Accelerated Failure Time Models: When Cox Doesn't Fit

Accelerated Failure Time (AFT) models offer a fundamentally different approach to survival analysis compared to Cox Proportional Hazards. Where Cox models the hazard rate, AFT models survival time directly.

When Cox Doesn't Work

The Cox regression requires proportional hazards: the hazard ratio between groups must be constant over time. When this assumption fails — for example, when a treatment effect wears off or when survival curves cross — Cox estimates become unreliable.

Common signs of non-proportional hazards:

• Schoenfeld residuals show a significant time trend
• Log-log survival plots are not parallel
• The effect makes clinical/product sense only for a limited period (e.g., an onboarding intervention helps early but not late)

How AFT Models Work

Instead of modeling the hazard rate, AFT models assume that covariates accelerate or decelerate the time to the event. The model takes the form:

$\log(T) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \ldots + \sigma \epsilon$

Where $T$ is survival time, $X$ are covariates, and $\epsilon$ follows a specified distribution. Exponentiating the coefficients gives time ratios.

Choosing a Distribution

Compare candidate distributions using AIC or BIC. The Weibull model is usually a safe starting point.

AFT vs. Cox: A Practical Comparison

Example

You are analyzing time from trial start to paid conversion. Cox regression shows Schoenfeld residual trends (non-proportional hazards) because the effect of a promotional offer diminishes after 14 days.

Fitting a Weibull AFT model, you find that users who received the offer have a time ratio of 0.72 — they convert 28% faster on average. This is more useful than a hazard ratio that changes over time.