statsmodels.stats.nonparametric.rank_compare_2indep(x1, x2, use_t=
Statistics and tests for the probability that x1 has larger values than x2.
p is the probability that a random draw from the population of the first sample has a larger value than a random draw from the population of the second sample, specifically
p = P(x1 > x2) + 0.5 * P(x1 = x2)
This is a measure underlying Wilcoxon-Mann-Whitney’s U test, Fligner-Policello test and Brunner-Munzel test, and Inference is based on the asymptotic distribution of the Brunner-Munzel test. The half probability for ties corresponds to the use of midranks and make it valid for discrete variables.
The Null hypothesis for stochastic equality is p = 0.5, which corresponds to the Brunner-Munzel test.
The results instance contains the results for the Brunner-Munzel test and has methods for hypothesis tests, confidence intervals and summary.
The Brunner-Munzel W statistic.
p-value assuming an t distribution. One-sided or two-sided, depending on the choice of alternative and use_t.
Wilcoxon-Mann-Whitney assumes equal variance or equal distribution under the Null hypothesis. Fligner-Policello test allows for unequal variances but assumes continuous distribution, i.e. no ties. Brunner-Munzel extend the test to allow for unequal variance and discrete or ordered categorical random variables.
Brunner and Munzel recommended to estimate the p-value by t-distribution when the size of data is 50 or less. If the size is lower than 10, it would be better to use permuted Brunner Munzel test (see ) for the test of stochastic equality.
This measure has been introduced in the literature under many different names relying on a variety of assumptions. In psychology, McGraw and Wong (1992) introduced it as Common Language effect size for the continuous, normal distribution case, Vargha and Delaney (2000)  extended it to the nonparametric continuous distribution case as in Fligner-Policello.
WMW and related tests can only be interpreted as test of medians or tests of central location only under very restrictive additional assumptions such as both distribution are identical under the equality null hypothesis (assumed by Mann-Whitney) or both distributions are symmetric (shown by Fligner-Policello). If the distribution of the two samples can differ in an arbitrary way, then the equality Null hypothesis corresponds to p=0.5 against an alternative p != 0.5. see for example Conroy (2012)  and Divine et al (2018)  .
Note: Brunner-Munzel and related literature define the probability that x1 is stochastically smaller than x2, while here we use stochastically larger. This equivalent to switching x1 and x2 in the two sample case.
Brunner, E. and Munzel, U. “The nonparametric Benhrens-Fisher problem: Asymptotic theory and a small-sample approximation”. Biometrical Journal. Vol. 42(2000): 17-25.
Neubert, K. and Brunner, E. “A studentized permutation test for the non-parametric Behrens-Fisher problem”. Computational Statistics and Data Analysis. Vol. 51(2007): 5192-5204.
Vargha, András, and Harold D. Delaney. 2000. “A Critique and Improvement of the CL Common Language Effect Size Statistics of McGraw and Wong.” Journal of Educational and Behavioral Statistics 25 (2): 101–32. https://doi.org/10.3102/10769986025002101.
Conroy, Ronán M. 2012. “What Hypotheses Do ‘Nonparametric’ Two-Group Tests Actually Test?” The Stata Journal: Promoting Communications on Statistics and Stata 12 (2): 182–90. https://doi.org/10.1177/1536867X1201200202.