Accelerated Failure Time Models Explained

Introduction

Clinical survival analysis is often presented through the Cox proportional hazards model. The Cox model is extremely useful because its hazard-ratio interpretation is familiar and it does not require specifying a particular baseline hazard.

But a hazard ratio is not always the most natural way to describe a treatment effect.

In many clinical settings, investigators may instead want to ask: How much longer or shorter does the treatment make patients remain event-free?

An accelerated failure time (AFT) model addresses this question directly by modeling the logarithm of survival time.

What Is an Accelerated Failure Time Model?

Let \(T\) denote the event time. An AFT model begins with the logarithm of the event time:

where:

• \(T\) is the event time.
• \(X_1,\ldots,X_p\) are covariates.
• \(\beta_0\) is an intercept or location parameter.
• \(\beta_j\) is the coefficient associated with covariate \(X_j\).
• \(\sigma\) is a scale parameter.
• \(\varepsilon\) follows a specified distribution.

The distribution assigned to \(\varepsilon\) determines the particular parametric AFT model.

The critical feature is that the regression operates on log survival time, rather than directly on the hazard.

The AFT Interpretation

Suppose \(X\) is a binary treatment indicator:

Consider the simple model:

Exponentiating gives:

For a treatment patient compared with an otherwise identical control patient, the multiplicative effect on survival time is:

This quantity is commonly called the acceleration factor or time ratio.

Why the Term "Accelerated" Failure Time?

The terminology can initially seem backward. If a treatment has an acceleration factor greater than 1, patients remain event-free longer.

The word acceleration refers to the underlying time scale. A covariate can accelerate or decelerate the progression toward failure.

An acceleration factor:

AFT Versus Cox Proportional Hazards Models

The most important comparison is between the AFT framework and the Cox proportional hazards model.

Hazard Ratios and Time Ratios Are Not the Same

A hazard ratio compares instantaneous event rates.

A time ratio compares the survival-time scale.

These are conceptually different quantities.

For example, a hazard ratio of 0.70 does not generally mean that patients survive exactly 1/0.70 times as long.

The relationship between a hazard ratio and a time ratio depends on the underlying survival distribution and model structure.

The Distribution Determines the AFT Model

AFT models require a probability distribution for the error term. Common choices include:

• Exponential
• Weibull
• Log-normal
• Log-logistic
• Generalized gamma in more flexible implementations

The choice is important because it determines the shape of the survival and hazard functions.

The Weibull AFT Model

The Weibull model is particularly important because it can be represented within both AFT and proportional-hazards frameworks.

A Weibull AFT model can be expressed as:

with an appropriate extreme-value distribution for \(\varepsilon\).

The survival function can be written in a parameterization such as:

where the scale parameter changes with the covariates.

Depending on parameterization, the regression coefficients may appear with different signs or scale transformations.

The Exponential AFT Model

The exponential model is a special case of the Weibull model.

Its defining characteristic is a constant hazard over time.

This can be useful for teaching and simple applications, but it is often too restrictive for clinical survival data because hazards commonly change over follow-up.

The Log-Normal AFT Model

The log-normal AFT model assumes:

with:

The log-normal distribution can accommodate a hazard that rises and then falls.

This can be attractive in settings where event risk is concentrated around a particular period of follow-up rather than continuously increasing or decreasing.

The Log-Logistic AFT Model

The log-logistic model assumes that the logarithm of survival time follows a logistic distribution.

Like the log-normal model, it can accommodate a non-monotone hazard.

It also has relatively heavy tails, which can be useful when long survival times are plausible.

Right Censoring

Clinical survival data are frequently right-censored.

Suppose a patient has not experienced the event by the end of follow-up. We know only that:

where \(C_i\) is the censoring time.

Define the event indicator:

The contribution to the likelihood is:

where \(f(t)\) is the density and \(S(t)\) is the survival function.

