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: v2 · 2026-08-30 · History
Domain-specific #: 1224
Origin domain: survival analysis
Subdomain: event time regression

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.^[1] 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.

Its autonomous residual is the covariate-dependent multiplication of event time and its log-linear representation, not survival analysis generally or any model whose predicted failure occurs sooner. The identity fails when a coefficient is interpreted as a hazard ratio by default, censoring is treated as an observed event, time zero changes across subjects without adjustment, the baseline distribution is hidden, competing events are ignored, or a time-varying acceleration factor is presented as the classical constant AFT model.

Recognition requires an analyst to define origin, event, and censoring; write the exact log-time and survival forms; map coefficients to time ratios; check whether covariate effects plausibly preserve the stated acceleration relation; inspect residuals and distributional fit; and separate AFT assumptions from proportional-hazards assumptions. Once established, it supports interpreting covariate effects on survival duration, fitting censored time-to-event data, comparing parametric lifetime families, extrapolating only under declared assumptions, and testing whether time scaling is more intelligible than hazard scaling without turning those uses into the definition.

Structural Signature¶

Carrier: a positive event time (T), covariates (x), possible right censoring, and a declared baseline log-time or survival distribution
Inputs or antecedent state: time origin and event definition, covariate vector, censoring indicators, log-time regression form, baseline error family or semiparametric assumption, scale parameter, sign convention, and estimation method
Constitutive operation: 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.
Invariant: 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
Recognition test: define origin, event, and censoring; write the exact log-time and survival forms; map coefficients to time ratios; check whether covariate effects plausibly preserve the stated acceleration relation; inspect residuals and distributional fit; and separate AFT assumptions from proportional-hazards assumptions
Output or consequence: interpreting covariate effects on survival duration, fitting censored time-to-event data, comparing parametric lifetime families, extrapolating only under declared assumptions, and testing whether time scaling is more intelligible than hazard scaling
Failure boundary: a coefficient is interpreted as a hazard ratio by default, censoring is treated as an observed event, time zero changes across subjects without adjustment, the baseline distribution is hidden, competing events are ignored, or a time-varying acceleration factor is presented as the classical constant AFT model

What It Is Not¶

It is not the whole field of survival analysis; many objects in that field do not satisfy its constitutive rule.
It is not its canonical example. 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. That is an instance, not a definition.
It is not Regression. Regression is the broad outcome-as-function-of-covariates family. AFT regression restricts the outcome to positive event time with censoring and imposes a multiplicative time-scale or additive log-time interpretation; ordinary linear or logistic regression does not supply those roles.
It is not an unrestricted metaphor. The Weibull family admits both proportional-hazards and AFT parameterizations, while lognormal and log-logistic AFT models generally do not have constant hazard ratios; notation can reverse coefficient signs depending on whether acceleration or deceleration is parameterized

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.^[2]

Recognition. define origin, event, and censoring; write the exact log-time and survival forms; map coefficients to time ratios; check whether covariate effects plausibly preserve the stated acceleration relation; inspect residuals and distributional fit; and separate AFT assumptions from proportional-hazards assumptions
Comparison. Compare legitimate instances through time origin, event definition, censoring mechanism, covariates, sign convention, time ratio, baseline family, scale parameter, residuals, extrapolation horizon, and competing risks.
Boundary. The Weibull family admits both proportional-hazards and AFT parameterizations, while lognormal and log-logistic AFT models generally do not have constant hazard ratios; notation can reverse coefficient signs depending on whether acceleration or deceleration is parameterized
Use. Preserve every assumption when using the identity for interpreting covariate effects on survival duration, fitting censored time-to-event data, comparing parametric lifetime families, extrapolating only under declared assumptions, and testing whether time scaling is more intelligible than hazard scaling.

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. The disciplined statement is that the object counts as Accelerated failure time model exactly when 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

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. Approximation or noisy evidence may weaken a classification without changing its definition.

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¶

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.
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.
Derive carefully. Infer interpreting covariate effects on survival duration, fitting censored time-to-event data, comparing parametric lifetime families, extrapolating only under declared assumptions, and testing whether time scaling is more intelligible than hazard scaling only under the stated assumptions.
Stress-test. Contrast the legitimate boundary case—The Weibull family admits both proportional-hazards and AFT parameterizations, while lognormal and log-logistic AFT models generally do not have constant hazard ratios; notation can reverse coefficient signs depending on whether acceleration or deceleration is parameterized—with this counterexample: a Cox proportional-hazards model with constant hazard ratio is not automatically an AFT model, because proportional hazards does not generally imply that every survival-time quantile is scaled by one common factor.

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.^[3]

Outside the domain, only the skeleton—multiply an entire temporal trajectory by a context-dependent scale so corresponding milestones occur earlier or later in fixed proportion—travels automatically. The terms survival time, censoring, log time, acceleration factor, time ratio, survival function, baseline distribution, likelihood, quantile, and proportional hazards retain domain-specific meanings, so every role and inference must be revalidated.

