""" Test functions for stats module

    WRITTEN BY LOUIS LUANGKESORN <lluang@yahoo.com> FOR THE STATS MODULE
    BASED ON WILKINSON'S STATISTICS QUIZ
    https://www.stanford.edu/~clint/bench/wilk.txt

    Additional tests by a host of SciPy developers.
"""
from __future__ import division, print_function, absolute_import

import os
import sys
import warnings
from collections import namedtuple

from numpy.testing import (assert_, assert_equal,
                           assert_almost_equal, assert_array_almost_equal,
                           assert_array_equal, assert_approx_equal,
                           assert_allclose)
import pytest
from pytest import raises as assert_raises
from scipy._lib._numpy_compat import suppress_warnings
import numpy.ma.testutils as mat
from numpy import array, arange, float32, float64, power
import numpy as np

import scipy.stats as stats
import scipy.stats.mstats as mstats
import scipy.stats.mstats_basic as mstats_basic
from scipy._lib._version import NumpyVersion
from scipy._lib.six import xrange
from .common_tests import check_named_results

""" Numbers in docstrings beginning with 'W' refer to the section numbers
    and headings found in the STATISTICS QUIZ of Leland Wilkinson.  These are
    considered to be essential functionality.  True testing and
    evaluation of a statistics package requires use of the
    NIST Statistical test data.  See McCoullough(1999) Assessing The Reliability
    of Statistical Software for a test methodology and its
    implementation in testing SAS, SPSS, and S-Plus
"""

#  Datasets
#  These data sets are from the nasty.dat sets used by Wilkinson
#  For completeness, I should write the relevant tests and count them as failures
#  Somewhat acceptable, since this is still beta software.  It would count as a
#  good target for 1.0 status
X = array([1,2,3,4,5,6,7,8,9], float)
ZERO = array([0,0,0,0,0,0,0,0,0], float)
BIG = array([99999991,99999992,99999993,99999994,99999995,99999996,99999997,
             99999998,99999999], float)
LITTLE = array([0.99999991,0.99999992,0.99999993,0.99999994,0.99999995,0.99999996,
                0.99999997,0.99999998,0.99999999], float)
HUGE = array([1e+12,2e+12,3e+12,4e+12,5e+12,6e+12,7e+12,8e+12,9e+12], float)
TINY = array([1e-12,2e-12,3e-12,4e-12,5e-12,6e-12,7e-12,8e-12,9e-12], float)
ROUND = array([0.5,1.5,2.5,3.5,4.5,5.5,6.5,7.5,8.5], float)

class TestTrimmedStats(object):
    # TODO: write these tests to handle missing values properly
    dprec = np.finfo(np.float64).precision

    def test_tmean(self):
        y = stats.tmean(X, (2, 8), (True, True))
        assert_approx_equal(y, 5.0, significant=self.dprec)

        y1 = stats.tmean(X, limits=(2, 8), inclusive=(False, False))
        y2 = stats.tmean(X, limits=None)
        assert_approx_equal(y1, y2, significant=self.dprec)

    def test_tvar(self):
        y = stats.tvar(X, limits=(2, 8), inclusive=(True, True))
        assert_approx_equal(y, 4.6666666666666661, significant=self.dprec)

        y = stats.tvar(X, limits=None)
        assert_approx_equal(y, X.var(ddof=1), significant=self.dprec)

    def test_tstd(self):
        y = stats.tstd(X, (2, 8), (True, True))
        assert_approx_equal(y, 2.1602468994692865, significant=self.dprec)

        y = stats.tstd(X, limits=None)
        assert_approx_equal(y, X.std(ddof=1), significant=self.dprec)

    def test_tmin(self):
        assert_equal(stats.tmin(4), 4)

        x = np.arange(10)
        assert_equal(stats.tmin(x), 0)
        assert_equal(stats.tmin(x, lowerlimit=0), 0)
        assert_equal(stats.tmin(x, lowerlimit=0, inclusive=False), 1)

        x = x.reshape((5, 2))
        assert_equal(stats.tmin(x, lowerlimit=0, inclusive=False), [2, 1])
        assert_equal(stats.tmin(x, axis=1), [0, 2, 4, 6, 8])
        assert_equal(stats.tmin(x, axis=None), 0)