Therefore:

• An observed event contributes the density \(f(t_i)\).
• A censored observation contributes the survival probability \(S(t_i)\).

The full likelihood is:

The model parameters are estimated by maximizing this likelihood, or equivalently its logarithm.

Why Censored Patients Are Still Informative

A censored patient does not provide an exact event time. But the observation still tells us that the patient survived at least until the censoring time.

For example, if a patient is censored at 18 months, the model knows:

That information contributes directly through the survival function.

The Core AFT Regression Equation

For \(p\) covariates, write:

Suppose \(X_j\) increases by one unit while all other covariates remain fixed. The expected or characteristic time scale is multiplied by:

Thus, the coefficient itself is on the log-time scale, while \(\exp(\beta_j)\) is usually much easier to communicate clinically.

Interpreting a Continuous Covariate

Suppose age has coefficient:

The time ratio for a one-year increase in age is:

Therefore, each additional year of age is associated with approximately a 1.5% reduction in the survival-time scale, holding the other covariates constant.

For a 10-year increase, the corresponding time ratio is:

or approximately a 13.9% shorter time scale.

Interpreting a Binary Treatment Effect

Suppose the treatment coefficient is:

Then:

The treatment is therefore associated with a survival-time scale approximately 1.50 times that of the reference group, conditional on the model and covariates.

Equivalently, the characteristic survival time is approximately 50% longer.

Interpreting a Time Ratio Below One

Suppose instead:

Then:

The treatment group has approximately 75% of the reference group's time scale.

That corresponds to approximately a:

or 25% shorter survival-time scale.

Confidence Intervals for the Time Ratio

Suppose the coefficient estimate is \(\hat\beta\) with standard error \(SE(\hat\beta)\). A large-sample 95% confidence interval on the coefficient scale is:

Exponentiating both endpoints gives the confidence interval for the time ratio:

Because the null value for a time ratio is 1, an interval that excludes 1 provides evidence of an association at the corresponding two-sided significance level.

A Complete Worked Example

Consider a randomized clinical trial comparing an experimental treatment with standard therapy. The primary endpoint is progression-free survival (PFS), measured in months. Suppose the investigators fit a Weibull AFT model with:

• Treatment group
• Age in years
• Baseline disease severity

The fitted treatment coefficient is:

with standard error:

The treatment time ratio is:

Thus the estimated PFS time scale for the treatment group is approximately 50% longer than that of the reference group, conditional on the other covariates.

Step 1: Calculate the Confidence Interval

The 95% confidence interval for the coefficient is:

The lower limit is:

and the upper limit is:

Exponentiating gives:

and:

Therefore the estimated time ratio is approximately:

The interval excludes 1, suggesting evidence that the treatment changes the PFS time scale.

Step 2: Translate the Result Clinically

A useful clinical interpretation is:

This is generally easier to explain to a clinical audience than a statement about an abstract log-time coefficient.

Relating the Time Ratio to a Median

Under an AFT model, the survival distribution for one covariate pattern is a time-scaled version of the baseline distribution.

If a covariate pattern has time ratio \(c\), then a corresponding quantile such as the median is also multiplied by \(c\), provided the same model parameterization applies.

For example, if the reference group's model-based median PFS is 8 months and the treatment time ratio is 1.50, then the corresponding model-based median is:

This illustrates why the AFT framework can provide an intuitive time-scale interpretation.

Survival Quantiles Under an AFT Model

More generally, if \(Q_0(q)\) is a baseline survival quantile, then the corresponding quantile for covariate vector \(\mathbf{x}\) can be expressed as a multiplicative time transformation:

This applies to the median as well as other quantiles, subject to the particular model parameterization.

This is one of the strongest conceptual advantages of AFT models: the covariate effect acts directly on the time axis.

What Does "Acceleration" Mean Mathematically?

