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Editing: _resampling.cpython-311.pyc
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When `method` is ``'percentile'``, a bootstrap confidence interval is computed according to the following procedure. 1. Resample the data: for each sample in `data` and for each of `n_resamples`, take a random sample of the original sample (with replacement) of the same size as the original sample. 2. Compute the bootstrap distribution of the statistic: for each set of resamples, compute the test statistic. 3. Determine the confidence interval: find the interval of the bootstrap distribution that is - symmetric about the median and - contains `confidence_level` of the resampled statistic values. While the ``'percentile'`` method is the most intuitive, it is rarely used in practice. Two more common methods are available, ``'basic'`` ('reverse percentile') and ``'BCa'`` ('bias-corrected and accelerated'); they differ in how step 3 is performed. If the samples in `data` are taken at random from their respective distributions :math:`n` times, the confidence interval returned by `bootstrap` will contain the true value of the statistic for those distributions approximately `confidence_level`:math:`\, \times \, n` times. Parameters ---------- data : sequence of array-like Each element of data is a sample from an underlying distribution. statistic : callable Statistic for which the confidence interval is to be calculated. `statistic` must be a callable that accepts ``len(data)`` samples as separate arguments 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`. n_resamples : int, default: ``9999`` The number of resamples performed to form the bootstrap distribution of the statistic. batch : int, optional The number of resamples to process in each vectorized call to `statistic`. Memory usage is O(`batch`*``n``), where ``n`` is the sample size. Default is ``None``, in which case ``batch = n_resamples`` (or ``batch = max(n_resamples, n)`` for ``method='BCa'``). vectorized : bool, 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. paired : bool, default: ``False`` Whether the statistic treats corresponding elements of the samples in `data` as paired. axis : int, default: ``0`` The axis of the samples in `data` along which the `statistic` is calculated. confidence_level : float, default: ``0.95`` The confidence level of the confidence interval. method : {'percentile', 'basic', 'bca'}, default: ``'BCa'`` Whether to return the 'percentile' bootstrap confidence interval (``'percentile'``), the 'basic' (AKA 'reverse') bootstrap confidence interval (``'basic'``), or the bias-corrected and accelerated bootstrap confidence interval (``'BCa'``). bootstrap_result : BootstrapResult, optional Provide the result object returned by a previous call to `bootstrap` to include the previous bootstrap distribution in the new bootstrap distribution. This can be used, for example, to change `confidence_level`, change `method`, or see the effect of performing additional resampling without repeating computations. random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional Pseudorandom number generator state used to generate resamples. If `random_state` is ``None`` (or `np.random`), 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 ------- res : BootstrapResult An object with attributes: confidence_interval : ConfidenceInterval The bootstrap confidence interval as an instance of `collections.namedtuple` with attributes `low` and `high`. bootstrap_distribution : ndarray The bootstrap distribution, that is, the value of `statistic` for each resample. The last dimension corresponds with the resamples (e.g. ``res.bootstrap_distribution.shape[-1] == n_resamples``). standard_error : float or ndarray The bootstrap standard error, that is, the sample standard deviation of the bootstrap distribution. Warns ----- `~scipy.stats.DegenerateDataWarning` Generated when ``method='BCa'`` and the bootstrap distribution is degenerate (e.g. all elements are identical). Notes ----- Elements of the confidence interval may be NaN for ``method='BCa'`` if the bootstrap distribution is degenerate (e.g. all elements are identical). In this case, consider using another `method` or inspecting `data` for indications that other analysis may be more appropriate (e.g. all observations are identical). References ---------- .. [1] B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap, Chapman & Hall/CRC, Boca Raton, FL, USA (1993) .. [2] Nathaniel E. Helwig, "Bootstrap Confidence Intervals", http://users.stat.umn.edu/~helwig/notes/bootci-Notes.pdf .. [3] Bootstrapping (statistics), Wikipedia, https://en.wikipedia.org/wiki/Bootstrapping_%28statistics%29 Examples -------- Suppose