scipy.stats.anderson_ksamp#

scipy.stats.anderson_ksamp(samples, midrank=True)[source]#

The Anderson-Darling test for k-samples.

The k-sample Anderson-Darling test is a modification of the one-sample Anderson-Darling test. It tests the null hypothesis that k-samples are drawn from the same population without having to specify the distribution function of that population. The critical values depend on the number of samples.

Parameters

samplessequence of 1-D array_like: Array of sample data in arrays.
midrankbool, optional: Type of Anderson-Darling test which is computed. Default (True) is the midrank test applicable to continuous and discrete populations. If False, the right side empirical distribution is used.

Returns

statisticfloat: Normalized k-sample Anderson-Darling test statistic.
critical_valuesarray: The critical values for significance levels 25%, 10%, 5%, 2.5%, 1%, 0.5%, 0.1%.
significance_levelfloat: An approximate significance level at which the null hypothesis for the provided samples can be rejected. The value is floored / capped at 0.1% / 25%.

Raises

ValueError: If less than 2 samples are provided, a sample is empty, or no distinct observations are in the samples.

See also

ks_2samp: 2 sample Kolmogorov-Smirnov test
anderson: 1 sample Anderson-Darling test

Notes

[1] defines three versions of the k-sample Anderson-Darling test: one for continuous distributions and two for discrete distributions, in which ties between samples may occur. The default of this routine is to compute the version based on the midrank empirical distribution function. This test is applicable to continuous and discrete data. If midrank is set to False, the right side empirical distribution is used for a test for discrete data. According to [1], the two discrete test statistics differ only slightly if a few collisions due to round-off errors occur in the test not adjusted for ties between samples.

The critical values corresponding to the significance levels from 0.01 to 0.25 are taken from [1]. p-values are floored / capped at 0.1% / 25%. Since the range of critical values might be extended in future releases, it is recommended not to test p == 0.25, but rather p >= 0.25 (analogously for the lower bound).

New in version 0.14.0.

References

1(1,2,3): Scholz, F. W and Stephens, M. A. (1987), K-Sample Anderson-Darling Tests, Journal of the American Statistical Association, Vol. 82, pp. 918-924.

Examples

>>> from scipy import stats
>>> rng = np.random.default_rng()

The null hypothesis that the two random samples come from the same distribution can be rejected at the 5% level because the returned test value is greater than the critical value for 5% (1.961) but not at the 2.5% level. The interpolation gives an approximate significance level of 3.2%:

>>> stats.anderson_ksamp([rng.normal(size=50),
... rng.normal(loc=0.5, size=30)])
(1.974403288713695,
  array([0.325, 1.226, 1.961, 2.718, 3.752, 4.592, 6.546]),
  0.04991293614572478)

The null hypothesis cannot be rejected for three samples from an identical distribution. The reported p-value (25%) has been capped and may not be very accurate (since it corresponds to the value 0.449 whereas the statistic is -0.731):

>>> stats.anderson_ksamp([rng.normal(size=50),
... rng.normal(size=30), rng.normal(size=20)])
(-0.29103725200789504,
  array([ 0.44925884,  1.3052767 ,  1.9434184 ,  2.57696569,  3.41634856,
  4.07210043, 5.56419101]),
  0.25)