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I'm conducting a meta-analysis that consists of 15 studies and ~40 construct effect sizes. Most of the constructs and studies use continuous variables and were straightforward in finding standardized mean differences (cohen's d). However, two studies use dichotomous variables and report the % of subjects who reported the construct. According to Lipsey & Wilson (2001), I'm leaning toward using the arcsine transformation over logit or probit due to the small sample size. For the below information "reported CSE", I got the arcsine transformations 1.115 for 28% and .644 for 10% according to the table of arcsine transformations in Lipsey & Wilson, making the estimated effect size = .471.

My main question is how do I compare the estimated effect sizes to the cohen's d in a forest plot? I have 95% CIs for all of the cohen's d calculations but only have the one point for the transformed ESs.

enter image description here

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  • $\begingroup$ At one point you mention "Lipskey". Do you mean Lipsey there? $\endgroup$
    – Glen_b
    Oct 11, 2022 at 21:59
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    $\begingroup$ I feel that a point of clarification is needed here, since it will lead to potential misunderstandings (this not in any sense the fault of the poster of the question, however). It is disturbingly common to see many authors (presumably mostly not mathematicians) write "arcsin transformation" when referring to the arcsine square root transformation, $\sin^{-1}(\sqrt{p})$ or sometimes $2\sin^{-1}(\sqrt{p})$, which in either form is the (asymptotic) variance-stabilizing transformation of a binomial proportion. ... ctd $\endgroup$
    – Glen_b
    Oct 11, 2022 at 22:22
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    $\begingroup$ ctd ... Dropping the square root (and the $2$ if present) will mislead people who take the term at face value. In this case the term "arcsine" in Lipsey and Wilson (which they get in turn from Jacob Cohen's book on power) is actually referring to the second version of the transformation: $2\sin^{-1}(\sqrt{p})$ (the "$2$" giving us variance $1/n$ asymptotically). While this contains an arcsine, to merely call it "arcsine" very likely to lead to confusion. Consequently, this comment is to explicitly draw attention to the transformation that is actually being used in the reference. $\endgroup$
    – Glen_b
    Oct 11, 2022 at 22:22
  • $\begingroup$ Thank you for the clarification. I was referring to the arcsine square root transformation from Lipsey and Wilson. With that in my mind, do you have any information on how I would compare the transformed effect size without CIs to the others? $\endgroup$
    – Elyssa S
    Oct 13, 2022 at 17:40

1 Answer 1

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According to Introduction to Meta-Analysis, 2nd Edition (Borenstein, 2009) you can calculate the variance of d as follows (Chapter 4, page 27):

enter image description here

Once you have the variance, you can easily get the confidence intervals based on it.

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