Is it legitimate to perform statistical test on test statistics such as t-values or z-scores?

[R_cubed]

@JeanChen

I wonder why summary statistics usually need to be on the original scale rather than a variance normalized “effect” such as z-score?

You might want to follow up on the references in this thread, especially by Senn and @Sander_Greenland

Nonparametric Effect Size estimation, Likelihood Methods, and Meta Analysis

The aggregation of effect sizes on anything other than interval and ratio scaled variables is much more complicated than I initially anticipated. You would not know this from most of the readily available texts on meta-analysis, as far as I can tell. This discussion hinted at the problems: Relevant article by Stephen Senn on Effect size measurements, particularly non-parametric ones. Stephen Senn (2011) U is for Unease: Reasons for Mistrusting Overlap Measures for Reporting Clinical Tri…

The essential critique is that the variance is largely affected by study design. In essence, you are “standardizing” by something that is variable, not a constant. That is how I understood Greenland’s numerous articles on standardized effect size.