Survival › Parametric fits

Parametric survival fit

A parametric accelerated-failure-time (AFT) model fits survival times to a chosen distribution — Weibull, exponential, log-normal, log-logistic, Gaussian or gamma.

What is Parametric survival fit?

Parametric survival models assume a specific distribution for survival times. The AFT parametrisation models log(T) as a linear function of covariates with a distributional error: log(T) = α + β'X + σ·W. Each distribution corresponds to a specific shape for W (e.g. extreme-value for Weibull, logistic for log-logistic).

Trade-offs vs Cox PH: parametric models can extrapolate beyond the observed time range (Cox can't) and are more efficient when the distribution is correctly specified. But you have to pick the distribution — compare candidate fits by AIC and check Q-Q plots of residuals.

Weibull is the most flexible monotonic-hazard option; log-normal and log-logistic accommodate non-monotonic hazards (rising then falling); exponential is constant-hazard and rarely realistic but sometimes acceptable for short windows.

When should I use Parametric survival fit?

Predicting beyond observed follow-up (Cox can't extrapolate).
When you want survival probabilities at any t, not just an HR.
Reliability / engineering applications where the underlying hazard family is known.

What data does it need?

Time + event status + distribution (Weibull / exponential / log-normal / log-logistic / gaussian / gamma).

What does it report?

Distribution-specific interpretable parameters (shape, rate, median, etc.) + native regression coefficients + logLik / AIC / BIC.

What does it assume?

Independent observations.
Non-informative censoring.
Survival times follow the assumed distribution (check via residual Q-Q or compare AIC across distributions).

How do I interpret the result?

Compare AIC across distributions; pick the lowest as the best-fitting family. Differences < 2 AIC ⇒ models are essentially indistinguishable.

Weibull shape k > 1 ⇒ increasing hazard, k < 1 ⇒ decreasing, k = 1 ⇒ exponential (constant).

References

Collett (2015). Modelling Survival Data in Medical Research, 3rd ed.