we have sampled data from an unknown distribution. >>> import numpy as np >>> rng = np.random.default_rng() >>> from scipy.stats import norm >>> dist = norm(loc=2, scale=4) # our "unknown" distribution >>> data = dist.rvs(size=100, random_state=rng) We are interested in the standard deviation of the distribution. >>> std_true = dist.std() # the true value of the statistic >>> print(std_true) 4.0 >>> std_sample = np.std(data) # the sample statistic >>> print(std_sample) 3.9460644295563863 The bootstrap is used to approximate the variability we would expect if we were to repeatedly sample from the unknown distribution and calculate the statistic of the sample each time. It does this by repeatedly resampling values *from the original sample* with replacement and calculating the statistic of each resample. This results in a "bootstrap distribution" of the statistic. >>> import matplotlib.pyplot as plt >>> from scipy.stats import bootstrap >>> data = (data,) # samples must be in a sequence >>> res = bootstrap(data, np.std, confidence_level=0.9, ... random_state=rng) >>> fig, ax = plt.subplots() >>> ax.hist(res.bootstrap_distribution, bins=25) >>> ax.set_title('Bootstrap Distribution') >>> ax.set_xlabel('statistic value') >>> ax.set_ylabel('frequency') >>> plt.show() The standard error quantifies this variability. It is calculated as the standard deviation of the bootstrap distribution. >>> res.standard_error 0.24427002125829136 >>> res.standard_error == np.std(res.bootstrap_distribution, ddof=1) True The bootstrap distribution of the statistic is often approximately normal with scale equal to the standard error. >>> x = np.linspace(3, 5) >>> pdf = norm.pdf(x, loc=std_sample, scale=res.standard_error) >>> fig, ax = plt.subplots() >>> ax.hist(res.bootstrap_distribution, bins=25, density=True) >>> ax.plot(x, pdf) >>> ax.set_title('Normal Approximation of the Bootstrap Distribution') >>> ax.set_xlabel('statistic value') >>> ax.set_ylabel('pdf') >>> plt.show() This suggests that we could construct a 90% confidence interval on the statistic based on quantiles of this normal distribution. >>> norm.interval(0.9, loc=std_sample, scale=res.standard_error) (3.5442759991341726, 4.3478528599786) Due to central limit theorem, this normal approximation is accurate for a variety of statistics and distributions underlying the samples; however, the approximation is not reliable in all cases. Because `bootstrap` is designed to work with arbitrary underlying distributions and statistics, it uses more advanced techniques to generate an accurate confidence interval. >>> print(res.confidence_interval) ConfidenceInterval(low=3.57655333533867, high=4.382043696342881) If we sample from the original distribution 1000 times and form a bootstrap confidence interval for each sample, the confidence interval contains the true value of the statistic approximately 90% of the time. >>> n_trials = 1000 >>> ci_contains_true_std = 0 >>> for i in range(n_trials): ... data = (dist.rvs(size=100, random_state=rng),) ... ci = bootstrap(data, np.std, confidence_level=0.9, n_resamples=1000, ... random_state=rng).confidence_interval ... if ci[0] < std_true < ci[1]: ... ci_contains_true_std += 1 >>> print(ci_contains_true_std) 875 Rather than writing a loop, we can also determine the confidence intervals for all 1000 samples at once. >>> data = (dist.rvs(size=(n_trials, 100), random_state=rng),) >>> res = bootstrap(data, np.std, axis=-1, confidence_level=0.9, ... n_resamples=1000, random_state=rng) >>> ci_l, ci_u = res.confidence_interval Here, `ci_l` and `ci_u` contain the confidence interval for each of the ``n_trials = 1000`` samples. >>> print(ci_l[995:]) [3.77729695 3.75090233 3.45829131 3.34078217 3.48072829] >>> print(ci_u[995:]) [4.88316666 4.86924034 4.32032996 4.2822427 4.59360598] And again, approximately 90% contain the true value, ``std_true = 4``. >>> print(np.sum((ci_l < std_true) & (std_true < ci_u))) 900 `bootstrap` can also be used to estimate confidence intervals of multi-sample statistics, including those calculated by hypothesis tests. `scipy.stats.mood` perform's Mood's test for equal scale parameters, and it returns two outputs: a statistic, and a p-value. To get a confidence interval for the test statistic, we first wrap `scipy.stats.mood` in a function that accepts two sample arguments, accepts an `axis` keyword argument, and returns only the statistic. >>> from scipy.stats import mood >>> def my_statistic(sample1, sample2, axis): ... statistic, _ = mood(sample1, sample2, axis=-1) ... return statistic Here, we use the 'percentile' method with the default 95% confidence level. >>> sample1 = norm.rvs(scale=1, size=100, random_state=rng) >>> sample2 = norm.rvs(scale=2, size=100, random_state=rng) >>> data = (sample1, sample2) >>> res = bootstrap(data, my_statistic, method='basic', random_state=rng) >>> print(mood(sample1, sample2)[0]) # element 0 is the statistic -5.521109549096542 >>> print(res.confidence_interval) ConfidenceInterval(low=-7.255994487314675, high=-4.016202624747605) The bootstrap estimate of the standard error is also available. >>> print(res.standard_error) 0.8344963846318795 Paired-sample statistics work, too. For example, consider the Pearson correlation coefficient. >>> from scipy.stats import pearsonr >>> n = 100 >>> x = np.linspace(0, 10, n) >>> y = x + rng.uniform(size=n) >>> print(pearsonr(x, y)[0]) # element 0 is the statistic 0.9962357936065914 We wrap `pearsonr` so that it returns only the statistic. >>> def my_statistic(x, y): ... return pearsonr(x, y)[0] We call `bootstrap` using ``paired=True``. Also, since ``my_statistic`` isn't vectorized to calculate the statistic along a given axis, we pass in ``vectorized=False``. >>> res = bootstrap((x, y), my_statistic, vectorized=False, paired=True, ... random_state=rng) >>> print(res.confidence_interval) ConfidenceInterval(low=0.9950085825848624, high=0.9971212407917498) The result object can be passed back into `bootstrap` to perform additional resampling: >>> len(res.bootstrap_distribution) 9999 >>> res = bootstrap((x, y), my_statistic, vectorized=False, paired=True, ... n_resamples=1001, random_state=rng, ... bootstrap_result=res) >>> len(res.bootstrap_distribution) 11000 or to change the confidence interval options: >>> res2 = bootstrap((x, y), my_statistic, vectorized=False, paired=True, ... n_resamples=0, random_state=rng, bootstrap_result=res, ... method='percentile', confidence_level=0.9) >>> np.testing.assert_equal(res2.bootstrap_distribution, ... res.bootstrap_distribution) >>> res.confidence_interval ConfidenceInterval(low=0.9950035351407804, high=0.9971170323404578) without repeating computation of the original bootstrap distribution. 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The null hypothesis is that the provided `sample` was drawn at random from the distribution for which `rvs` generates random variates. The value of the `statistic` for the given sample is compared against a Monte Carlo null distribution: the value of the statistic for each of `n_resamples` samples generated by `rvs`. This gives the p-value, the probability of observing such an extreme value of the test statistic under the null hypothesis. Parameters ---------- sample : array-like An array of observations. rvs : callable Generates random variates from the distribution against which `sample` will be tested. `rvs` must be a callable that accepts keyword argument ``size`` (e.g. ``rvs(size=(m, n))``) and returns an N-d array sample of that shape. statistic : callable Statistic for which the p-value of the hypothesis test is to be calculated. `statistic` must be a callable that accepts a sample (e.g. ``statistic(sample)``) 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 array. vectorized : bool, 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_resamples : int, default: 9999 Number of random permutations used to approximate the Monte Carlo null distribution. batch : int, optional The number of permutations to process in each call to `statistic`. Memory usage is O(`batch`*``sample.size[axis]``). Default is ``None``, in which case `batch` equals `n_resamples`. alternative : {'two-sided', 'less', 'greater'} The alternative hypothesis for which the p-value is calculated. For each alternative, the p-value is defined 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'`` : twice the smaller of the p-values above. axis : int, default: 0 The axis of `sample` over which to calculate the statistic. Returns ------- statistic : float or ndarray The observed test statistic of the sample. pvalue : float or ndarray The p-value for the given alternative. null_distribution : ndarray The values of the test statistic generated under the null hypothesis. References ---------- .. [1] 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). Examples -------- Suppose we wish to test whether a small sample has been drawn from a normal distribution. We decide that we will use the skew of the sample as a test statistic, and we will consider a p-value of 0.05 to be statistically significant. >>> import numpy as np >>> from scipy import stats >>> def statistic(x, axis): ... return stats.skew(x, axis) After collecting our data, we calculate the observed value of the test statistic. >>> rng = np.random.default_rng() >>> x = stats.skewnorm.rvs(a=1, size=50, random_state=rng) >>> statistic(x, axis=0) 0.12457412450240658 To determine the probability of observing such an extreme value of the skewness by