Estimating Latent Baseline-by-Treatment Interactions in Statistical Mediation Analysis

Abstract

Statistical mediation analysis is used to uncover intermediate variables, known as mediators [M], that explain how a treatment [X] changes an outcome [Y]. Often, researchers examine whether baseline levels of M and Y moderate the effect of X on posttest M or Y. However, there is limited guidance on how to estimate baseline-by-treatment interaction (BTI) effects when M and Y are latent variables, which entails the estimation of latent interaction effects. In this paper, we discuss two general approaches for estimating latent BTI effects in mediation analysis: using structural models or scoring latent variables prior to estimating observed BTIs and correcting for unreliability. We present simulation results describing bias, power, type 1 error rates, and interval coverage of the latent BTIs and mediated effects estimated using these approaches. These methods are also illustrated with an applied example. R and Mplus syntax are provided to facilitate the implementation of these approaches.

Keywords:

• Baseline-by-treatment interactions
• latent variables
• mediation analysis

Notes

1 The general term in the right hand side of the equation is ${{\left(B_w{\Lambda }_M\right)}^{-1}[{\Sigma }_{M_f}-B}_w{\Theta }_{eM}{B}_w^{\prime }] {\left({\Lambda }_M^{\prime }{B}_w^{\prime }\right)}^{-1}$ but Bartlett scores have the property that $B_w\mathrm{\Lambda }$ = I, which simplifies the expression. Regression scores do not have this property.

2 In Equations (7)(7) $${\mathrm{\Sigma }}_{M_{fc}}= {\mathrm{\Sigma }}_{M_f}-B_{wM}{\mathrm{\Theta }}_{eM}{B}_{wM}\prime$$(7) and (8)(8) $${\mathrm{\Sigma }}_{M_{sc}}= {\mathrm{\Sigma }}_{M_s}-S_M{\mathrm{\Theta }}_{eM}{S}_M^{\prime }.$$(8) we are summing the residual variances to estimate the error variance of the observed score. By subtracting the error variances from the observed score variance, what remains is the reliable variance.

3 We imposed longitudinal scalar invariance constraints by using the same parameter in different parts of the model. For example, a single posterior distribution represented the relation between M₁ and the item m₁₁ and M₂ and the item m₂₁. Also, we modeled the correlated residuals like the specific factors in bifactor models: the factor had two indicators, which were the same item across time, and these factors were orthogonal to all other variables.

4 Rescaled true values for the corrected summed score model were derived to serve as reference values for raw bias. Summed scores are on a different metric than the latent variable models, similar to Georgeson et al. (2021).

Additional information

Funding

This work was supported in part by National Institute on Drug Abuse of the National Institutes of Health under award number F32DA053137 and R37DA09757.