Accelerated Failure-Time Model Interpretation

Why an AFT Coefficient Describes Time

A covariate can affect survival in more than one interpretable way. In a proportional-hazards model, a coefficient describes a multiplicative change in the instantaneous event rate, conditional on being event-free up to that time. An accelerated failure-time (AFT) model instead describes how the event-time scale changes: a covariate contrast multiplies event times by a common factor. This makes the coefficient naturally interpretable in units of time, even though the coefficient itself is on a logarithmic scale.

The key is that the AFT model acts on the logarithm of event time. A positive coefficient stretches the event-time scale for an increase in that covariate; a negative coefficient compresses it. The interpretation is a model-based association, not by itself evidence that changing the covariate would cause event times to change.

Definition: Let $T>0$ be an event time and let $x$ be a vector of covariates. A parametric AFT model can be written $$ \log T = x^\mathsf{T}\beta + \sigma\varepsilon, $$ where $\beta$ is a vector of regression coefficients, $\sigma>0$ is a scale parameter, and the distribution of the error $\varepsilon$ is specified and does not vary with $x$. Equivalently, $T=\exp(x^\mathsf{T}\beta)T_0$, where $T_0=\exp(\sigma\varepsilon)$ has the reference event-time distribution. For a contrast from $x$ to $x^*$, the AFT time ratio is $\exp((x^*-x)^\mathsf{T}\beta)$.

The model requires a specified error distribution, such as one corresponding to a Weibull, lognormal, or log-logistic event-time distribution. Its covariate structure also assumes that covariates enter linearly on the log-time scale unless transformations or interactions are included. Most importantly for interpretation, the same error distribution and scale apply across covariate values: the covariates rescale time rather than changing the shape of the event-time distribution.

With right-censored data, let $Y=\min(T,C)$ be the observed time and let $\delta=1$ if the event is observed and $\delta=0$ if it is censored. Standard likelihood-based interpretation assumes censoring is independent of the event time conditional on the modeled covariates. Under that condition, a censored observation contributes information that the event time exceeds its observed censoring time. If censoring remains dependent on event time after conditioning on the covariates, the usual likelihood analysis need not give valid estimates.

From a Coefficient to a Time Ratio

For a one-unit increase in a single covariate $x_j$, holding the other covariates fixed, the log event time changes by $\beta_j$. Exponentiating gives the time ratio $\exp(\beta_j)$. A ratio above 1 means event times are longer under the covariate increase; a ratio below 1 means they are shorter. The percentage change in the event-time scale is $100\{\exp(\beta_j)-1\}\%$. This is not a percentage change in hazard.

For a change of $d$ units in $x_j$, the ratio is $\exp(d\beta_j)$, not $d\exp(\beta_j)$. For several covariates changing together, add their coefficient contributions on the log scale and then exponentiate. The covariate units matter: a coefficient per one year of age, for example, differs numerically from a coefficient per ten years of age.

Worked Example: Interpreting a One-Unit Increase

Suppose a hypothetical AFT model for time to a clinical event, measured in months, estimates $\beta=0.30$ for a one-unit increase in a treatment-intensity score. Assume the model’s distributional and censoring conditions are appropriate for this example. Interpret the coefficient.

The time ratio is $$ \exp(\beta)=\exp(0.30)=1.3499. $$

Thus, for two otherwise identical individuals whose scores differ by one unit, the model estimates that the higher-score individual’s event-time scale is about 1.35 times as long. The percentage change is $$ 100\{\exp(0.30)-1\}=34.99\%. $$

In context, the model associates a one-unit higher score with event times about 35.0% longer. This is a multiplicative comparison of modeled event times, not a claim that every individual’s event time is exactly 35.0% longer, nor a causal conclusion.

Worked Example: Comparing Two Covariate Profiles

Consider a hypothetical model with two covariates: a binary exposure $x_1$ and age $x_2$, measured in years. Let $\beta_1=0.40$ and $\beta_2=-0.02$. Compare a person with $x_1=1$ and age 60 with a person with $x_1=0$ and age 50, holding all other modeled characteristics equal.

