Accelerated Failure-Time Models

Why Model Event Time on a Log Scale?

Duration outcomes are positive and often strongly right-skewed: a small number of individuals may experience very long times before an event. An accelerated failure-time (AFT) model describes how covariates shift the scale of event times. Its central interpretation is direct: a covariate contrast multiplies a specified conditional event-time quantile by a constant factor.

The model is parametric because it specifies a distribution for the error on log time. That distribution matters: it determines the shape of survival and hazard functions, and therefore the implications of the fitted model beyond the time-ratio interpretation. As with other duration models, right-censored observations can contribute information even though their event times are not observed.

Definition: Let $T_i>0$ be the event time for individual $i$, and let $x_i$ be a vector of covariates. An AFT model has the form $$ \log T_i=x_i^\top\beta+\sigma\varepsilon_i,\qquad \sigma>0, $$ where $\beta$ is a coefficient vector and the error $\varepsilon_i$ has a specified distribution that does not depend on $x_i$. Conditional on the covariates, the model shifts log time by $x_i^\top\beta$; equivalently, covariate contrasts rescale event time.

The usual specification assumes independent individuals, a correctly specified linear predictor in the covariates, a common error distribution and scale $\sigma$, and independent right-censoring conditional on the included covariates. The error distribution is also assumed not to change across covariate values. These are modeling assumptions, not consequences of fitting the regression. In particular, a time ratio is a conditional distributional comparison; it is not automatically a causal effect.

Time Ratios and Conditional Quantiles

Let $F_\varepsilon$ be the error distribution’s cumulative distribution function, and let $F_\varepsilon^{-1}(\tau)$ be its $\tau$-quantile. The conditional $\tau$-quantile of event time follows directly from the model:

Q_\tau(T\mid x) =\exp\!\left(x^\top\beta+\sigma F_\varepsilon^{-1}(\tau)\right).

For two covariate values $x_a$ and $x_b$, the error quantile is the same under both conditions. Thus their conditional event-time quantile ratio is

\frac{Q_\tau(T\mid x_b)}{Q_\tau(T\mid x_a)} =\exp\!\left((x_b-x_a)^\top\beta\right).

This ratio does not depend on $\tau$. For a one-unit increase in a covariate $x_j$, holding other covariates fixed, the time acceleration factor is $\exp(\beta_j)$. A factor greater than one means longer conditional event times; a factor below one means shorter times. This language describes a multiplicative shift in event-time quantiles, not a change in hazard.

Worked Example: Interpreting a Treatment Coefficient

Suppose a hypothetical lognormal AFT analysis of time to symptom resolution, measured in days, estimates a treatment coefficient of $\hat\beta_{\text{treat}}=-0.25$. Treatment is coded 1 for the treatment group and 0 for the comparison group. Assume the model’s other covariates are held fixed.

The treatment-to-comparison time ratio at every conditional quantile is $$ \exp(\hat\beta_{\text{treat}})=\exp(-0.25)\approx 0.7788. $$

Thus the model estimates that each treatment-group conditional event-time quantile is about 0.7788 times the corresponding comparison-group quantile, or about $1-0.7788=0.2212$, a 22.12% shorter time. For example, if the comparison-group conditional median is 20 days, the treatment-group median is $$ 20\exp(-0.25)\approx 15.58\text{ days}. $$

The statement applies to corresponding quantiles because the model imposes a common multiplicative shift. It does not say that every individual’s resolution time is reduced by exactly 22.12%, nor does the coefficient alone establish a causal treatment effect.

Common Error Distributions

The AFT equation is a framework rather than a single distributional model. Three common choices produce different event-time distributions. In each case, $\eta=x^\top\beta$ is the conditional location on log time, and $\sigma>0$ is the error scale.

