scipy.stats.permutation_test#

scipy.stats.permutation_test(data, statistic, *, permutation_type='independent', vectorized=None, n_resamples=9999, batch=None, alternative='two-sided', axis=0, random_state=None)[source]#

Performs a permutation test of a given statistic on provided data.

For independent sample statistics, the null hypothesis is that the data are randomly sampled from the same distribution. For paired sample statistics, two null hypothesis can be tested: that the data are paired at random or that the data are assigned to samples at random.

Parameters:
dataiterable of array-like

Contains the samples, each of which is an array of observations. Dimensions of sample arrays must be compatible for broadcasting except along axis.

statisticcallable

Statistic for which the p-value of the hypothesis test is to be calculated. statistic must be a callable that accepts samples as separate arguments (e.g. statistic(*data)) and returns the resulting statistic. If vectorized is set True, statistic must also accept a keyword argument axis and be vectorized to compute the statistic along the provided axis of the sample arrays.

permutation_type{‘independent’, ‘samples’, ‘pairings’}, optional

The type of permutations to be performed, in accordance with the null hypothesis. The first two permutation types are for paired sample statistics, in which all samples contain the same number of observations and observations with corresponding indices along axis are considered to be paired; the third is for independent sample statistics.

  • 'samples' : observations are assigned to different samples but remain paired with the same observations from other samples. This permutation type is appropriate for paired sample hypothesis tests such as the Wilcoxon signed-rank test and the paired t-test.

  • 'pairings' : observations are paired with different observations, but they remain within the same sample. This permutation type is appropriate for association/correlation tests with statistics such as Spearman’s \(\rho\), Kendall’s \(\tau\), and Pearson’s \(r\).

  • 'independent' (default) : observations are assigned to different samples. Samples may contain different numbers of observations. This permutation type is appropriate for independent sample hypothesis tests such as the Mann-Whitney \(U\) test and the independent sample t-test.

    Please see the Notes section below for more detailed descriptions of the permutation types.

vectorizedbool, optional

If vectorized is set False, statistic will not be passed keyword argument axis and is expected to calculate the statistic only for 1D samples. If True, statistic will be passed keyword argument axis and is expected to calculate the statistic along axis when passed an ND sample array. If None (default), vectorized will be set True if axis is a parameter of statistic. Use of a vectorized statistic typically reduces computation time.

n_resamplesint or np.inf, default: 9999

Number of random permutations (resamples) used to approximate the null distribution. If greater than or equal to the number of distinct permutations, the exact null distribution will be computed. Note that the number of distinct permutations grows very rapidly with the sizes of samples, so exact tests are feasible only for very small data sets.

batchint, optional

The number of permutations to process in each call to statistic. Memory usage is O(batch`*``n`), where n is the total size of all samples, regardless of the value of vectorized. Default is None, in which case batch is the number of permutations.

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

The alternative hypothesis for which the p-value is calculated. For each alternative, the p-value is defined for exact tests as follows.

  • 'greater' : the percentage of the null distribution that is greater than or equal to the observed value of the test statistic.

  • 'less' : the percentage of the null distribution that is less than or equal to the observed value of the test statistic.

  • 'two-sided' (default) : twice the smaller of the p-values above.

Note that p-values for randomized tests are calculated according to the conservative (over-estimated) approximation suggested in [2] and [3] rather than the unbiased estimator suggested in [4]. That is, when calculating the proportion of the randomized null distribution that is as extreme as the observed value of the test statistic, the values in the numerator and denominator are both increased by one. An interpretation of this adjustment is that the observed value of the test statistic is always included as an element of the randomized null distribution. The convention used for two-sided p-values is not universal; the observed test statistic and null distribution are returned in case a different definition is preferred.

axisint, default: 0

The axis of the (broadcasted) samples over which to calculate the statistic. If samples have a different number of dimensions, singleton dimensions are prepended to samples with fewer dimensions before axis is considered.

random_state{None, int, numpy.random.Generator,

Pseudorandom number generator state used to generate permutations.

