An ANCOVA puzzler

Doing effect size calculations for meta-analysis is a good way to lose your faith in humanity—or at least your faith in researchers’ abilities to do anything like sensible statistical inference. Try it, and you’re surely encounter head-scratchingly weird ways that authors have reported even simple analyses, like basic group comparisons. When you encounter this sort of thing, you have two paths: you can despair, curse, and/or throw things, or you can view the studies as curious little puzzles—brain-teasers, if you will—to keep you awake and prevent you from losing track of those notes you took during your stats courses, back when. Here’s one of those curious little puzzles, which I recently encountered in helping a colleague with a meta-analysis project.

A researcher conducts a randomized experiment, assigning participants to each of \(G\) groups. Each participant is assessed on a variable \(Y\) at pre-test and at post-test (we can assume there’s no attrition). In their study write-up, the researcher reports sample sizes for each group, means and standard deviations for each group at pre-test and at post-test, and adjusted means at post-test, where the adjustment is done using a basic analysis of covariance, controlling for pre-test scores only. The data layout looks like this:

Note that the write-up does not provide an estimate of the correlation between the pre-test and the post-test, nor does it report a standard deviation or standard error for the mean change-score between pre-test and post-test within each group. All we have are the summary statistics, plus the adjusted post-test scores. We can assume that the adjustment was done according to the basic ANCOVA model, assuming a common slope across groups as well as homoskedasticity and so on. The model is then \[ y_{ig} = \alpha_g + \beta x_{ig} + e_{ig}, \] for \(i = 1,...,n_g\) and \(g = 1,...,G\), where \(e_{ig}\) is an independent error term that is assumed to have constant variance across groups.