Why does scipy.stats.anderson_ksamp give a p-value of over a million for these data?

Question

When I run

from scipy.stats import anderson_ksamp

a = [-1.8, -2.4, -2.4, -0.0, -1.5, -2.7, -1.8, -3.0, -1.8, -1.2, -3.0, -3.0, -2.8, -3.0, -2.1, -0.0, 0.6, -2.5, -2.4, -0.0, -2.7, -0.0, -2.5, -2.1, -0.9, -3.0, -0.6, -0.6, -1.5, -2.2, -1.2, -2.4, -2.4, -3.0, 1.5, -1.8, 1.5, -2.7, -3.0, -2.5, -2.5, -1.5, -1.5, -2.1, -2.1, -3.0, -0.6, -2.7, -3.0, -1.5, -0.6, -0.0, -2.1, 0.6, -2.0, -3.0, -3.0, -2.4, -3.0, -1.8, -0.0, -0.0, -3.0, -1.5, -3.0, -3.0, -1.5, -2.5, -3.0, -2.8, -3.0, -2.2, -0.6, -0.0, -1.5, -2.7, -2.1, -2.1, -2.2, -2.1, -0.6, -0.0, -2.5, -2.1, -1.5, -3.0, -2.2, -1.8, -2.7, -2.4, -1.5, -2.1, -2.9, -2.4, -0.9, -0.0, -0.0, -2.4, -2.7, -0.6, -2.2, -3.0, -1.5, -0.9, -3.0, -3.0, -0.0, -2.7, -2.7, -1.5, -2.2, -3.0, -0.0, 1.5, -3.0, -2.7, -2.2, -2.9, -2.2, -3.0, -1.8, -0.0, -3.0, -1.5, -2.7, -3.0, -3.0, -2.9, -3.0, -3.0, -3.0, -3.0, -2.5, -0.0, 0.9, -3.0, -0.0, -3.0, 3.0, -3.0, -3.0, -1.2, -2.1, 1.5, -0.0, -0.9, -3.0, -2.7, -1.5, -2.4, -2.1, -3.0, -0.9, -3.0, -0.8, -1.5, -2.1, -2.7, -0.0, -0.0, -2.2, -1.8, -2.1, -2.2, -3.0, 0.6, -2.4, -2.2, -2.4, -2.5, -1.5, -0.0, -2.7, -3.0, -3.0, -2.1, -0.0, -2.4, -2.4, -0.0, -2.0, -0.9, -2.4, -3.0, -1.4, -2.7, -2.7, -3.0, -3.0, -2.7, -1.2, -2.1, -3.0, -0.0, -3.0, -2.7, -2.7, -3.0, -3.0, -2.5, -3.0, -1.8, -1.5, -2.7, -2.4, -1.8, -3.0, -2.7, 2.1, -3.0, -2.2, -2.2, 0.6, -0.9, 6.0, -3.0, -2.1, -3.0, -2.1, -2.5, -3.0, -1.5, -2.5, 3.0, -2.1, -3.0, -3.0, -1.5, -2.1, -2.7, -2.5, 1.5, -2.1, -0.0, -0.0, -3.0, -0.6, 1.5, -2.7, -2.4, -2.1, -3.0, -2.7, -3.0, 9.0, -3.0, -1.7, -3.0, -0.0, -3.0, -2.2, -0.0, -0.6, -2.7, -2.7, -3.0, -3.0, -1.7, -2.1, -2.0, -3.0, -2.1, -3.0, -1.1, -3.0, -3.0, -2.4, -1.5, -3.0, -2.2, -3.0, -1.5, -2.7, -3.0, -3.0, -3.0, -2.2, -3.0, -2.1, -2.1, -2.4, -3.0, -3.0, -0.0, -3.0, -3.0, -2.1, -2.0, 1.5, -3.0, -3.0, -2.1, -2.9, -2.4, -3.0, -3.0, -1.5, -2.2, -0.9, -1.8, -2.1, -1.8, -1.5, -3.0, -1.5, -3.0, -1.5, -3.0, -2.4, 1.5, -2.7, -3.0, -1.8, -1.8, -1.5, -2.1, -2.7, -2.7, -2.7, -3.0, -1.5, -2.7, -3.0, -2.7, -3.0, -1.5, -1.5, -3.0, 0.6, -0.6, -3.0, -2.1, -2.4, -2.1, -3.0, -2.2, -3.0, -1.8, -1.2, -3.0, -0.8, -2.4, -2.5, -3.0, -1.5, -1.2, -0.0, -2.7, -2.4, -3.0, -3.0, -2.1, -2.1, -2.1, -3.0, -2.7, -2.4, -2.1, -1.5, -2.1, -0.6, -3.0, -3.0, -3.0, -3.0]
b = [-1.2, 6.0, -3.0, 1.5, 12.0, 1.5, 3.0, -0.0, -1.5, -0.0, 6.0, -0.0, 1.8, -1.5, -3.0, -3.0, 1.5, -0.0, 1.5, -3.0, -3.0, -0.0, -1.2, -3.0, 22.5, -0.0, -3.0, -3.0, -2.1, 3.0, 2.4, 1.5, -2.1, 4.5, -3.0, -3.0, 12.0, 6.0, -3.0, 3.0, -3.0, 12.0, -2.8, 0.6, 4.5, 3.0, -0.0, -3.0, -1.0, -3.0, -0.0, 3.0, 2.1, -0.6, -3.0, -3.0, 1.5, 1.5, -0.6, -0.0, 1.5, 6.0, -2.2, -2.1, -0.0, -0.8, 6.0, -3.0, -3.0, 9.0, -3.0, -0.9, 15.0, 1.5, -2.9, 19.5, 4.5, -3.0, 1.5, -1.8, 1.5, 0.9, -3.0, -3.0, 9.0, -0.3, 9.0, -2.1, -2.2, -3.0, 12.0, -2.1, -1.2, -3.0, 4.5, -1.2, -0.0, 12.0, -0.9, -3.0, -0.6, -3.0, -1.5, -0.0, 27.0, 4.5, -2.4, -2.7, -0.0, 3.0, 1.5, 3.0, -3.0, 16.5, -2.1, -0.6, -3.0, -1.5, -1.2, -3.0, 4.4, -2.5, -2.1, 3.0, -1.2, -3.0, 12.0, -1.8, -0.9, -3.0, -3.0, -0.0, -3.0, -1.2, -3.0, -3.0, -0.0, 1.5, -3.0, 7.5, -3.0, -2.1, 3.0, 1.5, 3.0, -3.0, 1.5, -3.0, 19.5, -3.0, -2.2, 27.0, 3.0, -1.5, -3.0, -3.0, 4.5, -1.2, 12.0, 3.0, 3.0, -2.1, 6.0, -2.2, -3.0, -2.4, 6.0, -3.0, -1.9, -0.6, -0.0, 3.6, 15.0, -3.0, -3.0, 7.5, -0.0, 4.5, -2.4, -3.0, -3.0, -3.0, -2.0, -3.0, -3.0, -3.0, -2.6, 3.0, -0.0, -3.0, 4.5, -1.2, -3.0, -3.0, 3.0, -3.0, 1.6, 1.5, -3.0, -3.0, -3.0, -3.0, 3.0, -3.0, -1.5, -3.0, -1.5, 12.0, -1.5, 3.0, 9.6, -0.0, -1.8, -3.0, -2.1, -0.6, 9.0, -3.0, 19.5, -2.4, -1.8, 15.0, -3.0, -3.0, 27.0, 7.5, 12.0, -3.0, -2.0, -3.0, 12.0, -3.0, -3.0, 0.9, -3.0, -2.5, 12.6, -1.5, -0.6, -3.0, -3.0, -0.0, -0.8, -3.0, 3.0, -0.0, 4.5, -3.0, -0.0, 0.6, 0.3, -2.7, -0.0, 4.5, -3.0, 3.0, -1.5, 9.0, -3.0, 1.5, -3.0, 6.0, -2.2, -0.0, -3.0, -3.0, 3.0, -3.0, -2.7, -0.0, -3.0, 0.6, 4.5, 1.5, 1.5, 1.5, -2.1, 7.5, -3.0, -3.0, -3.0, -2.1, -3.0, 6.0, -1.5, -2.0, -3.0, -0.6, 6.0, -1.5, -3.0, 6.0, -3.0, 3.0, 3.0, -3.0, 7.5, -3.0, -0.0, -3.0, -0.0, -3.0, -3.0, 0.6, -3.0, -3.0, -3.0, -3.0, -2.4, -0.0, 1.5, 3.0, 2.1, -3.0, 3.0, 4.5, -3.0, -2.4, -3.0, -2.2, 7.5, 2.1, -3.0, -0.0, -2.0, -3.0, -3.0, -1.8, -3.0, 4.5, -1.5, -3.0, 4.5, 3.0, 15.0, 4.5, -2.1, 12.0, 6.0, 4.5, 27.0, -3.0, 3.0, 0.6, -3.0, 0.6, -2.4, 4.5, -1.5, -2.2, 12.0, -2.0, 1.5, 9.0, -1.5, -1.2, -2.0, 1.5, -1.2, -1.8, -3.0, -0.0, -0.0, -3.0, -0.9]