If random_state is None (default), the numpy.random.RandomState singleton is used. If random_state is an int, a new RandomState instance is used, seeded with random_state. If random_state is already a Generator or RandomState instance then that instance is used.

Returns:
statisticfloat or ndarray

The observed test statistic of the data.

pvaluefloat or ndarray

The p-value for the given alternative.

null_distributionndarray

The values of the test statistic generated under the null hypothesis.

Notes

The three types of permutation tests supported by this function are described below.

Unpaired statistics (permutation_type='independent'):

The null hypothesis associated with this permutation type is that all observations are sampled from the same underlying distribution and that they have been assigned to one of the samples at random.

Suppose data contains two samples; e.g. a, b = data. When 1 < n_resamples < binom(n, k), where

  • k is the number of observations in a,

  • n is the total number of observations in a and b, and

  • binom(n, k) is the binomial coefficient (n choose k),

the data are pooled (concatenated), randomly assigned to either the first or second sample, and the statistic is calculated. This process is performed repeatedly, permutation times, generating a distribution of the statistic under the null hypothesis. The statistic of the original data is compared to this distribution to determine the p-value.

When n_resamples >= binom(n, k), an exact test is performed: the data are partitioned between the samples in each distinct way exactly once, and the exact null distribution is formed. Note that for a given partitioning of the data between the samples, only one ordering/permutation of the data within each sample is considered. For statistics that do not depend on the order of the data within samples, this dramatically reduces computational cost without affecting the shape of the null distribution (because the frequency/count of each value is affected by the same factor).

For a = [a1, a2, a3, a4] and b = [b1, b2, b3], an example of this permutation type is x = [b3, a1, a2, b2] and y = [a4, b1, a3]. Because only one ordering/permutation of the data within each sample is considered in an exact test, a resampling like x = [b3, a1, b2, a2] and y = [a4, a3, b1] would not be considered distinct from the example above.

permutation_type='independent' does not support one-sample statistics, but it can be applied to statistics with more than two samples. In this case, if n is an array of the number of observations within each sample, the number of distinct partitions is:

np.product([binom(sum(n[i:]), sum(n[i+1:])) for i in range(len(n)-1)])

Paired statistics, permute pairings (permutation_type='pairings'):

The null hypothesis associated with this permutation type is that observations within each sample are drawn from the same underlying distribution and that pairings with elements of other samples are assigned at random.

Suppose data contains only one sample; e.g. a, = data, and we wish to consider all possible pairings of elements of a with elements of a second sample, b. Let n be the number of observations in a, which must also equal the number of observations in b.

When 1 < n_resamples < factorial(n), the elements of a are randomly permuted. The user-supplied statistic accepts one data argument, say a_perm, and calculates the statistic considering a_perm and b. This process is performed repeatedly, permutation times, generating a distribution of the statistic under the null hypothesis. The statistic of the original data is compared to this distribution to determine the p-value.

When n_resamples >= factorial(n), an exact test is performed: a is permuted in each distinct way exactly once. Therefore, the statistic is computed for each unique pairing of samples between a and b exactly once.

For a = [a1, a2, a3] and b = [b1, b2, b3], an example of this permutation type is a_perm = [a3, a1, a2] while b is left in its original order.

permutation_type='pairings' supports data containing any number of samples, each of which must contain the same number of observations. All samples provided in data are permuted independently. Therefore, if m is the number of samples and n is the number of observations within each sample, then the number of permutations in an exact test is:

factorial(n)**m

Note that if a two-sample statistic, for example, does not inherently depend on the order in which observations are provided - only on the pairings of observations - then only one of the two samples should be provided in data. This dramatically reduces computational cost without affecting the shape of the null distribution (because the frequency/count of each value is affected by the same factor).

