Multivariate longitudinal/repeated measures models: change in latent variables vs. latent change

Question

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:

1. $g(y_{ik2}) - g(y_{ik1})$
2. $g(y_{ik2} - y_{ik1})$

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).

Answer 1

The advantage of latent change score models is that latent change scores are free of measurement error. Change scores based on observed (measured) variables are affected by measurement error (unreliability) at both Time 1 and Time 2. With latent change score models you can separate "true" changes from fluctuations that are merely due to random error. See, for example

Steyer, R., Eid, M., & Schwenkmezger, P. (1997). Modeling true intraindividual change: True change as a latent variable. Methods of Psychological Research, 2(1), 21–33.

https://www.psycharchives.org/en/item/e81728c7-926a-47a0-91a2-72d33de92360

And yes, this is particularly relevant when you want to study inter-individual differences (variance) in (and potential correlates of) change across time.