Proportional-Hazards and Accelerated-Failure-Time Models

Two Ways to Describe Covariate Effects on Event Time

A covariate can change the rate at which an event is occurring at a particular time, or it can stretch or compress the event-time distribution. These are related but distinct claims. A proportional-hazards (PH) model expresses the first claim: at every time, a covariate multiplies the hazard by the same factor. An accelerated-failure-time (AFT) model expresses the second: a covariate multiplies event times by a constant factor. In nonlinear mixed-effects survival models, these statements are usually made conditional on an individual effect, so a covariate effect for a given individual need not equal the corresponding marginal effect across the population.

Let \(T_i\) be the positive event time for individual \(i\), \(x_i\) a covariate vector, and \(b_i\) an individual random effect. Conditional on \(x_i\) and \(b_i\), define the survival function \(S_i(t)=P(T_i>t\mid x_i,b_i)\), the density \(f_i(t)\), and the hazard \(h_i(t)=f_i(t)/S_i(t)\). The hazard is an instantaneous event rate among individuals who have not yet experienced the event; it is not the probability of an event at time \(t\).

Definition: A conditional proportional-hazards model has hazard \(h_i(t\mid x_i,b_i)=h_0(t)\exp(x_i^\mathsf{T}\beta+z_i^\mathsf{T}b_i)\). Here \(h_0(t)\) is the baseline hazard, \(\beta\) is the covariate-effect vector, and \(z_i^\mathsf{T}b_i\) is the individual-effect contribution. For a one-unit increase in a covariate with coefficient \(\beta_j\), holding other covariates and the individual effect fixed, the hazard ratio is \(\exp(\beta_j)\), the same at every time.

This specification implies \(S_i(t\mid x_i,b_i)=\exp[-H_0(t)\exp(x_i^\mathsf{T}\beta+z_i^\mathsf{T}b_i)]\), where \(H_0(t)=\int_0^t h_0(u)\,du\) is the baseline cumulative hazard. The baseline hazard may be modeled parametrically or left more flexible; PH concerns the way covariates enter, not a particular baseline-hazard shape. A constant coefficient is essential: if a covariate effect changes with time, its hazard ratio is no longer constant.

Accelerated-Failure-Time Formulation

An AFT model starts from the event time rather than the hazard. One common mixed-effects form is a linear model for log event time:

\log T_i=\mu+x_i^\mathsf{T}\gamma+z_i^\mathsf{T}b_i+\sigma\varepsilon_i,\qquad \sigma>0.

Here \(\varepsilon_i\) follows a specified error distribution, which determines the baseline event-time distribution, and \(\gamma\) describes covariate effects on log time. Equivalently, if the covariate contribution is \(\eta_i=x_i^\mathsf{T}\gamma\), then, at fixed individual effect, the event-time distribution is the baseline distribution scaled by \(\exp(\eta_i)\). A positive \(\gamma_j\) therefore lengthens event times for a one-unit increase in covariate \(x_j\); a negative value shortens them.

Definition: In an AFT model, the event-time ratio for a one-unit increase in \(x_j\), holding other terms fixed, is \(\exp(\gamma_j)\). It multiplies every event-time quantile by that factor. For example, a ratio of 1.5 means that the median and other corresponding quantiles are 1.5 times as large, not that the event probability is multiplied by 1.5.

AFT models can use different distributions for \(\varepsilon_i\), including distributions that yield lognormal, Weibull, or log-logistic event times. The distributional choice matters: AFT specifies a common time scaling, but does not by itself determine the shape of the baseline event-time distribution. In a mixed-effects model, \(b_i\) accounts for individual heterogeneity conditional on the model. A random effect in an AFT model shifts log time; a random effect in a PH model shifts log hazard. Their signs and interpretations should not be casually equated.

What the AFT Assumption Implies for Hazards

The distinction between the formulations can be seen by deriving the hazard produced by an AFT time scaling. Fix an individual effect and write \(\eta=x^\mathsf{T}\gamma+z^\mathsf{T}b\). If \(T=\exp(\eta)T_0\), where \(T_0\) has baseline survival \(S_0\), then:

S(t\mid \eta)=P(\exp(\eta)T_0>t)=S_0(t\exp(-\eta)).