print(anderson_ksamp([a,b]))

I get the following output:

Anderson_ksampResult(
    statistic=53.560696338122263,
    critical_values=array([ 0.325,  1.226,  1.961,  2.718,  3.752]),
    significance_level=7238105.535194747
)

The "significance level" is far greater than 1. Why is this?

I'm mostly wondering whether it's an issue with the scipy implementation or is this issue inherent to numerical implementations of the Anderson-Darling test (as I don't know the details of how the p-value is actually computed). I'd also be interested in some info on how to get around this bug.

I'm aware of this issue, but that seems to be a slight error in extrapolation due to data from the same distribution (scipy.stats.ks_2samp gives a p-value of 0.962305 whereas anderson_ksamp gives 1.075969). On the other hand, with the above data we can run

from scipy.stats import ks_2samp
from statsmodels.distributions.empirical_distribution import ECDF
import matplotlib.pyplot as plt
import numpy as np

a = [-1.8, -2.4, -2.4, -0.0, -1.5, -2.7, -1.8, -3.0, -1.8, -1.2, -3.0, -3.0, -2.8, -3.0, -2.1, -0.0, 0.6, -2.5, -2.4, -0.0, -2.7, -0.0, -2.5, -2.1, -0.9, -3.0, -0.6, -0.6, -1.5, -2.2, -1.2, -2.4, -2.4, -3.0, 1.5, -1.8, 1.5, -2.7, -3.0, -2.5, -2.5, -1.5, -1.5, -2.1, -2.1, -3.0, -0.6, -2.7, -3.0, -1.5, -0.6, -0.0, -2.1, 0.6, -2.0, -3.0, -3.0, -2.4, -3.0, -1.8, -0.0, -0.0, -3.0, -1.5, -3.0, -3.0, -1.5, -2.5, -3.0, -2.8, -3.0, -2.2, -0.6, -0.0, -1.5, -2.7, -2.1, -2.1, -2.2, -2.1, -0.6, -0.0, -2.5, -2.1, -1.5, -3.0, -2.2, -1.8, -2.7, -2.4, -1.5, -2.1, -2.9, -2.4, -0.9, -0.0, -0.0, -2.4, -2.7, -0.6, -2.2, -3.0, -1.5, -0.9, -3.0, -3.0, -0.0, -2.7, -2.7, -1.5, -2.2, -3.0, -0.0, 1.5, -3.0, -2.7, -2.2, -2.9, -2.2, -3.0, -1.8, -0.0, -3.0, -1.5, -2.7, -3.0, -3.0, -2.9, -3.0, -3.0, -3.0, -3.0, -2.5, -0.0, 0.9, -3.0, -0.0, -3.0, 3.0, -3.0, -3.0, -1.2, -2.1, 1.5, -0.0, -0.9, -3.0, -2.7, -1.5, -2.4, -2.1, -3.0, -0.9, -3.0, -0.8, -1.5, -2.1, -2.7, -0.0, -0.0, -2.2, -1.8, -2.1, -2.2, -3.0, 0.6, -2.4, -2.2, -2.4, -2.5, -1.5, -0.0, -2.7, -3.0, -3.0, -2.1, -0.0, -2.4, -2.4, -0.0, -2.0, -0.9, -2.4, -3.0, -1.4, -2.7, -2.7, -3.0, -3.0, -2.7, -1.2, -2.1, -3.0, -0.0, -3.0, -2.7, -2.7, -3.0, -3.0, -2.5, -3.0, -1.8, -1.5, -2.7, -2.4, -1.8, -3.0, -2.7, 2.1, -3.0, -2.2, -2.2, 0.6, -0.9, 6.0, -3.0, -2.1, -3.0, -2.1, -2.5, -3.0, -1.5, -2.5, 3.0, -2.1, -3.0, -3.0, -1.5, -2.1, -2.7, -2.5, 1.5, -2.1, -0.0, -0.0, -3.0, -0.6, 1.5, -2.7, -2.4, -2.1, -3.0, -2.7, -3.0, 9.0, -3.0, -1.7, -3.0, -0.0, -3.0, -2.2, -0.0, -0.6, -2.7, -2.7, -3.0, -3.0, -1.7, -2.1, -2.0, -3.0, -2.1, -3.0, -1.1, -3.0, -3.0, -2.4, -1.5, -3.0, -2.2, -3.0, -1.5, -2.7, -3.0, -3.0, -3.0, -2.2, -3.0, -2.1, -2.1, -2.4, -3.0, -3.0, -0.0, -3.0, -3.0, -2.1, -2.0, 