Let \(T_0\) represent the event time under a reference covariate pattern. An AFT model implies a relationship of the form:

Therefore, if:

the time scale is stretched by a factor of 2. If the factor is 0.5, the time scale is compressed by half.

AFT Model and the Survival Function

An AFT model can also be described directly through the survival function. If \(S_0(t)\) is the baseline survival function, an AFT representation has the form:

This equation is particularly useful for understanding the word "accelerated."

If the time ratio is greater than 1, the argument of the baseline survival function is reduced, shifting the survival curve toward later times.

Thus AFT effects can be visualized as horizontal stretching or compression of the survival curve.

AFT Does Not Require Proportional Hazards

The standard Cox model assumes proportional hazards:

where the hazard ratio is constant over time.

An AFT model does not impose this proportional-hazards assumption.

In fact, depending on the selected distribution, the implied hazard ratio may vary with time.

When AFT Models Are Attractive

AFT models can be particularly useful when:

• The clinical question is naturally expressed in terms of time.
• Survival-time ratios are easier to communicate than hazard ratios.
• The proportional-hazards assumption is questionable.
• A clinically plausible parametric survival distribution is available.
• Model-based survival quantiles are useful.
• Long-term survival extrapolation is required.
• Health-economic or decision models require complete survival curves.

When AFT Models May Be Less Attractive

AFT models can be less suitable when:

• The chosen parametric distribution fits poorly.
• The survival distribution has complex features not captured by the selected family.
• There is little justification for a parametric extrapolation.
• The primary estimand is explicitly a hazard ratio.
• The study has complex time-varying treatment effects that are poorly represented by a simple AFT structure.

Weibull AFT Versus Log-Normal AFT

The choice of distribution can materially affect inference and extrapolation.

Model Selection Should Be Clinically Informed

It can be tempting to select the distribution with the smallest AIC and stop there.

That is often insufficient.

Model selection should consider:

• Likelihood-based criteria such as AIC and BIC
• Observed-versus-predicted survival curves
• Residual diagnostics
• Clinical plausibility
• Long-term extrapolation behavior
• External information about expected event patterns

AIC and BIC for AFT Models

AIC is defined as:

where \(\ell(\hat{\theta})\) is the maximized log-likelihood and \(k\) is the number of estimated parameters.

BIC is:

Lower values indicate better fit after accounting for model complexity.

AIC and BIC are useful for comparing candidate distributions fitted to the same data, but neither criterion by itself establishes that a model is clinically appropriate.

Residual Diagnostics for AFT Models

AFT models can be assessed using several types of residuals.

A common choice is the Cox-Snell residual:

where \(\hat H(t)\) is the estimated cumulative hazard.

If the model is correctly specified, Cox-Snell residuals should approximately follow an exponential distribution with rate 1.

Other useful residual approaches include:

• Deviance residuals
• Martingale-type residuals
• Quantile residuals
• Distribution-specific residual diagnostics

Visual Model Checking

Graphical diagnostics are particularly important for parametric survival models. Useful comparisons include:

• Kaplan-Meier versus model-based survival curves
• Observed versus predicted survival probabilities
• Observed versus predicted quantiles
• Residual survival plots
• Log-cumulative-hazard plots where appropriate

A model should reproduce the major features of the observed survival experience without producing implausible extrapolation.

AFT and Kaplan-Meier Curves

The Kaplan-Meier estimator is nonparametric and provides an empirical estimate of survival.

An AFT model instead specifies a complete survival distribution.

In practice, the Kaplan-Meier curve is often used as an important diagnostic benchmark for the fitted parametric model.

AFT Models With Multiple Covariates

Suppose the model is:

The treatment time ratio is:

holding age and baseline severity constant.

This is the usual adjusted interpretation of a regression coefficient.

Interactions in AFT Models

Suppose treatment effectiveness differs by disease severity. An interaction model could be:

where \(T\) is treatment and \(S\) is severity.

The treatment time ratio is then conditional on severity. For a one-unit increase in severity, the treatment effect on the log-time scale changes by \(\beta_3\).

