z-standardization for longitudinal data

Question

I have data gathered at two time points, T1 and T2, from the same individuals. I would like to see whether scores improved from T1 to T2 to measure growth. Each individual's score at each time point is actually made up of performance on several tests ($a$, $b$, $c$, $d$) that are measured on different scales and for the most part repeated across T1 and T2. Also, scores on ($a$, $b$, $c$, $d$) are highly correlated. Therefore, I plan to $z$-transform scores for each test and then combine $z(a)$, $z(b)$, $z(c)$ and $z(d)$ for each participant to get two composite scores: one at T1 and another at T2. Then I plan to compare the composite scores at T1 and T2 to assess growth.

What mean and standard deviation should I use for transforming ($a$, $b$, $c$, $d$)? Specifically, because I want to measure growth from T1 to T2, the means and SDs used for standardization need to be the same for T1 and T2. Here are some options that I had in mind:

• Use the mean and SD of each test at T1 to standardize scores for that test at both T1 and T2.
• Take the mean and SD of each test for T1 and T2 combined, and use that for z-standardization.

Answer 1

I strongly recommend to avoid $z$-standardization of longitudinal data for reasons explained in Moeller, J. (2015). A word on standardization in longitudinal studies: don't. Frontiers in psychology, 6, 1389.

Instead, you can use Proportion of Maximum Scaling transformation¹ to get all measures to the same metric (also explained in the link above).

¹ See for more information Little, Todd D. 2013. Longitudinal Structural Equation Modeling. New York: Guilford Press.