1.5, -3.0, -3.0, -2.1, -2.9, -2.4, -3.0, -3.0, -1.5, -2.2, -0.9, -1.8, -2.1, -1.8, -1.5, -3.0, -1.5, -3.0, -1.5, -3.0, -2.4, 1.5, -2.7, -3.0, -1.8, -1.8, -1.5, -2.1, -2.7, -2.7, -2.7, -3.0, -1.5, -2.7, -3.0, -2.7, -3.0, -1.5, -1.5, -3.0, 0.6, -0.6, -3.0, -2.1, -2.4, -2.1, -3.0, -2.2, -3.0, -1.8, -1.2, -3.0, -0.8, -2.4, -2.5, -3.0, -1.5, -1.2, -0.0, -2.7, -2.4, -3.0, -3.0, -2.1, -2.1, -2.1, -3.0, -2.7, -2.4, -2.1, -1.5, -2.1, -0.6, -3.0, -3.0, -3.0, -3.0]
b = [-1.2, 6.0, -3.0, 1.5, 12.0, 1.5, 3.0, -0.0, -1.5, -0.0, 6.0, -0.0, 1.8, -1.5, -3.0, -3.0, 1.5, -0.0, 1.5, -3.0, -3.0, -0.0, -1.2, -3.0, 22.5, -0.0, -3.0, -3.0, -2.1, 3.0, 2.4, 1.5, -2.1, 4.5, -3.0, -3.0, 12.0, 6.0, -3.0, 3.0, -3.0, 12.0, -2.8, 0.6, 4.5, 3.0, -0.0, -3.0, -1.0, -3.0, -0.0, 3.0, 2.1, -0.6, -3.0, -3.0, 1.5, 1.5, -0.6, -0.0, 1.5, 6.0, -2.2, -2.1, -0.0, -0.8, 6.0, -3.0, -3.0, 9.0, -3.0, -0.9, 15.0, 1.5, -2.9, 19.5, 4.5, -3.0, 1.5, -1.8, 1.5, 0.9, -3.0, -3.0, 9.0, -0.3, 9.0, -2.1, -2.2, -3.0, 12.0, -2.1, -1.2, -3.0, 4.5, -1.2, -0.0, 12.0, -0.9, -3.0, -0.6, -3.0, -1.5, -0.0, 27.0, 4.5, -2.4, -2.7, -0.0, 3.0, 1.5, 3.0, -3.0, 16.5, -2.1, -0.6, -3.0, -1.5, -1.2, -3.0, 4.4, -2.5, -2.1, 3.0, -1.2, -3.0, 12.0, -1.8, -0.9, -3.0, -3.0, -0.0, -3.0, -1.2, -3.0, -3.0, -0.0, 1.5, -3.0, 7.5, -3.0, -2.1, 3.0, 1.5, 3.0, -3.0, 1.5, -3.0, 19.5, -3.0, -2.2, 27.0, 3.0, -1.5, -3.0, -3.0, 4.5, -1.2, 12.0, 3.0, 3.0, -2.1, 6.0, -2.2, -3.0, -2.4, 6.0, -3.0, -1.9, -0.6, -0.0, 3.6, 15.0, -3.0, -3.0, 7.5, -0.0, 4.5, -2.4, -3.0, -3.0, -3.0, -2.0, -3.0, -3.0, -3.0, -2.6, 3.0, -0.0, -3.0, 4.5, -1.2, -3.0, -3.0, 3.0, -3.0, 1.6, 1.5, -3.0, -3.0, -3.0, -3.0, 3.0, -3.0, -1.5, -3.0, -1.5, 12.0, -1.5, 3.0, 9.6, -0.0, -1.8, -3.0, -2.1, -0.6, 9.0, -3.0, 19.5, -2.4, -1.8, 15.0, -3.0, -3.0, 27.0, 7.5, 12.0, -3.0, -2.0, -3.0, 12.0, -3.0, -3.0, 0.9, -3.0, -2.5, 12.6, -1.5, -0.6, -3.0, -3.0, -0.0, -0.8, -3.0, 3.0, -0.0, 4.5, -3.0, -0.0, 0.6, 0.3, -2.7, -0.0, 4.5, -3.0, 3.0, -1.5, 9.0, -3.0, 1.5, -3.0, 6.0, -2.2, -0.0, -3.0, -3.0, 3.0, -3.0, -2.7, -0.0, -3.0, 0.6, 4.5, 1.5, 1.5, 1.5, -2.1, 7.5, -3.0, -3.0, -3.0, -2.1, -3.0, 6.0, -1.5, -2.0, -3.0, -0.6, 6.0, -1.5, -3.0, 6.0, -3.0, 3.0, 3.0, -3.0, 7.5, -3.0, -0.0, -3.0, -0.0, -3.0, -3.0, 0.6, -3.0, -3.0, -3.0, -3.0, -2.4, -0.0, 1.5, 3.0, 2.1, -3.0, 3.0, 4.5, -3.0, -2.4, -3.0, -2.2, 7.5, 2.1, -3.0, -0.0, -2.0, -3.0, -3.0, -1.8, -3.0, 4.5, -1.5, -3.0, 4.5, 3.0, 15.0, 4.5, -2.1, 12.0, 6.0, 4.5, 27.0, -3.0, 3.0, 0.6, -3.0, 0.6, -2.4, 4.5, -1.5, -2.2, 12.0, -2.0, 1.5, 9.0, -1.5, -1.2, -2.0, 1.5, -1.2, -1.8, -3.0, -0.0, -0.0, -3.0, -0.9]

