from collections import namedtuple from dataclasses import make_dataclass import numpy as np import warnings from itertools import combinations import scipy.stats from scipy.optimize import shgo from . import distributions from ._continuous_distns import chi2, norm from scipy.special import gamma, kv, gammaln from . import _wilcoxon_data __all__ = ['epps_singleton_2samp', 'cramervonmises', 'somersd', 'barnard_exact', 'boschloo_exact', 'cramervonmises_2samp'] Epps_Singleton_2sampResult = namedtuple('Epps_Singleton_2sampResult', ('statistic', 'pvalue')) def epps_singleton_2samp(x, y, t=(0.4, 0.8)): """Compute the Epps-Singleton (ES) test statistic. Test the null hypothesis that two samples have the same underlying probability distribution. Parameters ---------- x, y : array-like The two samples of observations to be tested. Input must not have more than one dimension. Samples can have different lengths. t : array-like, optional The points (t1, ..., tn) where the empirical characteristic function is to be evaluated. It should be positive distinct numbers. The default value (0.4, 0.8) is proposed in [1]_. Input must not have more than one dimension. Returns ------- statistic : float The test statistic. pvalue : float The associated p-value based on the asymptotic chi2-distribution. See Also -------- ks_2samp, anderson_ksamp Notes ----- Testing whether two samples are generated by the same underlying distribution is a classical question in statistics. A widely used test is the Kolmogorov-Smirnov (KS) test which relies on the empirical distribution function. Epps and Singleton introduce a test based on the empirical characteristic function in [1]_. One advantage of the ES test compared to the KS test is that is does not assume a continuous distribution. In [1]_, the authors conclude that the test also has a higher power than the KS test in many examples. They recommend the use of the ES test for discrete samples as well as continuous samples with at least 25 observations each, whereas `anderson_ksamp` is recommended for smaller sample sizes in the continuous case. The p-value is computed from the asymptotic distribution of the test statistic which follows a `chi2` distribution. If the sample size of both `x` and `y` is below 25, the small sample correction proposed in [1]_ is applied to the test statistic. The default values of `t` are determined in [1]_ by considering various distributions and finding good values that lead to a high power of the test in general. Table III in [1]_ gives the optimal values for the distributions tested in that study. The values of `t` are scaled by the semi-interquartile range in the implementation, see [1]_. References ---------- .. [1] T. W. Epps and K. J. Singleton, "An omnibus test for the two-sample problem using the empirical characteristic function", Journal of Statistical Computation and Simulation 26, p. 177--203, 1986. .. [2] S. J. Goerg and J. Kaiser, "Nonparametric testing of distributions - the Epps-Singleton two-sample test using the empirical characteristic function", The Stata Journal 9(3), p. 454--465, 2009. """ x, y, t = np.asarray(x), np.asarray(y), np.asarray(t) # check if x and y are valid inputs if x.ndim > 1: raise ValueError('x must be 1d, but x.ndim equals {}.'.format(x.ndim)) if y.ndim > 1: raise ValueError('y must be 1d, but y.ndim equals {}.'.format(y.ndim)) nx, ny = len(x), len(y) if (nx < 5) or (ny < 5): raise ValueError('x and y should have at least 5 elements, but len(x) ' '= {} and len(y) = {}.'.format(nx, ny)) if not np.isfinite(x).all(): raise ValueError('x must not contain nonfinite values.') if not np.isfinite(y).all(): raise ValueError('y must not contain nonfinite values.') n = nx + ny # check if t is valid if t.ndim > 1: raise ValueError('t must be 1d, but t.ndim equals {}.'.format(t.ndim)) if np.less_equal(t, 0).any(): raise ValueError('t must contain positive elements only.') # rescale t with semi-iqr as proposed in [1]; import iqr here to avoid # circular import from scipy.stats import iqr sigma = iqr(np.hstack((x, y))) / 2 ts = np.reshape(t, (-1, 1)) / sigma # covariance estimation of ES test gx = np.vstack((np.cos(ts*x), np.sin(ts*x))).T # shape = (nx, 2*len(t)) gy = np.vstack((np.cos(ts*y), np.sin(ts*y))).T cov_x = np.cov(gx.T, bias=True) # the test uses biased cov-estimate cov_y = np.cov(gy.T, bias=True) est_cov = (n/nx)*cov_x + (n/ny)*cov_y est_cov_inv = np.linalg.pinv(est_cov) r = np.linalg.matrix_rank(est_cov_inv) if r < 2*len(t): warnings.warn('Estimated covariance matrix does not have full rank. ' 'This indicates a bad choice of the input t and the ' 'test might not be consistent.') # see p. 183 in [1]_ # compute test statistic w distributed asympt. as chisquare with df=r g_diff = np.mean(gx, axis=0) - np.mean(gy, axis=0) w = n*np.dot(g_diff.T, np.dot(est_cov_inv, g_diff)) # apply small-sample correction if (max(nx, ny) < 25): corr = 1.0/(1.0 + n**(-0.45) + 10.1*(nx**(-1.7) + ny**(-1.7))) w = corr * w p = chi2.sf(w, r) return Epps_Singleton_2sampResult(w, p) class CramerVonMisesResult: def __init__(self, statistic, pvalue): self.statistic = statistic self.pvalue = pvalue def __repr__(self): return (f"{self.__class__.