Identification, Estimation, and Sensitivity Analysis for Causal Mediation Effects Under Longitudinal Settings

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Abstract

Objectives or purposes
Addressing questions about causal mechanisms is of great interest in education and the social sciences. Causal mediation analysis is a statistical framework that enables researchers to investigate causal mechanisms. Although causal mediation analysis is a fast-growing research area, there is little extant literature investigating causal mechanisms under longitudinal settings (see e.g., VanderWeele and Tchetgen Tchetgen(2014) and Shpitser (2013)). Considering that mediation processes develop over time, ignoring the time effect on investigating causal mechanisms may be problematic. In addition, longitudinal data can shed more light on investigating mediation processes for variety of reasons (Mackinnon, 2008).

Under longitudinal settings, there exist statistical tools for investigating mediation processes such as auto-regressive models, difference score methods, covariance structural models, latent growth curve models and many others. Auto-regressive model is a special type of structural equation model where contemporaneous and longitudinal mediation effects are jointly evaluated. In contrast, difference score methods and latent growth curve models focus only on the trajectories of change across time. In other words, these models examine how change in the treatment affects the change in the mediator and, in turn, the change in the mediator affects the change in the outcome. Mackinnon (2008) and McArdle (2009) provide an overview of the methodologies in more detail.

However, a fundamental fact that is sometimes overlooked by these methodologies may involve questions about causation. While longitudinal data clearly has some benefits in examining causal relations, simply knowing the time order of a variable measured at different time points does not guarantee causality. Change over time is unlikely to be confounded by time-fixed covariates among individuals as each individual serves as a control for himself or herself (Mackinnon, 2008). However, there may be a time-varying variable that confounds the change. Therefore, the goal of this paper is to address questions concerning causation when investigating mediation processes in longitudinal settings.

Recently, VanderWeele and Tchetgen Tchetgen(2014) and Shpitser (2013) provided identification results for longitudinal mediation analysis under the potential outcomes theory of Rubin (1974). In their papers, VanderWeele and Tchetgen Tchetgen (2014) offered identification results based on several different causal structural models with time-varying treatments and mediators whereas Shpitser (2013) provided a theorem that is applicable to a general case of longitudinal mediation analysis. This paper will build on their related works. One of the causal structural models discussed in VanderWeele and Tchetgen Tchetgen (2014) includes a time-varying treatment, mediator and a final outcome with a fixed (i.e., time-invariant) set of pre-treatment covariates. In our paper, we use the similar causal structural model but allow also for time-varying covariates in addition to time-fixed covariates only. Assuming the ignorability of time-varying treatment and mediator selection, only conditioning on a set of fixed pre-treatment covariates may not be plausible in practice where confounding due to timevarying covariates is rather likely. Therefore, the paper aims to introduce a statistical framework of longitudinal causal mediation analysis that allows for time-varying confounding. We discuss the identifiability of Average Causal Mediation Effects (ACME) in relation to the time-varying covariates and provide a general estimation method using G-computation. Most importantly, a bias formula is presented that can be used for a sensitivity analysis to assess the mediation effects' sensitivity to unobserved time-varying confounders of the mediator-outcome relationships.

A Statistical Framework of Causal Mediation Analysis

Notation and Definition.
We introduce notation and definitions based on the potential outcomes framework of Rubin (1974). Let 𝑇𝑇1𝑖𝑖∈{0,1} and 𝑇𝑇2𝑖𝑖∈{0,1} be the first and second time measured binary treatments for individual i, where T=1 if assigned to the treatment condition and T=0 if assigned to the control condition, and let 𝑀𝑀1𝑖𝑖 and 𝑀𝑀2𝑖𝑖 be the time-varying mediators measured at the first and second time periods, respectively. Let 𝑌𝑌𝑖𝑖 be the outcome for individual i, and let 𝑋𝑋0𝑖𝑖, 𝑋𝑋1𝑖𝑖 and 𝑋𝑋2𝑖𝑖 be a set of pre-treatment covariates, measured before the first time point and then the first and second time periods thereafter.
The green path in Figure 1 represents the ACME where only two time points exist. We choose the simplest case where the mediation process is available where there exist two time points as a basis of the paper. Considering the complexity of longitudinal data, it may be the case that causal relations in reality are more complex than the causal diagram presented here. Thus, we further discuss the case where there are more than two time points in the paper.
Identification. This proposal describes the sequential ignorability assumption needed to identify the ACME under longitudinal settings and provide non-parametric identification results. The sequential ignorability is shown as below.

