Aitkin-Clayton (1980) The Fitting of Exponential, Weibull and Extreme Value Distributions to Complex Censored Survival Data using GLIM

Summary

This paper shows how to fit parametric proportional-hazards regression models — with exponential, Weibull, or extreme-value survival distributions — to right-censored survival data using ordinary Poisson generalized-linear-model software (GLIM). The key device is the "Poisson trick": the censored-data likelihood factorizes so that the censoring indicator behaves like a Poisson variate whose log-mean is the log-linear hazard model, so a survival regression can be estimated as a log-linear Poisson regression with $\log\Lambda(t)$ entered as a known offset. For the exponential case no iteration is needed; for Weibull and extreme-value a short iteration on the shape parameter suffices. A simultaneous test procedure is used to reduce a complex regression to a parsimonious model, illustrated on the Gehan (1965) leukaemia and Prentice (1973) cancer data.

Key Claims

• Log-linear (proportional) hazard. The hazard is $h(t_i)=\lambda(t_i)\exp(\beta'x_i)$, giving survival $S(t)=\exp(-\Lambda(t)e^{\beta'x})$ with $\Lambda(t)=\int_0^t\lambda(u)\,du$ — a parametric proportional-hazards model.
• The Poisson likelihood equivalence. With censoring indicator $w_i$ (1 = event, 0 = censored) and $\mu_i=\Lambda(t_i)\exp(\beta'x_i)$, the likelihood equals $\prod_i \mu_i^{w_i}e^{-\mu_i}\cdot[\lambda(t_i)/\Lambda(t_i)]^{w_i}$: the first factor is exactly the kernel of independent Poisson likelihoods for the $w_i$ with means $\mu_i$, and $\log\mu_i=\log\Lambda(t_i)+\beta'x_i$ is log-linear. So survival ML becomes Poisson GLM fitting.
• Exponential — no iteration. With $\Lambda(t)=t$, $\lambda/\Lambda=t^{-1}$ is parameter-free, so fitting reduces to a Poisson regression with $\log t_i$ as an offset (coefficient fixed at 1); a single GLM fit.
• Weibull and extreme value — one extra loop. With $\Lambda(t)=t^\alpha$ (Weibull) the ratio $\lambda/\Lambda=\alpha/t$ depends on the shape $\alpha$; alternate between the Poisson fit for $\beta$ (offset $\log\Lambda(t)$) and updating $\alpha$ until convergence. The extreme-value (Gumbel, log-Weibull) distribution for $\log t$ is handled the same way.
• Model reduction by simultaneous testing. Aitkin's (1978) simultaneous test procedure for complex cross-classifications is applied to choose a minimal adequate set of covariates without inflating error rates over many nested comparisons.

Concepts Introduced or Extended

• Parametric Survival Model
• Counting Process
• Link Function

Entities Mentioned

• Murray Aitkin
• David Clayton

Quotes

"The purpose of the present paper is to describe the use of GLIM… to fit exponential, Weibull or extreme value distributions, by expressing the likelihood in each case as a Poisson likelihood, with a log-linear model for the Poisson mean corresponding to the log-linear model for the hazard function."

My Take

This is a small paper with an outsized practical legacy: the observation that a censored parametric-survival likelihood is, up to a parameter-free factor, a Poisson likelihood is what let a generation of applied statisticians fit proportional-hazards models in whatever GLM software they already had, and it is the direct ancestor of the piecewise-exponential / Poisson-regression approach to survival still taught today. It sits a level above Aalen's martingale machinery in abstraction but complements it — Aalen gives the nonparametric cumulative hazard and its asymptotics, while Aitkin-Clayton give a parametric, regression-ready recipe. The obvious caveat is parametric risk: exponential/Weibull impose strong shape assumptions on the baseline hazard that Cox's partial likelihood avoids, so the method trades robustness for the convenience of standard GLM tooling.