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When doing a paired t-test, is it appropriate to use the paired t value to calculate $\eta^2$?
I ask because it is not appropriate to calculate Cohen's d based on the paired t value.

Thanks.

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I'm not so sure that you can't calculate Cohen's d based on paired data. I've done it myself. You just divide the mean of the differences by the SD of the differences. I.e.:
$$ d_{paired}=\frac{\bar{D}}{s_{\bar{D}}} $$
What you can't do is directly compare the paired version of Cohen's d with the Cohen's d for independent data. As is pointed out here, using Cohen's d is probably better than $\eta^2$. (That's a really good answer, by the way, it may be worth your time to read if you're interested in these issues.)

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As soon as you entered the paired test arena the question arises, "what should I do with that pesky within observational unit (subjects etc) variance". Your effect size is bound to reflect some opinion in this regard. $d_{paired}$ as in gung's answer contains an assumption that the effect size should be calculated disregarding the variance in observational units. The equivalent assumption is made in a partial $\eta^2$ calculation. In short, there is no reason not to calculate $\eta^2$ or $d_{paired}$. Remember your paired samples-t is the same as a one-sample t-test (this is why you can use the one-sample t equation for Cohen's d as in gung's answer) or a 1 factor 2 level within subjects ANOVA. So, effect sizes which are appropriate for those statistics can also be appropriate for your paired samples t.

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