The log-time contrast is $$ (1-0)(0.40)+(60-50)(-0.02)=0.40-0.20=0.20. $$

Therefore, the time ratio for the first profile relative to the second is $$ \exp(0.20)=1.221. $$

Under the fitted model, the first profile has an event-time scale about 1.22 times that of the second, or about 22.1% longer. The age difference contributes a time ratio of $\exp(10\times-0.02)=\exp(-0.20)$, which partly offsets the exposure contrast $\exp(0.40)$. Combining coefficients before exponentiating accounts for both changes together.

Why Every Event-Time Quantile Has the Same Ratio

The time-ratio interpretation is not limited to the median. It applies to every event-time quantile under the AFT model. To see why, write $\eta=x^\mathsf{T}\beta$. The model gives $T=\exp(\eta)T_0$, so for any $t>0$,

F(t\mid x)=P(T\leq t\mid x) =P\left(T_0\leq \frac{t}{\exp(\eta)}\right) =F_0\left(\frac{t}{\exp(\eta)}\right).

Here $F(t\mid x)$ is the conditional event-time distribution function and $F_0$ is the reference distribution function. Define the $q$-th event-time quantile $Q_q(x)$ as a value satisfying $F(Q_q(x)\mid x)=q$, for $0<q<1$. Substituting $t=\exp(\eta)Q_{q,0}$, where $F_0(Q_{q,0})=q$, gives

F(Q_q(x)\mid x) =F_0\left(\frac{\exp(\eta)Q_{q,0}}{\exp(\eta)}\right) =F_0(Q_{q,0})=q.

Consequently, $Q_q(x)=\exp(\eta)Q_{q,0}$. For a covariate contrast $x$ to $x^*$, the ratio of corresponding quantiles is therefore $\exp((x^*-x)^\mathsf{T}\beta)$, regardless of which quantile $q$ is chosen. This conclusion depends on the common-shape AFT assumption; it is not a general property of all survival models.

Worked Example: Translating a Time Ratio Into Quantiles

Suppose a hypothetical AFT model gives a time ratio of $\exp(0.25)$ for a one-unit covariate increase. For the reference profile, suppose the model’s 25th and 75th event-time quantiles are 6 months and 18 months. Calculate the corresponding quantiles after the increase.

The multiplier is $$ \exp(0.25)=1.2840. $$

The 25th quantile becomes $6\times1.2840=7.704$ months, and the 75th quantile becomes $18\times1.2840=23.112$ months. Rounded to three significant figures, these are 7.70 months and 23.1 months. Both quantiles have the same ratio, $1.284$, even though their absolute increases differ.

The model therefore predicts a stretched event-time distribution for the higher covariate value, not a fixed addition of 1.70 months at every point in the distribution.

Contrasting AFT Time Ratios With Cox Hazard Ratios

A Cox proportional-hazards model describes the conditional hazard through $h(t\mid x)=h_0(t)\exp(x^\mathsf{T}\gamma)$. For a one-unit increase in a covariate, $\exp(\gamma_j)$ is the hazard ratio: a ratio of instantaneous event rates among individuals still event-free at time $t$, under the proportional-hazards model. An AFT coefficient instead gives a ratio of event-time scales. These are different estimands, so a time ratio cannot generally be converted into one constant hazard ratio.

The difference can be seen directly from an AFT model. For $\eta=x^\mathsf{T}\beta$, its survival and hazard functions satisfy

S(t\mid x)=S_0\left(\frac{t}{\exp(\eta)}\right), \qquad h(t\mid x)=\exp(-\eta)\, h_0\left(\frac{t}{\exp(\eta)}\right).

The hazard expression follows by differentiating the survival function and using $h(t)=f(t)/S(t)$. In general, the baseline hazard evaluated at the rescaled time $t/\exp(\eta)$ changes with $t$. Thus the hazard ratio between covariate profiles need not be constant over time, even though their event-time quantiles have a constant ratio.

There is an important special case: the Weibull distribution can be parameterized as both an AFT model and a proportional-hazards model. If the baseline Weibull hazard is $h_0(t)=\lambda k t^{k-1}$, with shape $k>0$, then

h(t\mid x) =\exp(-\eta)\lambda k \left(\frac{t}{\exp(\eta)}\right)^{k-1} =h_0(t)\exp(-k\eta).

