Extensions of Fisher's method
# 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 $χ^2-distribution$ with 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 $χ^2-distribution,$ $cχ^2(k’),$ with k’ degrees of freedom.

The mean and variance of this scaled $χ^2$ variable are:

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

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

    $X=\sum _{i=1}^{k}\omega _{i}\tan[(0.5-p_{i})\pi ],$

where $\omega _{i}$ are non-negative weights, subject to $\sum _{i=1}^{k}\omega _{i}=1$. Under the null, $p_{i}$ are uniformly distributed, therefore $\tan[(0.5-p_{i})\pi ]$ are Cauchy distributed. Under some mild assumptions, but allowing for arbitrary dependency between the $p_{i}$, 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:

    $\lim _{t\to \infty }{\frac {P[X>t]}{P[W>t]}}=1.$

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

