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 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:
In the case that these p-values are not independent, Brown proposed the idea of approximating X using a scaled χ2-distribution, cχ2, with k’ degrees of freedom.
The mean and variance of this scaled χ2 variable are:
where and. This approximation is shown to be accurate up to two moments.

Unknown covariance

Harmonic mean ''p-''value

The harmonic mean p-value offers an alternative to Fisher's method for combining p-values when the dependency structure is unknown but the tests cannot be assumed to be independent.

Kost's method: ''t'' approximation">Student's t-distribution">''t'' approximation

This method requires the test statistics' covariance structure to be known up to a scalar multiplicative constant.