The causal interpretation of acceleration factors

In studies of time-to-event outcomes with unmeasured heterogeneity, the hazard ratio for treatment is known to have a complex causal interpretation. Accelerated failure time (AFT) models, which assess the effect on the survival time ratio scale, are often suggested as a better alternative because they model a parameter with direct causal interpretation while allowing straightforward adjustment for measured confounders. In this work, we formalize the causal interpretation of the acceleration factor in AFT models using structural causal models and data under independent censoring. We prove that the acceleration factor is a valid causal effect measure, even in the presence of frailty and treatment effect heterogeneity. Through simulations, we show that the acceleration factor better captures the causal effect than the hazard ratio when both AFT and conditional proportional hazards models apply. Additionally, we extend the interpretation to systems with time-dependent acceleration factors, illustrating the impossibility of distinguishing between a time-varying homogeneous effect and unmeasured effect heterogeneity. While the causal interpretation of acceleration factors is promising, we caution practitioners about potential challenges for the interpretation in the presence of effect heterogeneity.

💡 Research Summary

This paper addresses a fundamental problem in survival analysis: the hazard ratio (HR), the most common effect measure on the hazard scale, does not have a straightforward causal interpretation when unmeasured heterogeneity (frailty) or treatment‑effect heterogeneity is present. Conditioning on survival, which is implicit in the definition of the HR, induces a built‑in selection bias that can distort the causal meaning of the HR even in randomized trials.

Accelerated failure time (AFT) models provide an alternative by focusing on the survival‑time scale. The key parameter in an AFT model is the acceleration factor (θ), which quantifies how much the entire distribution of event times under treatment is stretched or compressed relative to the control distribution. The authors formalize the causal meaning of θ using structural causal models (SCM) and the standard independent‑censoring assumption.

First, they define a conditional causal acceleration factor θ_c(u₁,a,t) that links the quantiles of the potential outcome distributions for a given individual’s frailty (U₀) and treatment‑effect heterogeneity (U₁). By integrating over U₀ they obtain the marginal causal acceleration factor θ(a,t). Both definitions are expressed in terms of quantile functions, making the relationship between treated and untreated event times explicit: S_{T_a}(t)=S_{T₀}(t·θ(a,t)).

The central theoretical result (Theorem 3.1) shows that, under exchangeability (T_a ⟂ A) and causal consistency (T_A = T), the observed acceleration factor
θ_m(a,t)=t⁻¹ S_{T|A=0}^{‑1}(S_{T|A=a}(t))
coincides exactly with the causal acceleration factor θ(a,t) for every time point t. Importantly, this equality holds regardless of the distribution of frailty U₀ or the distribution of the individual‑level effect modifier U₁. Consequently, the AFT estimand is free from the selection bias that plagues the HR.

The authors extend the identification argument to right‑censored data. Assuming independent censoring, the censored survival function can be expressed via the observable hazard limit (Equation 12), and the same equality between θ_m and θ holds (Proposition 3.1). Thus, standard survival‑analysis software that estimates AFT models under independent censoring yields a consistent estimator of the causal effect.

To illustrate the practical implications, the paper presents simulation studies based on explicit SCMs. Two scenarios are compared: (i) a setting where the true causal effect is a constant acceleration factor (θ constant) and (ii) a setting where the true causal effect is a constant hazard ratio. In both cases the underlying frailty distribution is non‑degenerate and a heterogeneous treatment effect is introduced. The simulations confirm that the estimated θ accurately recovers the true causal effect in scenario (i), while the estimated HR is biased by the frailty and effect heterogeneity. Conversely, when the true effect is a hazard ratio, the HR performs well but the estimated θ deviates from the causal quantity, highlighting that each estimand is optimal only under its own model assumptions.

The paper also discusses time‑varying acceleration factors. By defining η(a,t)=∂/∂t S_{T₀}^{‑1}(S_{T_a}(t)), they show that when the conditional acceleration factor is homogeneous (θ_c does not depend on t), η and θ coincide. However, when the acceleration factor varies with time, η captures the instantaneous scaling of quantiles, and the observed θ_m still equals the causal θ(t). The authors point out an identifiability limitation: a genuinely time‑varying homogeneous effect and an unmeasured heterogeneous effect can generate the same observed θ(t), making it impossible to distinguish between them without additional assumptions or data.

In summary, the paper establishes that the acceleration factor in AFT models is a valid, interpretable causal effect measure even in the presence of frailty and treatment‑effect heterogeneity, provided that exchangeability and independent censoring hold. It offers a rigorous theoretical foundation, practical identification results, and simulation evidence that the AFT approach avoids the selection bias inherent to hazard‑ratio‑based analyses. Practitioners are encouraged to adopt AFT models when a causal interpretation on the time scale is desired, while remaining cautious about potential challenges when effect heterogeneity is strong or when the acceleration factor is suspected to be time‑varying.