This is analogous to interaction interpretation in ordinary regression: the treatment coefficient by itself is no longer the universal treatment effect.

Categorical Predictors

For a categorical variable with several levels, one level is designated as the reference. Suppose treatment has three categories:

The time ratios are:

and:

Each is interpreted relative to the control group.

Handling Continuous Predictors Carefully

For a continuous variable, the time ratio corresponds to the specified unit increase.

If age is measured in years, \(\exp(\beta)\) is the time ratio for one additional year.

If a more clinically meaningful contrast is 10 years, calculate:

This is often preferable to reporting an extremely small one-unit effect.

Time-Varying Covariates

Standard AFT models are generally formulated with covariates that are fixed or otherwise incorporated according to the model's assumptions.

If a treatment exposure changes over time, or if biomarkers are repeatedly measured, simply inserting time-updated values into a standard AFT model may not be appropriate.

Such settings may require more specialized approaches, such as:

• Time-dependent survival models
• Joint models
• Structural models
• Multi-state models
• Landmark analyses

Competing Risks

A standard AFT model for a single event time does not automatically solve a competing-risks problem.

For example, if the endpoint is cancer-specific death and patients may die from other causes, treating competing deaths as ordinary censoring can change the estimand and assumptions.

Competing-risk methods should therefore be considered when multiple mutually exclusive event types are clinically relevant.

Left Truncation and Delayed Entry

Some survival studies include patients who enter the risk set after time zero. For example, a registry may include only patients who survive long enough to enter the registry.

This is known as left truncation or delayed entry.

If delayed entry is present, the likelihood must account for the fact that the patient was known to have survived until entry.

Ignoring delayed entry can introduce selection bias.

R Implementation With Survival::survreg

The R package survival provides the survreg() function for several parametric accelerated failure time models.

A typical Weibull AFT model is:

library(survival)

fit_weibull <- survreg(
  Surv(pfs_time, pfs_event) ~ treatment + age + severity,
  data = trial_data,
  dist = "weibull"
)

summary(fit_weibull)

Here:

• pfs_time is the observed PFS time.
• pfs_event equals 1 for an observed event and 0 for censoring.
• treatment, age, and severity are predictors.
• dist = "weibull" specifies the Weibull model.

Log-Normal and Log-Logistic Models in R

The same modeling framework can be used to compare alternative distributions.

fit_lognormal <- survreg(
  Surv(pfs_time, pfs_event) ~ treatment + age + severity,
  data = trial_data,
  dist = "lognormal"
)

fit_loglogistic <- survreg(
  Surv(pfs_time, pfs_event) ~ treatment + age + severity,
  data = trial_data,
  dist = "loglogistic"
)

The models can then be compared using likelihood-based criteria and graphical diagnostics.

Extracting the Time Ratio

For an AFT coefficient from survreg(), the time ratio is obtained by exponentiating the coefficient:

beta_trt <- coef(fit_weibull)["treatmentTreatment"]

time_ratio <- exp(beta_trt)

time_ratio

For a positive coefficient, the time ratio is greater than 1. For a negative coefficient, it is less than 1.

Confidence Interval for the Time Ratio in R

beta <- coef(fit_weibull)["treatmentTreatment"]

se <- sqrt(
  vcov(fit_weibull)["treatmentTreatment",
                     "treatmentTreatment"]
)

ci_beta <- beta + c(-1, 1) * qnorm(0.975) * se

time_ratio <- exp(beta)
ci_time_ratio <- exp(ci_beta)

time_ratio
ci_time_ratio

This calculates the time ratio and its approximate 95% confidence interval by constructing the interval on the coefficient scale and exponentiating.

Comparing AFT Models With AIC

AIC(
  fit_weibull,
  fit_lognormal,
  fit_loglogistic
)

The model with the smallest AIC has the best AIC tradeoff among the candidate models being compared.