__name__}(statistic={self.statistic}, " f"pvalue={self.pvalue})") def _psi1_mod(x): """ psi1 is defined in equation 1.10 in Csorgo, S. and Faraway, J. (1996). This implements a modified version by excluding the term V(x) / 12 (here: _cdf_cvm_inf(x) / 12) to avoid evaluating _cdf_cvm_inf(x) twice in _cdf_cvm. Implementation based on MAPLE code of Julian Faraway and R code of the function pCvM in the package goftest (v1.1.1), permission granted by Adrian Baddeley. Main difference in the implementation: the code here keeps adding terms of the series until the terms are small enough. """ def _ed2(y): z = y**2 / 4 b = kv(1/4, z) + kv(3/4, z) return np.exp(-z) * (y/2)**(3/2) * b / np.sqrt(np.pi) def _ed3(y): z = y**2 / 4 c = np.exp(-z) / np.sqrt(np.pi) return c * (y/2)**(5/2) * (2*kv(1/4, z) + 3*kv(3/4, z) - kv(5/4, z)) def _Ak(k, x): m = 2*k + 1 sx = 2 * np.sqrt(x) y1 = x**(3/4) y2 = x**(5/4) e1 = m * gamma(k + 1/2) * _ed2((4 * k + 3)/sx) / (9 * y1) e2 = gamma(k + 1/2) * _ed3((4 * k + 1) / sx) / (72 * y2) e3 = 2 * (m + 2) * gamma(k + 3/2) * _ed3((4 * k + 5) / sx) / (12 * y2) e4 = 7 * m * gamma(k + 1/2) * _ed2((4 * k + 1) / sx) / (144 * y1) e5 = 7 * m * gamma(k + 1/2) * _ed2((4 * k + 5) / sx) / (144 * y1) return e1 + e2 + e3 + e4 + e5 x = np.asarray(x) tot = np.zeros_like(x, dtype='float') cond = np.ones_like(x, dtype='bool') k = 0 while np.any(cond): z = -_Ak(k, x[cond]) / (np.pi * gamma(k + 1)) tot[cond] = tot[cond] + z cond[cond] = np.abs(z) >= 1e-7 k += 1 return tot def _cdf_cvm_inf(x): """ Calculate the cdf of the Cramér-von Mises statistic (infinite sample size). See equation 1.2 in Csorgo, S. and Faraway, J. (1996). Implementation based on MAPLE code of Julian Faraway and R code of the function pCvM in the package goftest (v1.1.1), permission granted by Adrian Baddeley. Main difference in the implementation: the code here keeps adding terms of the series until the terms are small enough. The function is not expected to be accurate for large values of x, say x > 4, when the cdf is very close to 1. """ x = np.asarray(x) def term(x, k): # this expression can be found in [2], second line of (1.3) u = np.exp(gammaln(k + 0.5) - gammaln(k+1)) / (np.pi**1.5 * np.sqrt(x)) y = 4*k + 1 q = y**2 / (16*x) b = kv(0.25, q) return u * np.sqrt(y) * np.exp(-q) * b tot = np.zeros_like(x, dtype='float') cond = np.ones_like(x, dtype='bool') k = 0 while np.any(cond): z = term(x[cond], k) tot[cond] = tot[cond] + z cond[cond] = np.abs(z) >= 1e-7 k += 1 return tot def _cdf_cvm(x, n=None): """ Calculate the cdf of the Cramér-von Mises statistic for a finite sample size n. If N is None, use the asymptotic cdf (n=inf). See equation 1.8 in Csorgo, S. and Faraway, J. (1996) for finite samples, 1.2 for the asymptotic cdf. The function is not expected to be accurate for large values of x, say x > 2, when the cdf is very close to 1 and it might return values > 1 in that case, e.g. _cdf_cvm(2.0, 12) = 1.0000027556716846. """ x = np.asarray(x) if n is None: y = _cdf_cvm_inf(x) else: # support of the test statistic is [12/n, n/3], see 1.1 in [2] y = np.zeros_like(x, dtype='float') sup = (1./(12*n) < x) & (x < n/3.) # note: _psi1_mod does not include the term _cdf_cvm_inf(x) / 12 # therefore, we need to add it here y[sup] = _cdf_cvm_inf(x[sup]) * (1 + 1./(12*n)) + _psi1_mod(x[sup]) / n y[x >= n/3] = 1 if y.ndim == 0: return y[()] return y def cramervonmises(rvs, cdf, args=()): """Perform the one-sample Cramér-von Mises test for goodness of fit. This performs a test of the goodness of fit of a cumulative distribution function (cdf) :math:`F` compared to the empirical distribution function :math:`F_n` of observed random variates :math:`X_1, ..., X_n` that are assumed to be independent and identically distributed ([1]_). The null hypothesis is that the :math:`X_i` have cumulative distribution :math:`F`. Parameters ---------- rvs : array_like A 1-D array of observed values of the random variables :math:`X_i`. cdf : str or callable The cumulative distribution function :math:`F` to test the observations against. If a string, it should be the name of a distribution in `scipy.stats`. If a callable, that callable is used to calculate the cdf: ``cdf(x, *args) -> float``. args : tuple, optional Distribution parameters. These are assumed to be known; see Notes. Returns ------- res : object with attributes statistic : float Cramér-von Mises statistic. pvalue : float The p-value. See Also -------- kstest, cramervonmises_2samp Notes ----- .. versionadded:: 1.6.0 The p-value relies on the approximation given by equation 1.8 in [2]_. It is important to keep in mind that the p-value is only accurate if one tests a simple hypothesis, i.e. the parameters of the reference distribution are known. If the parameters are estimated from the data (composite hypothesis), the computed p-value is not reliable. References ---------- .. [1] Cramér-von Mises criterion, Wikipedia, https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion .. [2] Csorgo, S. and Faraway, J. (1996). The Exact and Asymptotic Distribution of Cramér-von Mises Statistics. Journal of the Royal Statistical Society, pp. 221-234. Examples -------- Suppose we wish to test whether data generated by ``scipy.stats.norm.rvs`` were, in fact, drawn from the standard normal distribution. We choose a significance level of alpha=0.05. >>> from scipy import stats >>> rng = np.random.default_rng() >>> x = stats.norm.rvs(size=500, random_state=rng) >>> res = stats.cramervonmises(x, 'norm') >>> res.statistic, res.pvalue (0.49121480855028343, 0.04189256516661377) The p-value 0.79 exceeds our chosen significance level, so we do not reject the null hypothesis that the observed sample is drawn from the standard normal distribution. Now suppose we wish to check whether the same samples shifted by 2.1 is consistent with being drawn from a normal distribution with a mean of 2. >>> y = x + 2.1 >>> res = stats.cramervonmises(y, 'norm', args=(2,)) >>> res.statistic, res.pvalue (0.07400330012187435, 0.7274595666160468) Here we have used the `args` keyword to specify the mean (``loc``) of the normal distribution to test the data against. This is equivalent to the following, in which we create a frozen normal distribution with mean 2.1, then pass its ``cdf`` method as an argument. >>> frozen_dist = stats.norm(loc=2) >>> res = stats.cramervonmises(y, frozen_dist.cdf) >>> res.statistic, res.pvalue (0.07400330012187435, 0.7274595666160468) In either case, we