Accelerated Failure Time Models Explained

An accelerated failure time (AFT) model asks a different question than a Cox model does. Cox regression asks how covariates multiply the instantaneous hazard at each moment. An AFT model asks how covariates multiply time itself — whether a covariate accelerates or decelerates a subject’s path toward the event, directly, on the same time scale the event was measured in. That is not a stylistic difference in how results get reported; it is a different model form, with a different assumption to check and a different failure mode when that assumption is wrong.

This guide covers the AFT parameterization itself — the model equation, why the time-ratio interpretation is not just a relabeled hazard ratio, when a violated proportional-hazards (PH) assumption makes AFT the better fit, and how to choose between the common AFT distributions (Weibull, log-normal, log-logistic). For the broader question of which survival method to reach for in the first place, see Survival Analysis: Choosing Between Kaplan-Meier, Cox, and Parametric Models; for the mechanics of the model AFT is usually compared against, see The Cox Proportional Hazards Model and How to Interpret a Hazard Ratio.

The AFT model, specified directly

An AFT model is a linear model for the logarithm of survival time:

log(T) = β₀ + β₁x + σε

where T is the event time, x is a covariate, β₁ is the coefficient to be estimated, σ is a scale parameter, and ε is a random error term whose distribution determines which AFT model you’re fitting (extreme-value error gives the Weibull AFT model, normal error gives log-normal, logistic error gives log-logistic). Exponentiate both sides and the covariate effect shows up as a direct multiplier on time:

T = exp(β₀) × exp(β₁x) × exp(σε)

exp(β₁) is the acceleration factor (sometimes “time ratio”): a one-unit increase in x multiplies expected survival time by exp(β₁). An acceleration factor of 1.5 means a one-unit increase in x is associated with survival time 1.5× as long — not a 1.5× change in risk at any instant, a 1.5× change in the time it takes to get there. An acceleration factor below 1 shortens time to the event (accelerates failure, hence the model’s name); above 1 lengthens it (decelerates failure).

Why this is a genuinely different question, not a relabeled hazard ratio

It’s tempting to treat a time ratio as just a hazard ratio with different units. It isn’t. A Cox hazard ratio is a statement about the instantaneous rate of the event at each moment, held constant in the PH model regardless of how long a subject has already survived. A time ratio is a statement about the whole survival distribution being rescaled on the time axis: if a covariate has acceleration factor 1.5, it stretches the entire survival curve by that factor — not just the median, every percentile. The 10th-percentile survival time is 1.5× as long, the 90th-percentile survival time is 1.5× as long, same constant factor throughout.

That “same ratio at every quantile” property is exactly what makes AFT coefficients easy to communicate to a non-statistical audience — “this exposure shortens median time-to-failure by 30%” is a directly meaningful sentence in a way “the hazard ratio is 1.8” usually isn’t outside a clinical-trial readership. It’s also why AFT is the natural framework whenever the deliverable actually is a time or probability at a specific horizon (equipment-warranty life, expected time to grant closeout, a health-economic model’s lifetime survival projection) rather than a relative-risk comparison.

When the proportional-hazards assumption fails and AFT fits better

Cox regression’s central assumption is that the hazard ratio between any two covariate values stays constant across the entire follow-up period. When it doesn’t — a treatment effect that fades or grows over time, Kaplan-Meier curves that visibly cross rather than staying separated, a Schoenfeld-residual test that flags a significant time trend — a standard Cox model is misspecified for that covariate. The Cox model guide covers the Schoenfeld-residual check itself and the within-Cox fixes (stratification, time-varying coefficients) worth trying first.

AFT becomes the better fit, rather than just an alternative, in a few recurring situations:

A non-monotonic hazard. Risk that rises then falls — common after surgery, or during an acute infection window — is exactly the shape a log-normal or log-logistic AFT model can represent natively and a standard Cox or Weibull-PH model cannot.
PH fixes don’t hold either. If stratifying or adding a time-by-covariate interaction to the Cox model still leaves a poor fit, the data may simply not be generated by a proportional-hazards process at all — forcing a PH structure onto it produces a hazard ratio that’s technically estimable but doesn’t summarize anything stable.
The deliverable is a time or probability, not a ratio. Even when PH holds fine, if the actual question is “how long until,” AFT (or any parametric model) answers it directly; Cox’s semi-parametric form deliberately leaves the baseline hazard unspecified, so it can’t produce an absolute time prediction without an extra estimation step.

The trade a Cox user makes by moving to AFT: you give up the “don’t have to guess the hazard shape” convenience of the semi-parametric Cox form in exchange for an interpretable, extrapolable time-scale model — but now the distributional choice itself is a real decision you have to defend, which the next section covers.

Choosing a distribution: Weibull, log-normal, log-logistic

Every AFT model needs an assumed distribution for the error term ε. The three that come up most often in practice, and what distinguishes them:

Weibull — the one distribution that is both AFT and PH

The Weibull distribution has a property no other common survival distribution shares: it can be parameterized as either an AFT model or a proportional-hazards model, describing the exact same data either way. That makes it the natural bridge between the two framings — and the relationship between the two sets of coefficients is a direct, derivable formula, not a coincidence.