        x = np.arange(10.)
        x[9] = np.nan
        with suppress_warnings() as sup:
            r = sup.record(RuntimeWarning, "invalid value*")
            assert_equal(stats.tmin(x), np.nan)
            assert_equal(stats.tmin(x, nan_policy='omit'), 0.)
            assert_raises(ValueError, stats.tmin, x, nan_policy='raise')
            assert_raises(ValueError, stats.tmin, x, nan_policy='foobar')
            msg = "'propagate', 'raise', 'omit'"
            with assert_raises(ValueError, match=msg):
                stats.tmin(x, nan_policy='foo')

    def test_tmax(self):
        assert_equal(stats.tmax(4), 4)

        x = np.arange(10)
        assert_equal(stats.tmax(x), 9)
        assert_equal(stats.tmax(x, upperlimit=9), 9)
        assert_equal(stats.tmax(x, upperlimit=9, inclusive=False), 8)

        x = x.reshape((5, 2))
        assert_equal(stats.tmax(x, upperlimit=9, inclusive=False), [8, 7])
        assert_equal(stats.tmax(x, axis=1), [1, 3, 5, 7, 9])
        assert_equal(stats.tmax(x, axis=None), 9)

        x = np.arange(10.)
        x[6] = np.nan
        with suppress_warnings() as sup:
            r = sup.record(RuntimeWarning, "invalid value*")
            assert_equal(stats.tmax(x), np.nan)
            assert_equal(stats.tmax(x, nan_policy='omit'), 9.)
            assert_raises(ValueError, stats.tmax, x, nan_policy='raise')
            assert_raises(ValueError, stats.tmax, x, nan_policy='foobar')

    def test_tsem(self):
        y = stats.tsem(X, limits=(3, 8), inclusive=(False, True))
        y_ref = np.array([4, 5, 6, 7, 8])
        assert_approx_equal(y, y_ref.std(ddof=1) / np.sqrt(y_ref.size),
                            significant=self.dprec)

        assert_approx_equal(stats.tsem(X, limits=[-1, 10]),
                            stats.tsem(X, limits=None),
                            significant=self.dprec)


class TestCorrPearsonr(object):
    """ W.II.D. Compute a correlation matrix on all the variables.

        All the correlations, except for ZERO and MISS, should be exactly 1.
        ZERO and MISS should have undefined or missing correlations with the
        other variables.  The same should go for SPEARMAN correlations, if
        your program has them.
    """
    def test_pXX(self):
        y = stats.pearsonr(X,X)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pXBIG(self):
        y = stats.pearsonr(X,BIG)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pXLITTLE(self):
        y = stats.pearsonr(X,LITTLE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pXHUGE(self):
        y = stats.pearsonr(X,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pXTINY(self):
        y = stats.pearsonr(X,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pXROUND(self):
        y = stats.pearsonr(X,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pBIGBIG(self):
        y = stats.pearsonr(BIG,BIG)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pBIGLITTLE(self):
        y = stats.pearsonr(BIG,LITTLE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pBIGHUGE(self):
        y = stats.pearsonr(BIG,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pBIGTINY(self):
        y = stats.pearsonr(BIG,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pBIGROUND(self):
        y = stats.pearsonr(BIG,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pLITTLELITTLE(self):
        y = stats.pearsonr(LITTLE,LITTLE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pLITTLEHUGE(self):
        y = stats.pearsonr(LITTLE,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pLITTLETINY(self):
        y = stats.pearsonr(LITTLE,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pLITTLEROUND(self):
        y = stats.pearsonr(LITTLE,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pHUGEHUGE(self):
        y = stats.pearsonr(HUGE,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pHUGETINY(self):
        y = stats.pearsonr(HUGE,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pHUGEROUND(self):
        y = stats.pearsonr(HUGE,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pTINYTINY(self):
        y = stats.pearsonr(TINY,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pTINYROUND(self):
        y = stats.pearsonr(TINY,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pROUNDROUND(self):
        y = stats.pearsonr(ROUND,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_r_exactly_pos1(self):
        a = arange(3.0)
        b = a
        r, prob = stats.pearsonr(a,b)
        assert_equal(r, 1.0)
        assert_equal(prob, 0.0)