Differentiating the survival function gives the density \(f(t\mid\eta)=\exp(-\eta)f_0(t\exp(-\eta))\). Dividing this density by the survival function gives:

h(t\mid\eta)=\exp(-\eta)h_0(t\exp(-\eta)).

This is generally not a proportional-hazards model: the covariate changes both the hazard multiplier and the time point at which the baseline hazard is evaluated. Thus, a constant event-time ratio does not generally imply a constant hazard ratio. Conversely, the PH survival function is \(S(t\mid\eta)=S_0(t)^{\exp(\eta)}\), which generally does not have a constant time-scaling interpretation.

There is an important special case. Suppose the baseline cumulative hazard is Weibull, \(H_0(t)=(t/\lambda)^k\), with shape \(k>0\) and scale \(\lambda>0\). Under AFT scaling by \(\exp(\eta)\),

H(t\mid\eta)=H_0(t\exp(-\eta)) =\left(\frac{t}{\lambda}\right)^k\exp(-k\eta) =H_0(t)\exp(-k\eta).

The Weibull AFT model is therefore also PH, with log hazard ratio \(-k\eta\). For a covariate coefficient \(\gamma_j\), the corresponding PH coefficient is \(-k\gamma_j\) under this parameterization. The negative sign is meaningful: increasing event time by a factor greater than one lowers the hazard in this Weibull case. This equivalence depends on the Weibull shape and parameterization; it is not a general identity between PH and AFT models.

Worked Example: Interpreting a Proportional-Hazards Effect

Suppose a hypothetical mixed-effects model for time to a clinical event has \(h_i(t)=h_0(t)\exp(0.693x_i+b_i)\), where \(x_i=1\) indicates a treatment and \(x_i=0\) indicates a comparator. At a specified time, let the baseline hazard be \(0.0800\) per month. Compare individuals with the same \(b_i=0\).

The treatment hazard ratio is \(\exp(0.693)=2.00\) (to three significant figures). For the comparator, the conditional hazard is \(0.0800\) per month. For treatment, it is \(0.0800\times 2.00=0.160\) per month.

Thus, at every time under this model, the treated individual’s instantaneous event rate is twice the comparator’s, conditional on the stated individual effect. This does not mean the probability of an event is twice as large, and it does not say that event times are half as long. The latter would require an AFT interpretation or additional distributional assumptions.

Worked Example: Interpreting an Accelerated-Failure-Time Effect

Suppose a hypothetical AFT model has \(\log T_i=\mu+0.405x_i+b_i+\sigma\varepsilon_i\), where \(x_i\) is an indicator for a supportive intervention. The estimated coefficient is \(0.405\), and the comparator median event time at a fixed individual effect is 8.00 weeks. Interpret the intervention effect and calculate its implied median.

The event-time ratio is \(\exp(0.405)=1.50\). Thus, at the same individual effect, the model predicts event-time quantiles under the intervention to be 1.50 times their comparator values. In particular, the median is \(8.00\times1.50=12.0\) weeks.

This is a model-based time-scale interpretation: the event-time distribution is stretched by 50% at that covariate contrast. It is not an assertion that every individual’s observed event time will equal exactly 12.0 weeks, nor does it describe a 50% reduction in hazard unless further assumptions—such as the Weibull structure above—are imposed.

Assumptions, Checks, and Model Choice

Both formulations require a clearly specified conditional event-time model and a defensible treatment of individual heterogeneity. Typical mixed-effects assumptions include that the random effects follow the specified distribution, are independent across individuals, and are appropriately related to the modeled covariates. The model also requires that the chosen conditional event-time distribution or hazard structure adequately represents the event process. Marginalizing over random effects can change the shape of a hazard over time, so a conditional PH model does not automatically imply marginal proportional hazards.