But AIC should be supplemented with graphical and clinical assessment.

Comparing Predicted Survival Curves

A useful diagnostic is to compare model-based survival predictions with the Kaplan-Meier estimate.

For example, model-based predictions can be generated for a representative covariate pattern.

new_patient <- data.frame(
  treatment = "Treatment",
  age = 60,
  severity = "Moderate"
)

pred <- predict(
  fit_weibull,
  newdata = new_patient,
  type = "quantile",
  p = c(0.25, 0.50, 0.75)
)

pred

The exact prediction options depend on the model and software implementation, so the software documentation should be consulted when producing final clinical analyses.

Understanding the Weibull Shape Parameter

The Weibull distribution has a shape parameter that controls how the hazard changes over time.

Conceptually:

This flexibility is one reason the Weibull model is widely used in clinical survival analysis.

Why Distribution Choice Matters

Imagine that two candidate models fit the first three years of a clinical trial similarly.

If the Weibull model predicts a gradually increasing hazard while the log-normal model predicts a hazard that eventually decreases, their long-term survival predictions may diverge substantially.

This matters when the model is used for:

• Economic evaluation
• Lifetime survival extrapolation
• Health technology assessment
• Long-term treatment benefit estimation
• Decision modeling

AFT Models and Restricted Mean Survival Time

Restricted mean survival time (RMST) is another useful way to summarize survival. For a restriction time \(\tau\):

RMST describes the average event-free time up to \(\tau\).

An AFT model and an RMST analysis answer different questions.

These approaches can be complementary rather than mutually exclusive.

AFT Models and Non-Proportional Hazards

Suppose the treatment effect changes over time. The proportional-hazards assumption may fail.

An AFT model may still provide a useful summary if the treatment effect is reasonably represented as a time-scale transformation.

However, failure of proportional hazards does not automatically imply that an AFT model is correct.

AFT and Hazard-Ratio Interpretation

For some distributions, including the Weibull distribution, an AFT model also has a proportional-hazards representation.

This is a special property of the distribution.

For example, the Weibull model belongs to an important class in which the same underlying distribution can be parameterized in both AFT and PH forms.

But the numerical relationship between the time ratio and hazard ratio depends on the parameterization and Weibull shape.

Consequently, investigators should not convert a time ratio to a hazard ratio using a generic reciprocal rule.

A Simple Conceptual Example

Suppose an AFT model estimates:

Imagine the reference group's median survival is 12 months. The corresponding model-based treatment median would be:

The model therefore describes the treatment as stretching the survival-time scale by 25%.

Now suppose another treatment has:

If the corresponding reference median is 12 months:

The event-time scale is compressed by 20%.

Common Mistake: Calling the Time Ratio a Hazard Ratio

A coefficient from an AFT model should not automatically be reported as a hazard ratio.

For example, if:

then:

In an AFT interpretation, 1.49 is a time ratio. It is not automatically a hazard ratio of 1.49.

Common Mistake: Forgetting the Direction of the Effect

In an AFT model:

means longer survival time.

By contrast, in a conventional hazard-ratio interpretation:

generally indicates lower hazard and therefore a favorable direction when the event is undesirable.

This difference in direction is a common source of confusion.

Common Mistake: Interpreting \(\beta\) Directly as a Percentage

Suppose:

It is not correct to say that survival increases by exactly 50%. Instead:

The model implies a time ratio of approximately 1.65, or a 64.9% longer time scale.

Common Mistake: Assuming Every AFT Model Has the Same Interpretation

The basic time-ratio interpretation is shared by standard AFT models, but software parameterizations differ.

The reported coefficient may correspond to a location parameter on the log time scale, and scale parameters may be reported separately.

Therefore:

• Read the software documentation.
• Check the distribution being used.
• Verify the sign convention.
• Confirm the interpretation with a simple known example.