    def test_r_exactly_neg1(self):
        a = arange(3.0)
        b = -a
        r, prob = stats.pearsonr(a,b)
        assert_equal(r, -1.0)
        assert_equal(prob, 0.0)

    def test_basic(self):
        # A basic test, with a correlation coefficient
        # that is not 1 or -1.
        a = array([-1, 0, 1])
        b = array([0, 0, 3])
        r, prob = stats.pearsonr(a, b)
        assert_approx_equal(r, np.sqrt(3)/2)
        assert_approx_equal(prob, 1.0/3)


class TestFisherExact(object):
    """Some tests to show that fisher_exact() works correctly.

    Note that in SciPy 0.9.0 this was not working well for large numbers due to
    inaccuracy of the hypergeom distribution (see #1218). Fixed now.

    Also note that R and Scipy have different argument formats for their
    hypergeometric distribution functions.

    R:
    > phyper(18999, 99000, 110000, 39000, lower.tail = FALSE)
    [1] 1.701815e-09
    """
    def test_basic(self):
        fisher_exact = stats.fisher_exact

        res = fisher_exact([[14500, 20000], [30000, 40000]])[1]
        assert_approx_equal(res, 0.01106, significant=4)
        res = fisher_exact([[100, 2], [1000, 5]])[1]
        assert_approx_equal(res, 0.1301, significant=4)
        res = fisher_exact([[2, 7], [8, 2]])[1]
        assert_approx_equal(res, 0.0230141, significant=6)
        res = fisher_exact([[5, 1], [10, 10]])[1]
        assert_approx_equal(res, 0.1973244, significant=6)
        res = fisher_exact([[5, 15], [20, 20]])[1]
        assert_approx_equal(res, 0.0958044, significant=6)
        res = fisher_exact([[5, 16], [20, 25]])[1]
        assert_approx_equal(res, 0.1725862, significant=6)
        res = fisher_exact([[10, 5], [10, 1]])[1]
        assert_approx_equal(res, 0.1973244, significant=6)
        res = fisher_exact([[5, 0], [1, 4]])[1]
        assert_approx_equal(res, 0.04761904, significant=6)
        res = fisher_exact([[0, 1], [3, 2]])[1]
        assert_approx_equal(res, 1.0)
        res = fisher_exact([[0, 2], [6, 4]])[1]
        assert_approx_equal(res, 0.4545454545)
        res = fisher_exact([[2, 7], [8, 2]])
        assert_approx_equal(res[1], 0.0230141, significant=6)
        assert_approx_equal(res[0], 4.0 / 56)

    def test_precise(self):
        # results from R
        #
        # R defines oddsratio differently (see Notes section of fisher_exact
        # docstring), so those will not match.  We leave them in anyway, in
        # case they will be useful later on. We test only the p-value.
        tablist = [
            ([[100, 2], [1000, 5]], (2.505583993422285e-001, 1.300759363430016e-001)),
            ([[2, 7], [8, 2]], (8.586235135736206e-002, 2.301413756522114e-002)),
            ([[5, 1], [10, 10]], (4.725646047336584e+000, 1.973244147157190e-001)),
            ([[5, 15], [20, 20]], (3.394396617440852e-001, 9.580440012477637e-002)),
            ([[5, 16], [20, 25]], (3.960558326183334e-001, 1.725864953812994e-001)),
            ([[10, 5], [10, 1]], (2.116112781158483e-001, 1.973244147157190e-001)),
            ([[10, 5], [10, 0]], (0.000000000000000e+000, 6.126482213438734e-002)),
            ([[5, 0], [1, 4]], (np.inf, 4.761904761904762e-002)),
            ([[0, 5], [1, 4]], (0.000000000000000e+000, 1.000000000000000e+000)),
            ([[5, 1], [0, 4]], (np.inf, 4.761904761904758e-002)),
            ([[0, 1], [3, 2]], (0.000000000000000e+000, 1.000000000000000e+000))
            ]
        for table, res_r in tablist:
            res = stats.fisher_exact(np.asarray(table))
            np.testing.assert_almost_equal(res[1], res_r[1], decimal=11,
                                           verbose=True)