A1. No treatment-mediator and treatment-outcome confounding at the 1st time point
A2. No treatment-mediator and treatment-outcome confounding at the 2nd time point
A3. No mediator-outcome confounding at the 1st and 2nd time periods

Estimation.
We provide a general estimation method that combines the G-computation with either frequentist or Bayesian inferential methods that are applicable to both single-level and multi-level data. We first demonstrate G-computation for the frequentist framework with single-level data and then discuss extensions to Bayesian and multi-level settings.
Sensitivity Analysis. We provide bias formulas for the ACME and ANDE in case of violating the sequential ignorability assumption that can be used for sensitivity analysis. We will only focus on the violation of ignorability of the mediator with respect to the outcome. Figure 2 represents the case in which the sequential ignorability assumption is violated due to an unobserved time-varying variables 𝑈𝑈1 and 𝑈𝑈2 that confound the respective mediator-outcome relationship. I extend sensitivity analysis proposed by VanderWeele (2010) that is suitable for a causal structural model. The idea is to calculate the difference between the ACME estimated ignoring the existence of unmeasured covariates and the true ACME.

Case Study.
We present the case study where our proposed method is applied to a simulated data based on Families And Student Together (FAST) (McDonald & Frey, 1999; McDonald, 2002). FAST is a school-level randomized intervention that is designed to increase the parent-parent and parent-school social capital among minorities. The program includes an eight-week session of weekly meetings and two years of follow-up monthly parent-led meetings. The intervention is given at the first year, and followed by parent-led meetings for two years. The level of parent social capital and student peer problems are measured at the first and third years. The program demonstrated the active engagement of culturally sensitive parents and improvement of academic and behavioral outcome of students.
Turley, Gamoran, Turner and Fish (2012) examined the causal effect of FAST on student peer problems and found that FAST has a significant effect on reducing student peer problems. Turley and her colleges went further, and attempted to identify the mechanisms through which FAST has its effect on reducing student peer problems. Parent improved social capital is hypothesized as a factor that mediate through the causal effect. Intergenerational closure is one of the indices that represents the measured parent social capital, and is constructed based on parent self-report about how many parents of your child's friends they know. Causal mediation analysis was conducted using the cross-sectional first-year data, and this paper aims to extend the analysis by using both first- and third-year data. Thus, our interest focuses on whether the FAST intervention reduces student peer problems through knowing more parents of children's friends in the longitudinal context.
Our simulated data loosely resembles the FAST data. We decided to simulate data rather than use real data for two reasons: 1) Sensitivity analysis is better illustrated when the true confounding structure is known, and 2) FAST has some complications that make our analysis challenging; including drop out, non-response, non-compliance to the FAST intervention, and time-invariant intervention. There are various ways to handle these issues but they are beyond the scope of this paper and would distract from our main purpose. Therefore, we generated a data set that mimics some of the FAST study variables but not its complex design, non-response, non-compliance, and invariant treatment.

Using the simulated data, we fit two models where 1) the sequential igonorability assumption is met by including all covariates, and 2) the sequential ignorability assumption is not met due to omitted time-varying covariates--i.e. parent depression levels. Then, we illustrate the use of sensitivity analysis for the second model where the sequential ignorability assumption is not satisfied due to omitted time-varying confounders. Our causal structural models are reflected in Figure 1 when sequential ignorability is met and Figure 2 when sequential ignorability is not met.

Scientific or scholarly significance of the study or work
In this article we discussed the definition and identification of ACMEs and ANDEs for longitudinal data sets with time-varying covariates (instead of fixed covariates only—as previous studies did). We also introduced sensitivity analyses for ACMEs at different time points and across the time points. Using a simulated data set, which we based on the FAST intervention, we demonstrated how one can implement a parametric mediation analysis and sensitivity analysis. Knowing the true mediation and confounding effects, we demonstrated that ACME and ANDE can be consistently estimated if the analysis controls for all fixed and time-varying confounding covariates and if the model is correctly specified. If some confounders are unobserved, then biased effect estimates result. However, sensitivity analysis can be used to probe the effect estimates sensitivity to unobserved confounders.

Future plan
I am planning to apply the proposed approach to the ECLS-K data. This case study aims to investigate causal mechanisms, underlying the relationship between the kindergarten retention policy and student performance for those who are retained. The proposed hypothesis is that the difference in motivational factors developed for those who are promoted and retained mediates through the relationship between the kindergarten retention policy and student outcome for those who are retained. Results are not available at this moment but will be available by the time that I present my work at 2016AERA annual meeting.

Author

• Soojin Park, The University of Wisconsin-Madison