Accelerated Failure-Time Models › Practical Implications

AFT models are most useful when the scientific or operational question is phrased on the time scale: how much longer until recovery, how much sooner until failure, or how much later until churn. In medicine, a time ratio can summarize how a treatment changes the typical time to an event. In reliability engineering, a Weibull AFT model is a standard way to describe how stress or operating conditions accelerate failure. For a stipulated churn example, let the event time run from subscription start to cancellation and let a baseline plan or prespecified onboarding-group indicator enter the covariate vector. Under the stated AFT assumptions, its exponentiated slope is a conditional time ratio. An observational association does not by itself identify the effect of changing the plan or onboarding process. In these settings, exponentiating a slope gives an effect size that collaborators can interpret without translating from instantaneous risk.

The model is a strong candidate when a full survival curve, event-time quantile, or prediction interval is needed and a parametric error distribution is defensible. It is less attractive when the proportional hazards assumption is credible and the event-time distribution is unknown, because Cox regression avoids the distributional commitment. A single acceleration factor is also too simple when the required conditional time ratios differ across subject groups or event-time quantiles. A hazard ratio that varies over time can still be compatible with a log-normal AFT model. When conditional time ratios differ across groups or event-time quantiles, interactions, stratified models, time-varying covariates, or a different modeling framework may be more appropriate than a single acceleration factor.

The main tradeoff is between interpretability and assumption strength. A time ratio describes how conditional quantiles scale, and the full likelihood supports survival-curve estimation, but the tail behavior is determined by the chosen distribution. If long-term extrapolation drives the decision, the AFT model should be treated as a structured hypothesis about the tail rather than as a neutral summary of the data.

The minimum data requirements are a positive observed time, an event indicator, and covariates measured before or at the start of follow-up. The censoring mechanism should be independent of the event time after conditioning on covariates. If subjects with worse unobserved prognosis are more likely to leave even after conditioning on the included covariates, the independent-censoring factorization can fail. Changing the event-time family alone does not establish a valid observation model. Time units should be consistent across subjects, zero or negative recorded times need investigation rather than automatic deletion or replacement, and tied event times should be handled with an explicit convention. Coarse measurement can create many ties; this is not fatal, but it can affect standard errors and should be documented.

Covariate scaling is not required for interpretation because a time ratio is defined per unit of the covariate in its original units. Centering changes the intercept but not the slope or time ratio. Standardizing changes the slope to an effect per standard deviation, which can be useful for comparing covariates but should be described explicitly in any report. Scaling is still helpful for numerical stability when covariates differ greatly in magnitude, and missing covariates must be addressed before fitting rather than hidden inside the likelihood. Do not substitute arbitrary event times for censored records in this likelihood. The survival contribution uses their known lower bounds; model-based imputation is a different procedure with its own assumptions.

The choice of error distribution should follow the hazard shape and the decision-relevant region of the survival curve. The Weibull model implies a monotone hazard: increasing when $k>1$, decreasing when $k<1$, and constant when $k=1$. The log-normal model can produce a non-monotone hazard, which is useful when risk rises and later falls. When both distributions are plausible, fitting both and overlaying their survival curves on Kaplan-Meier estimates is a useful sensitivity check. A small AIC difference in a small sample should not override a visible calibration failure in the time region that matters.

Optimization should respect the constrained parameter space. Parameterizing the scale as $\log\sigma$, using multiple starting values, and bounding $\log\sigma$ to a reasonable range such as $(-4,3)$ reduces the chance of exploring absurd scale values. In the implementation used here, three starts and maxiter=2000 are the settings for this small example, but larger or heavily censored data may require more starts or a different optimizer. Convergence should be checked explicitly, and fitted time ratios should be reported with uncertainty intervals and with the assumed distribution stated alongside them.

Model selection should combine numerical fit with subject-matter plausibility. AIC and BIC are useful when comparing full-likelihood AFT models on the same observations and censoring indicator, but they do not diagnose tail misspecification. A model with a slightly worse AIC may be preferable if it follows the Kaplan-Meier curve more closely in the region where decisions are made, or if its implied hazard shape is more credible. Reporting a single "best" model without that context overstates what the information criterion can decide.

An AFT slope is a log time ratio, not a log hazard ratio. Among the two families studied here, the Weibull model has the proportional-hazards conversion $\mathrm{HR}_j=\exp(-\beta_j/\sigma)$. Positive $\beta_j$ then gives a hazard ratio below one. The log-normal hazard ratio generally varies with time, so the same constant conversion does not apply. An observational time ratio is a conditional association unless the study design and assumptions support a causal interpretation.

Another pitfall is to treat the distributional choice as a computational detail. Weibull and log-normal models can agree near the median while diverging in the tail. If a decision horizon lies beyond most observed follow-up, that tail difference can change the prediction used for the decision. Extrapolating a fitted curve beyond the largest observed time therefore relies on the assumed event-time family. Similarly, selecting a distribution only because it has the smallest AIC in a small sample can favor a model that fits the middle of the data while missing the region of interest.

