Accelerated Failure Time (AFT) Model

Unlike hazard-based models, AFT models quantify how covariates accelerate or decelerate the event timeline. A covariate with a coefficient of 0.5 corresponds to an acceleration factor of exp(0.5) ≈ 1.65, meaning the subject experiences the event 65% faster than baseline.

• Interpretation: 'Treatment A extends median survival by 40%'
• Contrast: Hazard ratios describe instantaneous risk; AFT describes time displacement
• Clinical relevance: Directly answers 'how much longer' a patient is expected to live

AFT models require specifying a probability distribution for the baseline survival time. The choice of distribution fundamentally shapes the hazard function's behavior over time.

• Weibull: Monotonic hazard (increasing or decreasing), most common in medical applications
• Log-normal: Non-monotonic hazard that rises then falls, suitable for diseases with peak risk periods
• Log-logistic: Accommodates non-proportional hazards and crossing survival curves
• Gamma: Flexible shape with two parameters, often used as a robust alternative
• Exponential: Constant hazard (special case of Weibull), rarely realistic for biological processes

Model adequacy is assessed through standardized residuals that should approximate a known distribution if the parametric assumption holds.

• Cox-Snell residuals: Should follow a unit exponential distribution if the model is correctly specified
• Deviance residuals: Identify outliers and influential observations that disproportionately affect parameter estimates
• Martingale-like residuals: Detect non-linearity in covariate effects, guiding transformation decisions
• Q-Q plots: Visual comparison of ordered residuals against theoretical quantiles of the assumed distribution

AFT models naturally accommodate right-censored observations through the likelihood construction. Each censored subject contributes the probability of surviving beyond their observed follow-up time.

• Likelihood decomposition: Uncensored subjects contribute the probability density f(t); censored subjects contribute the survival function S(t)
• Assumption: Censoring must be non-informative — the censoring mechanism is independent of the event time given covariates
• Advantage: No special weighting or imputation required; censoring is handled intrinsically within the parametric framework

The choice between AFT and Cox models hinges on the research question and data characteristics.

• AFT advantage: Direct time interpretation; 'Treatment extends median survival by 6 months' is more clinically intuitive than 'Treatment reduces hazard by 30%'
• Cox advantage: No distributional assumption required; semiparametric flexibility
• When AFT fails: If the chosen distribution is misspecified, parameter estimates become biased and inconsistent
• Convergence: When the Weibull distribution is used and proportional hazards hold, AFT and Cox models are equivalent (differing only by a sign transformation on coefficients)
• Crossing survival curves: AFT models with log-normal or log-logistic distributions can accommodate scenarios where treatment effects reverse over time

AFT models are available across major statistical computing environments with varying distribution options.

• R survival package: survreg() function supports Weibull, exponential, log-normal, log-logistic, and Gaussian distributions
• Python lifelines: WeibullAFTFitter(), LogNormalAFTFitter(), and LogLogisticAFTFitter() classes with scikit-learn-compatible APIs
• SAS PROC LIFEREG: Enterprise implementation with CLASS statements for categorical variables and CONTRAST for hypothesis testing
• Stan/PyMC: Bayesian implementations allowing prior specification on acceleration factors and distribution parameters for full uncertainty quantification