Accommodating a Latent XM Interaction in Statistical Mediation Analysis

Abstract

Statistical mediation analysis is used in the social sciences and public health to uncover potential mechanisms, known as mediators, by which a treatment led to a change in an outcome. Recently, the estimation of the treatment-by-mediator interaction (i.e., the XM interaction) has been shown to play a pivotal role in understanding the equivalence between the traditional mediation effects in linear models and the causal mediation effects in the potential outcomes framework. However, there is limited guidance on how to estimate the XM interaction when the mediator is latent. In this article, we discuss eight methods to accommodate latent XM interactions in statistical mediation analysis, which fall in two categories: using structural models (e.g., latent moderated structural equations, Bayesian mediation, unconstrained product indicator method, multiple-group models) or scoring the mediator prior to estimating the XM interaction (e.g., summed scores and factor scores, with and without attenuation correction). Simulation results suggest that finite-sample bias is low, type 1 error rates and coverage of percentile bootstrap confidence intervals and Bayesian credible intervals are close to the nominal values, and statistical power is similar across approaches. The methods are demonstrated with an applied example, syntax is provided for their implementation, and general considerations are discussed.

Keywords:

• Statistical mediation
• latent interactions
• latent variable scores
• structural models

Notes

1 When X is centered, these expressions must be adjusted. This case is also shown in the supplement (part 1).

2 Regression scores, a popular alternative, are linearly related to Bartlett scores (Lawley & Maxwell, 1971).

3 Note that the reliability-adjusted product indicator (RAPI) approach uses similar adjustments (Hsiao et al., 2018).

4 In the typical UPI implementation, indicators of the variables that compose the interaction are only included in one product indicator (Marsh et al., 2012), but we had to reuse X because X does not have multiple indicators.

5 We can use this rescaling to obtain the expected parameter values when Bartlett scores for M are used – parameters would be biased because M is both an independent and dependent variable in the model (Skrondal & Laake, 2001).