Extensions of Fisher's method

In statistics, extensions of Fisher's method are a group of approaches that allow approximately valid statistical inferences to be made when the assumptions required for the direct application of Fisher's method are not valid. Fisher's method is a way of combining the information in the p-values from different statistical tests so as to form a single overall test: this method requires that the individual test statistics (or, more immediately, their resulting p-values) should be statistically independent.

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

Fisher's method showed that the log-sum of k independent p-values follow a χ²-distribution with 2k degrees of freedom: ^[1][2]

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

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:

${\displaystyle \operatorname {E} [c\chi ^{2}(k')]=ck',}$

${\displaystyle \operatorname {Var} [c\chi ^{2}(k')]=2c^{2}k'.}$

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

Unknown covariance

Kost's method: t approximation

It should be noted that the method does require the test statistics' covariance structure to be known up to a scalar multiplicative constant. See reference 2.

References

1. Brown, M. (1975). "A method for combining non-independent, one-sided tests of significance". Biometrics. 31: 987–992. doi:10.2307/2529826.
2. Kost, J.; McDermott, M. (2002). "Combining dependent P-values". Statistics & Probability Letters. 60: 183–190. doi:10.1016/S0167-7152(02)00310-3.