    @pytest.mark.slow
    def test_large_numbers(self):
        # Test with some large numbers. Regression test for #1401
        pvals = [5.56e-11, 2.666e-11, 1.363e-11]  # from R
        for pval, num in zip(pvals, [75, 76, 77]):
            res = stats.fisher_exact([[17704, 496], [1065, num]])[1]
            assert_approx_equal(res, pval, significant=4)

        res = stats.fisher_exact([[18000, 80000], [20000, 90000]])[1]
        assert_approx_equal(res, 0.2751, significant=4)

    def test_raises(self):
        # test we raise an error for wrong shape of input.
        assert_raises(ValueError, stats.fisher_exact,
                      np.arange(6).reshape(2, 3))

    def test_row_or_col_zero(self):
        tables = ([[0, 0], [5, 10]],
                  [[5, 10], [0, 0]],
                  [[0, 5], [0, 10]],
                  [[5, 0], [10, 0]])
        for table in tables:
            oddsratio, pval = stats.fisher_exact(table)
            assert_equal(pval, 1.0)
            assert_equal(oddsratio, np.nan)

    def test_less_greater(self):
        tables = (
            # Some tables to compare with R:
            [[2, 7], [8, 2]],
            [[200, 7], [8, 300]],
            [[28, 21], [6, 1957]],
            [[190, 800], [200, 900]],
            # Some tables with simple exact values
            # (includes regression test for ticket #1568):
            [[0, 2], [3, 0]],
            [[1, 1], [2, 1]],
            [[2, 0], [1, 2]],
            [[0, 1], [2, 3]],
            [[1, 0], [1, 4]],
            )
        pvals = (
            # from R:
            [0.018521725952066501, 0.9990149169715733],
            [1.0, 2.0056578803889148e-122],
            [1.0, 5.7284374608319831e-44],
            [0.7416227, 0.2959826],
            # Exact:
            [0.1, 1.0],
            [0.7, 0.9],
            [1.0, 0.3],
            [2./3, 1.0],
            [1.0, 1./3],
            )
        for table, pval in zip(tables, pvals):
            res = []
            res.append(stats.fisher_exact(table, alternative="less")[1])
            res.append(stats.fisher_exact(table, alternative="greater")[1])
            assert_allclose(res, pval, atol=0, rtol=1e-7)

    def test_gh3014(self):
        # check if issue #3014 has been fixed.
        # before, this would have risen a ValueError
        odds, pvalue = stats.fisher_exact([[1, 2], [9, 84419233]])


class TestCorrSpearmanr(object):
    """ W.II.D. Compute a correlation matrix on all the variables.

        All the correlations, except for ZERO and MISS, should be exactly 1.
        ZERO and MISS should have undefined or missing correlations with the
        other variables.  The same should go for SPEARMAN corelations, if
        your program has them.
    """
    def test_scalar(self):
        y = stats.spearmanr(4., 2.)
        assert_(np.isnan(y).all())

    def test_uneven_lengths(self):
        assert_raises(ValueError, stats.spearmanr, [1, 2, 1], [8, 9])
        assert_raises(ValueError, stats.spearmanr, [1, 2, 1], 8)

    def test_nan_policy(self):
        x = np.arange(10.)
        x[9] = np.nan
        assert_array_equal(stats.spearmanr(x, x), (np.nan, np.nan))
        assert_array_equal(stats.spearmanr(x, x, nan_policy='omit'),
                           (1.0, 0.0))
        assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='raise')
        assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='foobar')