Model	Error specification	Implication for event time
Lognormal AFT	$\varepsilon\sim N(0,1)$	$\log T\mid x\sim N(\eta,\sigma^2)$
Weibull AFT	$F_\varepsilon(w)=1-\exp[-\exp(w)]$	Weibull survival with shape $1/\sigma$ and scale $\exp(\eta)$
Log-logistic AFT	$F_\varepsilon(w)=1/[1+\exp(-w)]$	Log-logistic survival; the conditional median is $\exp(\eta)$

For the lognormal model, $\log T$ is normal conditional on covariates. Its conditional median is $\exp(\eta)$, while its conditional mean is $\exp(\eta+\sigma^2/2)$; the mean and median are not interchangeable. For the Weibull specification shown, the conditional survival function is $$ S(t\mid x)=\exp\!\left[-\left(\frac{t}{\exp(\eta)}\right)^{1/\sigma}\right],\qquad t>0. $$ For the log-logistic specification, it is $$ S(t\mid x)=\frac{1}{1+\exp\!\left((\log t-\eta)/\sigma\right)},\qquad t>0. $$

The log-logistic distribution can have a nonmonotone hazard, whereas the Weibull hazard is monotone or constant depending on its shape. Consequently, choosing a distribution is not merely a computational choice: it imposes a particular pattern on the event-time distribution. A fit should be assessed in light of the data and the scientific setting.

Right Censoring and Likelihood Contributions

For right-censored data, write the observed duration as $Y_i=\min(T_i,C_i)$, where $C_i$ is the censoring time, and let $\delta_i=1$ if the event is observed and $0$ if the observation is right-censored. Under conditional independence of event and censoring times, the event-time contribution to the likelihood is $$ L_i=f_T(Y_i\mid x_i)^{\delta_i}S_T(Y_i\mid x_i)^{1-\delta_i}, $$ where $f_T$ and $S_T$ are the conditional event-time density and survival function. An observed event contributes a density; a censored duration contributes the probability that the event time exceeds the censoring time. The censoring mechanism itself is not modeled in this likelihood when it is conditionally independent and its parameters are not of interest.

Worked Example: A Lognormal Likelihood Contribution

Suppose, hypothetically, that a lognormal AFT model for time to a device failure, measured in months, has conditional log-time mean $\eta=\log(10)$ and scale $\sigma=0.5$. Consider one device that fails at 8 months and another that remains event-free through 12 months and is then censored. Assume censoring is independent of failure time conditional on the model covariates.

For the observed failure at $t=8$, the standardized log time is $$ z=\frac{\log(8)-\log(10)}{0.5} =\frac{\log(0.8)}{0.5}\approx -0.4463. $$

The lognormal density is $f_T(t\mid x)=\phi(z)/(t\sigma)$, where $\phi$ is the standard normal density. Since $\phi(-0.4463)\approx0.3611$, this observation contributes $$ f_T(8\mid x)\approx\frac{0.3611}{8(0.5)} =\frac{0.3611}{4}\approx0.09028 $$ per month to the likelihood.

For the device censored at 12 months, the standardized log time is $$ z=\frac{\log(12)-\log(10)}{0.5} =\frac{\log(1.2)}{0.5}\approx0.3646. $$

Its contribution is the probability of surviving beyond 12 months: $$ S_T(12\mid x)=1-\Phi(0.3646)\approx1-0.6423=0.3577, $$ where $\Phi$ is the standard normal cumulative distribution function. The censored observation does not contribute a failure density at 12 months: it contributes the probability of an event time greater than 12 months.

When AFT and Proportional Hazards Coincide

An AFT model does not generally imply proportional hazards. A proportional-hazards (PH) model requires the hazard ratio between two covariate values to be constant over time. The Weibull distribution is a special case in which the AFT and PH formulations coincide when the shape parameter is common across individuals.

Result: Under the Weibull AFT specification above, let $k=1/\sigma$ be the common shape and $\lambda(x)=\exp(x^\top\beta)$ the scale. The hazard is $$ h(t\mid x)=k t^{k-1}\exp(-k x^\top\beta). $$ Consequently, for covariate values $x_a$ and $x_b$, $$ \frac{h(t\mid x_b)}{h(t\mid x_a)} =\exp\!\left[-k(x_b-x_a)^\top\beta\right], $$ which does not depend on $t$.

