Extensions of Fisher'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:

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

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

where ${\displaystyle c=\operatorname {Var} (X)/(2\operatorname {E} [X])}$ and ${\displaystyle k'=2(\operatorname {E} [X])^{2}/\operatorname {Var} (X)}$. This approximation is shown to be accurate up to two moments.

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.[3][4]

This method requires the test statistics' covariance structure to be known up to a scalar multiplicative constant.[2]

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:

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

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

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

This leads to a combined hypothesis test, in which X is compared to the quantiles of the Cauchy distribution.[5]