# scipy.stats.combine_pvalues

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

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

  These methods are intended only for combining p-values from hypothesis tests based upon continuous distributions.

  Each method assumes that under the null hypothesis, the p-values are sampled independently and uniformly from the interval [0, 1]. A test statistic (different for each method) is computed and a combined p-value is calculated based upon the distribution of this test statistic under the null hypothesis.

  Parameters:
    pvalues array_like, 1-D

      Array of p-values assumed to come from independent tests based on continuous distributions.

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

      Name of method to use to combine p-values.

      The available methods are (see Notes for details):

      - ‘fisher’: Fisher’s method (Fisher’s combined probability test)

      - ‘pearson’: Pearson’s method

      - ‘mudholkar_george’: Mudholkar’s and George’s method

      - ‘tippett’: Tippett’s method

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

    weights array_like, 1-D, optional

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

  Returns:
    res SignificanceResult

      An object containing attributes:

      statisticfloat

        The statistic calculated by the specified method.

      pvaluefloat

        The combined p-value.

  Notes

  If this function is applied to tests with a discrete statistics such as any rank test or contingency-table test, it will yield systematically wrong results, e.g. Fisher’s method will systematically overestimate the p-value [1]. This problem becomes less severe for large sample sizes when the discrete distributions become approximately continuous.

  The differences between the methods can be best illustrated by their statistics and what aspects of a combination of p-values they emphasise when considering significance [2]. For example, methods emphasising large p-values are more sensitive to strong false and true negatives; conversely methods focussing on small p-values are sensitive to positives.

  - The statistics of Fisher’s method (also known as Fisher’s combined probability test) [3] is \(-2\sum_i \log(p_i)\), which is equivalent (as a test statistics) to the product of individual p-values: \(\prod_i p_i\). Under the null hypothesis, this statistics follows a \(\chi^2\) distribution. This method emphasises small p-values.

  - Pearson’s method uses \(-2\sum_i\log(1-p_i)\), which is equivalent to \(\prod_i \frac{1}{1-p_i}\) [2]. It thus emphasises large p-values.

  - Mudholkar and George compromise between Fisher’s and Pearson’s method by averaging their statistics [4]. Their method emphasises extreme p-values, both close to 1 and 0.

  - Stouffer’s method [5] uses Z-scores and the statistic: \(\sum_i \Phi^{-1} (p_i)\), where \(\Phi\) is the CDF of the standard normal distribution. The advantage of this 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].

  - Tippett’s method uses the smallest p-value as a statistic. (Mind that this minimum is not the combined p-value.)

  Fisher’s method may be extended to combine p-values from dependent tests [8]. Extensions such as Brown’s method and Kost’s method are not currently implemented.

  New in version 0.15.0.

  References

  [1]

  Kincaid, W. M., “The Combination of Tests Based on Discrete Distributions.” Journal of the American Statistical Association 57, no. 297 (1962), 10-19.

  [2] (1,2)

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

  [3]

  https://en.wikipedia.org/wiki/Fisher%27s_method

  [4]

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

  [5]

  https://en.wikipedia.org/wiki/Fisher%27s_method#Relation_to_Stouffer.27s_Z-score_method

  [6]

  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.

  [7]

  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.

  [8]

  https://en.wikipedia.org/wiki/Extensions_of_Fisher%27s_method