    def test_sXX(self):
        y = stats.spearmanr(X,X)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sXBIG(self):
        y = stats.spearmanr(X,BIG)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sXLITTLE(self):
        y = stats.spearmanr(X,LITTLE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sXHUGE(self):
        y = stats.spearmanr(X,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sXTINY(self):
        y = stats.spearmanr(X,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sXROUND(self):
        y = stats.spearmanr(X,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sBIGBIG(self):
        y = stats.spearmanr(BIG,BIG)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sBIGLITTLE(self):
        y = stats.spearmanr(BIG,LITTLE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sBIGHUGE(self):
        y = stats.spearmanr(BIG,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sBIGTINY(self):
        y = stats.spearmanr(BIG,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sBIGROUND(self):
        y = stats.spearmanr(BIG,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sLITTLELITTLE(self):
        y = stats.spearmanr(LITTLE,LITTLE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sLITTLEHUGE(self):
        y = stats.spearmanr(LITTLE,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sLITTLETINY(self):
        y = stats.spearmanr(LITTLE,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sLITTLEROUND(self):
        y = stats.spearmanr(LITTLE,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sHUGEHUGE(self):
        y = stats.spearmanr(HUGE,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sHUGETINY(self):
        y = stats.spearmanr(HUGE,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sHUGEROUND(self):
        y = stats.spearmanr(HUGE,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sTINYTINY(self):
        y = stats.spearmanr(TINY,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sTINYROUND(self):
        y = stats.spearmanr(TINY,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sROUNDROUND(self):
        y = stats.spearmanr(ROUND,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_spearmanr_result_attributes(self):
        res = stats.spearmanr(X, X)
        attributes = ('correlation', 'pvalue')
        check_named_results(res, attributes)

def test_spearmanr():
    # Cross-check with R:
    # cor.test(c(1,2,3,4,5),c(5,6,7,8,7),method="spearmanr")
    x1 = [1, 2, 3, 4, 5]
    x2 = [5, 6, 7, 8, 7]
    expected = (0.82078268166812329, 0.088587005313543798)
    res = stats.spearmanr(x1, x2)
    assert_approx_equal(res[0], expected[0])
    assert_approx_equal(res[1], expected[1])

    attributes = ('correlation', 'pvalue')
    res = stats.spearmanr(x1, x2)
    check_named_results(res, attributes)

    # with only ties in one or both inputs
    with np.errstate(invalid="ignore"):
        assert_equal(stats.spearmanr([2,2,2], [2,2,2]), (np.nan, np.nan))
        assert_equal(stats.spearmanr([2,0,2], [2,2,2]), (np.nan, np.nan))
        assert_equal(stats.spearmanr([2,2,2], [2,0,2]), (np.nan, np.nan))

    # empty arrays provided as input
    assert_equal(stats.spearmanr([], []), (np.nan, np.nan))

    np.random.seed(7546)
    x = np.array([np.random.normal(loc=1, scale=1, size=500),
                np.random.normal(loc=1, scale=1, size=500)])
    corr = [[1.0, 0.3],
            [0.3, 1.0]]
    x = np.dot(np.linalg.cholesky(corr), x)
    expected = (0.28659685838743354, 6.579862219051161e-11)
    res = stats.spearmanr(x[0], x[1])
    assert_approx_equal(res[0], expected[0])
    assert_approx_equal(res[1], expected[1])

    assert_approx_equal(stats.spearmanr([1,1,2], [1,1,2])[0], 1.0)

    # test nan_policy
    x = np.arange(10.)
    x[9] = np.nan
    assert_array_equal(stats.spearmanr(x, x), (np.nan, np.nan))
    assert_allclose(stats.spearmanr(x, x, nan_policy='omit'),
                    (1.0, 0))
    assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='raise')
    assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='foobar')

    # test unequal length inputs
    x = np.arange(10.)
    y = np.arange(20.)
    assert_raises(ValueError, stats.spearmanr, x, y)

    #test paired value
    x1 = [1, 2, 3, 4]
    x2 = [8, 7, 6, np.nan]
    res1 = stats.spearmanr(x1, x2, nan_policy='omit')
    res2 = stats.spearmanr(x1[:3], x2[:3], nan_policy='omit')
    assert_equal(res1, res2)

    # Regression test for GitHub issue #6061 - Overflow on Windows
    x = list(range(2000))
    y = list(range(2000))
    y[0], y[9] = y[9], y[0]
    y[10], y[434] = y[434], y[10]
    y[435], y[1509] = y[1509], y[435]
    # rho = 1 - 6 * (2 * (9^2 + 424^2 + 1074^2))/(2000 * (2000^2 - 1))
    #     = 1 - (1 / 500)
    #     = 0.998
    x.append(np.nan)
    y.append(3.0)
    assert_almost_equal(stats.spearmanr(x, y, nan_policy='omit')[0], 0.998)

class TestCorrSpearmanrTies(object):
    """Some tests of tie-handling by the spearmanr function."""

