p = ranksum (x, y) returns the p-value of a two-sided Wilcoxon rank sum test. It tests the null hypothesis that two independent samples, in the vectors X and Y, come from continuous distributions with equal medians, against the alternative hypothesis that they are not. x and y can have different lengths and the test assumes that they are independent.

ranksum treats NaN in x, y as missing values. The two-sided p-value is computed by doubling the most significant one-sided value.

[p, h] = ranksum (x, y) also returns the result of the hypothesis test with h = 1 indicating a rejection of the null hypothesis at the default alpha = 0.05 significance level, and h = 0 indicating a failure to reject the null hypothesis at the same significance level.

[p, h, stats] = ranksum (x, y) also returns the structure stats with information about the test statistic. It contains the field ranksum with the value of the rank sum test statistic, and the field zval with the value of the z-statistic when computed with the "approximate" method, and empty otherwise. The field is always there. Two further fields, an Octave extension, hold the effect size: RankBiserial, the rank-biserial correlation, which is Cliff’s delta, the share of pairs (x_i, y_j) with x_i > y_j less the share with x_i < y_j, and RankBiserialCI, its two-sided confidence interval at the level 100 (1 - alpha) percent, whatever the tail. Both are those of meanEffectSize (x, y, "Effect", "cliff"): by default the interval is Cliff’s, and the options "ConfidenceIntervalType", "bootstrap" for a BCa bootstrap interval, "NumBootstraps" and "Resampling" are passed on to meanEffectSize.

[…] = ranksum (x, y, alpha) or alternatively […] = ranksum (x, y, "alpha", alpha) returns the result of the hypothesis test performed at the significance level ALPHA.

[…] = ranksum (x, y, "method", M) defines the computation method of the p-value specified in M, which can be "exact", "approximate", or "oldexact". M must be a single string. When "method" is unspecified, the default is: "exact" when min (length (x), length (y)) < 10 and length (x) + length (y) < 20, otherwise the "approximate" method is used.

• "exact" method uses full enumeration for small total sample size (< 10), otherwise the network algorithm is used for larger samples.
• "approximate" uses normal approximation method for computing the p-value.
• "oldexact" uses full enumeration for any sample size. Note, that this option can lead to out of memory error for large samples. Use with caution!

[…] = ranksum (x, y, "tail", tail) defines the type of test, which can be "both", "right", or "left". tail must be a single string.

• "both" – "medians are not equal" (two-tailed test, default)
• "right" – "median of X is greater than median of Y" (right-tailed test)
• "left" – "median of X is less than median of Y" (left-tailed test)

Note: the rank sum statistic is based on the smaller sample of vectors x and y.

See also: signrank, kruskalwallis, meanEffectSize

Source Code: ranksum