Causal parameter moderation: Applying moderated nonlinear factor analysis to causal inference with latent outcomes

When studying latent outcomes in causal inference, causal effects can occur not only on the mean of the latent variable, but on all parts of the measurement model: the mean and residual variance of the latent variable, as well as the item parameters. This paper proposes causal parameter moderation, the application of moderated nonlinear factor analysis (MNLFA) to causal inference with latent outcomes. The proposed MNLFA-based approach is presented in contrast to another method for handling item-level deviations from a main treatment effect, item-level heterogeneous treatment effect (IL-HTE) analysis. The models are best suited to handle different types of DIF, and thus the circumstances under which each is appropriate differ: IL-HTE models are highly confirmatory in the sense that they assume a distribution of DIF across all items, so the more exploratory MNLFA approach may make sense as a precursor or alternative to IL-HTE analysis, especially when there is reason to believe that a subset of items on the outcome might be particularly sensitive to the treatment. Because MNLFA can handle an arbitrary number of continuous or categorical covariates, it is also uniquely well-suited to handling heterogenous treatment effects; this is demonstrated in a reanalysis of data from a randomized controlled trial of a reading intervention’s effects on science vocabulary where there is substantial evidence of both a main effect on the latent variable and additional effects on a subset of items.