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I am performing a meta-regression using various types of effect size from two independent samples design study; pre- and post-test for one sample design study; and frequency comparison design study and so on.

Can I include all those various types of effect size into one meta-regression?

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If you can convert them all onto a common scale then the answer is yes, otherwise no. If you do do this then include as one of your moderators a factor with levels corresponding to different study designs to check if the difference makes a difference.

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