Parametric Survival Model

Definition

A parametric survival model specifies the distribution of a time-to-event $T$ through a fully parametric hazard (or, equivalently, survival) function and estimates covariate effects by maximum likelihood, accounting for censoring. In the proportional-hazards form the hazard is $h(t\mid x)=\lambda(t)\exp(\beta'x)$ with a parametric baseline $\lambda(t)$ (exponential, Weibull, or extreme-value), giving survival $S(t\mid x)=\exp\!\big(-\Lambda(t)e^{\beta'x}\big)$, $\Lambda(t)=\int_0^t\lambda(u)\,du$. Aitkin-Clayton (1980) showed such models can be fit with ordinary Poisson generalized-linear-model software.

Key Ideas

• Baseline hazard families. Exponential: $\Lambda(t)=t$ (constant hazard, memoryless). Weibull: $\Lambda(t)=t^{\alpha}$ (monotone hazard, increasing/decreasing with the shape $\alpha$). Extreme value (Gumbel): the distribution of $\log T$ under a Weibull $T$ — a log-location-scale survival model.
• Proportional hazards, parametrically. Covariates act multiplicatively on the hazard, $h=\lambda(t)e^{\beta'x}$; $e^{\beta_j}$ is a hazard ratio. Unlike Cox's semiparametric model, the baseline $\lambda(t)$ is specified, so the full survival curve is estimated and extrapolation is possible.
• The Poisson trick. Writing the censored likelihood with event indicator $w_i$ and $\mu_i=\Lambda(t_i)e^{\beta'x_i}$ gives $\prod_i \mu_i^{w_i}e^{-\mu_i}$ times a parameter-free factor — the kernel of independent Poisson likelihoods with $\log\mu_i=\log\Lambda(t_i)+\beta'x_i$. Survival ML is thus a log-linear Poisson regression with $\log\Lambda(t)$ as an offset.
• Fitting. Exponential needs a single Poisson fit ($\log t$ offset, coefficient 1); Weibull/extreme-value alternate a Poisson fit for $\beta$ with an update of the shape $\alpha$ until convergence. Any GLM package suffices — the historical appeal of the method.
• Censoring. Right-censoring enters only through the indicator $w_i$ under the standard assumption that censoring is independent of the covariates, so the same machinery handles complex censored designs.
• Model selection. A simultaneous test procedure (Aitkin 1978) reduces a complex covariate cross-classification to a minimal adequate model while controlling the error rate over many nested comparisons.
• Bayesian nonparametric baseline (Kalbfleisch 1978). Instead of forcing an exponential/Weibull shape or leaving the baseline unspecified (Cox's partial likelihood), put a gamma-process prior on the cumulative baseline hazard $\Lambda_0$ — a random-measure (Bayesian nonparametric) prior — and integrate it out. This gives a marginal posterior for $\beta$ and an estimate of the baseline survival curve with honest uncertainty; Cox's partial likelihood is recovered as the diffuse-prior limit. The gamma process is the survival cousin of the Ferguson/Dirichlet-process priors, and the object it endows with a prior is exactly the cumulative hazard that Nelson-Aalen estimates frequentistically.

Why It Matters

• GLM tooling for survival. By recasting parametric survival as Poisson regression, the approach made proportional-hazards modeling accessible in general-purpose statistical software and is the direct ancestor of the piecewise-exponential / Poisson-regression survival methods used today.
• Interpretable, extrapolatable. A specified baseline hazard yields hazard ratios and absolute survival predictions beyond the observed follow-up — useful for actuarial and health-economic projection where Cox's baseline is left unspecified.
• Complements the counting-process view. It gives a parametric, regression-ready recipe sitting alongside the nonparametric counting-process / Nelson-Aalen estimator and its martingale asymptotics.

Open Questions

• Parametric misspecification. Exponential/Weibull impose strong shape assumptions on the baseline hazard; if wrong, hazard ratios are biased — the robustness cost relative to Cox's partial likelihood.
• Time-varying and non-proportional effects. Extending the Poisson formulation to time-dependent covariates or non-proportional hazards requires expanding the data into time intervals (the piecewise-exponential layout).
• Shape-parameter uncertainty. Standard errors for $\beta$ that ignore estimation of the Weibull shape $\alpha$ can be optimistic.

Related

• Counting Process
• Interval Censoring
• Link Function
• Cure Rate Model
• Sequential Ordinal Model
• Dirichlet Process Mixture
• Murray Aitkin
• David Clayton
• John D. Kalbfleisch
• Aitkin-Clayton (1980) The Fitting of Exponential, Weibull and Extreme Value Distributions to Complex Censored Survival Data using GLIM
• Kalbfleisch (1978) Non-parametric Bayesian Analysis of Survival Time Data
• Event History Analysis
• Lewis-Raftery (1999) Bayesian Analysis of Event History Models With Unobserved Heterogeneity via MCMC