Examples¶

Canonical¶

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. The statement concerns a time ratio, not a claim that the treatment divides the hazard by 1.5 at every time. Weibull models can satisfy both AFT and proportional-hazards forms, but that overlap is special. It is canonical because the carrier, rule, invariant, and consequence are all inspectable.^[1]

Mapped back: a positive event time (T), covariates (x), possible right censoring, and a declared baseline log-time or survival distribution → 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. → 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 → interpreting covariate effects on survival duration, fitting censored time-to-event data, comparing parametric lifetime families, extrapolating only under declared assumptions, and testing whether time scaling is more intelligible than hazard scaling

Applied / In Practice¶

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. Long-horizon extrapolation remains distribution-dependent even when in-sample fit is similar; engineering decisions therefore carry model-family uncertainty and censoring coverage. It qualifies only after the same diagnostic and failure boundary are checked.^[2]

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Exact identity vs. practical recognition. The constitutive condition may be exact while evidence is indirect. Diagnostic: Can the reviewer state both the condition and the warrant?
T2: Canonical form vs. variants. Weibull, exponential, lognormal, log-logistic, and generalized-gamma parametric AFT models; rank-based semiparametric AFT estimation; frailty or clustered extensions; and time-varying departures can preserve or change the identity. Diagnostic: Which named role is invariant across the variants?
T3: Compression vs. hidden assumptions. The label is useful only while prerequisites remain visible. Diagnostic: Can each downstream inference be traced to a declared assumption?
T4: Autonomy vs. reduction. The candidate uses broader structures but claims the covariate-dependent multiplication of event time and its log-linear representation, not survival analysis generally or any model whose predicted failure occurs sooner. Diagnostic: Does that residual still support independent recognition after the parent and neighbors are subtracted?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is multiply an entire temporal trajectory by a context-dependent scale so corresponding milestones occur earlier or later in fixed proportion; its identity-bearing terms are survival time, censoring, log time, acceleration factor, time ratio, survival function, baseline distribution, likelihood, quantile, and proportional hazards. Those terms determine admissible objects, evidence, and consequences inside survival analysis.

Structural Core vs. Domain Accent¶

The structural core is a carrier governed by 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. and tested by define origin, event, and censoring; write the exact log-time and survival forms; map coefficients to time ratios; check whether covariate effects plausibly preserve the stated acceleration relation; inspect residuals and distributional fit; and separate AFT assumptions from proportional-hazards assumptions. The domain accent is constitutive rather than decorative, so an analogy that preserves only the skeleton is not another instance of Accelerated failure time model.

The proposed strict upward parent is prime:proportionality. The defining invariant is that corresponding survival-time quantiles are multiplied by a covariate-dependent constant; censoring, log-time errors, and baseline survival supply the statistical residual. The edge is proposal-only and points to a frozen prior-baseline Prime.

The entry does not collapse into the parent because the covariate-dependent multiplication of event time and its log-linear representation, not survival analysis generally or any model whose predicted failure occurs sooner A thematic neighbor is declined whenever it does not literally subsume that rule.

The prospective workspace queue contains one strict upward edge to prime:proportionality. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Accelerated failure time model Domain-specific

Parents (1) — more general patterns this builds on

Accelerated failure time model is a kind of Proportionality Prime

The proposed strict upward parent is prime:proportionality.

Hierarchy paths (2) — routes to 2 parentless roots

Accelerated failure time model → Proportionality → Normativity → Constraint

Show alternative path (1)

Neighborhood in Abstraction Space¶

Accelerated failure time model sits in a sparse region of the domain-specific corpus (62^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Longitudinal Models & Time-Series Structure (8 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 · 2026-09-08

Not to Be Confused With¶

Cox proportional-hazards model. Multiplies hazard rather than the time scale and leaves the baseline hazard unspecified.
Linear regression on observed log times. Ignores censoring unless the fitting procedure incorporates it and may lack a valid survival distribution.
Proportional-odds survival model. Constrains survival odds rather than time ratios.
Mean residual life model. Models expected remaining lifetime under a different estimand and link.

References¶

[1] Lee-Jen Wei, 'The Accelerated Failure Time Model: A Useful Alternative to the Cox Regression Model in Survival Analysis,' Statistics in Medicine 11(14–15), 1871–1879 (1992), DOI 10.1002/sim.4780111409. registry ↩a ↩b

[2] John D. Kalbfleisch and Ross L. Prentice, The Statistical Analysis of Failure Time Data, 2^nd ed., Wiley, 2002, DOI 10.1002/9781118032985. registry ↩a ↩b

[3] John P. Klein and Melvin L. Moeschberger, Survival Analysis: Techniques for Censored and Truncated Data, 2^nd ed., Springer, 2003, DOI 10.1007/b97377. registry ↩