Inverting a negative correlation 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?
 A: 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.
