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"""Some functions for working with contingency tables (i.e. cross tabulations). """
"""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 -------- >>> 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
""" 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 -------- >>> observed = np.array([[10, 10, 20],[20, 20, 20]]) >>> from scipy.stats import expected_freq >>> 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
"""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 `power_divergence` for details.
Returns ------- chi2 : float The test statistic. p : float The p-value of the test dof : int Degrees of freedom expected : ndarray, same shape as `observed` The expected frequencies, based on the marginal sums of the table.
See Also -------- contingency.expected_freq fisher_exact chisquare power_divergence
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 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 stats.chisquare. That is, if one calls::
chi2, p, dof, ex = chi2_contingency(obs, correction=False)
then the following is true::
(chi2, p) == 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", http://en.wikipedia.org/wiki/Contingency_table .. [2] "Pearson's chi-squared test", http://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.
Examples -------- A two-way example (2 x 3):
>>> from scipy.stats import chi2_contingency >>> obs = np.array([[10, 10, 20], [20, 20, 20]]) >>> chi2_contingency(obs) (2.7777777777777777, 0.24935220877729619, 2, 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.
>>> g, p, dof, expctd = chi2_contingency(obs, lambda_="log-likelihood") >>> g, p (2.7688587616781319, 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]]]]) >>> chi2_contingency(obs) (8.7584514426741897, 0.64417725029295503, 11, array([[[[ 14.15462386, 14.15462386], [ 16.49423111, 16.49423111]], [[ 11.2461395 , 11.2461395 ], [ 13.10500554, 13.10500554]]], [[[ 19.5591166 , 19.5591166 ], [ 22.79202844, 22.79202844]], [[ 15.54012004, 15.54012004], [ 18.10873492, 18.10873492]]]])) """ 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.where(expected == 0)))[0] raise ValueError("The internally computed table of expected " "frequencies has a zero element at %s." % (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. observed = observed + 0.5 * np.sign(expected - observed)
chi2, p = power_divergence(observed, expected, ddof=observed.size - 1 - dof, axis=None, lambda_=lambda_)
return chi2, p, dof, expected |