would reject the null hypothesis that the observed sample is drawn from a normal distribution with a mean of 2 (and default variance of 1) because the p-value 0.04 is less than our chosen significance level. """ if isinstance(cdf, str): cdf = getattr(distributions, cdf).cdf vals = np.sort(np.asarray(rvs)) if vals.size <= 1: raise ValueError('The sample must contain at least two observations.') if vals.ndim > 1: raise ValueError('The sample must be one-dimensional.') n = len(vals) cdfvals = cdf(vals, *args) u = (2*np.arange(1, n+1) - 1)/(2*n) w = 1/(12*n) + np.sum((u - cdfvals)**2) # avoid small negative values that can occur due to the approximation p = max(0, 1. - _cdf_cvm(w, n)) return CramerVonMisesResult(statistic=w, pvalue=p) def _get_wilcoxon_distr(n): """ Distribution of counts of the Wilcoxon ranksum statistic r_plus (sum of ranks of positive differences). Returns an array with the counts/frequencies of all the possible ranks r = 0, ..., n*(n+1)/2 """ cnt = _wilcoxon_data.COUNTS.get(n) if cnt is None: raise ValueError("The exact distribution of the Wilcoxon test " "statistic is not implemented for n={}".format(n)) return np.array(cnt, dtype=int) def _Aij(A, i, j): """Sum of upper-left and lower right blocks of contingency table.""" # See [2] bottom of page 309 return A[:i, :j].sum() + A[i+1:, j+1:].sum() def _Dij(A, i, j): """Sum of lower-left and upper-right blocks of contingency table.""" # See [2] bottom of page 309 return A[i+1:, :j].sum() + A[:i, j+1:].sum() def _P(A): """Twice the number of concordant pairs, excluding ties.""" # See [2] bottom of page 309 m, n = A.shape count = 0 for i in range(m): for j in range(n): count += A[i, j]*_Aij(A, i, j) return count def _Q(A): """Twice the number of discordant pairs, excluding ties.""" # See [2] bottom of page 309 m, n = A.shape count = 0 for i in range(m): for j in range(n): count += A[i, j]*_Dij(A, i, j) return count def _a_ij_Aij_Dij2(A): """A term that appears in the ASE of Kendall's tau and Somers' D.""" # See [2] section 4: Modified ASEs to test the null hypothesis... m, n = A.shape count = 0 for i in range(m): for j in range(n): count += A[i, j]*(_Aij(A, i, j) - _Dij(A, i, j))**2 return count def _tau_b(A): """Calculate Kendall's tau-b and p-value from contingency table.""" # See [2] 2.2 and 4.2 # contingency table must be truly 2D if A.shape[0] == 1 or A.shape[1] == 1: return np.nan, np.nan NA = A.sum() PA = _P(A) QA = _Q(A) Sri2 = (A.sum(axis=1)**2).sum() Scj2 = (A.sum(axis=0)**2).sum() denominator = (NA**2 - Sri2)*(NA**2 - Scj2) tau = (PA-QA)/(denominator)**0.5 numerator = 4*(_a_ij_Aij_Dij2(A) - (PA - QA)**2 / NA) s02_tau_b = numerator/denominator if s02_tau_b == 0: # Avoid divide by zero return tau, 0 Z = tau/s02_tau_b**0.5 p = 2*norm.sf(abs(Z)) # 2-sided p-value return tau, p def _somers_d(A): """Calculate Somers' D and p-value from contingency table.""" # See [3] page 1740 # contingency table must be truly 2D if A.shape[0] <= 1 or A.shape[1] <= 1: return np.nan, np.nan NA = A.sum() NA2 = NA**2 PA = _P(A) QA = _Q(A) Sri2 = (A.sum(axis=1)**2).sum() d = (PA - QA)/(NA2 - Sri2) S = _a_ij_Aij_Dij2(A) - (PA-QA)**2/NA if S == 0: # Avoid divide by zero return d, 0 Z = (PA - QA)/(4*(S))**0.5 p = 2*norm.sf(abs(Z)) # 2-sided p-value return d, p SomersDResult = make_dataclass("SomersDResult", ("statistic", "pvalue", "table")) def somersd(x, y=None): r"""Calculates Somers' D, an asymmetric measure of ordinal association. Like Kendall's :math:`\tau`, Somers' :math:`D` is a measure of the correspondence between two rankings. Both statistics consider the difference between the number of concordant and discordant pairs in two rankings :math:`X` and :math:`Y`, and both are normalized such that values close to 1 indicate strong agreement and values close to -1 indicate strong disagreement. They differ in how they are normalized. To show the relationship, Somers' :math:`D` can be defined in terms of Kendall's :math:`\tau_a`: .. math:: D(Y|X) = \frac{\tau_a(X, Y)}{\tau_a(X, X)} Suppose the first ranking :math:`X` has :math:`r` distinct ranks and the second ranking :math:`Y` has :math:`s` distinct ranks. These two lists of :math:`n` rankings can also be viewed as an :math:`r \times s` contingency table in which element :math:`i, j` is the number of rank pairs with rank :math:`i` in ranking :math:`X` and rank :math:`j` in ranking :math:`Y`. Accordingly, `somersd` also allows the input data to be supplied as a single, 2D contingency table instead of as two separate, 1D rankings. Note that the definition of Somers' :math:`D` is asymmetric: in general, :math:`D(Y|X) \neq D(X|Y)`. ``somersd(x, y)`` calculates Somers' :math:`D(Y|X)`: the "row" variable :math:`X` is treated as an independent variable, and the "column" variable :math:`Y` is dependent. For Somers' :math:`D(X|Y)`, swap the input lists or transpose the input table. Parameters ---------- x: array_like 1D array of rankings, treated as the (row) independent variable. Alternatively, a 2D contingency table. y: array_like If `x` is a 1D array of rankings, `y` is a 1D array of rankings of the same length, treated as the (column) dependent variable. If `x` is 2D, `y` is ignored. Returns ------- res : SomersDResult A `SomersDResult` object with the following fields: correlation : float The Somers' :math:`D` statistic. pvalue : float The two-sided p-value for a hypothesis test whose null hypothesis is an absence of association, :math:`D=0`. See notes for more information. table : 2D array The contingency table formed from rankings `x` and `y` (or the provided contingency table, if `x` is a 2D array) See Also -------- kendalltau : Calculates Kendall's tau, another correlation measure. weightedtau : Computes a weighted version of Kendall's tau. spearmanr : Calculates a Spearman rank-order correlation coefficient. pearsonr : Calculates a Pearson correlation coefficient. Notes ----- This function follows the contingency table approach of [2]_ and [3]_. *p*-values are computed based on an asymptotic approximation of the test statistic distribution under the null hypothesis :math:`D=0`. Theoretically, hypothesis tests based on Kendall's :math:`tau` and Somers' :math:`D` should be identical. However, the *p*-values returned by `kendalltau` are based on the null hypothesis of *independence* between :math:`X` and :math:`Y` (i.e. the population from which pairs in :math:`X` and :math:`Y` are sampled contains equal numbers of all possible pairs), which is more specific than the null hypothesis :math:`D=0` used here. If the null hypothesis of independence is desired, it is acceptable to use the *p*-value returned by `kendalltau` with the statistic returned by `somersd` and vice versa. For more information, see [2]_. Contingency tables are formatted according to the convention used by SAS and R: the first ranking supplied (``x``) is the "row" variable, and the second ranking supplied (``y``) is the "column" variable. This is opposite the convention of Somers' original paper [1]_. References ---------- .. [1] Robert H. Somers, "A New Asymmetric Measure of Association for Ordinal Variables", *American Sociological Review*, Vol. 27, No. 6, pp. 799--811, 1962. .. [2] Morton B. Brown and Jacqueline K. Benedetti, "Sampling Behavior of Tests for Correlation in Two-Way Contingency Tables", *Journal of the American Statistical Association* Vol. 72, No. 358, pp. 309--315, 1977. .. [3] SAS Institute, Inc., "The FREQ Procedure (Book Excerpt)", *SAS/STAT 9.2 User's Guide, Second Edition*, SAS Publishing, 2009. .. [4] Laerd Statistics, "Somers' d using SPSS Statistics", *SPSS Statistics Tutorials and Statistical Guides*, https://statistics.laerd.com/spss-tutorials/somers-d-using-spss-statistics.php, Accessed July 31, 2020. Examples -------- We calculate Somers' D for the example given in [4]_, in which a hotel chain owner seeks to determine the association between hotel room cleanliness and customer satisfaction. The independent variable, hotel room cleanliness, is ranked on an ordinal scale: "below average (1)", "average (2)", or "above average (3)". The dependent variable, customer satisfaction, is ranked on a second scale: "very dissatisfied (1)", "moderately dissatisfied (2)", "neither dissatisfied nor satisfied (3)", "moderately satisfied (4)", or "very satisfied (5)". 189 customers respond to the survey, and the results are cast into a contingency table with the hotel room cleanliness as the "row" variable and customer satisfaction as the "column" variable. +-----+-----+-----+-----+-----+-----+ | | (1) | (2) | (3) | (4) | (5) | +=====+=====+=====+=====+=====+=====+ | (1) | 27 | 25 | 14 | 7 | 0 | +-----+-----+-----+-----+-----+-----+ | (2) | 7 | 14 | 18 | 35 | 12 | +-----+-----+-----+-----+-----+-----+ | (3) | 1 | 3 | 2 | 7 | 17 | +-----+-----+-----+-----+-----+-----+ For example, 27 customers assigned their room a cleanliness ranking of "below average (1)" and a corresponding satisfaction of "very dissatisfied (1)". We perform the analysis as follows. >>> from scipy.stats import somersd >>> table = [[27, 25, 14, 7, 0], [7, 14, 18, 35, 12], [1, 3, 2, 7, 17]] >>> res = somersd(table) >>> res.statistic 0.6032766111513396 >>> res.pvalue 1.0007091191074533e-27 The value of the Somers' D statistic is approximately 0.6, indicating a positive correlation between room cleanliness and customer satisfaction in the sample. The *p*-value is very small, indicating a very small probability of observing such an extreme value of the statistic under the null hypothesis that the statistic of the entire population (from which our sample of 189 customers is drawn) is zero. This supports the alternative hypothesis that the true value of Somers' D for the population is nonzero. """ x, y = np.array(x), np.array(y) if x.ndim == 1: if x.size != y.size: raise ValueError("Rankings must be of equal length.") table = scipy.stats.contingency.crosstab(x, y)[1] elif x.ndim == 2: if np.any(x < 0): raise ValueError("All elements of the contingency table must be " "non-negative.") if np.any(x != x.astype(int)): raise ValueError("All elements of the contingency table must be " "integer.") if x.nonzero()[0].size < 2: raise ValueError("At least two elements of the contingency table " "must be nonzero.") table = x else: raise ValueError("x must be either a 1D or 2D array") d, p = _somers_d(table) return SomersDResult(d, p, table) def _all_partitions(nx, ny): """ Partition a set of indices into two fixed-length sets in all possible ways Partition a set of indices 0 ... nx + ny - 1 into two sets of length nx and ny in all possible ways (ignoring order of elements). """ z = np.arange(nx+ny) for c in combinations(z, nx): x = np.array(c) mask = np.ones(nx+ny, bool) mask[x] = False y = z[mask] yield x, y def _compute_log_combinations(n): """Compute all log combination of C(n, k).""" gammaln_arr = gammaln(np.arange(n + 1) + 1) return gammaln(n + 1) - gammaln_arr - gammaln_arr[::-1] BarnardExactResult = make_dataclass( "BarnardExactResult", [("statistic", float), ("pvalue", float)] ) def barnard_exact(table, alternative="two-sided", pooled=True, n=32): r"""Perform a Barnard exact test on a 2x2 contingency table. Parameters ---------- table : array_like of ints A 2x2 contingency table. Elements should be non-negative integers. alternative : {'two-sided', 'less', 'greater'}, optional Defines the null and alternative hypotheses. Default is 'two-sided'. Please see explanations in the Notes section below. pooled : bool, optional Whether to compute score statistic with pooled variance (as in Student's t-test, for example) or unpooled variance (as in Welch's t-test). Default is ``True``. n : int, optional Number of sampling points used in the construction of the sampling method. Note that this argument will automatically be converted to the next higher power of 2 since `scipy.stats.qmc.Sobol` is used to select sample points. Default is 32. Must be positive. In most cases, 32 points is enough to reach good precision. More points comes at performance cost. Returns ------- ber : BarnardExactResult A result object with the following attributes. statistic : float The Wald statistic with pooled or unpooled variance, depending on the user choice of `pooled`. pvalue : float P-value, the probability of obtaining a distribution at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. See Also -------- chi2_contingency : Chi-square test of independence of variables in a contingency table. fisher_exact : Fisher exact test on a 2x2 contingency table. boschloo_exact : Boschloo's exact test on a 2x2 contingency table, which is an uniformly more powerful alternative to Fisher's exact test. Notes ----- Barnard's test is an exact test used in the analysis of contingency tables. It examines the association of two categorical variables, and is a more powerful alternative than Fisher's exact test for 2x2 contingency tables. Let's define :math:`X_0` a 2x2 matrix representing the observed sample, where each column stores the binomial experiment, as in the example below. Let's also define :math:`p_1, p_2` the theoretical binomial probabilities for :math:`x_{11}` and :math:`x_{12}`. When using Barnard exact test, we can assert three different null hypotheses : - :math:`H_0 : p_1 \geq p_2` versus :math:`H_1 : p_1 < p_2`, with `alternative` = "less" - :math:`H_0 : p_1 \leq p_2` versus :math:`H_1 : p_1 > p_2`, with `alternative` = "greater" - :math:`H_0 : p_1 = p_2` versus :math:`H_1 : p_1 \neq p_2`, with `alternative` = "two-sided" (default one) In order to compute Barnard's exact test, we are using the Wald statistic [3]_ with pooled or unpooled variance. Under the default assumption that both variances are equal (``pooled = True``), the statistic is computed as: .. math:: T(X) = \frac{ \hat{p}_1 - \hat{p}_2 }{ \sqrt{ \hat{p}(1 - \hat{p}) (\frac{1}{c_1} + \frac{1}{c_2}) } } with :math:`\hat{p}_1, \hat{p}_2` and :math:`\hat{p}` the estimator of :math:`p_1, p_2` and :math:`p`, the latter being the combined probability, given the assumption that :math:`p_1 = p_2`. If this assumption is invalid (``pooled = False``), the statistic is: .. math:: T(X) = \frac{ \hat{p}_1 - \hat{p}_2 }{ \sqrt{ \frac{\hat{p}_1 (1 - \hat{p}_1)}{c_1} + \frac{\hat{p}_2 (1 - \hat{p}_2)}{c_2} } } The p-value is then computed as: .. math:: \sum \binom{c_1}{x_{11}} \binom{c_2}{x_{12}} \pi^{x_{11} + x_{12}} (1 - \pi)^{t - x_{11} - x_{12}} where the sum is over all 2x2 contingency tables :math:`X` such that: * :math:`T(X) \leq T(X_0)` when `alternative` = "less", * :math:`T(X) \geq T(X_0)` when `alternative` = "greater", or * :math:`T(X) \geq |T(X_0)|` when `alternative` = "two-sided". Above, :math:`c_1, c_2` are the sum of the columns 1 and 2, and :math:`t` the total (sum of the 4 sample's element). The returned p-value is the maximum p-value taken over the nuisance parameter :math:`\pi`, where :math:`0 \leq \pi \leq 1`. This function's complexity is :math:`O(n c_1 c_2)`, where `n` is the number of sample points. References ---------- .. [1] Barnard, G. A. "Significance Tests for 2x2 Tables". *Biometrika*. 34.1/2 (1947): 123-138. :doi:`dpgkg3` .. [2] Mehta, Cyrus R., and Pralay Senchaudhuri. "Conditional versus unconditional exact tests for comparing two binomials." *Cytel Software Corporation* 675 (2003): 1-5. .. [3] "Wald Test". *Wikipedia*. https://en.wikipedia.org/wiki/Wald_test Examples -------- An example use of Barnard's test is presented in [2]_. Consider the following example of a vaccine efficacy study (Chan, 1998). In a randomized clinical trial of 30 subjects, 15 were inoculated with a recombinant DNA influenza vaccine and the 15 were inoculated with a placebo. Twelve of the 15 subjects in the placebo group (80%) eventually became infected with influenza whereas for the vaccine group, only 7 of the 15 subjects (47%) became infected. The data are tabulated as a 2 x 2 table:: Vaccine Placebo Yes 7 12 No 8 3 When working with statistical hypothesis testing, we usually use a threshold probability or significance level upon which we decide to reject the null hypothesis :math:`H_0`. Suppose we choose the common significance level of 5%. Our alternative hypothesis is that the vaccine will lower the chance of becoming infected with the virus; that is, the probability :math:`p_1` of catching the virus with the vaccine will be *less than* the probability :math:`p_2` of catching the virus without the vaccine. Therefore, we call `barnard_exact` with the ``alternative="less"`` option: >>> import scipy.stats as stats >>> res = stats.barnard_exact([[7, 12], [8, 3]], alternative="less") >>> res.statistic -1.894... >>> res.pvalue 0.03407... Under the null hypothesis that the vaccine will not lower the chance of becoming infected, the probability of obtaining test results at least as extreme as the observed data is approximately 3.4%. Since this p-value is less than our chosen significance level, we have evidence to reject :math:`H_0` in favor of the alternative. Suppose we had used Fisher's exact test instead: >>> _, pvalue = stats.fisher_exact([[7, 12], [8, 3]], alternative="less") >>> pvalue 0.0640... With the same threshold significance of 5%, we would not have been able to reject the null hypothesis in favor of the alternative. As stated in [2]_, Barnard's test is uniformly more powerful than Fisher's exact test because Barnard's test does not condition on any margin. Fisher's test should only be used when both sets of marginals are fixed. """ if n <= 0: raise