Researchers often employ multivariate longitudinal/repeated meatures models when they collect multiple measures ($k\in K$) on each of multiple occasions ($t\in T$) for each of multiple units ($i \in N$). Such model include longitudinal/multi-group item response models, structural equation models, and multi-group confirmatory factor analysis.
It is sometimes useful to model univariate longitudinal outcomes using the change score $y_{it} - y_{it-1}$ instead of $y_{it}$ when one is interested in modeling within-unit change-over-time. While within-unit change-over-time is also of interest when outcomes are multivariate, there are two different "types" of change scores in the multivariate case. Assuming measurement equation $g()$ and multivariate longitudinal outcome $y_{ikt}$, both of the below are possible:
- $g(y_{ik2}) - g(y_{ik1})$
- $g(y_{ik2} - y_{ik1})$
Question
Are these approaches equivalent? When might I use one versus the other, i.e. what are the differences in assumptions and trade-offs (e.g. efficiency)?
By linearity of expectations, I'd expect them to have the same expectation but possibly different variances...but intuitively the change in two latent variables (1) seems different from the latent change in observed variables (2).