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Causal moderated mediation analysis: Methods and software

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Abstract

Research questions regarding how, for whom, and where a treatment achieves its effect on an outcome have become increasingly valued in substantive research. Such questions can be answered by causal moderated mediation analysis, which assesses the heterogeneity of the mediation mechanism underlying the treatment effect across individual and contextual characteristics. Various moderated mediation analysis methods have been developed under the traditional path analysis/structural equation modeling framework. One challenge is that the definitions of moderated mediation effects depend on statistical models of the mediator and the outcome, and no solutions have been provided when either the mediator or the outcome is binary, or when the mediator or outcome model is nonlinear. In addition, it remains unclear to empirical researchers how to make causal arguments of moderated mediation effects due to a lack of clarifications of the underlying assumptions and methods for assessing the sensitivity to violations of the assumptions. This article overcomes the limitations by developing general definition, identification, estimation, and sensitivity analysis for causal moderated mediation effects under the potential outcomes framework. We also developed a user-friendly R package moderate.mediation (https://cran.r-project.org/web/packages/moderate.mediation/index.html) that allows applied researchers to easily implement the proposed methods and visualize the initial analysis results and sensitivity analysis results. We illustrated the application of the proposed methods and the package implementation with a re-analysis of the National Evaluation of Welfare-to-Work Strategies (NEWWS) Riverside data.

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Notes

  1. If a variable that moderates the mediation mechanism is posttreatment (i.e., affected by the treatment), it is essentially an additional mediator that interacts with the treatment and/or the focal mediator.

  2. \(Y_{t}(t^{\prime}, m)\) stands for the potential outcome if the treatment condition is at t′ and the potential mediator takes the value of m. A proof of why Assumptions 1 and 2 are needed for the identification of the conditional and moderated mediation effects can be found in Appendix A.

  3. For the notations, the subscript indicates the predictor that the coefficient corresponds to, and the superscript indicates the model that the predictor belongs to. This can avoid misreading especially when there are a lot of predictors and/or moderators in the model. Here \({\beta}_0^m\), \({\beta}_t^m\), \({\boldsymbol{\beta}}_x^m\), \({\beta}_0^y\), \({\beta}_t^y\), \({\beta}_m^y\), \({\beta}_{tm}^y\), and \({\boldsymbol{\beta}}_x^y\) respectively correspond to β0, β1, β2, θ0, θ1, θ2, θ3, and θ4 in VanderWeele (2015) and α2, β2, ξ2, α3, β3, γ, κ, and ξ3 in Imai et al. (2010a).

  4. This is because the sampling distribution of the regression coefficient estimates is asymptotically multivariate normal (King et al., 2000).

  5. We compute the final point estimates based on the mean rather than the median of the Q sets of point estimates as Imai et al. (2010a) did. This is because we found through simulations that the mean estimators are mostly less biased than the median estimators across different scenarios, especially when the sample size is relatively small. Simulations were conducted under the settings described in Appendix C. Simulation results are available upon request.

  6. We compute the final point estimates based on the mean rather than the median of the post-burn-in samples. This is because, similar to what Wang and Preacher (2015) reported, we found through simulations that the mean estimators are mostly less biased than the median estimators across different scenarios, especially when the sample size is relatively small. Simulations were conducted under the settings described in Appendix C. Simulation results are available upon request.

  7. Because U is generated from the frequentist perspective, this sensitivity analysis algorithm applies to the bootstrapping method and Monte Carlo method but does not directly apply to the Bayesian method. To implement the sensitivity analysis with the Bayesian method, U should be generated from the Bayesian perspective in Step 2.1.

  8. To obtain mediation effect estimates that are comparable across studies, some researchers reported mediation effect estimates in the standard deviation of the outcome in the control group (Kraft, 2020), or the standard deviation of the outcome  in the whole sample (e.g., Hong et al., ), while some fully standardized the mediation effects by standardizing all the variables (e.g., Preacher & Hayes, 2008). If a researcher would like to report such effect size measures in the following graphical summary and sensitivity analysis, they may run the analysis based on the data with the corresponding variables standardized in the desired way. We chose to report the estimates in the standard deviation of the outcome in the control group to eliminate the influence of the treatment on the standard deviation of the outcome. As Kraft (2020) argued on page 245, “it is preferable to use the standard deviation of the control group outcome rather than the pooled sample because the intervention may have affected the variation in outcomes among the treatment group”.

  9. It is inappropriate to simply plot \(\hat{\delta}_i\) predicted from Step 2.4 based on one replication against the moderator, because it ignores the uncertainty of \(\hat{\delta}_i\) when constructing the confidence band. The cost of the proposed approach is that it is more time-consuming. To speed up the calculation, the package allows parallel computing by setting the number of CPU cores via ncore. Its default value is 2. A progress bar is displayed as the program runs. An application of the above syntax to the NEWWS data took 2.5 minutes. In the cases when Step 2.5 does not involve a spline regression, it usually takes around 30 seconds with only one core.

  10. The default range is symmetric, and the upper limit of range.b.m (range.b.y) is twice the magnitude of the largest coefficient of the observed covariates in the standardized mediator (outcome) model.

  11. The sensitivity analysis plot looks similar to the L.O.V.E. method (Mauro, 1990; Cox et al., 2013), while we allow more scenarios (e.g., inclusion of the treatment-by-mediator interaction) for assessing sensitivity of conditional and moderated causal mediation effects.

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Acknowledgements

This study received support from Pitt Momentum Funds, a U.S. Department of Education Institute of Education Sciences Grant (R305D200031) an National Institutes of Health Grant (R01AG080590), and the 2022 National Academy of Education (NAEd)/Spencer Postdoctoral Fellowship Program. Lijuan Wang is grateful for the support from IES grant R305D210023 during the study. The authors thank Guanglei Hong and Fan Yang for their insightful comments on previous versions of the manuscript. In addition, comments from the Associate Editor and the reviewers have led to major improvements in this article.

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  1. Department of Health and Human Development at the School of Education, University of Pittsburgh, 5312 Wesley W. Posvar Hall, 230 South Bouquet Street, Pittsburgh, PA, 15260, USA

    Xu Qin

  2. University of Notre Dame, Notre Dame, IN, USA

    Lijuan Wang

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Correspondence to Xu Qin.

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The data and R package are available on CRAN: https://cran.r-project.org/web/packages/moderate.mediation/index.html. The data were not preregistered.

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Qin, X., Wang, L. Causal moderated mediation analysis: Methods and software. Behav Res 56, 1314–1334 (2024). https://doi.org/10.3758/s13428-023-02095-4

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