Fisher's Method - Application To Independent Test Statistics

Application To Independent Test Statistics

Fisher's method combines extreme value probabilities from each test, commonly known as "p-values", into one test statistic (X2) using the formula

where p_i is the p-value for the ith hypothesis test. When the p-values tend to be small, the test statistic X2 will be large, which suggests that the null hypotheses are not true for every test.

When all the null hypotheses are true, and the p_i (or their corresponding test statistics) are independent, X2 has a chi-squared distribution with 2k degrees of freedom, where k is the number of tests being combined. This fact can be used to determine the p-value for X2.

The distribution of X2 is a chi-squared distribution for the following reason. Under the null hypothesis for test i, the p-value p_i follows a uniform distribution on the interval . The negative natural logarithm of a uniformly distributed value follows an exponential distribution. Scaling a value that follows an exponential distribution by a factor of two yields a quantity that follows a chi-squared distribution with two degrees of freedom. Finally, the sum of k independent chi-squared values, each with two degrees of freedom, follows a chi-squared distribution with 2k degrees of freedom.