The Impact of Omitting Confounders in Parallel Process Latent Growth Curve Mediation Models: Three Sensitivity Analysis Approaches

Abstract

Parallel process latent growth curve mediation models (PP-LGCMMs) are frequently used to longitudinally investigate the mediation effects of treatment on the level and change of outcome through the level and change of mediator. An important but often violated assumption in empirical PP-LGCMM analysis is the absence of omitted confounders of the relationships among treatment, mediator, and outcome. In this study, we analytically examined how omitting pretreatment confounders impacts the inference of mediation from the PP-LGCMM. Using the analytical results, we developed three sensitivity analysis approaches for the PP-LGCMM, including the frequentist, Bayesian, and Monte Carlo approaches. The three approaches help investigate different questions regarding the robustness of mediation results from the PP-LGCMM, and handle the uncertainty in the sensitivity parameters differently. Applications of the three sensitivity analyses are illustrated using a real-data example. A user-friendly Shiny web application is developed to conduct the sensitivity analyses.

Keywords:

• Bayesian methods
• latent growth curve models
• mediation analysis
• sensitivity analysis
• unmeasured confounding

Notes

1 Besides the product-of-coefficients approach, mediation effects can be defined using approaches proposed in causal inference literature, which often involves the use of the potential outcomes framework (e.g., Pearl, 2001; Robins & Greenland, 1992; Valeri & VanderWeele, 2013). In this article, we consider the product-of-coefficients approach because it is widely used in empirical PP-LGCMM analysis in psychology and behavioral sciences and is relatively easy for substantive researchers to interpret.

2 In addition to the no-omitted-confounders assumption, mediation inference from a statistical mediation model requires the model to have correct functional form specification and correct assumptions on the residual distributions, the causal ordering of treatment, mediator, and outcome, and measurement errors in observed variables (e.g., MacKinnon, 2008; Mackinnon et al., 2007).

3 ${\widehat{\Psi }}_{vM}^{-1}=$ ${({\widehat{\Psi }}_{vM})}^{-1}=\frac{1}{{\widehat{\psi }}_{v_{IM}{v}_{IM}}{\widehat{\psi }}_{v_{SM}{v}_{SM}}-{\widehat{\psi }}_{v_{IM}{v}_{SM}}^2}(\begin{array}{cc}{\widehat{\psi }}_{v_{SM}{v}_{SM}}& -{\widehat{\psi }}_{v_{IM}{v}_{SM}}\\ -{\widehat{\psi }}_{v_{IM}{v}_{SM}}& {\widehat{\psi }}_{v_{IM}{v}_{IM}}\end{array}),$ where ${\widehat{\Psi }}_{vM}$ is the estimated residual covariance matrix of ${(v_{IM},v_{SM})}^{\prime }$ yielded by the original model.

4 The joint significance test can be used because the analytical results in Yuan and Bentler (2006) suggest that for the PP-LGCMM with known slope loadings (i.e., Time_Mt and Time_Yt are known), the ML estimates of the a-paths ${\widehat{\alpha }}_{IM.X}^{*}$ and ${\widehat{\alpha }}_{SM.X}^{*}$ from Model M1 (or the counterparts from Model M0) are asymptotically independent of the ML estimates of the b-paths, ${\widehat{\beta }}_{IY.IM}^{*},$ ${\widehat{\beta }}_{IY.SM}^{*},\hspace{0.17em}{\widehat{\beta }}_{SY.IM}^{*},$ and ${\widehat{\beta }}_{SY.SM}^{*}$ from Model M1 (or the counterparts from Model M0).

5 In the SEM literature, model identification is about whether it is possible to find unique estimates of the model parameters (Bollen, 1989). Under the sensitivity analysis model (Model M1), the maximum likelihood of the observed data can be obtained with non-unique values of Model M1’s parameters. Thus, with observed data only, Model M1 is not identified if there are no additional information (e.g., informative priors) about the model parameters.

6 In Speidel et al. (2020), the linear LGC model did not have adequate fit for the mediator. Thus, they freely estimated the mediator slope loading at T3, with those at T2 and T1 fixed respectively at 0 and −0.33 (i.e., the 2-month interval between T1 and T2 divided by the 6-month total study duration). For the outcome, the linear LGC model did not have adequate fit. Thus, they freely estimated the outcome slope loading at T2, with those at T3 and T1 fixed respectively at 0 and −1. In the current study, the slope loadings of mediator and outcome from Speidel et al. (2020) are used in both the original and sensitivity analysis models to illustrate the developed sensitivity analyses.

7 That is, ${\Psi }_{vM}\sim IW(\mathrm{d}{\widehat{\Psi }}_{vM}^{\text{ML}},\mathrm{d}=2)$ where the ML estimate ${\widehat{\Psi }}_{vM}^{\text{ML}}$ was $(\begin{array}{cc}0.54& 0.37\\ 0.37& 2.33\end{array}),$ and ${\Psi }_{vY}\sim IW(\mathrm{d}{\widehat{\Psi }}_{vY}^{\text{ML}},\mathrm{d}=2)$ where ${\widehat{\Psi }}_{vY}^{\text{ML}}$ was $(\begin{array}{cc}32.68& 10.90\\ 10.90& 13.29\end{array}).$

8 In causal mediation analysis, the natural indirect effect of a treatment on an outcome via a mediator cannot be causally identified, when there exists a posttreatment confounder(s) of the mediator-outcome relation, either omitted or not, that is itself affected by the treatment (e.g., VanderWeele & Vansteelandt, 2009; VanderWeele & Vansteelandt, 2014). This issue is different from the omitted-pretreatment-confounders issue considered in our current study. The omitted-pretreatment-confounders issue can be potentially addressed by collecting data on and adjusting for all (or a sufficiently rich set of) pretreatment confounding variables in the analysis. The post-treatment confounding issue, however, is the issue with the existence of such posttreatment confounders, irrespective of whether they are omitted from or included in the analysis.

Additional information

Funding

Lijuan Wang and Zhiyong Zhang are grateful for the financial support from IES grant R305D210023 during the study. Lijuan Wang is also grateful for the financial support from NIH grant R01HD091235 during the study.