ValueError( "Number of points `n` must be strictly positive, " f"found {n!r}" ) table = np.asarray(table, dtype=np.int64) if not table.shape == (2, 2): raise ValueError("The input `table` must be of shape (2, 2).") if np.any(table < 0): raise ValueError("All values in `table` must be nonnegative.") if 0 in table.sum(axis=0): # If both values in column are zero, the p-value is 1 and # the score's statistic is NaN. return BarnardExactResult(np.nan, 1.0) total_col_1, total_col_2 = table.sum(axis=0) x1 = np.arange(total_col_1 + 1, dtype=np.int64).reshape(-1, 1) x2 = np.arange(total_col_2 + 1, dtype=np.int64).reshape(1, -1) # We need to calculate the wald statistics for each combination of x1 and # x2. p1, p2 = x1 / total_col_1, x2 / total_col_2 if pooled: p = (x1 + x2) / (total_col_1 + total_col_2) variances = p * (1 - p) * (1 / total_col_1 + 1 / total_col_2) else: variances = p1 * (1 - p1) / total_col_1 + p2 * (1 - p2) / total_col_2 # To avoid warning when dividing by 0 with np.errstate(divide="ignore", invalid="ignore"): wald_statistic = np.divide((p1 - p2), np.sqrt(variances)) wald_statistic[p1 == p2] = 0 # Removing NaN values wald_stat_obs = wald_statistic[table[0, 0], table[0, 1]] if alternative == "two-sided": index_arr = np.abs(wald_statistic) >= abs(wald_stat_obs) elif alternative == "less": index_arr = wald_statistic <= wald_stat_obs elif alternative == "greater": index_arr = wald_statistic >= wald_stat_obs else: msg = ( "`alternative` should be one of {'two-sided', 'less', 'greater'}," f" found {alternative!r}" ) raise ValueError(msg) x1_sum_x2 = x1 + x2 x1_log_comb = _compute_log_combinations(total_col_1) x2_log_comb = _compute_log_combinations(total_col_2) x1_sum_x2_log_comb = x1_log_comb[x1] + x2_log_comb[x2] result = shgo( _get_binomial_log_p_value_with_nuisance_param, args=(x1_sum_x2, x1_sum_x2_log_comb, index_arr), bounds=((0, 1),), n=n, sampling_method="sobol", ) # result.fun is the negative log pvalue and therefore needs to be # changed before return p_value = np.clip(np.exp(-result.fun), a_min=0, a_max=1) return BarnardExactResult(wald_stat_obs, p_value) BoschlooExactResult = make_dataclass( "BoschlooExactResult", [("statistic", float), ("pvalue", float)] ) def boschloo_exact(table, alternative="two-sided", n=32): r"""Perform Boschloo's exact test on a 2x2 contingency table. Parameters ---------- table : array_like of ints A 2x2 contingency table. Elements should be non-negative integers. alternative : {'two-sided', 'less', 'greater'}, optional Defines the null and alternative hypotheses. Default is 'two-sided'. Please see explanations in the Notes section below. n : int, optional Number of sampling points used in the construction of the sampling method. Note that this argument will automatically be converted to the next higher power of 2 since `scipy.stats.qmc.Sobol` is used to select sample points. Default is 32. Must be positive. In most cases, 32 points is enough to reach good precision. More points comes at performance cost. Returns ------- ber : BoschlooExactResult A result object with the following attributes. statistic : float The statistic used in Boschloo's test; that is, the p-value from Fisher's exact test. pvalue : float P-value, the probability of obtaining a distribution at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. See Also -------- chi2_contingency : Chi-square test of independence of variables in a contingency table. fisher_exact : Fisher exact test on a 2x2 contingency table. barnard_exact : Barnard's exact test, which is a more powerful alternative than Fisher's exact test for 2x2 contingency tables. Notes ----- Boschloo's test is an exact test used in the analysis of contingency tables. It examines the association of two categorical variables, and is a uniformly more powerful alternative to Fisher's exact test for 2x2 contingency tables. Let's define :math:`X_0` a 2x2 matrix representing the observed sample, where each column stores the binomial experiment, as in the example below. Let's also define :math:`p_1, p_2` the theoretical binomial probabilities for :math:`x_{11}` and :math:`x_{12}`. When using Boschloo exact test, we can assert three different null hypotheses : - :math:`H_0 : p_1=p_2` versus :math:`H_1 : p_1 < p_2`, with `alternative` = "less" - :math:`H_0 : p_1=p_2` versus :math:`H_1 : p_1 > p_2`, with `alternative` = "greater" - :math:`H_0 : p_1=p_2` versus :math:`H_1 : p_1 \neq p_2`, with `alternative` = "two-sided" (default one) Boschloo's exact test uses the p-value of Fisher's exact test as a statistic, and Boschloo's p-value is the probability under the null hypothesis of observing such an extreme value of this statistic. Boschloo's and Barnard's are both more powerful than Fisher's exact test. .. versionadded:: 1.7.0 References ---------- .. [1] R.D. Boschloo. "Raised conditional level of significance for the 2 x 2-table when testing the equality of two probabilities", Statistica Neerlandica, 24(1), 1970 .. [2] "Boschloo's test", Wikipedia, https://en.wikipedia.org/wiki/Boschloo%27s_test .. [3] Lise M. Saari et al. "Employee attitudes and job satisfaction", Human Resource Management, 43(4), 395-407, 2004, :doi:`10.1002/hrm.20032`. Examples -------- In the following example, we consider the article "Employee attitudes and job satisfaction" [3]_ which reports the results of a survey from 63 scientists and 117 college professors. Of the 63 scientists, 31 said they were very satisfied with their jobs, whereas 74 of the college professors were very satisfied with their work. Is this significant evidence that college professors are happier with their work than scientists? The following table summarizes the data mentioned above:: college professors scientists Very Satisfied 74 31 Dissatisfied 43 32 When working with statistical hypothesis testing, we usually use