# Testing two-tailed p-values using Stouffer's approach

About combining multiple test statistics, Wikipedia says

This Z-score (for the overall meta-analysis) is appropriate for one-sided right-tailed p-values; minor modifications can be made if two-sided or left-tailed p-values are being analyzed.

What are those minor modifications?

1) If two-sided p-values are being analyzed, you use the two-sided p-value in the calculation of the $Z_i$. The two-sided p-value is $\tilde{p}_i = p_i/2$.
2) If left-tailed p-values are used, you use $1-p_i$ instead of $p_i$ in the calculation of the $Z_i$.
• I don’t get the rationale of your definition of $\widetilde p_i$. I would have put $Z_i = \Phi^{-1}(1-p_i/2)$, and use as a score $\sum Z_i^2$ which follows a χ²(k) distribution. Dec 21 '11 at 19:35
• I was apparently suffering the same kind of syndrome, as I now don’t understand why I wanted to take the same of squared $Z_i$. I think your answer is now perfect (+1). Dec 21 '11 at 20:58
• If the test is two-sided, then low p-value can correspond as well to Z=10 as to Z=-10. Why are always keeping the sign positive? Jan 6 '19 at 22:00