## Survival models v. GLMs?

At some point you may be challenged to decide whether to use survival models or the older generalised linear models (GLMs). You could be forgiven for thinking that the two were mutually exclusive, especially since some commercial commentators have tried to frame the debate that way.

In fact, survival models and GLMs are not necessarily mutually exclusive. It is true that GLMs are more commonly used for modelling the rate of mortality, *q*_{x}, whereas survival models are always used for modelling the force of mortality, *μ*_{x}. Indeed, a survival model can be defined as a model for *μ*_{x}.

However, there are GLMs for the force of mortality as well. One notable example is the Poisson model for the number of deaths, *D*_{x}, in a group with a total exposure time, *E*_{x} (the *waiting time* for statistician readers). This is written as follows:

* D*_{x} ∼ Poisson(*E*_{x}μ_{x})

and can be fitted as a standard generalised linear model (GLM) in any package, including the free software R. We thus have a model for the force of mortality, *μ*_{x}, which is therefore both a survival model *and* a GLM.

So we have a link from GLMs to survival models, but what about in the other direction? Well, there is a class of survival models known as *accelerated failure-time models*, where the lifetime of each individual is assumed to come from an exponential, Weibull, Lognormal or Log-logistic distribution. What unites these survival models is that the algorithm for fitting them is the same one which fits Poisson GLMs.

In short, there are clear and strong connections between survival models and GLMs, and to suggest that the two are separate is demonstrably false.

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