Fisher's Method - Relation To Stouffer's Z-score Method

Relation To Stouffer's Z-score Method

A closely related approach to Fisher's method is based on Z-scores rather than p-values. If we let Z_i = Φ − 1(1−p_i), where Φ is the standard normal cumulative distribution function, then

$Z = \frac{\sum_{i=1}^k Z_i}{\sqrt{k}},$

is a Z-score for the overall meta-analysis. This Z-score 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. This method is named for the sociologist Samuel A. Stouffer.

Since Fisher's method is based on the average of −log(p_i) values, and the Z-score method is based on the average of the Z_i values, the relationship between these two approaches follows from the relationship between z and −log(p) = −log(1−Φ(z)). For the normal distribution, these two values are not perfectly linearly related, but they follow a highly linear relationship over the range of Z-values most often observed, from 1 to 5. As a result, the power of the Z-score method is nearly identical to the power of Fisher's method.

One advantage of the Z-score approach is that it is straightforward to introduce weights. If the ith Z-score is weighted by w_i, then the meta-analysis Z-score is

$Z = \frac{\sum_{i=1}^k w_iZ_i}{\sqrt{\sum_{i=1}^k w_i^2}},$

which follows a standard normal distribution under the null hypothesis. While weighted versions of Fisher's statistic can be derived, the null distribution becomes a weighted sum of independent chi-squared statistics, which is less convenient to work with.

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