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I am working on a meta-regression model with a moderator variable (a factor), and so am using the QE and QM tests to explore the importance of the the moderator. I understand the QE tests for residual heterogeneity in the data (i.e. that not explained by the moderators) while the QM tests whether the moderator explains any heterogeneity in the data.

What I am unclear on is what exactly 'heterogeneity' is in this context. For example, in the meta-analysis I am working on, we compare log mean ratios between a control and a treatment. If I run a model without the moderator variable, then the model estimate for the mean ratio is significant (P<0.05) and QE P>0.05. This indicates, I think, that the difference between the control and the treatment is significantly different from zero, and there is no significant heterogeneity left to explain. There is of course though still some variance in the mean ratios ... how does the QE test decide this variance not 'heterogeneity'?

If I do include the moderator in the model, then QE P is still >0.05 but now QM P<0.05 ... indicating the moderator does explain some heterogeneity, even though in the previous version of the model with no moderator, the QE said there was no heterogeneity left to explain. Some levels of the moderator (a factor) are significant in the Z tests. How do I interpret the importance of the moderator in this situation?

On a related note, I am also curious if the ratio of the QM to QE is effectively an R2? i.e. does QE + QM = total heterogeneity in the dataset, and thus QM/QE = proportion of heterogeneity explained by the moderators?

(In case it's relevant, I am using rma.mv in metafor in R)

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  • $\begingroup$ Those are very good questions. I wonder the same as your post. $\endgroup$ Jul 26, 2021 at 11:32

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I'm sure you got your answer by now from somewhere else.

That said, QE is the test for residual heterogeneity while Qm is the omnibus test for moderators in your model.

You do have to realise that the a mixed model is just a random model with moderators. In your first question I will think the heterogeneity left is that due to variance.

If the p-value of Qm is significant then it means the moderator you included explains a great propotion of the heterogeneity and that the differeces in mean ratios are likely not due to chance.

I'm not sure about your last question but read this http://dx.doi.org/10.1027/0044-3409.215.2.104 by Prof Wolfgang Viechtbauer himself.

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