Moderation of compound parameters: New opportunities for testing group- and individual-level heterogeneity

Whereas traditional moderation analyses examine moderation in individual model parameters, we propose an expanded approach for investigating moderation within statistical models that accommodates compound parameters: theoretically interesting quantities that can be expressed as functions of individual model parameters. We define the concept of moderation of compound parameters and illustrate it through four examples: moderated indirect effects (within measured variable path models), moderated intraclass correlation (within multilevel models), moderated reliability (within confirmatory factor models), and moderated total accumulation (within latent growth models). Using bootstrap resampling and posterior simulation-based methods, we demonstrate how to estimate conditional effects on these compound parameters and construct confidence intervals for novel features (e.g., the maximum indirect effect, or reliability minimum). This framework broadens the scope of moderation analysis, allowing researchers to probe how more complex model characteristics vary across covariates. We discuss implications for model interpretation and provide recommendations for applied researchers to leverage these techniques in examining group- and individual-level heterogeneity.