For a Weibull model with shape parameter k (constant across covariate values, the AFT assumption), the hazard function under covariate x is h(t|x) = (k/λ(x)) × (t/λ(x))^k−1, where λ(x) = λ₀ × exp(β₁x) is the AFT scale — exactly the “covariate multiplies time” form above. Dividing that hazard by the baseline hazard h₀(t) at x=0 cancels every t-dependent term and leaves:

Hazard ratio = exp(β₁)^−k = AF^−k

which is constant in t — confirming the Weibull model is simultaneously a valid PH model with hazard ratio AF^−k. A verified numeric illustration (Weibull shape k=1.5, illustrative acceleration factor AF=1.5, computed directly from the formulas above, not fitted to any real dataset):

Time (months)	Hazard, baseline	Hazard, AF=1.5 group	Ratio (h1/h0)
2	0.0671	0.0365	0.5443
10	0.1500	0.0816	0.5443
20	0.2121	0.1155	0.5443
30	0.2598	0.1414	0.5443

The ratio holds constant across every time point checked, matching the closed-form prediction exp(β₁)^−k = 1.5^−1.5 = 0.5443 exactly. Meanwhile the AFT reading of the same model holds too: median survival time is λ(ln 2)^1/k, giving 7.83 months at baseline and 11.75 months in the AF=1.5 group — a ratio of exactly 1.5, matching the acceleration factor, at the median and (by the same algebra) at every other percentile.

Log-normal — AFT-native, no PH equivalent

Log-normal error (T is log-normally distributed) produces a hazard that characteristically rises from zero, peaks, then declines — a shape that cannot be written as a proportional-hazards model in closed form. It’s a common choice when the failure mechanism has a genuine “wear-in then decline” or acute-risk-then-recovery shape and there’s no reason to expect a monotonic hazard.

Log-logistic — AFT-native, with a proportional-odds property

Log-logistic error gives a similarly non-monotonic hazard (rising then falling, or, for some parameter values, monotonically declining), and is the AFT distribution used most often in practice for exactly that flexibility. It carries a second useful property that Weibull’s PH duality has as its counterpart: log-logistic AFT models are also proportional-odds models on the survival scale. For survival function S(t|x) = 1 / (1 + (t/μ(x))^p), the odds of surviving past t are (μ(x)/t)^p, and the ratio of those odds between two covariate values reduces to AF^p — constant in t, the same cancellation trick as the Weibull case above. A verified numeric check (log-logistic shape p=2, illustrative AF=1.4, again computed directly, not fitted):

Time (months)	Odds, baseline	Odds, AF=1.4 group	Odds ratio
2	16.00	31.36	1.96
8	1.00	1.96	1.96
20	0.16	0.3136	1.96

The odds ratio holds at 1.96 — matching AF^p = 1.4² = 1.96 — across every time point, confirming the proportional-odds reading of the same underlying AFT model.

Picking one in practice

Start from the hazard shape the failure mechanism implies: roughly constant risk points to exponential; monotonically rising or falling risk points to Weibull; a rise-then-fall pattern points to log-normal or log-logistic.
Compare candidate distributions on AIC/BIC once fitted, and check the fit visually — a quantile-quantile plot of the fitted distribution against the nonparametric Kaplan-Meier estimate, or a linearity check on the appropriate transformed scale (log-time vs. standard-normal quantiles for log-normal, log-odds vs. log-time for log-logistic).
If Weibull fits reasonably well, it’s often the default choice specifically because it’s reportable in either the AFT or PH framing, which matters when the audience is used to reading hazard ratios.
Don’t let convenience pick the distribution silently — a model that fits the observed follow-up window well can still extrapolate very differently depending on the tail behavior of the chosen distribution, which matters a great deal if the model’s job includes projecting beyond the data.

Fitting and reporting an AFT model

AFT models are a standard option in the same software already used for Cox and Kaplan-Meier: R’s survreg() (in the survival package) and the flexsurv package fit Weibull, log-normal, log-logistic, and other AFT forms; Stata’s streg supports the same set with a time option for the AFT metric; Python’s lifelines library provides WeibullAFTFitter, LogNormalAFTFitter, and LogLogisticAFTFitter directly. Report the acceleration factor (exp(β)) with its confidence interval, state the distribution used and why, and show the diagnostic check (AIC/BIC comparison or a fit plot) that justified it — the distributional choice is the one extra piece of accountability AFT carries that a Cox write-up doesn’t need.

Frequently asked questions

Is an acceleration factor the same thing as a hazard ratio, just inverted?

Only in the special Weibull case, and even then it’s not a simple inversion — it’s exp(β)^−k, scaled by the shape parameter k, not just 1/exp(β). For log-normal or log-logistic AFT models there is no corresponding constant hazard ratio at all; the hazard ratio between two covariate values actually changes over time under those models, which is precisely why they can represent non-proportional-hazards data that Cox can’t.

Can I fit both a Cox model and an AFT model to the same data and compare?

Yes, and it’s a reasonable diagnostic in its own right: if a Cox model and a Weibull AFT model (converted to its PH form) give noticeably different hazard-ratio estimates, that’s a sign the proportional-hazards assumption is under strain even if a formal Schoenfeld test didn’t flag it clearly. Log-normal or log-logistic AFT results aren’t directly comparable to a Cox hazard ratio at all, for the reason in the FAQ above — compare survival curves or time-ratio predictions instead.

Do I need a large sample size to fit an AFT model reliably?

AFT models are fully parametric, so they generally need enough events to estimate both the covariate effects and the distributional shape parameter(s) with reasonable precision — a small-sample dataset that would give a roughly stable Cox hazard ratio can give an unstable shape-parameter estimate in a parametric fit. There’s no single threshold; check the confidence interval width on the shape parameter itself, not just on the covariate coefficients, before trusting extrapolated predictions from a small sample.

Which AFT distribution should I default to if I have no strong reason to prefer one?

Weibull is the reasonable default starting point precisely because of its dual AFT/PH reporting and because a constant (monotonic) hazard shape is a common enough real-world pattern. Fit it, check the shape parameter and a fit diagnostic against the Kaplan-Meier curve, and move to log-normal or log-logistic specifically if the diagnostic shows a rise-then-fall hazard shape Weibull can’t capture.

Accelerated Failure Time Models: The Time-Ratio Alternative to Cox