ecdf1, ecdf2 = ECDF(a), ECDF(b)

xs = np.linspace(min(a+b),max(a+b), num=10000)
plt.figure(figsize=(12,8))
plt.plot(xs,ecdf1(xs), xs,ecdf2(xs))
plt.show()

print(ks_2samp(a,b))

to get a p-value of 2.6169e-17, with the following ECDF plot showing that the distributions which a and b were sampled from are very different:

KstestResult(statistic=0.3232876712328767, pvalue=2.8586245432606456e-17)

It's obviously a bug and appears to be a result of gross extrapolation. Are you looking for any other kind of answer? — whuber, Jul 25 '16 at 20:20
I'm mostly wondering whether it's an issue with the scipy implementation or is this issue inherent to numerical implementations of the Anderson-Darling test (as I don't know the details of how the p-value is actually computed). I'd also be interested in some info on how to get around this bug. — Esteemator, Jul 25 '16 at 20:24
Please include that clarification prominently in an edit to the question, lest others vote to close it (as being solely a computing issue). — whuber, Jul 25 '16 at 20:26
In case someone encounters the same issue, I'd like to update that as of Sept 2018 the extrapolation problem mentioned in this post has been fixed in the current [scipy github](https://github.com/scipy/scipy/blob/master/scipy/stats/morestats.py). The critical value is now capped/floored at confidence interval 25% and 0.1%. This update should be available in the next version (1.2.0) — user12075, Sep 29 '18 at 01:59
I’m voting to close this question because it is about a programming bug that was fixed a few years ago, not statistics. — Sycorax, Feb 27 '22 at 18:39

