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I am trying to understand the relationship between AFT and Cox regression in terms of the Weibull distribution. Specifically, one can compute a factor for h(t) from the coefficients of an AFT using the shape parameter of the Weibull distribution, e.g. see https://rpubs.com/kaz_yos/aft. Does this mean that the application of AFT with Weibull is limited to cases where the proportional hazard assumption for CoxPH holds?

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As this source notes:

the Weibull is the only distribution that is closed under both the accelerated life and proportional hazards families.

So if you are working with a distribution in the Weibull family (that includes simple exponential models), then you can model with either a proportional hazard (PH) or an accelerated failure time (AFT) model, and you will still end up with a Weibull distribution. In that sense, whenever you use an AFT model with an underlying Weibull distribution, the PH assumption will hold--and vice versa. For distributions outside the Weibull family, at most of one of AFT or PH can hold.

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  • $\begingroup$ In other words, if I know PH is violated, I cannot use AFT with Weibull? $\endgroup$ Oct 16, 2020 at 14:33
  • $\begingroup$ @user11130854 if you know that PH doesn't hold, then the underlying survival distribution, as you parameterized the model, can't be Weibull. But sometimes an apparent violation of PH is due to mis-specification of the model, for example in terms of transformations of predictors. See this discussion for example. PH might hold if you model the predictors differently. You also have to consider, as a judgment call, whether the violation of PH is sufficiently large to negate the simplifications that a parametric Weibull model might provide. $\endgroup$
    – EdM
    Oct 16, 2020 at 14:55
  • $\begingroup$ @user11130854 if you think that an AFT model is appropriate, you could consider some of the distributions for which closure under AFT works but PH does not hold, such as the log-normal. There is, however, no semi-parametric equivalent to Cox PH regression for AFT models; you must specify a distribution for AFT and then determine how well the model represents the data and the underlying population. $\endgroup$
    – EdM
    Oct 16, 2020 at 14:59

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