For a one-unit increase in a covariate with AFT coefficient $\beta_j$, the corresponding Weibull hazard ratio is $\exp(-k\beta_j)$. The shape parameter is essential to this conversion, and the negative sign reflects the fact that longer event times correspond to a lower hazard in this parameterization. Outside such a specific distributional relationship, do not translate an AFT time ratio into a Cox hazard ratio.

Worked Example: A Weibull-Specific Hazard Ratio

Suppose a hypothetical Weibull AFT model has shape $k=1.5$ and a covariate coefficient $\beta=0.25$. The AFT time ratio for a one-unit increase is $$ \exp(0.25)=1.284. $$

Because this example assumes a Weibull distribution, the corresponding hazard ratio is $$ \exp(-k\beta)=\exp(-1.5\times0.25) =\exp(-0.375)=0.6873. $$

Thus, in this Weibull model, the covariate increase is associated with event times scaled up by about 1.28 and a hazard about 0.687 times as large at a given time. The hazard-ratio conclusion follows from the Weibull assumption and its shape parameter; the AFT coefficient alone does not establish it.

Diagnostics, Limitations, and a Common Mistake

A time ratio is only as credible as the model that produced it. The assumed parametric distribution may fit poorly, the log-time covariate relationship may be misspecified, or a covariate may change distributional shape rather than simply rescale time. Signs of trouble include systematic disagreement between fitted and observed survival patterns, strong patterns in residual-based checks, or markedly different conclusions under reasonable alternative model specifications. Such evidence calls for reviewing the model, not for treating the coefficient’s exponential as automatically meaningful.

The AFT structure also imposes a strong quantile implication: the same covariate contrast must multiply the 25th percentile, median, and 75th percentile by the same amount. If the scientific question concerns a constant hazard ratio, a Cox model is a direct alternative; if the goal is a parametric time-scale interpretation, AFT models provide that instead. Neither framework is universally preferable. The choice depends on the target interpretation and whether its assumptions are plausible.

Common mistake: calling $\exp(\beta)$ a hazard ratio. In an AFT model it is a time ratio. A hazard-ratio interpretation requires a proportional-hazards formulation or a specific additional model relationship, such as the Weibull case derived above.
Common mistake: describing a multiplicative effect as a fixed time difference. A time ratio of 1.2 means 20% longer event-time quantiles, not 20 extra days or months. The absolute difference depends on the starting quantile.
Common mistake: ignoring the covariate scale. A coefficient per one unit may be difficult to interpret if one unit is tiny or unusually large. State the unit and calculate the ratio for a meaningful contrast.

Key takeaway: In an AFT model, exponentiating a coefficient contrast gives a multiplicative event-time ratio. The same ratio applies to every event-time quantile under the model’s common-shape assumption. It is not generally a Cox hazard ratio.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

An AFT model estimates $\beta=-0.20$ for a one-year increase in a covariate. What is the time ratio, and what is its interpretation?
A covariate contrast has time ratio 1.5. If the reference median event time is 8 months, what is the modeled median for the contrasted profile? Is the difference a fixed shift at all quantiles?
Why can an AFT time ratio not generally be reported as a Cox hazard ratio?
For a Weibull AFT model with shape $k=2$ and coefficient $\beta=-0.10$, calculate the corresponding hazard ratio under the Weibull PH representation.

Solutions. (1) $\exp(-0.20)=0.8187$. The model associates a one-year increase with event-time quantiles about 0.819 times as long, or about 18.1% shorter, holding other covariates fixed. (2) The modeled median is $8\times1.5=12$ months. The 50% increase is multiplicative; the absolute difference is 4 months here and need not be 4 months at other quantiles. (3) The AFT model rescales event times, whereas a Cox model specifies a constant multiplicative ratio of hazards. An AFT model generally produces a time-varying hazard ratio; a conversion needs additional distributional structure. (4) $\exp(-k\beta)=\exp(-2\times-0.10)=\exp(0.20)=1.221$. Under this Weibull model, the hazard ratio is about 1.22.