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 principal 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:

[math]\displaystyle{ X = -2\sum_{i=1}^k \log_e(p_i) \sim \chi^2(2k) . }[/math]

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:

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

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

where [math]\displaystyle{ c=\operatorname{Var}(X)/(2\operatorname{E}[X]) }[/math] and [math]\displaystyle{ k'=2(\operatorname{E}[X])^2/\operatorname{Var}(X) }[/math]. 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

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

Cauchy combination test

This is conceptually similar to Fisher's method: it computes a sum of transformed p-values. Unlike Fisher's method, which uses a log transformation to obtain a test statistic which has a chi-squared distribution under the null, the Cauchy combination test uses a tan transformation to obtain a test statistic whose tail is asymptotic to that of a Cauchy distribution under the null. The test statistic is:

[math]\displaystyle{ X = \sum_{i=1}^k \omega_i \tan[(0.5-p_i)\pi] , }[/math]

where [math]\displaystyle{ \omega_i }[/math] are non-negative weights, subject to [math]\displaystyle{ \sum_{i=1}^k \omega_i = 1 }[/math]. Under the null, [math]\displaystyle{ p_i }[/math] are uniformly distributed, therefore [math]\displaystyle{ \tan[(0.5-p_i)\pi] }[/math] are Cauchy distributed. Under some mild assumptions, but allowing for arbitrary dependency between the [math]\displaystyle{ p_i }[/math], the tail of the distribution of X is asymptotic to that of a Cauchy distribution. More precisely, letting W denote a standard Cauchy random variable:

[math]\displaystyle{ \lim_{t \to \infty} \frac{P[X \gt t]}{P[W \gt t]} = 1. }[/math]

This leads to a combined hypothesis test, in which X is compared to the quantiles of the Cauchy distribution.