The causal interpretation of acceleration factors

TL;DR

The paper addresses the causal interpretation of time-to-event effects under unmeasured heterogeneity, where hazard ratios confound treatment effects with survivor selection. It develops a structural causal framework for accelerated failure time (AFT) models, defining both conditional and marginal causal acceleration factors and proving identifiability under independent censoring and no unmeasured confounding. Through simulations, it shows that the acceleration factor more faithfully captures causal effects than hazard-based measures, particularly in the presence of frailty and effect heterogeneity, and demonstrates time-varying acceleration factors when effects vary over time. A case study on gastric cancer data illustrates how time-varying acceleration can arise from heterogeneous effects and cautions about interpretation with small samples. Overall, the work argues for using time-varying AFT models to obtain interpretable causal estimands, while noting practical challenges in estimation and confounding control.

Abstract

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.

Figures (17)

• Figure 1: Cox estimator (purple markers) evaluated at follow-up times $t_{FU}$ and $\hat{\theta}_m$ (green curves) evaluated on $(0,t_{FU}]$ with follow-up times corresponding to increasing quantiles of the $T^{1}$ distribution (x-axis) and independent censoring times $T_C$ with varying means. Marker shape and line type indicate expected censoring, $\lambda_i^a(t) = \tfrac{t^2}{20} U_{0i} \exp(\beta a)$, $U_0 \sim \Gamma(1,1)$, $\exp(\beta)=1/3$, $\theta=\exp(\beta)^{1/3}$ and $n_{\mathrm{obs}}=10^6$.
• Figure 2: $\theta$ when $T^1 = T^0 / U_1$, $\lambda^a_{i}(t) = \frac{t^2}{20} U_{0i} \exp (\beta a), U_0 \sim \Gamma(1,1)$ and $U_1$ follows a BHN distribution with $\rho_1 = 1$, $\mathbb{E}[U_1] = 3^{1/3}$ (($p_1$, $\mu_1$, $p_2$, $\mu_2$) = $(0.05, 0.5, 0.18, 3.53)$) (green) and $\mathbb{E}[U_1] = (1/3)^{1/3}$ (($p_1$, $\mu_1$, $p_2$, $\mu_2$) = $(0.7, 0.3, 0.05, 5.10)$) (orange) (left); when $T^0 \sim \mathrm{Weibull}(\Lambda,2), \Lambda \sim X / \Gamma(1 + 1/2), X$ categorical ($\mathbb{P}(X = 1 ) = \mathbb{P}(X=10) = 0.5$) and $U_1$ as specified for left hand side (right). See associated $S_{T^0}$ and $S_{T^1}$ in \ref{['fig:sc_fig2']}. When $\theta(Q_{T^1}(p))>1$, the $p$-th quantile of $T^1$ is smaller than that of $T^0$ and $\theta(Q_{T^1}(p))<1$ the opposite holds.
• Figure 3: $\theta$ when $T^1 = T^0 / U_1$, $T^0$ as specified in \ref{['fig:effect_hetero']}, $U_1$ follows a Gamma distribution with $\mathrm{Var}(U_1) = 0.5,1,2$ (dashed, solid, dotted), $\mathbb{E}[U_1] = 3^{1/3}$ (green) and $\mathbb{E}[U_1] = (1/3)^{1/3}$ (orange). See associated $S_{T^0}$, $S_{T^1}$ in \ref{['fig:sc_fig3']}.
• Figure 4: Estimated $\theta_m$ (solid black) and corresponding $95$% confidence intervals (dashed black) on the quantile scale (left) and time scale (right). Furthermore, $\theta$ for $S_{T^0}(t) = S_{T | A = 0}(t)$ and $S_{T^{1}}(t) = 0.5 S_{T^{0}}(0.9t) + 0.5 S_{T^{0}}(0.45t)$ is presented (purple).
• Figure 5: SWIG for SCM (\ref{['SCAFT']}) extended with a (potential) confounder $L$. In \ref{['swig:label1']} the undirected dashed line indicate the presence or absence of arrows in either directions. In \ref{['swig:label2']} the undirected dashed lines indicate the presence or absence of arrows in either directions under the restriction $U_0 \perp \!\!\! \perp U_1$, i.e. the three possible scenarios: ($i$) $U_0 \rightarrow L \leftarrow U_1$, ($ii$) $L \perp \!\!\! \perp U_1$ and $U_0 \rightarrow L$ or $L \rightarrow U_0$ and ($iii$) $L \perp \!\!\! \perp U_0$ and $U_1 \rightarrow L$ or $L \rightarrow U_1$.

Theorems & Definitions (8)

• Definition 1: Causal acceleration factor
• Lemma 2.1
• Definition 2: Observed acceleration factor
• Theorem 3.1
• Proposition 3.1
• proof
• proof
• proof