Survival Regression

Cox regression is semi-parametric: it models covariate effects without specifying the baseline hazard shape, which makes it robust but prevents extrapolation. Survival regression is fully parametric: it specifies the hazard family (e.g., Weibull), enabling extrapolation and yielding interpretable AFT time-ratio coefficients at the cost of requiring a distributional assumption.

Censoring occurs when the event time is only partially observed — a patient leaves the study before experiencing the event, so only a minimum event time is known. Dropping censored observations biases results toward shorter survival times. Survival regression handles censoring correctly by including censored cases in the likelihood through the survival function S(t).

Compare competing distributions using AIC or BIC, and check fit with Cox-Snell residual plots: well-fitted models produce residuals that follow an exponential(1) distribution. The Weibull family is a common default because it nests the exponential and allows monotone hazard shapes; log-normal or log-logistic suit humped or declining hazard patterns.

Standard parametric survival regression assumes fixed covariates. Time-varying covariates can be accommodated by reformatting the data into interval-split episodes and using a piecewise exponential or similar model, but this substantially increases data complexity.

An AFT model re-parameterises the survival regression so that covariates multiply (or divide) the expected survival time rather than the hazard. A time-ratio coefficient of 1.5, for instance, means the covariate extends median survival by 50%. This is often more interpretable than a hazard ratio in clinical or policy contexts.