Numerical and design problems also matter. With heavy censoring or few events, the likelihood can be flat, and an optimizer may report convergence at an unstable solution. Models with many covariates and few events can produce large time ratios with wide or unreliable intervals; event count alone does not guarantee stability, particularly with collinear or weakly varying covariates. Finally, informative censoring can invalidate this reduced likelihood regardless of how well the model converges, so the censoring process deserves the same scrutiny as the event process.

For the likelihoods used here, each dense objective evaluation costs $O(n(p+1))$ for $p$ covariates plus an intercept: the model matrix is multiplied by the coefficient vector, residuals are formed, and the density and survival contributions are summed over subjects. There is no pairwise distance matrix, so the method scales linearly in the number of observations per evaluation rather than quadratically. Total fitting cost depends on the number and cost of objective and gradient evaluations across all starts. Fitting both families requires separate optimization, and their evaluation counts can differ. With finite-difference gradients, an optimizer iteration can require multiple objective calls, so iteration count alone is not a runtime bound.

Memory is dominated by the model matrix and the vectors used to form residuals and likelihood contributions. Dense matrices are convenient for small or moderate problems; for large data with categorical covariates, sparse design matrices can reduce storage substantially. When $n$ is very large, it can be helpful to compute the log-likelihood in chunks rather than materializing many temporary arrays at once. The implementation bounds log scale and rejects nonfinite likelihood evaluations instead of clipping the model exponent. These numerical safeguards do not make an unidentified model identifiable.

A useful diagnostic is the Cox-Snell residual. For each subject, define

$$r_i = -\log S_T(y_i\mid \mathbf{x}_i).$$

With the true continuous conditional survival function, $-\log S_T(T_i\mid\mathbf{x}_i)$ has an exponential distribution with rate one. Applying the same transformation to $y_i$ gives censored residuals that retain the event indicators. A Kaplan-Meier residual-survival curve can be compared with $\exp(-r)$ over the supported range. When parameters are estimated from the same sample, this is an approximate fitted-model diagnostic, not an exact independent sample from the reference law. Standardized log-time residuals also remain censored; the observed-event subset is selected and should not simply be compared with an uncensored sample from the assumed error distribution. Censoring-aware residual methods require the corresponding observation assumptions.

Model comparison should distinguish between models with the same likelihood and models with different likelihoods. The log-normal and Weibull AFT models are both full likelihoods on the same observed times and event indicators, so AIC and BIC are directly comparable, with the caveat that they reward different complexity penalties. Under regular nested-model asymptotics with identifiable interior parameters, the likelihood-ratio statistic has a chi-square reference law. Log-normal and Weibull are not nested, so the usual nested-model chi-square likelihood-ratio test is not appropriate for choosing between them.

Comparing an AFT model with a Cox model requires more care. The Cox model maximizes a partial likelihood, while the AFT model maximizes a full likelihood. Their AIC values are not on the same scale and should not be compared directly. Instead, we compare out-of-sample predictive performance: concordance, calibration of predicted survival curves, integrated Brier score, or a survival log score. Score models on common held-out records with an appropriate censoring treatment, and assess horizon-specific probabilities only where follow-up has the required support. We can also compare fitted survival curves at clinically or operationally relevant times. If a Weibull AFT model is used, we can translate its coefficients to hazard ratios through $\exp(-\beta_j/\sigma)$ and check whether that proportional hazards structure is plausible. A time-varying hazard ratio is outside the Weibull AFT model's proportional-hazards restriction. Proportional-hazards diagnostics from a Cox fit can motivate checking that possibility; a log-normal or log-logistic AFT model may fit the pattern better.

Check influence and extrapolation explicitly. A single early event or a very late censoring time can have substantial leverage in a parametric model. Refitting with and without influential observations, checking convergence from multiple starting values, and plotting fitted curves against Kaplan-Meier estimates are simple but effective safeguards. If the fitted curve is stable in the observed range but sensitive to one observation in the tail, the extrapolated portion should not be presented as if it were estimated with the same confidence as the interior of the distribution.

Performance should be judged on the time scale whenever possible. Concordance measures ranking, but it does not show whether predicted survival probabilities are calibrated. Calibration curves at clinically or operationally relevant horizons, the integrated Brier score, and a survival log score provide complementary views. These metrics are especially useful when comparing an AFT model with a Cox model, because they avoid the invalid comparison of partial and full likelihood AIC values. A model that ranks subjects well but assigns systematically optimistic survival probabilities may still be a poor choice for planning.

Deployment requires storing more than the fitted coefficients. The error distribution, the scale parameter, the covariate preprocessing rules, the time unit, and the censoring convention all affect predictions and should be versioned together. Monitoring should track the event rate, the censoring rate, and calibration at the horizons used for decisions, not just overall concordance. If the population or the censoring process shifts, the fitted baseline location and scale may need recalibration or refitting even when the covariate effects remain stable.

The fitted AFT model can supply survival curves, event-time quantiles, and prediction intervals, but those outputs inherit the distributional assumption. In production, it is therefore useful to distinguish between interpolation within the observed follow-up range and extrapolation beyond it. The former is supported mainly by the fit to the observed data; the latter is supported mainly by the assumed error distribution. Treating the two as equally reliable hides how strongly the extrapolated prediction depends on the chosen tail model.