Paired statistics, permute samples (permutation_type='samples'):

The null hypothesis associated with this permutation type is that observations within each pair are drawn from the same underlying distribution and that the sample to which they are assigned is random.

Suppose data contains two samples; e.g. a, b = data. Let n be the number of observations in a, which must also equal the number of observations in b.

When 1 < n_resamples < 2**n, the elements of a are b are randomly swapped between samples (maintaining their pairings) and the statistic is calculated. This process is performed repeatedly, permutation times, generating a distribution of the statistic under the null hypothesis. The statistic of the original data is compared to this distribution to determine the p-value.

When n_resamples >= 2**n, an exact test is performed: the observations are assigned to the two samples in each distinct way (while maintaining pairings) exactly once.

For a = [a1, a2, a3] and b = [b1, b2, b3], an example of this permutation type is x = [b1, a2, b3] and y = [a1, b2, a3].

permutation_type='samples' supports data containing any number of samples, each of which must contain the same number of observations. If data contains more than one sample, paired observations within data are exchanged between samples independently. Therefore, if m is the number of samples and n is the number of observations within each sample, then the number of permutations in an exact test is:

factorial(m)**n

Several paired-sample statistical tests, such as the Wilcoxon signed rank test and paired-sample t-test, can be performed considering only the difference between two paired elements. Accordingly, if data contains only one sample, then the null distribution is formed by independently changing the sign of each observation.

Warning

The p-value is calculated by counting the elements of the null distribution that are as extreme or more extreme than the observed value of the statistic. Due to the use of finite precision arithmetic, some statistic functions return numerically distinct values when the theoretical values would be exactly equal. In some cases, this could lead to a large error in the calculated p-value. permutation_test guards against this by considering elements in the null distribution that are “close” (within a factor of 1+1e-14) to the observed value of the test statistic as equal to the observed value of the test statistic. However, the user is advised to inspect the null distribution to assess whether this method of comparison is appropriate, and if not, calculate the p-value manually. See example below.

References

[1]

    1. Fisher. The Design of Experiments, 6th Ed (1951).

[2]

B. Phipson and G. K. Smyth. “Permutation P-values Should Never Be Zero: Calculating Exact P-values When Permutations Are Randomly Drawn.” Statistical Applications in Genetics and Molecular Biology 9.1 (2010).

[3]

M. D. Ernst. “Permutation Methods: A Basis for Exact Inference”. Statistical Science (2004).

[4]

B. Efron and R. J. Tibshirani. An Introduction to the Bootstrap (1993).

Examples

Suppose we wish to test whether two samples are drawn from the same distribution. Assume that the underlying distributions are unknown to us, and that before observing the data, we hypothesized that the mean of the first sample would be less than that of the second sample. We decide that we will use the difference between the sample means as a test statistic, and we will consider a p-value of 0.05 to be statistically significant.

For efficiency, we write the function defining the test statistic in a vectorized fashion: the samples x and y can be ND arrays, and the statistic will be calculated for each axis-slice along axis.

>>> import numpy as np
>>> def statistic(x, y, axis):
...     return np.mean(x, axis=axis) - np.mean(y, axis=axis)

After collecting our data, we calculate the observed value of the test statistic.

>>> from scipy.stats import norm
>>> rng = np.random.default_rng()
>>> x = norm.rvs(size=5, random_state=rng)
>>> y = norm.rvs(size=6, loc = 3, random_state=rng)
>>> statistic(x, y, 0)
-3.5411688580987266

Indeed, the test statistic is negative, suggesting that the true mean of the distribution underlying x is less than that of the distribution underlying y. To determine the probability of this occuring by chance if the two samples were drawn from the same distribution, we perform a permutation test.

