Dr Cheung:

I have used your tssem1 to deal with the meta analysis work. And I pooled 11 correlation matrix and the tssem1 reports RMSEA=.0915, SRMR=.0492, TLI=.976 and CFI=.978. I think it should be homogeneous. According to this result, it should be ok. When I use the conventional meta analysis, the criterion to judge heterogeneity is Q (Cochran Q) and its p value. What is the theoretical base to judge the pooled correlation matrices? Because you use SEM, there must be a model you try to compare with, what is that model? Because I try to use regression of sqrt weight * EF on sqrt weight, the model did indicate the synthesized EF (slope), but I check the model fit index, it is not the one for homogeneous test.

And I also used Variance and covariance matrix to tssem1, and the result is very similar, but only one effect size changes (.591 in correlation and .672 in var-covariance matrix, matrix is 4X4), I wonder how to interpret the change, because the rest doesn't change much.

I assume that you are referring to the multiple-group SEM approach in the first stage of analysis.

In the SEM literature, goodness-of-fit indices, e.g., RMSEA, SRMR, and CFI, are used to test approximate fit whereas the chi-square statistic (Q test in meta-analysis) is used to test the exact fit (a null hypothesis with equality constraints).

In the meta-analysis literature, the Q test is used to test the homogeneity of effect sizes. However, most researchers prefer to use a random-effects model regardless of whether the Q test is statistically significant.

There is plenty of literature on these topics. You should have no problems in finding some.