""" Contingency table functions (:mod:`scipy.stats.contingency`) ============================================================ Functions for creating and analyzing contingency tables. .. currentmodule:: scipy.stats.contingency .. autosummary:: :toctree: generated/ chi2_contingency relative_risk odds_ratio crosstab association expected_freq margins """ from functools import reduce import math import numpy as np from ._stats_py import power_divergence from ._relative_risk import relative_risk from ._crosstab import crosstab from ._odds_ratio import odds_ratio from scipy._lib._bunch import _make_tuple_bunch __all__ = ['margins', 'expected_freq', 'chi2_contingency', 'crosstab', 'association', 'relative_risk', 'odds_ratio'] def margins(a): """Return a list of the marginal sums of the array `a`. Parameters ---------- a : ndarray The array for which to compute the marginal sums. Returns ------- margsums : list of ndarrays A list of length `a.ndim`. `margsums[k]` is the result of summing `a` over all axes except `k`; it has the same number of dimensions as `a`, but the length of each axis except axis `k` will be 1. Examples -------- >>> import numpy as np >>> from scipy.stats.contingency import margins >>> a = np.arange(12).reshape(2, 6) >>> a array([[ 0, 1, 2, 3, 4, 5], [ 6, 7, 8, 9, 10, 11]]) >>> m0, m1 = margins(a) >>> m0 array([[15], [51]]) >>> m1 array([[ 6, 8, 10, 12, 14, 16]]) >>> b = np.arange(24).reshape(2,3,4) >>> m0, m1, m2 = margins(b) >>> m0 array([[[ 66]], [[210]]]) >>> m1 array([[[ 60], [ 92], [124]]]) >>> m2 array([[[60, 66, 72, 78]]]) """ margsums = [] ranged = list(range(a.ndim)) for k in ranged: marg = np.apply_over_axes(np.sum, a, [j for j in ranged if j != k]) margsums.append(marg) return margsums def expected_freq(observed): """ Compute the expected frequencies from a contingency table. Given an n-dimensional contingency table of observed frequencies, compute the expected frequencies for the table based on the marginal sums under the assumption that the groups associated with each dimension are independent. Parameters ---------- observed : array_like The table of observed frequencies. (While this function can handle a 1-D array, that case is trivial. Generally `observed` is at least 2-D.) Returns ------- expected : ndarray of float64 The expected frequencies, based on the marginal sums of the table. Same shape as `observed`. Examples -------- >>> import numpy as np >>> from scipy.stats.contingency import expected_freq >>> observed = np.array([[10, 10, 20],[20, 20, 20]]) >>> expected_freq(observed) array([[ 12., 12., 16.], [ 18., 18., 24.]]) """ # Typically `observed` is an integer array. If `observed` has a large # number of dimensions or holds large values, some of the following # computations may overflow, so we first switch to floating point. observed = np.asarray(observed, dtype=np.float64) # Create a list of the marginal sums. margsums = margins(observed) # Create the array of expected frequencies. The shapes of the # marginal sums returned by apply_over_axes() are just what we # need for broadcasting in the following product. d = observed.ndim expected = reduce(np.multiply, margsums) / observed.sum() ** (d - 1) return expected Chi2ContingencyResult = _make_tuple_bunch( 'Chi2ContingencyResult', ['statistic', 'pvalue', 'dof', 'expected_freq'], [] ) def chi2_contingency(observed, correction=True, lambda_=None): """Chi-square test of independence of variables in a contingency table. This function computes the chi-square statistic and p-value for the hypothesis test of independence of the observed frequencies in the contingency table [1]_ `observed`. The expected frequencies are computed based on the marginal sums under the assumption of independence; see `scipy.stats.contingency.expected_freq`. The number of degrees of freedom is (expressed using numpy functions and attributes):: dof = observed.size - sum(observed.shape) + observed.ndim - 1 Parameters ---------- observed : array_like The contingency table. The table contains the observed frequencies (i.e. number of occurrences) in each category. In the two-dimensional case, the table is often described as an "R x C table". correction : bool, optional If True, *and* the degrees of freedom is 1, apply Yates' correction for continuity. The effect of the correction is to adjust each observed value by 0.5 towards the corresponding expected value. lambda_ : float or str, optional By default, the statistic computed in this test is Pearson's chi-squared statistic [2]_. `lambda_` allows a statistic from the Cressie-Read power divergence family [3]_ to be used instead. See `scipy.stats.power_divergence` for