How to interpret standardized simple slopes and indirect effects in R with semTools and lavaan?

Question

I am attempting to obtain a standardized solution for simple slopes and indirect effects in a SEM in R using lavaan and semTools::probe2WayMC, following the methods of Schoemann & Jorgensen (2021) (https://doi.org/10.3390/psych3030024). I found another post that provided some insight on the topic (How to use probe2WayMC() with fully standardized solution with semTools / Lavaan R packages?). The solution given was to standardize numeric variables before creating product indicators, then fitting the model, probing the interaction, and interpreting the Est column as standardized. Unfortunately, I'm having a hard time understanding the implications of this.

1. If the data is standardized before running the model, am I then to interpret all of the loadings and coefficients from the SEM output under the Estimate column as fully standardized, as if it were the Std.all column? What then would the Std.all column represent here?

2. Why is the data scaled, but not centered? scale(dat2way, center = FALSE, scale = TRUE) Simple slopes would not be affected by centering, but wouldn't the rest of the parameter estimates in the model output be affected? If the indirect effect is found by multiplying the simple slope by the other coefficients in the path diagram, would that effect still be standardized?

3. Finally, could I be mistaken in wanting to find a standardized solution in the first place? My understanding was that they were easier to interpret than estimates the scales of latent factors. Is there any advantage of non-standard solutions?

Thanks very much.

Answer 1

I found another post that provided some insight on the topic

Please note that I just edited that 2022 response (because I am not allowed to rescind an accepted answer). Please read there about the complexities involved.

Regarding your questions, it is a matter of understanding which variables can be interpreted in units of SD (i.e., because their SD = 1). Regardless of whether some observed variables are functions of other observed variables (e.g., product terms), lavaan applies the same formula to calculate solutions that can be interpreted in units of SD for the latent variables only (Std.lv) or both latent and observed variables (Std.all).

1. The Std.all gives you a result on the condition that all observed variables (including product indicators, which is wrong) and all latent variables (including the product/interaction factor(s), which is also wrong) have SD = 1.

2. I don't know what you mean by "the rest", but it is not relevant where a variable is "located" on the number line (i.e., its central tendency). Here is a primer on interaction effects. If your estimates are in units of SD, then generally speaking, yes: your indirect effect would also be in SD units.

3. The scale of the latent variables is arbitrary, so units of SD are as meaningful as anything else. But standardized slopes are only one type of standardized effect size. You could report (change in) R-squared to quantify the unique effect of the interaction-factor on the outcome: https://doi.org/10.3758/s13428-020-01532-y