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is it appropriate to take the absolute value of a correlation coefficient to include in analyses? I am planning a meta-analysis and some of the hypothesized outcome variables might have a negative relationship. To include in one model, would it be okay to use the absolute value of the correlations to be consistent with other coefficients?

Edit: to clarify, I want to include correlation coefficients in a random-effects model for meta-analysis. If the majority of the correlations are positive, could I use the absolute values of the negative correlations if such correlations would have been positive had one of the measures been reverse scored?

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    $\begingroup$ Can you tell us a little more about what you are trying to use these correlation coefficients in? You mention a model -- are you using correlation coefficient as a covariate in another model, and if so, can you say a bit more about that? $\endgroup$
    – Izzy
    Jan 6 at 3:26
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    $\begingroup$ If they would have been reversed, then yes, reverse them. E.g. if one study measure is 'happiness' and another measure is 'depression' then you should reverse them so they are all the same direction. But you shouldn't reverse them just because they are negative. $\endgroup$ Jan 6 at 17:35
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    $\begingroup$ Yes, so long as you explain what you are doing. $\endgroup$
    – Nick Cox
    Jan 6 at 18:22
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If you know that one of the measures was reverse scored, you can do this, because $-\mathit{cor}(X, Y) = \mathit{cor}(-X, Y) = \mathit{cor}(X, -Y)$.

First, notice that the variance of a random variable is unchanged by negation, due to $x^2 = (-x)^2$ and linearity of expectation:

\begin{align} \text{var}(X) & = \mathbb{E}\left[X^2\right] - \mathbb{E}\left[X\right]^2 \\ & = \mathbb{E}\left[(-X)^2\right] - (-\mathbb{E}\left[X\right])^2 \\ & = \mathbb{E}\left[(-X)^2\right] - \mathbb{E}\left[-X\right]^2 \\ & = \text{var}(-X)\\ \end{align}

Combining this with the definition of correlation coefficient and linearity of expectation again, we have:

\begin{align} -\text{cor}(X, Y) & = -\frac{ \mathbb{E}\left[(X - \mathbb{E}\left[X\right])(Y - \mathbb{E}\left[Y\right])\right] }{ \text{var}(X)\text{var}(Y) } \\ & = \frac{ \mathbb{E}\left[(-X - \mathbb{E}\left[-X\right])(Y - \mathbb{E}\left[Y\right])\right] }{ \text{var}(-X)\text{var}(Y) }\\ & = \text{cor}(-X, Y) \end{align}

However, I would not blindly take the absolute value; this only goes through if you know that $X$ from one study was measured as $-X$ in another study.

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