details. Returns ------- res : Chi2ContingencyResult An object containing attributes: statistic : float The test statistic. pvalue : float The p-value of the test. dof : int The degrees of freedom. expected_freq : ndarray, same shape as `observed` The expected frequencies, based on the marginal sums of the table. See Also -------- scipy.stats.contingency.expected_freq scipy.stats.fisher_exact scipy.stats.chisquare scipy.stats.power_divergence scipy.stats.barnard_exact scipy.stats.boschloo_exact Notes ----- An often quoted guideline for the validity of this calculation is that the test should be used only if the observed and expected frequencies in each cell are at least 5. This is a test for the independence of different categories of a population. The test is only meaningful when the dimension of `observed` is two or more. Applying the test to a one-dimensional table will always result in `expected` equal to `observed` and a chi-square statistic equal to 0. This function does not handle masked arrays, because the calculation does not make sense with missing values. Like `scipy.stats.chisquare`, this function computes a chi-square statistic; the convenience this function provides is to figure out the expected frequencies and degrees of freedom from the given contingency table. If these were already known, and if the Yates' correction was not required, one could use `scipy.stats.chisquare`. That is, if one calls:: res = chi2_contingency(obs, correction=False) then the following is true:: (res.statistic, res.pvalue) == stats.chisquare(obs.ravel(), f_exp=ex.ravel(), ddof=obs.size - 1 - dof) The `lambda_` argument was added in version 0.13.0 of scipy. References ---------- .. [1] "Contingency table", https://en.wikipedia.org/wiki/Contingency_table .. [2] "Pearson's chi-squared test", https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test .. [3] Cressie, N. and Read, T. R. C., "Multinomial Goodness-of-Fit Tests", J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984), pp. 440-464. .. [4] Berger, Jeffrey S. et al. "Aspirin for the Primary Prevention of Cardiovascular Events in Women and Men: A Sex-Specific Meta-analysis of Randomized Controlled Trials." JAMA, 295(3):306-313, :doi:`10.1001/jama.295.3.306`, 2006. Examples -------- In [4]_, the use of aspirin to prevent cardiovascular events in women and men was investigated. The study notably concluded: ...aspirin therapy reduced the risk of a composite of cardiovascular events due to its effect on reducing the risk of ischemic stroke in women [...] The article lists studies of various cardiovascular events. Let's focus on the ischemic stoke in women. The following table summarizes the results of the experiment in which participants took aspirin or a placebo on a regular basis for several years. Cases of ischemic stroke were recorded:: Aspirin Control/Placebo Ischemic stroke 176 230 No stroke 21035 21018 Is there evidence that the aspirin reduces the risk of ischemic stroke? We begin by formulating a null hypothesis :math:`H_0`: The effect of aspirin is equivalent to that of placebo. Let's assess the plausibility of this hypothesis with a chi-square test. >>> import numpy as np >>> from scipy.stats import chi2_contingency >>> table = np.array([[176, 230], [21035, 21018]]) >>> res = chi2_contingency(table) >>> res.statistic 6.892569132546561 >>> res.pvalue 0.008655478161175739 Using a significance level of 5%, we would reject the null hypothesis in favor of the alternative hypothesis: "the effect of aspirin is not equivalent to the effect of placebo". Because `scipy.stats.contingency.chi2_contingency` performs a two-sided test, the alternative hypothesis does not indicate the direction of the effect. We can use `stats.contingency.odds_ratio` to support the conclusion that aspirin *reduces* the risk of ischemic stroke. Below are further examples showing how larger contingency tables can be tested. A two-way example (2 x 3): >>> obs = np.array([[10, 10, 20], [20, 20, 20]]) >>> res = chi2_contingency(obs) >>> res.statistic 2.7777777777777777 >>> res.pvalue 0.24935220877729619 >>> res.dof 2 >>> res.expected_freq array([[ 12., 12., 16.], [ 18., 18., 24.]]) Perform the test using the log-likelihood ratio (i.e. the "G-test") instead of Pearson's chi-squared statistic. >>> res = chi2_contingency(obs, lambda_="log-likelihood") >>> res.statistic 2.7688587616781319 >>> res.pvalue 0.25046668010954165 A four-way example (2 x 2 x 2 x 2): >>> obs = np.array( ... [[[[12, 17], ... [11, 16]], ... [[11, 12], ... [15, 16]]], ... [[[23, 15], ... [30, 22]], ... [[14, 17], ... [15, 16]]]]) >>> res = chi2_contingency(obs) >>> res.statistic 