How should we model the effect of "change" - or should we?

Abstract

There have been long and bitter debates between those who advocate for the use of residualized change (regressing a variable on itself measured at some time lag prior) as the foundation of longitudinal models versus those who utilize difference scores (subtracting prior from current status). However, most of the methodological work on this topic has focused on the outcome variable in different models. Here, we extend these same issues to the covariates -- or predictors -- in longitudinal models of change and find that similar issues arise when using lagged versus difference score predictors. We show how apparently distinct models using different versions of time-varying covariates are, in fact, simply repackaged versions of the same predictive information and are related through a set of equations that we lay out. We then work through several applied examples across traditional and multilevel regression models. We conclude by considering the issues that arise where a time-varying variable acts as both outcome and predictor - with a specific focus on mediation analysis within multivariate longitudinal models. Our results suggest that users should exercise caution when using change scores as time-varying covariates – not because they are wrong per se, but because they can introduce apparent inferential inversions that can mislead researchers when drawing substantive conclusions.