Estimation of separable direct and indirect effects in a continuous-time illness-death model

Abstract

In this article we study the effect of a baseline exposure on a terminal time-to-event outcome either directly or mediated by the illness state of a continuous-time illness-death process with baseline covariates. We propose a definition of the corresponding direct and indirect effects using the concept of separable (interventionist) effects (Robins and Richardson in Causality and psychopathology: finding the determinants of disorders and their cures, Oxford University Press, 2011; Robins et al. in arXiv:2008.06019 , 2021; Stensrud et al. in J Am Stat Assoc 117:175-183, 2022). Our proposal generalizes Martinussen and Stensrud (Biometrics 79:127-139, 2023) who consider similar causal estimands for disentangling the causal treatment effects on the event of interest and competing events in the standard continuous-time competing risk model. Unlike natural direct and indirect effects (Robins and Greenland in Epidemiology 3:143-155, 1992; Pearl in Proceedings of the seventeenth conference on uncertainty in artificial intelligence, Morgan Kaufmann, 2001) which are usually defined through manipulations of the mediator independently of the exposure (so-called cross-world interventions), separable direct and indirect effects are defined through interventions on different components of the exposure that exert their effects through distinct causal mechanisms. This approach allows us to define meaningful mediation targets even though the mediating event is truncated by the terminal event. We present the conditions for identifiability, which include some arguably restrictive structural assumptions on the treatment mechanism, and discuss when such assumptions are valid. The identifying functionals are used to construct plug-in estimators for the separable direct and indirect effects. We also present multiply robust and asymptotically efficient estimators based on the efficient influence functions. We verify the theoretical properties of the estimators in a simulation study, and we demonstrate the use of the estimators using data from a Danish registry study.

Keywords: Causal inference; Illness-death model; Mediation analysis; Separable effects; Survival analysis.

Fig. 1

Illness-death model without recovery

Fig. 2

An informal causal diagram illustrating the relationship between the treatment components and the counting processes. The thick edges indicate a deterministic relationship

Fig. 3

Comparison of the G-computation (white rectangles) and one-step (black triangles) estimators of the SDE computed at time points $t\in \{2,5,10,15,20,25\}$ in terms of bias, empirical standard error, coverage of 95% confidence intervals and accuracy of the standard error estimator. This figure contains scenarios (i)–(iv) (Color figure online)

Fig. 4

Comparison of the G-computation (white rectangles) and one-step (black triangles) estimators of the SDE computed at time points $t\in \{2,5,10,15,20,25\}$ in terms of bias, empirical standard error coverage of 95% confidence intervals and accuracy of the standard error estimator. This figure contains scenarios (v)–(vii) (Color figure online)

Fig. 5

Bias of the plug-in (white rectangles) and one-step (black triangles) estimators of the SDE computed at time points $t=15$ under violation of the identification assumption

Fig. 6

Nelson–Aalen estimates of the cumulative hazards of MI (top left), overall mortality (top right) and death without recurrent MI (bottom) in our cohort. The red curves are the treatment arm and the black curves are the placebo arm. Along with the hazards (solid lines) are shown 95% confidence intervals (dashed lines) (Color figure online)

Fig. 7

Estimates of the separable direct effect (SDE), separable indirect effect (SIE) and total effect (TE) using the one-step estimator. Solid lines represent effect estimates and dashed lines the corresponding 95 % point-wise confidence intervals

Fig. 8

Estimates of the separable direct effect (SDE), separable indirect effect (SIE) and total effect (TE) using the plug-in estimator. Solid lines represent effect estimates and dashed lines the corresponding 95 % point-wise confidence intervals