Ali Cirik · Answer 1 · 2022-02-27T18:18:19.367

February 27, 2022 update for someone who may encounter the same issue and find this post:

Note: This answer is intended to point to a scipy bug when the question was originally asked on Jul 25, 2016. This bug led to p-values greater than 1 and was fixed in this commit and looks like scipy versions scipy>=1.2.0 are not impacted by this. Code examples and outputs are available here for reproducibility. Following are tested on Python 3.9.9 and scipy==1.8.0

The first code chunk:

from scipy.stats import anderson_ksamp

a = [-1.8, -2.4, -2.4, -0.0, -1.5, -2.7, -1.8, -3.0, -1.8, -1.2, -3.0,
     -3.0, -2.8, -3.0, -2.1, -0.0, 0.6, -2.5, -2.4, -0.0, -2.7, -0.0,
     -2.5, -2.1, -0.9, -3.0, -0.6, -0.6, -1.5, -2.2, -1.2, -2.4, -2.4,
     -3.0, 1.5, -1.8, 1.5, -2.7, -3.0, -2.5, -2.5, -1.5, -1.5, -2.1,
     -2.1, -3.0, -0.6, -2.7, -3.0, -1.5, -0.6, -0.0, -2.1, 0.6, -2.0,
     -3.0, -3.0, -2.4, -3.0, -1.8, -0.0, -0.0, -3.0, -1.5, -3.0, -3.0,
     -1.5, -2.5, -3.0, -2.8, -3.0, -2.2, -0.6, -0.0, -1.5, -2.7, -2.1,
     -2.1, -2.2, -2.1, -0.6, -0.0, -2.5, -2.1, -1.5, -3.0, -2.2, -1.8,
     -2.7, -2.4, -1.5, -2.1, -2.9, -2.4, -0.9, -0.0, -0.0, -2.4, -2.7,
     -0.6, -2.2, -3.0, -1.5, -0.9, -3.0, -3.0, -0.0, -2.7, -2.7, -1.5,
     -2.2, -3.0, -0.0, 1.5, -3.0, -2.7, -2.2, -2.9, -2.2, -3.0, -1.8,
     -0.0, -3.0, -1.5, -2.7, -3.0, -3.0, -2.9, -3.0, -3.0, -3.0, -3.0,
     -2.5, -0.0, 0.9, -3.0, -0.0, -3.0, 3.0, -3.0, -3.0, -1.2, -2.1,
     1.5, -0.0, -0.9, -3.0, -2.7, -1.5, -2.4, -2.1, -3.0, -0.9, -3.0,
     -0.8, -1.5, -2.1, -2.7, -0.0, -0.0, -2.2, -1.8, -2.1, -2.2, -3.0,
     0.6, -2.4, -2.2, -2.4, -2.5, -1.5, -0.0, -2.7, -3.0, -3.0, -2.1,
     -0.0, -2.4, -2.4, -0.0, -2.0, -0.9, -2.4, -3.0, -1.4, -2.7, -2.7,
     -3.0, -3.0, -2.7, -1.2, -2.1, -3.0, -0.0, -3.0, -2.7, -2.7, -3.0,
     -3.0, -2.5, -3.0, -1.8, -1.5, -2.7, -2.4, -1.8, -3.0, -2.7, 2.1,
     -3.0, -2.2, -2.2, 0.6, -0.9, 6.0, -3.0, -2.1, -3.0, -2.1, -2.5,
     -3.0, -1.5, -2.5, 3.0, -2.1, -3.0, -3.0, -1.5, -2.1, -2.7, -2.5,
     1.5, -2.1, -0.0, -0.0, -3.0, -0.6, 1.5, -2.7, -2.4, -2.1, -3.0,
     -2.7, -3.0, 9.0, -3.0, -1.7, -3.0, -0.0, -3.0, -2.2, -0.0, -0.6,
     -2.7, -2.7, -3.0, -3.0, -1.7, -2.1, -2.0, -3.0, -2.1, -3.0, -1.1,
     -3.0, -3.0, -2.4, -1.5, -3.0, -2.2, -3.0, -1.5, -2.7, -3.0, -3.0,
     -3.0, -2.2, -3.0, -2.1, -2.1, -2.4, -3.0, -3.0, -0.0, -3.0, -3.0,
     -2.1, -2.0, 1.5, -3.0, -3.0, -2.1, -2.9, -2.4, -3.0, -3.0, -1.5,
     -2.2, -0.9, -1.8, -2.1, -1.8, -1.5, -3.0, -1.5, -3.0, -1.5, -3.0,
     -2.4, 1.5, -2.7, -3.0, -1.8, -1.8, -1.5, -2.1, -2.7, -2.7, -2.7,
