Why is Stouffer's method so often performed on z's that correspond to one-tailed $p$-values when the mathematics allows for z's that correspond to two-tailed $p$-values?
Suppose the null hypothesis $\mu = 0$ is considered in two $2$-tailed studies. Suppose that one study rejects the null hypothesis because all data are strongly positive (supporting the alternative hypothesis $\mu > 0$ as well as the alternative hypothesis $\mu \neq 0$), while the other study rejects the null because all the data are strongly negative (supporting the alternative hypothesis $\mu < 0$ as well as the alternative hypothesis $\mu \neq 0$). Clearly, if the data from the two studies were combined, the null hypothesis would be rejected because all the data differ significantly from $0$ and thus support the alternative hypothesis $\mu \neq 0$ corresponding to a $2$-tailed study. However, the $z$'s from the studies will be positive and negative respectively, and Stouffer's method will add the two $z$ scores to get an answer close to $0$ and thus say that the null hypothesis should not be rejected at all.
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