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I'm looking into finding a way to calculate Cohen's $d$ for correlated samples.

Assuming pooled variances, we end up getting $$\text{SE}\left( \Delta \text{ of means}\right) = S_p\sqrt{\dfrac{1}{n_1}+\dfrac{1}{n_2} - \dfrac{2}{\sqrt{n_1n_2}}\rho_{\bar{X}, \bar{Y}}}\text{.}$$ The derivation here for the case of independence shows the formula for $S_p$ quite clearly, and it makes a lot of sense. But here, I'm working with a correlated sample. What is the formula for $S_p$ when the two samples are correlated?

The derivation seems to make use of the fact that independent $\chi^2$-variables can be written as $\dfrac{(n-1)s^2}{\sigma^2}$ and then adds them up to eventually get the formula for $S_p$, but we have dependent $\chi^2$-variables in this case.

Edit: I found

Stanley, J. and Glass, G. Statistical Methods in Education and Psychology. Englewood Cliffs, N.J.: Prentice-Hall, 1970.

in a journal article apparently having a formula for a pooled variance. If anyone happens to have this, it would be helpful.