    def test_tie1(self):
        # Data
        x = [1.0, 2.0, 3.0, 4.0]
        y = [1.0, 2.0, 2.0, 3.0]
        # Ranks of the data, with tie-handling.
        xr = [1.0, 2.0, 3.0, 4.0]
        yr = [1.0, 2.5, 2.5, 4.0]
        # Result of spearmanr should be the same as applying
        # pearsonr to the ranks.
        sr = stats.spearmanr(x, y)
        pr = stats.pearsonr(xr, yr)
        assert_almost_equal(sr, pr)

    def test_tie2(self):
        # Test tie-handling if inputs contain nan's
        # Data without nan's
        x1 = [1, 2, 2.5, 2]
        y1 = [1, 3, 2.5, 4]
        # Same data with nan's
        x2 = [1, 2, 2.5, 2, np.nan]
        y2 = [1, 3, 2.5, 4, np.nan]

        # Results for two data sets should be the same if nan's are ignored
        sr1 = stats.spearmanr(x1, y1)
        sr2 = stats.spearmanr(x2, y2, nan_policy='omit')
        assert_almost_equal(sr1, sr2)


#    W.II.E.  Tabulate X against X, using BIG as a case weight.  The values
#    should appear on the diagonal and the total should be 899999955.
#    If the table cannot hold these values, forget about working with
#    census data.  You can also tabulate HUGE against TINY.  There is no
#    reason a tabulation program should not be able to distinguish
#    different values regardless of their magnitude.

# I need to figure out how to do this one.


def test_kendalltau():
    # simple case without ties
    x = np.arange(10)
    y = np.arange(10)
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (1.0, 5.511463844797e-07)
    res = stats.kendalltau(x, y)
    assert_approx_equal(res[0], expected[0])
    assert_approx_equal(res[1], expected[1])

    # swap a couple of values
    b = y[1]
    y[1] = y[2]
    y[2] = b
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (0.9555555555555556, 5.511463844797e-06)
    res = stats.kendalltau(x, y)
    assert_approx_equal(res[0], expected[0])
    assert_approx_equal(res[1], expected[1])

    # swap a couple more
    b = y[5]
    y[5] = y[6]
    y[6] = b
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (0.9111111111111111, 2.976190476190e-05)
    res = stats.kendalltau(x, y)
    assert_approx_equal(res[0], expected[0])
    assert_approx_equal(res[1], expected[1])

    # same in opposite direction
    x = np.arange(10)
    y = np.arange(10)[::-1]
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (-1.0, 5.511463844797e-07)
    res = stats.kendalltau(x, y)
    assert_approx_equal(res[0], expected[0])
    assert_approx_equal(res[1], expected[1])

    # swap a couple of values
    b = y[1]
    y[1] = y[2]
    y[2] = b
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (-0.9555555555555556, 5.511463844797e-06)
    res = stats.kendalltau(x, y)
    assert_approx_equal(res[0], expected[0])
    assert_approx_equal(res[1], expected[1])

    # swap a couple more
    b = y[5]
    y[5] = y[6]
    y[6] = b
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (-0.9111111111111111, 2.976190476190e-05)
    res = stats.kendalltau(x, y)
    assert_approx_equal(res[0], expected[0])
    assert_approx_equal(res[1], expected[1])

    # check exception in case of ties
    y[2] = y[1]
    assert_raises(ValueError, stats.kendalltau, x, y, method='exact')

    # check exception in case of invalid method keyword
    assert_raises(ValueError, stats.kendalltau, x, y, method='banana')

    # with some ties
    # Cross-check with R:
    # cor.test(c(12,2,1,12,2),c(1,4,7,1,0),method="kendall",exact=FALSE)
    x1 = [12, 2, 1, 12, 2]
    x2 = [1, 4, 7, 1, 0]
    expected = (-0.47140452079103173, 0.28274545993277478)
    res = stats.kendalltau(x1, x2)
    assert_approx_equal(res[0], expected[0])
    assert_approx_equal(res[1], expected[1])

    # test for namedtuple attribute results
    attributes = ('correlation', 'pvalue')
    res = stats.kendalltau(x1, x2)
    check_named_results(res, attributes)