     -3.0, -1.5, -2.7, -3.0, -2.7, -3.0, -1.5, -1.5, -3.0, 0.6, -0.6,
     -3.0, -2.1, -2.4, -2.1, -3.0, -2.2, -3.0, -1.8, -1.2, -3.0, -0.8,
     -2.4, -2.5, -3.0, -1.5, -1.2, -0.0, -2.7, -2.4, -3.0, -3.0, -2.1,
     -2.1, -2.1, -3.0, -2.7, -2.4, -2.1, -1.5, -2.1, -0.6, -3.0, -3.0,
     -3.0, -3.0]
b = [-1.2, 6.0, -3.0, 1.5, 12.0, 1.5, 3.0, -0.0, -1.5, -0.0, 6.0,
     -0.0, 1.8, -1.5, -3.0, -3.0, 1.5, -0.0, 1.5, -3.0, -3.0, -0.0,
     -1.2, -3.0, 22.5, -0.0, -3.0, -3.0, -2.1, 3.0, 2.4, 1.5, -2.1,
     4.5, -3.0, -3.0, 12.0, 6.0, -3.0, 3.0, -3.0, 12.0, -2.8, 0.6,
     4.5, 3.0, -0.0, -3.0, -1.0, -3.0, -0.0, 3.0, 2.1, -0.6, -3.0,
     -3.0, 1.5, 1.5, -0.6, -0.0, 1.5, 6.0, -2.2, -2.1, -0.0, -0.8,
     6.0, -3.0, -3.0, 9.0, -3.0, -0.9, 15.0, 1.5, -2.9, 19.5, 4.5,
     -3.0, 1.5, -1.8, 1.5, 0.9, -3.0, -3.0, 9.0, -0.3, 9.0, -2.1,
     -2.2, -3.0, 12.0, -2.1, -1.2, -3.0, 4.5, -1.2, -0.0, 12.0, -0.9,
     -3.0, -0.6, -3.0, -1.5, -0.0, 27.0, 4.5, -2.4, -2.7, -0.0, 3.0,
     1.5, 3.0, -3.0, 16.5, -2.1, -0.6, -3.0, -1.5, -1.2, -3.0, 4.4,
     -2.5, -2.1, 3.0, -1.2, -3.0, 12.0, -1.8, -0.9, -3.0, -3.0, -0.0,
     -3.0, -1.2, -3.0, -3.0, -0.0, 1.5, -3.0, 7.5, -3.0, -2.1, 3.0,
     1.5, 3.0, -3.0, 1.5, -3.0, 19.5, -3.0, -2.2, 27.0, 3.0, -1.5,
     -3.0, -3.0, 4.5, -1.2, 12.0, 3.0, 3.0, -2.1, 6.0, -2.2, -3.0,
     -2.4, 6.0, -3.0, -1.9, -0.6, -0.0, 3.6, 15.0, -3.0, -3.0, 7.5,
     -0.0, 4.5, -2.4, -3.0, -3.0, -3.0, -2.0, -3.0, -3.0, -3.0, -2.6,
     3.0, -0.0, -3.0, 4.5, -1.2, -3.0, -3.0, 3.0, -3.0, 1.6, 1.5,
     -3.0, -3.0, -3.0, -3.0, 3.0, -3.0, -1.5, -3.0, -1.5, 12.0, -1.5,
     3.0, 9.6, -0.0, -1.8, -3.0, -2.1, -0.6, 9.0, -3.0, 19.5, -2.4,
     -1.8, 15.0, -3.0, -3.0, 27.0, 7.5, 12.0, -3.0, -2.0, -3.0, 12.0,
     -3.0, -3.0, 0.9, -3.0, -2.5, 12.6, -1.5, -0.6, -3.0, -3.0, -0.0,
     -0.8, -3.0, 3.0, -0.0, 4.5, -3.0, -0.0, 0.6, 0.3, -2.7, -0.0,
     4.5, -3.0, 3.0, -1.5, 9.0, -3.0, 1.5, -3.0, 6.0, -2.2, -0.0,
     -3.0, -3.0, 3.0, -3.0, -2.7, -0.0, -3.0, 0.6, 4.5, 1.5, 1.5, 1.5,
     -2.1, 7.5, -3.0, -3.0, -3.0, -2.1, -3.0, 6.0, -1.5, -2.0, -3.0,
     -0.6, 6.0, -1.5, -3.0, 6.0, -3.0, 3.0, 3.0, -3.0, 7.5, -3.0,
     -0.0, -3.0, -0.0, -3.0, -3.0, 0.6, -3.0, -3.0, -3.0, -3.0, -2.4,
     -0.0, 1.5, 3.0, 2.1, -3.0, 3.0, 4.5, -3.0, -2.4, -3.0, -2.2, 7.5,
     2.1, -3.0, -0.0, -2.0, -3.0, -3.0, -1.8, -3.0, 4.5, -1.5, -3.0,
     4.5, 3.0, 15.0, 4.5, -2.1, 12.0, 6.0, 4.5, 27.0, -3.0, 3.0, 0.6,
     -3.0, 0.6, -2.4, 4.5, -1.5, -2.2, 12.0, -2.0, 1.5, 9.0, -1.5,
     -1.2, -2.0, 1.5, -1.2, -1.8, -3.0, -0.0, -0.0, -3.0, -0.9]

print(anderson_ksamp([a, b]))

Outputs:

Anderson_ksampResult(
    statistic=53.56069633812226,
    critical_values=array([0.325, 1.226, 1.961, 2.718, 3.752, 4.592, 6.546]),
    significance_level=0.001
)

And generates a warning:

UserWarning: p-value floored: true value smaller than 0.001

The second code chunk:

 from scipy.stats import ks_2samp
 from statsmodels.distributions.empirical_distribution import ECDF
 import matplotlib.pyplot as plt
 import numpy as np
 
 a = [-1.8, -2.4, -2.4, -0.0, -1.5, -2.7, -1.8, -3.0, -1.8, -1.2, -3.0,
      -3.0, -2.8, -3.0, -2.1, -0.0, 0.6, -2.5, -2.4, -0.0, -2.7, -0.0,
      -2.5, -2.1, -0.9, -3.0, -0.6, -0.6, -1.5, -2.2, -1.2, -2.4, -2.4,
      -3.0, 1.5, -1.8, 1.5, -2.7, -3.0, -2.5, -2.5, -1.5, -1.5, -2.1,
      -2.1, -3.0, -0.6, -2.7, -3.0, -1.5, -0.6, -0.0, -2.1, 0.6, -2.0,
      -3.0, -3.0, -2.4, -3.0, -1.8, -0.0, -0.0, -3.0, -1.5, -3.0, -3.0,
      -1.5, -2.5, -3.0, -2.8, -3.0, -2.2, -0.6, -0.0, -1.5, -2.7, -2.1,
      -2.1, -2.2, -2.1, -0.6, -0.0, -2.5, -2.1, -1.5, -3.0, -2.2, -1.8,
      -2.7, -2.4, -1.5, -2.1, -2.9, -2.4, -0.9, -0.0, -0.0, -2.4, -2.7,
      -0.6, -2.2, -3.0, -1.5, -0.9, -3.0, -3.0, -0.0, -2.7, -2.7, -1.5,
      -2.2, -3.0, -0.0, 1.5, -3.0, -2.7, -2.2, -2.9, -2.2, -3.0, -1.8,
      -0.0, -3.0, -1.5, -2.7, -3.0, -3.0, -2.9, -3.0, -3.0, -3.0, -3.0,
      -2.5, -0.0, 0.9, -3.0, -0.0, -3.0, 3.0, -3.0, -3.0, -1.2, -2.1,
      1.5, -0.0, -0.9, -3.0, -2.7, -1.5, -2.4, -2.1, -3.0, -0.9, -3.0,
      -0.8, -1.5, -2.1, -2.7, -0.0, -0.0, -2.2, -1.8, -2.1, -2.2, -3.0,
      0.6, -2.4, -2.2, -2.4, -2.5, -1.5, -0.0, -2.7, -3.0, -3.0, -2.1,
      -0.0, -2.4, -2.4, -0.0, -2.0, -0.9, -2.4, -3.0, -1.4, -2.7, -2.7,
      -3.0, -3.0, -2.7, -1.2, -2.1, -3.0, -0.0, -3.0, -2.7, -2.7, -3.0,
      -3.0, -2.5, -3.0, -1.8, -1.5, -2.7, -2.4, -1.8, -3.0, -2.7, 2.1,
      -3.0, -2.2, -2.2, 0.6, -0.9, 6.0, -3.0, -2.1, -3.0, -2.1, -2.5,
      -3.0, -1.5, -2.5, 3.0, -2.1, -3.0, -3.0, -1.5, -2.1, -2.7, -2.5,
      1.5, -2.1, -0.0, -0.0, -3.0, -0.6, 1.5, -2.7, -2.4, -2.1, -3.0,
      -2.7, -3.0, 9.0, -3.0, -1.7, -3.0, -0.0, -3.0, -2.2, -0.0, -0.6,
      -2.7, -2.7, -3.0, -3.0, -1.7, -2.1, -2.0, -3.0, -2.1, -3.0, -1.1,
      -3.0, -3.0, -2.4, -1.5, -3.0, -2.2, -3.0, -1.5, -2.7, -3.0, -3.0,
      -3.0, -2.2, -3.0, -2.1, -2.1, -2.4, -3.0, -3.0, -0.0, -3.0, -3.0,
      -2.1, -2.0, 1.5, -3.0, -3.0, -2.1, -2.9, -2.4, -3.0, -3.0, -1.5,
      -2.2, -0.9, -1.8, -2.1, -1.8, -1.5, -3.0, -1.5, -3.0, -1.5, -3.0,
      -2.4, 1.5, -2.7, -3.0, -1.8, -1.8, -1.5, -2.1, -2.7, -2.7, -2.7,
      -3.0, -1.5, -2.7, -3.0, -2.7, -3.0, -1.5, -1.5, -3.0, 0.6, -0.6,
      -3.0, -2.1, -2.4, -2.1, -3.0, -2.2, -3.0, -1.8, -1.2, -3.0, -0.8,
      -2.4, -2.5, -3.0, -1.5, -1.2, -0.0, -2.7, -2.4, -3.0, -3.0, -2.1,