8.7584514426741897 >>> res.pvalue 0.64417725029295503 """ observed = np.asarray(observed) if np.any(observed < 0): raise ValueError("All values in `observed` must be nonnegative.") if observed.size == 0: raise ValueError("No data; `observed` has size 0.") expected = expected_freq(observed) if np.any(expected == 0): # Include one of the positions where expected is zero in # the exception message. zeropos = list(zip(*np.nonzero(expected == 0)))[0] raise ValueError("The internally computed table of expected " "frequencies has a zero element at {}.".format(zeropos)) # The degrees of freedom dof = expected.size - sum(expected.shape) + expected.ndim - 1 if dof == 0: # Degenerate case; this occurs when `observed` is 1D (or, more # generally, when it has only one nontrivial dimension). In this # case, we also have observed == expected, so chi2 is 0. chi2 = 0.0 p = 1.0 else: if dof == 1 and correction: # Adjust `observed` according to Yates' correction for continuity. # Magnitude of correction no bigger than difference; see gh-13875 diff = expected - observed direction = np.sign(diff) magnitude = np.minimum(0.5, np.abs(diff)) observed = observed + magnitude * direction chi2, p = power_divergence(observed, expected, ddof=observed.size - 1 - dof, axis=None, lambda_=lambda_) return Chi2ContingencyResult(chi2, p, dof, expected) def association(observed, method="cramer", correction=False, lambda_=None): """Calculates degree of association between two nominal variables. The function provides the option for computing one of three measures of association between two nominal variables from the data given in a 2d contingency table: Tschuprow's T, Pearson's Contingency Coefficient and Cramer's V. Parameters ---------- observed : array-like The array of observed values method : {"cramer", "tschuprow", "pearson"} (default = "cramer") The association test statistic. correction : bool, optional Inherited from `scipy.stats.contingency.chi2_contingency()` lambda_ : float or str, optional Inherited from `scipy.stats.contingency.chi2_contingency()` Returns ------- statistic : float Value of the test statistic Notes ----- Cramer's V, Tschuprow's T and Pearson's Contingency Coefficient, all measure the degree to which two nominal or ordinal variables are related, or the level of their association. This differs from correlation, although many often mistakenly consider them equivalent. Correlation measures in what way two variables are related, whereas, association measures how related the variables are. As such, association does not subsume independent variables, and is rather a test of independence. A value of 1.0 indicates perfect association, and 0.0 means the variables have no association. Both the Cramer's V and Tschuprow's T are extensions of the phi coefficient. Moreover, due to the close relationship between the Cramer's V and Tschuprow's T the returned values can often be similar or even equivalent. They are likely to diverge more as the array shape diverges from a 2x2. References ---------- .. [1] "Tschuprow's T", https://en.wikipedia.org/wiki/Tschuprow's_T .. [2] Tschuprow, A. A. (1939) Principles of the Mathematical Theory of Correlation; translated by M. Kantorowitsch. W. Hodge & Co. .. [3] "Cramer's V", https://en.wikipedia.org/wiki/Cramer's_V .. [4] "Nominal Association: Phi and Cramer's V", http://www.people.vcu.edu/~pdattalo/702SuppRead/MeasAssoc/NominalAssoc.html .. [5] Gingrich, Paul, "Association Between Variables", http://uregina.ca/~gingrich/ch11a.pdf Examples -------- An example with a 4x2 contingency table: >>> import numpy as np >>> from scipy.stats.contingency import association >>> obs4x2 = np.array([[100, 150], [203, 322], [420, 700], [320, 210]]) Pearson's contingency coefficient >>> association(obs4x2, method="pearson") 0.18303298140595667 Cramer's V >>> association(obs4x2, method="cramer") 0.18617813077483678 Tschuprow's T >>> association(obs4x2, method="tschuprow") 0.14146478765062995 """ arr = np.asarray(observed) if not np.issubdtype(arr.dtype, np.integer): raise ValueError("`observed` must be an integer array.") if len(arr.shape) != 2: raise ValueError("method only accepts 2d arrays") chi2_stat = chi2_contingency(arr, correction=correction, lambda_=lambda_) phi2 = chi2_stat.statistic / arr.sum() n_rows, n_cols = arr.shape if method == "cramer": value = phi2 / min(n_cols - 1, n_rows - 1) elif method == "tschuprow": value = phi2 / math.sqrt((n_rows - 1) * (n_cols - 1)) elif method == 'pearson': value = phi2 / (1 + phi2) else: raise ValueError("Invalid argument value: 'method' argument must " "be 'cramer', 'tschuprow', or 'pearson'") return math.sqrt(value)