a threshold probability or significance level upon which we decide to reject the null hypothesis :math:`H_0`. Suppose we choose the common significance level of 5%. Our alternative hypothesis is that college professors are truly more satisfied with their work than scientists. Therefore, we expect :math:`p_1` the proportion of very satisfied college professors to be greater than :math:`p_2`, the proportion of very satisfied scientists. We thus call `boschloo_exact` with the ``alternative="greater"`` option: >>> import scipy.stats as stats >>> res = stats.boschloo_exact([[74, 31], [43, 32]], alternative="greater") >>> res.statistic 0.0483... >>> res.pvalue 0.0355... Under the null hypothesis that scientists are happier in their work than college professors, the probability of obtaining test results at least as extreme as the observed data is approximately 3.55%. Since this p-value is less than our chosen significance level, we have evidence to reject :math:`H_0` in favor of the alternative hypothesis. """ hypergeom = distributions.hypergeom if n <= 0: raise ValueError( "Number of points `n` must be strictly positive," f" found {n!r}" ) table = np.asarray(table, dtype=np.int64) if not table.shape == (2, 2): raise ValueError("The input `table` must be of shape (2, 2).") if np.any(table < 0): raise ValueError("All values in `table` must be nonnegative.") if 0 in table.sum(axis=0): # If both values in column are zero, the p-value is 1 and # the score's statistic is NaN. return BoschlooExactResult(np.nan, np.nan) total_col_1, total_col_2 = table.sum(axis=0) total = total_col_1 + total_col_2 x1 = np.arange(total_col_1 + 1, dtype=np.int64).reshape(1, -1) x2 = np.arange(total_col_2 + 1, dtype=np.int64).reshape(-1, 1) x1_sum_x2 = x1 + x2 if alternative == 'less': pvalues = hypergeom.cdf(x1, total, x1_sum_x2, total_col_1).T elif alternative == 'greater': # Same formula as the 'less' case, but with the second column. pvalues = hypergeom.cdf(x2, total, x1_sum_x2, total_col_2).T elif alternative == 'two-sided': boschloo_less = boschloo_exact(table, alternative="less", n=n) boschloo_greater = boschloo_exact(table, alternative="greater", n=n) res = ( boschloo_less if boschloo_less.pvalue < boschloo_greater.pvalue else boschloo_greater ) # Two-sided p-value is defined as twice the minimum of the one-sided # p-values pvalue = 2 * res.pvalue return BoschlooExactResult(res.statistic, pvalue) else: msg = ( f"`alternative` should be one of {'two-sided', 'less', 'greater'}," f" found {alternative!r}" ) raise ValueError(msg) fisher_stat = pvalues[table[0, 0], table[0, 1]] # fisher_stat * (1+1e-13) guards us from small numerical error. It is # equivalent to np.isclose with relative tol of 1e-13 and absolute tol of 0 # For more throughout explanations, see gh-14178 index_arr = pvalues <= fisher_stat * (1+1e-13) x1, x2, x1_sum_x2 = x1.T, x2.T, x1_sum_x2.T x1_log_comb = _compute_log_combinations(total_col_1) x2_log_comb = _compute_log_combinations(total_col_2) x1_sum_x2_log_comb = x1_log_comb[x1] + x2_log_comb[x2] result = shgo( _get_binomial_log_p_value_with_nuisance_param, args=(x1_sum_x2, x1_sum_x2_log_comb, index_arr), bounds=((0, 1),), n=n, sampling_method="sobol", ) # result.fun is the negative log pvalue and therefore needs to be # changed before return p_value = np.clip(np.exp(-result.fun), a_min=0, a_max=1) return BoschlooExactResult(fisher_stat, p_value) def _get_binomial_log_p_value_with_nuisance_param( nuisance_param, x1_sum_x2, x1_sum_x2_log_comb, index_arr ): r""" Compute the log pvalue in respect of a nuisance parameter considering a 2x2 sample space. Parameters ---------- nuisance_param : float nuisance parameter used in the computation of the maximisation of the p-value. Must be between 0 and 1 x1_sum_x2 : ndarray Sum of x1 and x2 inside barnard_exact x1_sum_x2_log_comb : ndarray sum of the log combination of x1 and x2 index_arr : ndarray of boolean Returns ------- p_value : float Return the maximum p-value considering every nuisance paramater between 0 and 1 Notes ----- Both Barnard's test and Boschloo's test iterate over a nuisance parameter :math:`\pi \in [0, 1]` to find the maximum p-value. To search this maxima, this function return the negative log pvalue with respect to the nuisance parameter passed in params. This negative log p-value is then used in `shgo` to find the minimum negative pvalue which is our maximum pvalue. Also, to compute the different combination used in the p-values' computation formula, this function uses `gammaln` which is more tolerant for large value than `scipy.special.comb`. `gammaln` gives a log combination. For the little precision loss, performances are improved a lot. """ t1, t2 = x1_sum_x2.shape n = t1 + t2 - 2 with np.errstate(divide="ignore", invalid="ignore"): log_nuisance = np.log( nuisance_param, out=np.zeros_like(nuisance_param), where=nuisance_param >= 0, ) log_1_minus_nuisance = np.log( 1 - nuisance_param, out=np.zeros_like(nuisance_param), where=1 - nuisance_param >= 0, ) nuisance_power_x1_x2 = log_nuisance * x1_sum_x2 nuisance_power_x1_x2[(x1_sum_x2 == 0)[:, :]] = 0 nuisance_power_n_minus_x1_x2 = log_1_minus_nuisance * (n - x1_sum_x2) nuisance_power_n_minus_x1_x2[(x1_sum_x2 == n)[:, :]] = 0 tmp_log_values_arr = ( x1_sum_x2_log_comb + nuisance_power_x1_x2 + nuisance_power_n_minus_x1_x2 ) tmp_values_from_index = tmp_log_values_arr[index_arr] # To avoid dividing by zero in log function and getting inf value, # values are centered according to the max max_value = tmp_values_from_index.max() # To have better result's precision, the log pvalue is taken here. # Indeed, pvalue is included inside [0, 1] interval. Passing the # pvalue to log makes the interval a lot bigger ([-inf, 0]), and thus # help us to achieve better precision with np.errstate(divide="ignore", invalid="ignore"): log_probs = np.exp(tmp_values_from_index - max_value).sum() log_pvalue = max_value + np.log( log_probs, out=np.full_like(log_probs, -np.inf), where=log_probs > 0, ) # Since shgo find the minima, minus log pvalue is returned return -log_pvalue def _pval_cvm_2samp_exact(s, nx, ny): """ Compute the exact p-value of the Cramer-von Mises two-sample test for a given value s (float) of the test statistic by enumerating all possible combinations. nx and ny are the sizes of the samples. """ rangex = np.arange(nx) rangey = np.arange(ny) us = [] # x and y are all possible partitions of ranks from 0 to nx + ny - 1 # into two sets of length nx and ny # Here, ranks are from 0 to nx + ny - 1 instead of 1 to nx + ny, but # this does not change the value of the statistic. for x, y in _all_partitions(nx, ny): # compute the statistic u = nx * np.sum((x - rangex)**2) u += ny * np.sum((y - rangey)**2) us.append(u) # compute the values of u and the frequencies u, cnt = np.unique(us, return_counts=True) return np.sum(cnt[u >= s]) / np.sum(cnt) def cramervonmises_2samp(x, y, method='auto'): """Perform the two-sample Cramér-von Mises test for goodness of fit. This is the two-sample version of the Cramér-von Mises test ([1]_): for two independent samples :math:`X_1, ..., X_n` and :math:`Y_1, ..., Y_m`, the null hypothesis is that the samples come from the same (unspecified) continuous distribution. Parameters ---------- x : array_like A 1-D array of observed values of the random variables :math:`X_i`. y : array_like A 1-D array of observed values of the random variables :math:`Y_i`. method : {'auto', 'asymptotic', 'exact'}, optional The method used to compute the p-value, see Notes for details. The default is 'auto'. Returns ------- res : object with attributes statistic : float Cramér-von Mises statistic. pvalue : float The p-value. See Also -------- cramervonmises, anderson_ksamp, epps_singleton_2samp, ks_2samp Notes ----- .. versionadded:: 1.7.0 The statistic is computed according to equation 9 in [2]_. The calculation of the p-value depends on the keyword `method`: - ``asymptotic``: The p-value is approximated by using the limiting distribution of the test statistic. - ``exact``: The exact p-value is computed by enumerating all possible combinations of the test statistic, see [2]_. The exact calculation will be very slow even for moderate sample sizes as the number of combinations increases rapidly with the size of the samples. If ``method=='auto'``, the exact approach is used if both samples contain less than 10 observations, otherwise the asymptotic distribution is used. If the underlying distribution is not continuous, the p-value is likely to be conservative (Section 6.2 in [3]_). When ranking the data to compute the test statistic, midranks are used if there are ties. References ---------- .. [1] https://en.wikipedia.org/wiki/Cramer-von_Mises_criterion .. [2] Anderson, T.W. (1962). On the distribution of the two-sample Cramer-von-Mises criterion. The Annals of Mathematical Statistics, pp. 1148-1159. .. [3] Conover, W.J., Practical Nonparametric Statistics, 1971. Examples -------- Suppose we wish to test whether two samples generated by ``scipy.stats.norm.rvs`` have the same distribution. We choose a significance level of alpha=0.05. >>> from scipy import stats >>> rng = np.random.default_rng() >>> x = stats.norm.rvs(size=100, random_state=rng) >>> y = stats.norm.rvs(size=70, random_state=rng) >>> res = stats.cramervonmises_2samp(x, y) >>> res.statistic, res.pvalue (0.29376470588235293, 0.1412873014573014) The p-value exceeds our chosen significance level, so we do not reject the null hypothesis that the observed samples are drawn from the same distribution. For small sample sizes, one can compute the exact p-values: >>> x = stats.norm.rvs(size=7, random_state=rng) >>> y = stats.t.rvs(df=2, size=6, random_state=rng) >>> res = stats.cramervonmises_2samp(x, y, method='exact') >>> res.statistic, res.pvalue (0.197802197802198, 0.31643356643356646) The p-value based on the asymptotic distribution is a good approximation even though the sample size is small. >>> res = stats.cramervonmises_2samp(x, y, method='asymptotic') >>> res.statistic, res.pvalue (0.197802197802198, 0.2966041181527128) Independent of the method, one would not reject the null hypothesis at the chosen significance level in this example. """ xa = np.sort(np.asarray(x)) ya = np.sort(np.asarray(y)) if xa.size <= 1 or ya.size <= 1: raise ValueError('x and y must contain at least two observations.') if xa.ndim > 1 or ya.ndim > 1: raise ValueError('The samples must be one-dimensional.') if method not in ['auto', 'exact', 'asymptotic']: raise ValueError('method must be either auto, exact or asymptotic.') nx = len(xa) ny = len(ya) if method == 'auto': if max(nx, ny) > 10: method = 'asymptotic' else: method = 'exact' # get ranks of x and y in the pooled sample z = np.concatenate([xa, ya]) # in case of ties, use midrank (see [1]) r = scipy.stats.rankdata(z, method='average') rx = r[:nx] ry = r[nx:] # compute U (eq. 10 in [2]) u = nx * np.sum((rx - np.arange(1, nx+1))**2) u += ny * np.sum((ry - np.arange(1, ny+1))**2) # compute T (eq. 9 in [2]) k, N = nx*ny, nx + ny t = u / (k*N) - (4*k - 1)/(6*N) if method == 'exact': p = _pval_cvm_2samp_exact(u, nx, ny) else: # compute expected value and variance of T (eq. 11 and 14 in [2]) et = (1 + 1/N)/6 vt = (N+1) * (4*k*N - 3*(nx**2 + ny**2) - 2*k) vt = vt / (45 * N**2 * 4 * k) # computed the normalized statistic (eq. 15 in [2]) tn = 1/6 + (t - et) / np.sqrt(45 * vt) # approximate distribution of tn with limiting distribution # of the one-sample test statistic # if tn < 0.003, the _cdf_cvm_inf(tn) < 1.28*1e-18, return 1.0 directly if tn < 0.003: p = 1.0 else: p = max(0, 1. - _cdf_cvm_inf(tn)) return CramerVonMisesResult(statistic=t, pvalue=p)