Proportional Hazards and Accelerated Failure Time Models in PK/PD Models

Two Ways to Express an Exposure Effect

For a time-to-event endpoint, an exposure effect can be described in terms of the event’s instantaneous hazard or in terms of how long event times tend to be. These are different questions. A hazard ratio compares instantaneous event rates among individuals who have remained event-free up to a given time. An accelerated failure time model instead asks whether exposure stretches or contracts the event-time scale.

The distinction matters in PK/PD work because a statement such as “higher exposure delays the event” is not itself a model. A proportional-hazards model quantifies the exposure contrast as a hazard ratio. An accelerated failure time (AFT) model quantifies it as a time ratio. Neither number can generally be substituted for the other.

The hazard is an instantaneous event rate among individuals who have not yet experienced the event; it is not the probability of an event over a specified interval. Thus, a hazard ratio of 0.60 means the instantaneous hazard is 40% lower for one exposure level than the comparison level, at each time under the PH assumption. It does not mean the probability of an event is 40% lower.

An AFT model makes a different assumption: exposure multiplies event times by a common factor. One common parameterization is a linear model for log event time, \(\log T=\mu+\gamma z+\sigma\epsilon\), where \(T\) is the event time, \(\epsilon\) has a specified distribution, and \(\sigma\) is a scale parameter. For a contrast \(\Delta z\), the time ratio is \(a=\exp(\gamma\Delta z)\). This formulation requires an appropriate distributional assumption for the log-time residuals.

How Each Model Transforms Survival

Let \(S_0(t)\) denote the survival function at the comparison exposure, and let \(H_0(t)=-\log S_0(t)\) denote its cumulative hazard. Under PH, the exposure contrast multiplies the cumulative hazard by the hazard ratio. Under AFT, it rescales the time argument of the baseline survival curve.

The PH survival transformation follows from \(S(t)=\exp[-H(t)]\). Multiplying the cumulative hazard by the hazard ratio \(r\) gives \(S_{\text{high}}(t)=\exp[-rH_{\text{low}}(t)]=S_{\text{low}}(t)^r\). For AFT, if the higher-exposure event time is distributed as \(aT_{\text{low}}\), then \(P(aT_{\text{low}}>t)=P(T_{\text{low}}>t/a)\), which gives the time-rescaled survival curve.

The AFT model also implies a hazard transformation. Differentiating the transformed survival curve gives \(h_{\text{high}}(t)=h_{\text{low}}(t/a)/a\). Therefore its hazard ratio is

This ratio can vary with time because the baseline hazard is evaluated at two different times. An AFT model therefore does not generally imply proportional hazards. The models can sometimes produce equivalent predictions for particular baseline survival shapes, but this is not a general property.

Comparing Curves and Model Implications

A parameter comparison is most informative when the implied survival curves are also examined. Consider a hypothetical baseline cumulative hazard, with time measured in days, defined as \(H_0(t)=0.05t+0.0025t^2\). This increasing hazard shape is used only to illustrate the transformations. Compare a PH model with hazard ratio 0.50 and an AFT model with time ratio 1.50.

Assumptions, Diagnostics, and Choice of Scale

Both models require a clearly defined event, time origin, exposure contrast, and covariate set. For ordinary right-censored data, the event-time model also relies on censoring being independent of the event time conditional on the modeled information. If loss to follow-up is related to unobserved event risk after conditioning, standard interpretations may be biased. Exposure coefficients are conditional model associations; a causal interpretation requires additional design and confounding assumptions.

The central PH assumption is that the hazard ratio for a given exposure contrast is constant over follow-up. Diagnostics include assessing whether estimated exposure effects vary systematically with time and checking survival-pattern displays for departures from the PH structure. A visibly changing hazard contrast, or residual patterns that indicate a time trend, calls the constant-ratio interpretation into question. A single average hazard ratio may conceal such variation.

An AFT model requires both a suitable log-time distribution and a common multiplicative time effect. Assess whether the selected distribution reproduces observed survival patterns and whether predicted time quantiles are calibrated across clinically relevant parts of follow-up. If the exposure shifts early and late quantiles by materially different factors, a single AFT time ratio is inadequate. Sparse late follow-up, heavy censoring, and limited overlap in exposure can make these checks weak.

The main alternative is therefore not “hazard ratio versus time ratio” as competing numerical summaries of one fixed effect. It is a choice of model structure and scientific scale. PH is useful when a constant relative instantaneous hazard is meaningful and a flexible baseline hazard is desirable. AFT is useful when multiplicative changes in event time are scientifically interpretable and a credible event-time distribution can be specified. Compare fitted survival curves and relevant quantiles, not only coefficient magnitudes or model fit statistics.