# How to write a Gompertz model as an accelerated failure time model?

Accelerated failure time models are usually defined as a model for $$X>0$$ with (constants) covariates $$Z\in \mathbb{R}^p$$ such that

$$$$\log(X) = \mu + \beta^*Z + \sigma W,$$$$ where $$W$$ is a random variable. The usual AFT models are Weibull, Loglogistic and Lognormal*. I am trying to implement Gompertz as an AFT model to get a model with exponential hazards, and since my data is atypically censored and truncated, I have given up trying to find an R package and just doing it myself. Some R packages have an implementation of AFT models that are Gompertz distributed, but reading the documentation and code; I have not been able to figure out what distribution $$W$$ needs to have.

I'm pretty sure that the Gompertz model is not exactly in the same form as above but on the form $$$$\log(X) = \mu + \beta^*Z + W$$$$ where $$W$$ is a distribution depending on a parameter $$\sigma$$ but not in the usual scaling way. Whether or not this is an AFT model seems to be up to definition (or taste), but it still has the interpretation of a proportional median survival time model (and all other quantiles), which is what I am interested in. I hope someone can help me, and it would be perfect if someone had a reference for an answer.

*See e.g. Survival analysis, techniques for censored and truncated data, John P. Klein & Melvin L. Moeschberger 2003 second edition, Science+Business Media, Inc series: Statistics for Biology and Health Springer.

• What is the question? Commented Feb 15 at 21:28

Although you can fit a parametric Gompertz regression model, it won't have a simple interpretation in terms of the proportional survival quantiles that AFT models provide. That type of interpretation depends on the form of $$\sigma W$$ being independent of the covariate values, with (1) effects of all of the covariate values incorporated into $$\beta^* Z$$ and (2) $$\beta^* Z$$ being linearly related to $$\log X$$. Once you make $$\sigma W$$ depend on covariate values (as seems implicit in your question), that simple interpretation in terms of acceleration of time goes away.
The Gompertz model does have a proportional hazards interpretation. Among parametric survival families, only the Weibull distribution is closed under both proportional-hazards and accelerated-failure-time models. The Gompertz model can be thought of as a log-Weibull distribution, with the log-hazard linear in $$X$$ rather than in $$\log X$$ as for the Weibull. See the Rodríguez course notes.