Accelerated failure time model

Model covariates as multiplying an event-time scale—equivalently shifting log survival time—so coefficients are interpreted through time ratios rather than the constant hazard ratios of proportional-hazards regression.

Version

v3 · 2026-09-19 · History

Domain-specific #

1224

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Survival Analysis, Event Time Regression → Mathematics

Core Idea

An accelerated failure time model writes \(log T=x^{\mathsf T}\beta+\sigma\varepsilon\), or equivalently relates survival curves by \(S(t\mid x)=S_0(t\exp(-x^{\mathsf T}\beta))\) under this sign convention. Thus \(exp(\beta_j)\) is a multiplicative time ratio for a one-unit covariate change. A covariate shifts the location of log event time, which stretches or compresses the entire time axis. Choosing an extreme-value, normal, logistic, or other error distribution yields Weibull, lognormal, log-logistic, or related parametric survival families. Censoring contributes a survival rather than density term to the likelihood.

How would you explain it like I'm…

Fast Clock, Slow Clock

Imagine two toy robots whose batteries run down. One lives in a hot room, and it's as if its clock ticks twice as fast, so everything that happens to the cool robot by day ten happens to the hot robot by day five. An accelerated failure time model is a way of measuring how much things like heat speed up or slow down the clock for how long something lasts.

Stretching the Waiting Time

Scientists often study how long it takes until something happens, like a light bulb burning out or a patient getting better. An accelerated failure time model says that certain factors stretch or squeeze time itself. If a factor doubles the speed, then whatever would have happened at 10 years now happens at 5 years, for every stage along the way. The model gives each factor a number that says how many times longer or shorter the waiting time becomes. It also handles cases where the study ended before the event happened, by counting them as 'it hadn't happened yet' instead of pretending it did.

Time-Ratio Survival Model

An accelerated failure time (AFT) model is a survival-analysis model for time until an event. It assumes the logarithm of the event time equals a linear combination of covariates plus a random error term, so each covariate multiplies the event time by a fixed factor. In effect a covariate stretches or compresses the entire time axis, and the coefficient, once exponentiated, is a time ratio: how many times longer (or shorter) the time to event is per unit change. The choice of error distribution gives familiar families such as Weibull, lognormal or log-logistic. Subjects whose event hasn't happened by the end of observation (censored) still contribute information. A common mistake is reading AFT coefficients as hazard ratios, which belong to a different kind of model, the proportional hazards model.

 

An accelerated failure time model specifies log T = x'beta + sigma*epsilon, or equivalently S(t | x) = S0(t exp(-x'beta)) with this sign convention, so that exp(beta_j) is a multiplicative time ratio for a one-unit change in covariate j. A covariate shifts the location of log event time, which rescales the whole time axis rather than acting on the instantaneous risk. The error distribution determines the parametric family: extreme-value errors give the Weibull model, normal errors the lognormal, logistic errors the log-logistic. In the likelihood, observed events contribute a density term while censored observations contribute a survival term. The model's identity is this covariate-driven multiplication of event time, not survival analysis in general and not any model that predicts earlier failure. Typical errors include reading coefficients as hazard ratios by default, treating censoring as an event, shifting time zero across subjects without adjustment, hiding the baseline distribution, ignoring competing events, and calling a model with time-varying acceleration the classical constant AFT model. Good practice defines origin, event and censoring, writes the log-time and survival forms, checks whether the acceleration assumption is plausible via residuals and fit, and keeps it distinct from proportional-hazards assumptions.

Scope of Application

Accelerated failure time model applies when the analyst can specify a positive event time (T), covariates (x), possible right censoring, and a declared baseline log-time or survival distribution and establish that event time is positive and well defined, covariates enter through a multiplicative time-scale relation or additive log-time form, censoring is represented correctly, coefficient signs and time-ratio interpretation are declared, and any parametric baseline family is part of the model rather than inferred from the acronym. The entry is a descriptive statistical specification. It does not recommend a treatment, predict an individual's survival, or license extrapolation beyond the supported population and follow-up.

Clarity

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because authors differ on whether a positive coefficient lengthens time or increases an acceleration parameter, so every interpretation must be derived from the displayed equation rather than the word accelerated.

Identity and measurement remain separate. Model assessment reports censoring, residual diagnostics, distributional comparisons, coefficient uncertainty, time-ratio intervals, calibration horizon, and sensitivity to influential observations and extrapolation.

Manages Complexity

The abstraction compresses Weibull, exponential, lognormal, log-logistic, and generalized-gamma parametric AFT models; rank-based semiparametric AFT estimation; frailty or clustered extensions; and time-varying departures into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Compression can hide assumptions. A responsible use therefore declares time origin, event definition, censoring mechanism, covariates, sign convention, time ratio, baseline family, scale parameter, residuals, extrapolation horizon, and competing risks and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning

1. Type the carrier. Establish a positive event time (T), covariates (x), possible right censoring, and a declared baseline log-time or survival distribution and reject examples from a different problem.
2. Lock the rule. Express that event time is positive and well defined, covariates enter through a multiplicative time-scale relation or additive log-time form, censoring is represented correctly, coefficient signs and time-ratio interpretation are declared, and any parametric baseline family is part of the model rather than inferred from the acronym independently of one notation or implementation.

Knowledge Transfer

Transfer within survival analysis is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from If a binary treatment coefficient is \(\beta=\log 1.5\) under \(\log T=x\beta+\varepsilon\), the fitted event-time quantiles for the treated group are 1.5 times the corresponding baseline quantiles, subject to the model and covariate conditions. to A reliability analysis fits Weibull, lognormal, and log-logistic AFT models to right-censored component lifetimes and compares residual behavior, information criteria, and sensitivity of mission-time predictions. demonstrates that continuity.

Relationships to Other Abstractions

Current abstraction Accelerated failure time model Domain-specific

Parents (1) — outgoing typed relations

• Accelerated failure time model is a kind of Proportionality Prime

  The defining invariant is that corresponding survival-time quantiles are multiplied by a covariate-dependent constant; What makes it its own entry: the covariate-dependent multiplication of event time and its log-linear representation, not survival analysis generally or any model whose predicted failure occurs sooner.

Hierarchy paths (2) — routes to 2 parentless roots

• Accelerated failure time model → is a kind of Proportionality → presupposes Normativity → is a decomposition of Constraint

Show alternative path (1)

Neighborhood in Abstraction Space

Accelerated failure time model sits in a moderately populated region (49^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Survival Analysis & Demographic Rates (16 abstractions)

Nearest neighbors

• Logrank test — 0.90
• Discrete-time proportional hazards — 0.90
• Recurrent event analysis — 0.87
• Failure rate — 0.85
• Unevenly spaced time series — 0.84

Computed from structural-signature embeddings · 2026-10-25