Common Mistake: Choosing the Distribution Only by AIC

AIC is useful but should not be treated as the sole criterion.

For example, suppose Weibull and log-normal models have very similar AIC values. The choice may then depend on:

• Graphical fit
• Residual behavior
• Clinical plausibility
• Expected long-term hazard behavior
• External evidence

A tiny AIC difference should not automatically dictate a clinically implausible extrapolation.

Common Mistake: Ignoring Censoring

AFT models are survival models and must account properly for censoring.

Treating censored patients as though their censoring time were their actual event time can bias the analysis.

For example, a patient censored at 24 months did not necessarily experience the event at 24 months. The data only establish:

Common Mistake: Ignoring Informative Censoring

The usual likelihood assumes a suitable censoring mechanism, commonly described as independent or non-informative censoring conditional on the model variables.

If patients with worse prognosis are systematically more likely to leave follow-up, this assumption may fail.

Sensitivity analyses may then be needed.

Clinical Interpretation of an AFT Result

A strong clinical report should generally provide:

• The fitted distribution
• The estimated time ratio
• The confidence interval
• The covariates included
• The event definition
• The censoring definition
• The reference categories

For example:

Model Diagnostics Workflow

AFT Model Selection in a Clinical Trial

Suppose a Phase III trial has PFS as its primary endpoint. A reasonable model-development process might be:

1. Estimate Kaplan-Meier curves.
2. Examine the empirical event pattern.
3. Fit Weibull, log-normal, and log-logistic AFT models.
4. Compare likelihood-based fit statistics.
5. Compare predicted survival against Kaplan-Meier estimates.
6. Inspect extrapolated hazards and survival curves.
7. Assess whether the time-ratio interpretation is clinically meaningful.
8. Predefine or justify the final model used for inference.

AFT Models for Health Economic Modeling

AFT models are especially important when survival must be extrapolated beyond the period directly observed in a clinical trial.

For example, a cost-effectiveness model may need survival probabilities over 10 or 20 years even though the randomized trial has only several years of follow-up.

A parametric AFT model can generate a complete survival curve:

which can then be used to estimate:

• Life expectancy
• Quality-adjusted life years
• Time in progression-free states
• Expected treatment duration
• Long-term event probabilities

Comparing Extrapolated Survival

Suppose two models fit the observed data similarly:

The differences may appear modest within observed follow-up but become substantial over longer horizons.

This is why model uncertainty should be explicitly evaluated when AFT models are used for extrapolation.

Interpreting the Baseline Intercept

The intercept \(\beta_0\) represents the baseline location of log survival time when all covariates equal their reference or zero values.

Its clinical interpretation depends heavily on how the predictors are coded.

For example, if age is centered at 60 years, then the intercept corresponds to a patient aged 60 rather than age 0.

Centering Covariates

Suppose age is transformed as:

Then:

corresponds to a 60-year-old patient. The intercept therefore describes the log-time scale at age 60 rather than age zero.

Sample Size and Power for AFT Models

Unlike a simple binary endpoint, survival-model sample-size calculations depend on factors such as:

• Expected number of events
• Follow-up duration
• Accrual period
• Censoring
• Covariate distribution
• Effect size
• Chosen survival distribution

For many clinical trials, event count is more informative than total enrollment when planning survival analyses.

However, parametric AFT sample-size calculations require assumptions about the underlying survival distribution and should be validated carefully.

Hypothesis Testing in an AFT Model

For a treatment coefficient \(\beta_T\), the null hypothesis of no time-scale effect is:

Equivalently, on the time-ratio scale:

A two-sided Wald statistic can be constructed as:

The corresponding p-value is based on the assumed large-sample distribution.

Confidence Intervals Are Often More Informative

For clinical interpretation, the estimated time ratio and its confidence interval are usually more informative than the p-value alone.

For example:

communicates both the estimated magnitude and uncertainty.

A p-value alone would not communicate the size of the survival-time effect.

