scipy.stats.combine_pvalues(pvalues, method='fisher', weights=None)[source]

Combine p-values from independent tests bearing upon the same hypothesis.

pvaluesarray_like, 1-D

Array of p-values assumed to come from independent tests.

method{‘fisher’, ‘pearson’, ‘tippett’, ‘stouffer’,

‘mudholkar_george’}, optional

Name of method to use to combine p-values. The following methods are available (default is ‘fisher’):

  • ‘fisher’: Fisher’s method (Fisher’s combined probability test), the sum of the logarithm of the p-values

  • ‘pearson’: Pearson’s method (similar to Fisher’s but uses sum of the complement of the p-values inside the logarithms)

  • ‘tippett’: Tippett’s method (minimum of p-values)

  • ‘stouffer’: Stouffer’s Z-score method

  • ‘mudholkar_george’: the difference of Fisher’s and Pearson’s methods divided by 2

weightsarray_like, 1-D, optional

Optional array of weights used only for Stouffer’s Z-score method.

statistic: float

The statistic calculated by the specified method.

pval: float

The combined p-value.


Fisher’s method (also known as Fisher’s combined probability test) [1] uses a chi-squared statistic to compute a combined p-value. The closely related Stouffer’s Z-score method [2] uses Z-scores rather than p-values. The advantage of Stouffer’s method is that it is straightforward to introduce weights, which can make Stouffer’s method more powerful than Fisher’s method when the p-values are from studies of different size [6] [7]. The Pearson’s method uses \(log(1-p_i)\) inside the sum whereas Fisher’s method uses \(log(p_i)\) [4]. For Fisher’s and Pearson’s method, the sum of the logarithms is multiplied by -2 in the implementation. This quantity has a chi-square distribution that determines the p-value. The mudholkar_george method is the difference of the Fisher’s and Pearson’s test statistics, each of which include the -2 factor [4]. However, the mudholkar_george method does not include these -2 factors. The test statistic of mudholkar_george is the sum of logisitic random variables and equation 3.6 in [3] is used to approximate the p-value based on Student’s t-distribution.

Fisher’s method may be extended to combine p-values from dependent tests [5]. Extensions such as Brown’s method and Kost’s method are not currently implemented.

New in version 0.15.0.





George, E. O., and G. S. Mudholkar. “On the convolution of logistic random variables.” Metrika 30.1 (1983): 1-13.


Heard, N. and Rubin-Delanchey, P. “Choosing between methods of combining p-values.” Biometrika 105.1 (2018): 239-246.


Whitlock, M. C. “Combining probability from independent tests: the weighted Z-method is superior to Fisher’s approach.” Journal of Evolutionary Biology 18, no. 5 (2005): 1368-1373.


Zaykin, Dmitri V. “Optimally weighted Z-test is a powerful method for combining probabilities in meta-analysis.” Journal of Evolutionary Biology 24, no. 8 (2011): 1836-1841.