    # with only ties in one or both inputs
    assert_equal(stats.kendalltau([2,2,2], [2,2,2]), (np.nan, np.nan))
    assert_equal(stats.kendalltau([2,0,2], [2,2,2]), (np.nan, np.nan))
    assert_equal(stats.kendalltau([2,2,2], [2,0,2]), (np.nan, np.nan))

    # empty arrays provided as input
    assert_equal(stats.kendalltau([], []), (np.nan, np.nan))

    # check with larger arrays
    np.random.seed(7546)
    x = np.array([np.random.normal(loc=1, scale=1, size=500),
                np.random.normal(loc=1, scale=1, size=500)])
    corr = [[1.0, 0.3],
            [0.3, 1.0]]
    x = np.dot(np.linalg.cholesky(corr), x)
    expected = (0.19291382765531062, 1.1337095377742629e-10)
    res = stats.kendalltau(x[0], x[1])
    assert_approx_equal(res[0], expected[0])
    assert_approx_equal(res[1], expected[1])

    # and do we get a tau of 1 for identical inputs?
    assert_approx_equal(stats.kendalltau([1,1,2], [1,1,2])[0], 1.0)

    # test nan_policy
    x = np.arange(10.)
    x[9] = np.nan
    assert_array_equal(stats.kendalltau(x, x), (np.nan, np.nan))
    assert_allclose(stats.kendalltau(x, x, nan_policy='omit'),
                    (1.0, 5.5114638e-6), rtol=1e-06)
    assert_allclose(stats.kendalltau(x, x, nan_policy='omit', method='asymptotic'),
                    (1.0, 0.00017455009626808976), rtol=1e-06)
    assert_raises(ValueError, stats.kendalltau, x, x, nan_policy='raise')
    assert_raises(ValueError, stats.kendalltau, x, x, nan_policy='foobar')

    # test unequal length inputs
    x = np.arange(10.)
    y = np.arange(20.)
    assert_raises(ValueError, stats.kendalltau, x, y)

    # test all ties
    tau, p_value = stats.kendalltau([], [])
    assert_equal(np.nan, tau)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.kendalltau([0], [0])
    assert_equal(np.nan, tau)
    assert_equal(np.nan, p_value)

    # Regression test for GitHub issue #6061 - Overflow on Windows
    x = np.arange(2000, dtype=float)
    x = np.ma.masked_greater(x, 1995)
    y = np.arange(2000, dtype=float)
    y = np.concatenate((y[1000:], y[:1000]))
    assert_(np.isfinite(stats.kendalltau(x,y)[1]))

def test_kendalltau_vs_mstats_basic():
    np.random.seed(42)
    for s in range(2,10):
        a = []
        # Generate rankings with ties
        for i in range(s):
            a += [i]*i
        b = list(a)
        np.random.shuffle(a)
        np.random.shuffle(b)
        expected = mstats_basic.kendalltau(a, b)
        actual = stats.kendalltau(a, b)
        assert_approx_equal(actual[0], expected[0])
        assert_approx_equal(actual[1], expected[1])


def test_kendalltau_nan_2nd_arg():
    # regression test for gh-6134: nans in the second arg were not handled
    x = [1., 2., 3., 4.]
    y = [np.nan, 2.4, 3.4, 3.4]

    r1 = stats.kendalltau(x, y, nan_policy='omit')
    r2 = stats.kendalltau(x[1:], y[1:])
    assert_allclose(r1.correlation, r2.correlation, atol=1e-15)


def test_weightedtau():
    x = [12, 2, 1, 12, 2]
    y = [1, 4, 7, 1, 0]
    tau, p_value = stats.weightedtau(x, y)
    assert_approx_equal(tau, -0.56694968153682723)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau(x, y, additive=False)
    assert_approx_equal(tau, -0.62205716951801038)
    assert_equal(np.nan, p_value)
    # This must be exactly Kendall's tau
    tau, p_value = stats.weightedtau(x, y, weigher=lambda x: 1)
    assert_approx_equal(tau, -0.47140452079103173)
    assert_equal(np.nan, p_value)

