Extensions of Fisher's method

(Introductory block)

Dependent statistics

A principle limitation of Fisher's method is its exclusive design to combine independent p-values, which renders it an unreliable technique to combine dependent p-values. To overcome this limitation, a number of methods were developed to extend its utility.

Known covariance

Brown's method: Gaussian approximation

Fisher's method showed that the log-sum of k independent p-values follow a χ²-distribution of 2k degrees of freedom:

$X=-2\sum _{i=1}^{k}\log _{e}(p_{i})\sim \chi ^{2}(2k)$

In the case that these p-values are not independent, Brown proposed the idea of approximating X using a scaled χ²-distribution, cχ²(k’), with k’ degrees of freedom.

The mean and variance of this scaled χ² variable are:

Failed to parse (syntax error): {\displaystyle E(cχ²(k’)) = ck'}

Failed to parse (syntax error): {\displaystyle Var(cχ²(k’)) = 2c<sup>2<sup>k' }

This approximation is shown to be accurate up to two moments.