scipy.stats.ansari#

scipy.stats.ansari(x, y, alternative='two-sided')[source]#

Perform the Ansari-Bradley test for equal scale parameters.

The Ansari-Bradley test ([1], [2]) is a non-parametric test for the equality of the scale parameter of the distributions from which two samples were drawn. The null hypothesis states that the ratio of the scale of the distribution underlying x to the scale of the distribution underlying y is 1.

Parameters

x, yarray_like

Arrays of sample data.

alternative{‘two-sided’, ‘less’, ‘greater’}, optional

Defines the alternative hypothesis. Default is ‘two-sided’. The following options are available:

‘two-sided’: the ratio of scales is not equal to 1.
‘less’: the ratio of scales is less than 1.
‘greater’: the ratio of scales is greater than 1.

New in version 1.7.0.

Returns

statisticfloat: The Ansari-Bradley test statistic.
pvaluefloat: The p-value of the hypothesis test.

See also

fligner: A non-parametric test for the equality of k variances
mood: A non-parametric test for the equality of two scale parameters

Notes

The p-value given is exact when the sample sizes are both less than 55 and there are no ties, otherwise a normal approximation for the p-value is used.

References

1: Ansari, A. R. and Bradley, R. A. (1960) Rank-sum tests for dispersions, Annals of Mathematical Statistics, 31, 1174-1189.
2: Sprent, Peter and N.C. Smeeton. Applied nonparametric statistical methods. 3rd ed. Chapman and Hall/CRC. 2001. Section 5.8.2.
3: Nathaniel E. Helwig “Nonparametric Dispersion and Equality Tests” at http://users.stat.umn.edu/~helwig/notes/npde-Notes.pdf

Examples

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

For these examples, we’ll create three random data sets. The first two, with sizes 35 and 25, are drawn from a normal distribution with mean 0 and standard deviation 2. The third data set has size 25 and is drawn from a normal distribution with standard deviation 1.25.

>>> x1 = rng.normal(loc=0, scale=2, size=35)
>>> x2 = rng.normal(loc=0, scale=2, size=25)
>>> x3 = rng.normal(loc=0, scale=1.25, size=25)

First we apply ansari to x1 and x2. These samples are drawn from the same distribution, so we expect the Ansari-Bradley test should not lead us to conclude that the scales of the distributions are different.

>>> ansari(x1, x2)
AnsariResult(statistic=541.0, pvalue=0.9762532927399098)

With a p-value close to 1, we cannot conclude that there is a significant difference in the scales (as expected).

Now apply the test to x1 and x3:

>>> ansari(x1, x3)
AnsariResult(statistic=425.0, pvalue=0.0003087020407974518)

The probability of observing such an extreme value of the statistic under the null hypothesis of equal scales is only 0.03087%. We take this as evidence against the null hypothesis in favor of the alternative: the scales of the distributions from which the samples were drawn are not equal.

We can use the alternative parameter to perform a one-tailed test. In the above example, the scale of x1 is greater than x3 and so the ratio of scales of x1 and x3 is greater than 1. This means that the p-value when alternative='greater' should be near 0 and hence we should be able to reject the null hypothesis:

>>> ansari(x1, x3, alternative='greater')
AnsariResult(statistic=425.0, pvalue=0.0001543510203987259)

As we can see, the p-value is indeed quite low. Use of alternative='less' should thus yield a large p-value:

>>> ansari(x1, x3, alternative='less')
AnsariResult(statistic=425.0, pvalue=0.9998643258449039)

scipy.stats.tukey_hsd

scipy.stats.bartlett