Feedback and Mediation in Causal Inference Illustrated by Stochastic Process Models

Abstract

The concept of causality is naturally related to processes developing over time. Central ideas of causal inference like time-dependent confounding (feedback) and mediation should be viewed as dynamic concepts. We shall study these concepts in the context of simple dynamic systems. Time-dependent confounding and its implications are illustrated in a Markov model. We emphasize the distinction between average treatment effect, ATE, and treatment effect of the treated, ATT. These effects could be quite different, and we discuss the relationship between them. Mediation is studied in a stochastic differential equation model. A type of natural direct and indirect effects is considered for this model. Mediation analysis of discrete measurements from such processes may give misleading results, and one needs to consider the underlying continuous process. The dynamic and time-continuous view of causality and mediation is an essential feature, and more attention should be payed to the time aspect in causal inference.

Time‐dependent mediators in survival analysis: Modeling direct and indirect effects with the additive hazards model

“…A time‐continuous stochastic process view of mediation is studied in Aalen, Røysland, Gran, Kouyos, and Lange (); Aalen, Gran, Røysland, Stensrud, and Strohmaier () and it is shown how misleading results may follow when the time aspect is not properly taken care of. Although we here only have time‐discrete mediator measurements available, we are still able to represent key aspects of of the time‐varying structure.…”

Interventions in dynamic systems: A causal approach to continuous-time mediation analysis

“…This argument follows the same reasoning as above when we took it that the DT-VAR(1) model with interval ∆t was the true data-generating model in comparison to the DT model with 2∆t: Any Φ(∆t) represents the combination of all pathways between a pair of variables through unobserved latent values of X, M and Y , as depicted in Figure 3a. As such, in the general case, two processes which are time-locally independent may still produce lagged values that are dependent, due to indirect pathways (A jk = 0 Φ(∆t) jk = 0) (see also Aalen et al, 2012Aalen et al, , 2016Aalen et al, , 2018. Deboeck & Preacher (2016), and equivalently Aalen et al (2012) and Aalen et al (2018), thus offer an alternative method for calculating the direct and indirect effects based on path-tracing.…”

“…As such, in the general case, two processes which are time-locally independent may still produce lagged values that are dependent, due to indirect pathways (A jk = 0 Φ(∆t) jk = 0) (see also Aalen et al, 2012Aalen et al, , 2016Aalen et al, , 2018. Deboeck & Preacher (2016), and equivalently Aalen et al (2012) and Aalen et al (2018), thus offer an alternative method for calculating the direct and indirect effects based on path-tracing. They suggest calculating path-specific effects by first altering the A matrix to remove pathways between variables which you wish to exclude, then applying the matrix exponential function e A∆t to this altered drift matrix to obtain the path-specific lagged effect of interest.…”

“…However, while Deboeck & Preacher (2016) and Aalen et al (2012Aalen et al ( , 2018 both offer alternative conceptualizations of total, direct and indirect effects for CT-VAR(1) systems, they do so strictly within the path-tracing framework. In the case of Deboeck & Preacher (2016), no connection is drawn to interventionist causal framework.…”

“…CT models are appealing for both practical reasons, in that they deal well with unequal time-intervals between observation waves, and conceptual reasons, in that they may better reflect our substantive notions about the underlying data-generating process (Boker, 2002;Aalen et al, 2016;Voelkle et al, 2012). Deboeck & Preacher (2016) and Aalen et al (2012Aalen et al ( , 2018 have specifically discussed CT models for mediation analyses, demonstrating the application of path-tracing rules in this setting. These authors argue that very different conclusions about the mediation mechanism result when using CT, rather than DT models: for instance, it is argued that the total effect derived from the usual DT-VAR(1) should actually be interpreted as a direct effect.…”