      -2.1, -2.1, -3.0, -2.7, -2.4, -2.1, -1.5, -2.1, -0.6, -3.0, -3.0,
      -3.0, -3.0]
 b = [-1.2, 6.0, -3.0, 1.5, 12.0, 1.5, 3.0, -0.0, -1.5, -0.0, 6.0,
      -0.0, 1.8, -1.5, -3.0, -3.0, 1.5, -0.0, 1.5, -3.0, -3.0, -0.0,
      -1.2, -3.0, 22.5, -0.0, -3.0, -3.0, -2.1, 3.0, 2.4, 1.5, -2.1,
      4.5, -3.0, -3.0, 12.0, 6.0, -3.0, 3.0, -3.0, 12.0, -2.8, 0.6,
      4.5, 3.0, -0.0, -3.0, -1.0, -3.0, -0.0, 3.0, 2.1, -0.6, -3.0,
      -3.0, 1.5, 1.5, -0.6, -0.0, 1.5, 6.0, -2.2, -2.1, -0.0, -0.8,
      6.0, -3.0, -3.0, 9.0, -3.0, -0.9, 15.0, 1.5, -2.9, 19.5, 4.5,
      -3.0, 1.5, -1.8, 1.5, 0.9, -3.0, -3.0, 9.0, -0.3, 9.0, -2.1,
      -2.2, -3.0, 12.0, -2.1, -1.2, -3.0, 4.5, -1.2, -0.0, 12.0, -0.9,
      -3.0, -0.6, -3.0, -1.5, -0.0, 27.0, 4.5, -2.4, -2.7, -0.0, 3.0,
      1.5, 3.0, -3.0, 16.5, -2.1, -0.6, -3.0, -1.5, -1.2, -3.0, 4.4,
      -2.5, -2.1, 3.0, -1.2, -3.0, 12.0, -1.8, -0.9, -3.0, -3.0, -0.0,
      -3.0, -1.2, -3.0, -3.0, -0.0, 1.5, -3.0, 7.5, -3.0, -2.1, 3.0,
      1.5, 3.0, -3.0, 1.5, -3.0, 19.5, -3.0, -2.2, 27.0, 3.0, -1.5,
      -3.0, -3.0, 4.5, -1.2, 12.0, 3.0, 3.0, -2.1, 6.0, -2.2, -3.0,
      -2.4, 6.0, -3.0, -1.9, -0.6, -0.0, 3.6, 15.0, -3.0, -3.0, 7.5,
      -0.0, 4.5, -2.4, -3.0, -3.0, -3.0, -2.0, -3.0, -3.0, -3.0, -2.6,
      3.0, -0.0, -3.0, 4.5, -1.2, -3.0, -3.0, 3.0, -3.0, 1.6, 1.5,
      -3.0, -3.0, -3.0, -3.0, 3.0, -3.0, -1.5, -3.0, -1.5, 12.0, -1.5,
      3.0, 9.6, -0.0, -1.8, -3.0, -2.1, -0.6, 9.0, -3.0, 19.5, -2.4,
      -1.8, 15.0, -3.0, -3.0, 27.0, 7.5, 12.0, -3.0, -2.0, -3.0, 12.0,
      -3.0, -3.0, 0.9, -3.0, -2.5, 12.6, -1.5, -0.6, -3.0, -3.0, -0.0,
      -0.8, -3.0, 3.0, -0.0, 4.5, -3.0, -0.0, 0.6, 0.3, -2.7, -0.0,
      4.5, -3.0, 3.0, -1.5, 9.0, -3.0, 1.5, -3.0, 6.0, -2.2, -0.0,
      -3.0, -3.0, 3.0, -3.0, -2.7, -0.0, -3.0, 0.6, 4.5, 1.5, 1.5, 1.5,
      -2.1, 7.5, -3.0, -3.0, -3.0, -2.1, -3.0, 6.0, -1.5, -2.0, -3.0,
      -0.6, 6.0, -1.5, -3.0, 6.0, -3.0, 3.0, 3.0, -3.0, 7.5, -3.0,
      -0.0, -3.0, -0.0, -3.0, -3.0, 0.6, -3.0, -3.0, -3.0, -3.0, -2.4,
      -0.0, 1.5, 3.0, 2.1, -3.0, 3.0, 4.5, -3.0, -2.4, -3.0, -2.2, 7.5,
      2.1, -3.0, -0.0, -2.0, -3.0, -3.0, -1.8, -3.0, 4.5, -1.5, -3.0,
      4.5, 3.0, 15.0, 4.5, -2.1, 12.0, 6.0, 4.5, 27.0, -3.0, 3.0, 0.6,
      -3.0, 0.6, -2.4, 4.5, -1.5, -2.2, 12.0, -2.0, 1.5, 9.0, -1.5,
      -1.2, -2.0, 1.5, -1.2, -1.8, -3.0, -0.0, -0.0, -3.0, -0.9]
 
 ecdf1, ecdf2 = ECDF(a), ECDF(b)
 
 xs = np.linspace(min(a + b), max(a + b), num=10000)
 plt.figure(figsize=(12, 8))
 plt.plot(xs, ecdf1(xs), xs, ecdf2(xs))
 plt.show()
 
 print(ks_2samp(a, b))

Outputs:

KstestResult(statistic=0.3232876712328767, pvalue=2.8586245432606456e-17)

Why does scipy.stats.anderson_ksamp give a p-value of over a million for these data?

1 Answers1