Fisher's method

In its basic form, it is used to combine the results from several independence tests bearing upon the same overall hypothesis (H0).

However, if positive dependence is not accounted for, and the meta-analysis p-value is found to be small, the evidence against the null hypothesis is generally overstated.

reduced for k independent or positively correlated tests, may suffice to control alpha for useful comparison to an over-small p-value from Fisher's X2.

A common strategy is to approximate the null distribution with a scaled χ2-distribution random variable.

Brown's method[3] can be used to combine dependent p-values whose underlying test statistics have a multivariate normal distribution with a known covariance matrix.

Kost's method[4] extends Brown's to allow one to combine p-values when the covariance matrix is known only up to a scalar multiplicative factor.

[5][6] Fisher's method is typically applied to a collection of independent test statistics, usually from separate studies having the same null hypothesis.

For example, consider a collection of medical studies looking at the risk of a high glucose diet for developing type II diabetes.

Due to genetic or environmental factors, the true risk associated with a given level of glucose consumption may be greater in some human populations than in others.

Any discrepancies among the results from separate studies or experiments must be due to chance, possibly driven by differences in power.

A closely related approach to Fisher's method is Stouffer's Z, based on Z-scores rather than p-values, allowing incorporation of study weights.