What If the Time Ratio Is Exactly One?

If:

then:

and the covariate has no multiplicative effect on the modeled survival-time scale.

What If the Confidence Interval Crosses One?

Suppose:

The point estimate suggests a longer survival-time scale, but the confidence interval includes 1.

Therefore the data do not provide strong evidence of a nonzero time-scale effect at the conventional two-sided 5% level.

The estimate should still be reported rather than simply labeling the result "negative."

AFT Models With Stratification

Stratification is straightforward conceptually in survival analysis, but standard AFT implementations differ in how stratification and multiple baseline distributions are handled.

If important heterogeneity exists across sites or disease strata, investigators should carefully consider whether the chosen parametric model adequately represents that heterogeneity.

AFT Models and Frailty

If patients are clustered within centers, physicians, families, or other groups, observations may not be independent.

Possible approaches include:

• Cluster-robust inference
• Frailty models
• Hierarchical survival models

The appropriate method depends on the scientific question and software implementation.

AFT Models and Missing Covariates

Missing baseline covariates can complicate AFT analyses.

Complete-case analysis may be acceptable in some circumstances but can be inefficient or biased if missingness is informative.

Multiple imputation may be considered when appropriate.

Importantly, the imputation model should respect the survival outcome and censoring structure rather than treating survival time as an ordinary uncensored continuous variable.

Reporting an AFT Analysis

A statistical report should identify:

• The AFT distribution used
• The survival endpoint
• The event definition
• The censoring rule
• The covariates
• The reference categories
• The coefficient estimates
• The time ratios
• Confidence intervals
• P-values where appropriate
• Model-selection criteria
• Diagnostic assessments
• Extrapolation assumptions if relevant

Example Results Table

The table is considerably easier to interpret when the regression coefficients are accompanied by their exponentiated time ratios.

A Practical Interpretation of the Example

The treatment time ratio of 1.50 indicates a 50% increase in the modeled survival-time scale relative to control.

The age time ratio of 0.985 indicates that each additional year of age is associated with approximately a 1.5% shorter survival-time scale, holding the other covariates constant.

The severe-disease time ratio of 0.803 indicates approximately a 19.7% shorter survival-time scale compared with the moderate-disease reference category.

Why AFT Models Can Be Clinically Intuitive

Consider the question: "How much longer does this treatment keep patients event-free?"

A time ratio answers that question directly.

For example:

can be described as a 40% longer survival-time scale.

This is often easier for non-statistical audiences to visualize than:

which describes a relative instantaneous event rate rather than a direct multiplication of survival time.

But AFT Is Not Always More Interpretable

The time-ratio interpretation is attractive, but it relies on the AFT model being a reasonable representation of the data.

If the survival curves cross in a way that cannot reasonably be represented by a time-scale transformation, a single AFT time ratio may be misleading.

Model adequacy therefore comes before interpretation.

Crossing Survival Curves

Crossing survival curves are often a warning sign for simple proportional hazards models.

They can also raise questions about a simple AFT interpretation.

An AFT model implies that one survival distribution can be transformed into another primarily by stretching or compressing the time axis.

If treatment effects change qualitatively over time, that assumption may not be adequate.

Comparing AFT With a Cox Model in Practice

A useful analysis can fit both models when scientifically appropriate.

For example:

cox_fit <- coxph(
  Surv(pfs_time, pfs_event) ~ treatment + age + severity,
  data = trial_data
)

aft_fit <- survreg(
  Surv(pfs_time, pfs_event) ~ treatment + age + severity,
  data = trial_data,
  dist = "weibull"
)

summary(cox_fit)
summary(aft_fit)

The models answer related but distinct questions.

When Both Models Are Reasonable

For a Weibull model, the PH and AFT formulations are mathematically connected.

If both frameworks appear plausible, reporting both can sometimes provide a useful sensitivity analysis.

