Fisher's method

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In statistics, Fisher's method, also known as Fisher's combined probability test, developed by and named for Ronald Fisher, is a data fusion or "meta-analysis" (analysis after analysis) technique for combining the results from a variety of independent tests bearing upon the same overall hypothesis (H0) as if in a single large test.

Fisher's method combines extreme value probabilities, P(results at least as extreme, assuming H0 true) from each test, called "p-values", into one test statistic (X2) having a chi-square distribution using the formula

X^2_{2k} = -2\sum_{i=1}^k \log_e(p_i).

The p-value for X2 itself can then be interpolated from a chi-square table using 2k "degrees of freedom", where k is the number of tests being combined. As in any similar test, H0 is rejected for small p-values, usually < 0.05.

This figure shows how two p-values ~0.10 (or ~0.04 and ~0.25) combine into one ~0.05.
This figure shows how two p-values ~0.10 (or ~0.04 and ~0.25) combine into one ~0.05.

In the case that the tests are not independent, the null distribution of X2 is more complicated. If the correlations between the loge(pi) are known, these can be used to form an approximation.

[edit] References

  • Fisher, R. A. (1948) "Combining independent tests of significance", American Statistician, vol. 2, issue 5, page 30. (In response to Question 14)

[edit] See also