To verify the result, differentiate the Weibull cumulative hazard $\left(t/\lambda(x)\right)^k$ with respect to $t$. The resulting hazard is $k t^{k-1}/\lambda(x)^k$. Substituting $\lambda(x)=\exp(x^\top\beta)$ gives the displayed hazard and ratio. Outside this Weibull case, a time ratio can still be constant across quantiles in an AFT model even though the hazard ratio varies with time.

Worked Example: A Weibull Time Ratio and Hazard Ratio

Consider a hypothetical Weibull AFT model for time to discharge, in days: $$ \log T=2.3+0.4x+0.5\varepsilon, $$ where $x$ is a binary indicator and $\varepsilon$ has the Weibull AFT error distribution defined above. The common shape is $k=1/0.5=2$.

Changing $x$ from 0 to 1 multiplies each conditional event-time quantile by $$ \exp(0.4)\approx1.492. $$ Thus, under this model, the $x=1$ group has conditional quantiles about 1.492 times those of the $x=0$ group. The Weibull scales are $\exp(2.3)\approx9.974$ days and $\exp(2.7)\approx14.88$ days, respectively; their ratio is also $\exp(0.4)$.

The corresponding hazard ratio for $x=1$ relative to $x=0$ is $$ \exp[-k(0.4)]=\exp(-0.8)\approx0.4493. $$ This is a constant hazard ratio under the shared-shape Weibull model. The time ratio, approximately 1.492, and hazard ratio, approximately 0.4493, describe different comparisons and are not reciprocals in general.

Common Pitfalls and Model Checks

A frequent mistake is to read $\exp(\beta_j)$ as a hazard ratio. In an AFT model it is a time ratio for corresponding conditional quantiles. Only in special distributional cases, notably Weibull with a common shape, does the same specification also imply proportional hazards; even then, the hazard ratio is $\exp(-k\beta_j)$, not $\exp(\beta_j)$.

Another mistake is to treat the parametric form as harmless because the coefficient interpretation is simple. A misspecified error distribution can yield poor survival-tail estimates and misleading predictions, even when a time-ratio summary seems plausible. Evaluate distributional fit and the consequences of the chosen tail shape, and distinguish conditional model comparisons from causal claims. If censoring depends on unmodeled information about event time, the stated likelihood contributions may not be valid without additional modeling.

Key takeaway: An AFT model places covariates on log event time. Exponentiating a coefficient gives a multiplicative conditional time-quantile contrast. The error distribution determines the survival and hazard shapes; proportional hazards follows only for particular specifications, including the common-shape Weibull model.

Check Your Understanding

Answer each question using the model assumptions and definitions above.

In an AFT model, what does $\exp(\beta_j)=0.8$ mean for corresponding conditional event-time quantiles when $x_j$ increases by one unit?
Why does a right-censored observation contribute a survival probability rather than an event-time density at its censoring time?
Under what condition does the Weibull AFT specification imply a hazard ratio that is constant over time?
For a lognormal AFT model, which quantity is $\exp(\eta)$: the conditional mean or the conditional median?
Why should $\exp(\beta_j)$ not generally be interpreted as a hazard ratio or a causal effect?

Model	Error specification	Implication for event time
Lognormal AFT	\(\varepsilon\sim N(0,1)\)	\(\log T\mid x\sim N(\eta,\sigma^2)\)
Weibull AFT	\(F_\varepsilon(w)=1-\exp[-\exp(w)]\)	Weibull survival with shape \(1/\sigma\) and scale \(\exp(\eta)\)
Log-logistic AFT	\(F_\varepsilon(w)=1/[1+\exp(-w)]\)	Log-logistic survival; the conditional median is \(\exp(\eta)\)