However, the primary model should be selected based on the prespecified estimand and scientific rationale rather than simply choosing whichever produces the more favorable treatment result.

Model-Based Survival Prediction

One advantage of a parametric AFT model is the ability to generate predicted survival probabilities.

For a covariate vector \(\mathbf{x}\):

can be evaluated at clinically meaningful time points.

For example:

These model-based estimates can be used for clinical interpretation and decision modeling, provided the underlying model is appropriate.

Long-Term Extrapolation Requires Sensitivity Analysis

Suppose observed trial follow-up ends at 36 months but the analysis requires 10-year survival estimates.

The extrapolated region may contain little direct information.

A responsible analysis should therefore consider alternative plausible distributions.

For example:

• Primary Weibull AFT model
• Log-normal sensitivity analysis
• Log-logistic sensitivity analysis
• Alternative clinically informed hazard assumptions

The goal is not to select the model with the most favorable treatment effect. The goal is to characterize uncertainty caused by the survival model.

AFT Model Checklist

Common AFT Model Mistakes

1. Calling an AFT time ratio a hazard ratio. The two measures represent different treatment effects.
2. Interpreting \(\beta\) itself as a percentage. The percentage interpretation should be based on \(\exp(\beta)\).
3. Forgetting that the null time ratio is 1. The null coefficient is 0, but the null time ratio is 1.
4. Ignoring the distributional assumption. An AFT model is parametric and therefore depends on its chosen distribution.
5. Choosing a model solely by AIC. Clinical plausibility and graphical fit are also important.
6. Ignoring censoring. Censored observations contribute survival information and must be handled correctly.
7. Assuming that non-proportional hazards automatically justify AFT. AFT and PH are different modeling assumptions.
8. Ignoring extrapolation uncertainty. Long-term survival predictions can depend strongly on the selected distribution.
9. Reporting only a p-value. The estimated time ratio and confidence interval communicate the magnitude much better.
10. Failing to state the software parameterization. Weibull and other distributions can be parameterized differently across software.

What Should Be Reported in a Statistical Analysis Plan?

If an AFT model is prespecified as the primary or sensitivity analysis, the statistical analysis plan should specify:

• Survival endpoint
• Event definition
• Censoring definition
• Primary estimand
• AFT distribution
• Covariates
• Reference categories
• Handling of missing covariates
• Model diagnostics
• Time-ratio interpretation
• Confidence interval method
• Model-selection strategy, if applicable
• Extrapolation strategy, if applicable
• Sensitivity analyses

AFT Versus Other Survival Effect Measures

Each measure answers a somewhat different clinical question.

The Most Important Concept

The central idea behind an accelerated failure time model is simple: covariates multiply the survival-time scale.

The model starts with:

and exponentiating the relevant coefficient gives:

A time ratio greater than 1 means that event times are stretched longer. A time ratio less than 1 means that event times are compressed toward earlier events.

The distribution determines the shape of the survival and hazard functions, while the regression coefficients describe how covariates transform the time scale.

References

Kalbfleisch, J.D. & Prentice, R.L. (2002). The Statistical Analysis of Failure Time Data. 2nd ed. Wiley.
Klein, J.P. & Moeschberger, M.L. (2003). Survival Analysis: Techniques for Censored and Truncated Data. 2nd ed. Springer.
Collett, D. (2015). Modelling Survival Data in Medical Research. 3rd ed. CRC Press.
Hosmer, D.W., Lemeshow, S. & May, S. (2008). Applied Survival Analysis: Regression Modeling of Time-to-Event Data. 2nd ed. Wiley.
Royston, P. & Parmar, M.K.B. (2002). Flexible parametric proportional-hazards and proportional-odds models for censored survival data, with application to prognostic modelling and estimation of treatment effects. Statistics in Medicine, 21, 2175–2197.
Therneau, T.M. & Grambsch, P.M. (2000). Modeling Survival Data: Extending the Cox Model. Springer.