>>> from scipy.stats import permutation_test
>>> # because our statistic is vectorized, we pass `vectorized=True`
>>> # `n_resamples=np.inf` indicates that an exact test is to be performed
>>> res = permutation_test((x, y), statistic, vectorized=True,
...                        n_resamples=np.inf, alternative='less')
>>> print(res.statistic)
-3.5411688580987266
>>> print(res.pvalue)
0.004329004329004329

The probability of obtaining a test statistic less than or equal to the observed value under the null hypothesis is 0.4329%. This is less than our chosen threshold of 5%, so we consider this to be significant evidence against the null hypothesis in favor of the alternative.

Because the size of the samples above was small, permutation_test could perform an exact test. For larger samples, we resort to a randomized permutation test.

>>> x = norm.rvs(size=100, random_state=rng)
>>> y = norm.rvs(size=120, loc=0.3, random_state=rng)
>>> res = permutation_test((x, y), statistic, n_resamples=100000,
...                        vectorized=True, alternative='less',
...                        random_state=rng)
>>> print(res.statistic)
-0.5230459671240913
>>> print(res.pvalue)
0.00016999830001699983

The approximate probability of obtaining a test statistic less than or equal to the observed value under the null hypothesis is 0.0225%. This is again less than our chosen threshold of 5%, so again we have significant evidence to reject the null hypothesis in favor of the alternative.

For large samples and number of permutations, the result is comparable to that of the corresponding asymptotic test, the independent sample t-test.

>>> from scipy.stats import ttest_ind
>>> res_asymptotic = ttest_ind(x, y, alternative='less')
>>> print(res_asymptotic.pvalue)
0.00012688101537979522

The permutation distribution of the test statistic is provided for further investigation.

>>> import matplotlib.pyplot as plt
>>> plt.hist(res.null_distribution, bins=50)
>>> plt.title("Permutation distribution of test statistic")
>>> plt.xlabel("Value of Statistic")
>>> plt.ylabel("Frequency")
>>> plt.show()
../../_images/scipy-stats-permutation_test-1_00_00.png

Inspection of the null distribution is essential if the statistic suffers from inaccuracy due to limited machine precision. Consider the following case:

>>> from scipy.stats import pearsonr
>>> x = [1, 2, 4, 3]
>>> y = [2, 4, 6, 8]
>>> def statistic(x, y):
...     return pearsonr(x, y).statistic
>>> res = permutation_test((x, y), statistic, vectorized=False,
...                        permutation_type='pairings',
...                        alternative='greater')
>>> r, pvalue, null = res.statistic, res.pvalue, res.null_distribution

In this case, some elements of the null distribution differ from the observed value of the correlation coefficient r due to numerical noise. We manually inspect the elements of the null distribution that are nearly the same as the observed value of the test statistic.

>>> r
0.8
>>> unique = np.unique(null)
>>> unique
array([-1. , -0.8, -0.8, -0.6, -0.4, -0.2, -0.2,  0. ,  0.2,  0.2,  0.4,
        0.6,  0.8,  0.8,  1. ]) # may vary
>>> unique[np.isclose(r, unique)].tolist()
[0.7999999999999999, 0.8]

If permutation_test were to perform the comparison naively, the elements of the null distribution with value 0.7999999999999999 would not be considered as extreme or more extreme as the observed value of the statistic, so the calculated p-value would be too small.

>>> incorrect_pvalue = np.count_nonzero(null >= r) / len(null)
>>> incorrect_pvalue
0.1111111111111111  # may vary

Instead, permutation_test treats elements of the null distribution that are within max(1e-14, abs(r)*1e-14) of the observed value of the statistic r to be equal to r.

>>> correct_pvalue = np.count_nonzero(null >= r - 1e-14) / len(null)
>>> correct_pvalue
0.16666666666666666
>>> res.pvalue == correct_pvalue
True

This method of comparison is expected to be accurate in most practical situations, but the user is advised to assess this by inspecting the elements of the null distribution that are close to the observed value of the statistic. Also, consider the use of statistics that can be calculated using exact arithmetic (e.g. integer statistics).