my parameters. The idea would be to obtain the survival function $S(t) = e^{(-h(t) \cdot t)}$ from the hazard function $h(t)$ of a fitted PH model. Isolate t in the formula and express it in function of $S(t)$ and $h(t)$: $T(t)=- \frac{ln(S(t))}{h(t)}$. Then I fix t at the last point in time, so the function doesn't depend on time but is using parameter estimates that were derived by taking time into account. It is important to note that I'm not actually fitting the AFT model. The reason why I'm trying to use the estimates obtained with the fully-parameterized PH model with time-varying covariates into a non-fitted AFT model is that I believe these betas are "better" since they were derived based on a larger data set that takes the time dimension into account.

My second question is: can I do that or is there a silly step in the above development?

Thanks in advance for your help and sorry for the long question!

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