Handling Measurement Error and Omitted Confounders Considering Informativeness of the Confounding Effect under Mediation Modeling

Abstract

In the article, we focused on the issues of measurement error and omitted confounders while conducting mediation analysis under experimental studies. Depending on informativeness of the confounders between the mediator (M) and outcome (Y), we described two approaches. When researchers are confident that primary confounders are included (e.g., based on theory, literature), mediation effects can be estimated using Bayesian estimation with informative priors for the correlation of residuals between latent M and Y (ρ). Simulation study results showed that mediation analysis without accounting for secondary confounders yielded inaccurate inferences about mediation effects. Without prior knowledge about the primary confounders between M and Y, we suggest using sensitivity analysis to probe robustness of mediation analysis results. ρ was used as the sensitivity parameter and an R shiny app was developed to conduct sensitivity analysis: https://qwang17.shinyapps.io/sensitivity/. Both unidimensional and multidimensional models are allowed under our developed R shiny app to conduct sensitivity analysis. Two real data examples were used for illustrating the procedure of sensitivity analysis. We ended the article by making recommendations and entertaining future directions in mediation analysis.

Keywords:

• Confounders
• measurement error
• mediation analysis
• sensitivity analysis

Notes

1 Without loss of generality, we assume the means of M and Y are zero. Therefore, intercepts are not included in the models of M or Y.

2 In the special case when ${\mathit{C}}_U$ impacts M and does not impact Y, the effect estimates are not impacted as X and ${\mathit{C}}_U$ are independent due to random assignment to levels of X. Thus, estimate of α is the same with and without ${\mathit{C}}_U.$ When ${\mathit{C}}_U$ impacts Y only and does not impact M, the results would not be impacted either because this suggests that ${\mathit{C}}_U$ is independent of M and X. Thus, when ${\mathit{C}}_U$ impacts at most one variable between M and Y, estimates of β and γ are the same with and without ${\mathit{C}}_U.$

3 There are two recommended approaches to mediation analysis: product of coefficients and difference in coefficients (MacKinnon, 2000). Here we adopted the product of coefficients approach to estimate mediation effect in one step based on the latent variable model in Figure 2a. To apply the difference-in-coefficients approach, one needs to obtain two coefficients: the effect of X on Y without controlling for M (${\gamma }^{*}$) and the other effect of X on Y controlling for M (γ). The difference ${\gamma }^{*}-\gamma$ would be the parameter of interest. With random assignment of X, ${\gamma }^{*}$ estimate is not impacted by ${\mathit{C}}_U.$ The γ estimate, however, may be impacted if ${\mathit{C}}_U$ is omitted due to the correlated X and M. Thus, we argue that our proposed approaches would also apply to estimating or conducting sensitivity analysis for ${\gamma }^{*}-\gamma$ based on informativeness of ρ. Given a relatively precise prior information about ρ, ${\gamma }^{*}-\gamma$ can be estimated; without precise prior information about ρ, one can still use ρ as a sensitivity parameter to check whether the inference about ${\gamma }^{*}-\gamma$ is robust to variation in ρ.

4 One needs to create dummy variable(s) for any categorical variable(s) first before importing data to the R shiny app.

Additional information

Funding

The research reported in the article was made possible (in part) by a grant from the Spencer Foundation (202300078, “Handling Covariate Selection Uncertainty and Measurement Error in Causal Mediation Analysis: A Two-Step Approach to More Accurate Intervention Studies”). The views expressed are those of the authors and do not necessarily reflect the views of the Spencer Foundation.