chance if the sample were drawn from the normal distribution, we can perform a Monte Carlo hypothesis test. The test will draw many samples at random from their normal distribution, calculate the skewness of each sample, and compare our original skewness against this distribution to determine an approximate p-value. >>> from scipy.stats import monte_carlo_test >>> # because our statistic is vectorized, we pass `vectorized=True` >>> rvs = lambda size: stats.norm.rvs(size=size, random_state=rng) >>> res = monte_carlo_test(x, rvs, statistic, vectorized=True) >>> print(res.statistic) 0.12457412450240658 >>> print(res.pvalue) 0.7012 The probability of obtaining a test statistic less than or equal to the observed value under the null hypothesis is ~70%. This is greater than our chosen threshold of 5%, so we cannot consider this to be significant evidence against the null hypothesis. Note that this p-value essentially matches that of `scipy.stats.skewtest`, which relies on an asymptotic distribution of a test statistic based on the sample skewness. >>> stats.skewtest(x).pvalue 0.6892046027110614 This asymptotic approximation is not valid for small sample sizes, but `monte_carlo_test` can be used with samples of any size. >>> x = stats.skewnorm.rvs(a=1, size=7, random_state=rng) >>> # stats.skewtest(x) would produce an error due to small sample >>> res = monte_carlo_test(x, rvs, statistic, vectorized=True) The Monte Carlo distribution of the test statistic is provided for further investigation. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> ax.hist(res.null_distribution, bins=50) >>> ax.set_title("Monte Carlo distribution of test statistic") >>> ax.set_xlabel("Value of Statistic") >>> ax.set_ylabel("Frequency") >>> plt.show() r" r r r �r� r c �R �� | |k }|� d�� � dz �dz z }|S �Nr r r ru �r� �observed�cmps�pvaluesrG s �r r� z monte_carlo_test.<locals>.less= �3 �� � �H�,���8�8��8�#�#�a�'�K�!�O�<���r! c �R �� | |k }|� d�� � dz �dz z }|S r� ru r� s �r r� z!monte_carlo_test.<locals>.greaterB r� r! c �h �� �| |� � } �| |� � }t j ||� � dz }|S �NrK �r% �minimum�r� r� � pvalues_less�pvalues_greaterr� r� r� s ��r � two_sidedz#monte_carlo_test.<locals>.two_sidedG �B �� ��t� -�x�8�8� �!�'�"3�X�>�>���*�\�?�;�;�a�?���r! �r� r� r� ) r� r% rz r r5 r6 r| rh r< �ndim�clipr� )r r� r* r� rG r= r� r r� r� �n_observationsr? r� r@ rA rD r� �comparer� r� r� s ` @@r r r � s� ���� �\ ���Y� � +�U�K�� G� G�D� /3�,�V�S�)�Z��%��d� �z�)�)�F��4�4�4�5�5�b�9�H��\�"�%�N��(�[�M��� �1�k�=� 1� 1� @� @���=�+�a�-�8�8� ��C�l�N� ;�<�<�<� �� � ���9�2�!>�!>�!>�?�?�?�?���'8�9�9��)�1�1�2�$�!��X�]�9J�2J�K�K��� � � � � � � � � � � � � � � � �!� %�'� '�G� #�g�k�"�#4�h�?�?�G��g�g�q�!�$�$�G� ��'�3D� E� E�Er! �PermutationTestResultc # � ��K � d� ���fd��t t t j | � � � � � � } �|| dd� � � D ]<}t j d� |D � � � � � t � � }|V � �=dS )zu Generate all partitions of indices of groups of given sizes, concatenated `ns` is an iterable of ints. c 3 �d K � t | |� � D ] }t |� � }| |z }||gV � � d S r$ )r �set)r' r> �c�x0�x1s r �all_partitionsz4_all_partitions_concatenated.<locals>.all_partitionsa sL � � � � �a��#�#� � �A��Q���B��R��B��r�(�N�N�N�N� � r! c 3 �� �K � t |� � dk r| gV � d S �| |d � � D ].} �|d |dd � � � D ]}|dd� |z V � ��/d S )Nr r )r� )r' �nsr� �dr� �all_partitions_ns ��r r� z6_all_partitions_concatenated.<locals>.all_partitions_ng s� �� � � � �r�7�7�a�<�<��#�I�I�I� �F� � ��2�a�5�)�)� !� !�A�%�%�a��d�B�q�r�r�F�3�3� !� !����!��f�q�j� � � � � !� !� !r! 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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 ---------- data : iterable 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`. statistic : callable 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 :math:`\rho`, Kendall's :math:`\tau`, and Pearson's :math:`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 :math:`U` test and the independent sample t-test. Please see the Notes section below for more detailed descriptions of the permutation types. vectorized : bool, 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_resamples : int 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. batch : int, 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. axis : int, 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`, `numpy.random.RandomState`}, optional 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 ------- statistic : float or ndarray The observed test statistic of the data. pvalue : float or ndarray The p-value for the given alternative. null_distribution : ndarray 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] R. A. 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() 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. 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