How Should We Model the Effect of "Change"-Or Should We?

There have been long and bitter debates between those who advocate for the use of residualized change as the foundation of longitudinal models versus those who utilize difference scores. However, these debates have focused primarily on modeling change in the outcome variable. Here, we extend these same ideas to the covariate side of the change equation, finding similar issues arise when using lagged versus difference scores as covariates of interest in models of change. We derive a system of relationships that emerge across models differing in how time-varying covariates are represented, and then demonstrate how the set of logical transformations emerges in applied longitudinal settings. We conclude by considering the practical implications of a synthesized understanding of the effects of difference scores as both outcomes and predictors, with specific consequences for mediation analysis within multivariate longitudinal models. Our results suggest that there is reason for caution when using difference scores as time-varying covariates, given their propensity for inducing apparent inferential inversions within different analyses.