The distinguishing assumption is how covariates act. PH assumes a time-constant hazard ratio conditional on the model terms. AFT assumes a constant event-time ratio, so all corresponding quantiles scale together. These assumptions can be implausible in different ways: a covariate may have an early effect that fades, violating simple PH, or it may shift early and late quantiles by different factors, violating simple AFT. Repeated measurements or a more complex covariate history may require extensions; neither basic form makes a time-varying covariate effect automatic.

Useful checks should target the assumption at issue. For PH, examine whether the estimated covariate effect appears to vary over time, rather than treating one constant hazard ratio as self-evident. For AFT, compare observed and model-implied event-time distributions or quantiles across covariate groups, asking whether a common time stretch is plausible. Also check whether the selected baseline distribution and random-effects structure reproduce important features of the data. These are checks of model adequacy, not proofs that an assumption is true; detailed survival diagnostics require additional tools.

Worked Example: Comparing Weibull PH and AFT Interpretations

Consider a hypothetical Weibull AFT model with shape \(k=2.00\) and covariate coefficient \(\gamma= \log(1.40)=0.336\). Calculate the event-time ratio and the equivalent conditional hazard ratio under the Weibull model.

The AFT event-time ratio is \(\exp(\gamma)=\exp(0.336)=1.40\). A one-unit increase in the covariate multiplies each event-time quantile by 1.40.

Using the Weibull relation, the PH log-hazard coefficient is \(-k\gamma=-2.00(0.336)=-0.672\). The conditional hazard ratio is \(\exp(-0.672)=0.511\), equivalently \(1/(1.40)^2=0.510\) up to rounding. This model therefore combines longer event times with a lower hazard. The calculation depends on the Weibull shape being 2.00; it cannot be transferred to an arbitrary AFT distribution.

PH is often a natural choice when the scientific question concerns relative instantaneous event rates and a constant hazard ratio is plausible. AFT is often more direct when the question concerns how much longer or shorter event times are expected to be. A flexible baseline hazard can avoid committing to a narrow baseline shape in a PH model, while a parametric AFT model supplies a full event-time distribution through its chosen error distribution. Neither approach is uniformly preferable: the relevant covariate assumption, baseline fit, and prediction target should guide the choice.

Common Pitfalls

Calling a hazard ratio a risk ratio. A hazard is an instantaneous rate among those still event-free, not a cumulative probability. Report a hazard ratio as a relative hazard, with its conditional context.
Reversing the AFT direction. A time ratio of 1.40 means event times are multiplied by 1.40 and are longer, not shorter. A ratio below 1 indicates shorter event times.
Assuming PH and AFT are equivalent. They generally impose different covariate-effect structures. The Weibull case is a notable special case, with a parameter-dependent coefficient mapping.
Ignoring individual effects in interpretation. A conditional coefficient compares individuals at the same random-effect value. Averaging over random effects can yield a different population-level pattern.

Key takeaway: PH multiplies the conditional hazard by a constant covariate-dependent factor; AFT multiplies conditional event times by a constant factor. State which quantity is scaled, identify whether the effect is conditional on individual effects, and verify that the corresponding assumption is scientifically plausible.

Check Your Understanding

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

A PH coefficient is \(\log(0.70)\). What is the hazard ratio, and what does it mean?
An AFT coefficient is \(\log(0.80)\). Interpret the event-time ratio and the direction of the effect.
Why does a constant AFT time ratio not generally imply a constant hazard ratio?
For a Weibull AFT model with shape \(k=1.5\) and time ratio 2.00, calculate the corresponding hazard ratio.

Solutions. (1) The hazard ratio is \(\exp[\log(0.70)]=0.70\): at each time, the covariate group has 30% lower conditional hazard, holding other terms and individual effect fixed. (2) The time ratio is 0.80, so corresponding event-time quantiles are 20% shorter under the covariate contrast. (3) AFT scaling gives \(h(t\mid\eta)=\exp(-\eta)h_0(t\exp(-\eta))\); the shifted time argument generally changes the hazard ratio over time. (4) The Weibull hazard ratio is \(1/(2.00)^{1.5}=0.354\) (three significant figures), because the Weibull log-hazard coefficient equals \(-k\gamma\).