    # Asymmetric, ranked version
    tau, p_value = stats.weightedtau(x, y, rank=None)
    assert_approx_equal(tau, -0.4157652301037516)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau(y, x, rank=None)
    assert_approx_equal(tau, -0.7181341329699029)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau(x, y, rank=None, additive=False)
    assert_approx_equal(tau, -0.40644850966246893)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau(y, x, rank=None, additive=False)
    assert_approx_equal(tau, -0.83766582937355172)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau(x, y, rank=False)
    assert_approx_equal(tau, -0.51604397940261848)
    assert_equal(np.nan, p_value)
    # This must be exactly Kendall's tau
    tau, p_value = stats.weightedtau(x, y, rank=True, weigher=lambda x: 1)
    assert_approx_equal(tau, -0.47140452079103173)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau(y, x, rank=True, weigher=lambda x: 1)
    assert_approx_equal(tau, -0.47140452079103173)
    assert_equal(np.nan, p_value)
    # Test argument conversion
    tau, p_value = stats.weightedtau(np.asarray(x, dtype=np.float64), y)
    assert_approx_equal(tau, -0.56694968153682723)
    tau, p_value = stats.weightedtau(np.asarray(x, dtype=np.int16), y)
    assert_approx_equal(tau, -0.56694968153682723)
    tau, p_value = stats.weightedtau(np.asarray(x, dtype=np.float64), np.asarray(y, dtype=np.float64))
    assert_approx_equal(tau, -0.56694968153682723)
    # All ties
    tau, p_value = stats.weightedtau([], [])
    assert_equal(np.nan, tau)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau([0], [0])
    assert_equal(np.nan, tau)
    assert_equal(np.nan, p_value)
    # Size mismatches
    assert_raises(ValueError, stats.weightedtau, [0, 1], [0, 1, 2])
    assert_raises(ValueError, stats.weightedtau, [0, 1], [0, 1], [0])
    # NaNs
    x = [12, 2, 1, 12, 2]
    y = [1, 4, 7, 1, np.nan]
    tau, p_value = stats.weightedtau(x, y)
    assert_approx_equal(tau, -0.56694968153682723)
    x = [12, 2, np.nan, 12, 2]
    tau, p_value = stats.weightedtau(x, y)
    assert_approx_equal(tau, -0.56694968153682723)


def test_weightedtau_vs_quadratic():
    # Trivial quadratic implementation, all parameters mandatory
    def wkq(x, y, rank, weigher, add):
        tot = conc = disc = u = v = 0
        for i in range(len(x)):
            for j in range(len(x)):
                w = weigher(rank[i]) + weigher(rank[j]) if add else weigher(rank[i]) * weigher(rank[j])
                tot += w
                if x[i] == x[j]:
                    u += w
                if y[i] == y[j]:
                    v += w
                if x[i] < x[j] and y[i] < y[j] or x[i] > x[j] and y[i] > y[j]:
                    conc += w
                elif x[i] < x[j] and y[i] > y[j] or x[i] > x[j] and y[i] < y[j]:
                    disc += w
        return (conc - disc) / np.sqrt(tot - u) / np.sqrt(tot - v)

    np.random.seed(42)
    for s in range(3,10):
        a = []
        # Generate rankings with ties
        for i in range(s):
            a += [i]*i
        b = list(a)
        np.random.shuffle(a)
        np.random.shuffle(b)
        # First pass: use element indices as ranks
        rank = np.arange(len(a), dtype=np.intp)
        for _ in range(2):
            for add in [True, False]:
                expected = wkq(a, b, rank, lambda x: 1./(x+1), add)
                actual = stats.weightedtau(a, b, rank, lambda x: 1./(x+1), add).correlation
                assert_approx_equal(expected, actual)
            # Second pass: use a random rank
            np.random.shuffle(rank)


class TestFindRepeats(object):

    def test_basic(self):
        a = [1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 5]
        res, nums = stats.find_repeats(a)
        assert_array_equal(res, [1, 2, 3, 4])
        assert_array_equal(nums, [3, 3, 2, 2])

    def test_empty_result(self):
        # Check that empty arrays are returned when there are no repeats.
        for a in [[10, 20, 50, 30, 40], []]:
            repeated, counts = stats.find_repeats(a)
            assert_array_equal(repeated, [])
            assert_array_equal(counts, [])


class TestRegression(object):
    def test_linregressBIGX(self):
        # W.II.F.  Regress BIG on X.
        # The constant should be 99999990 and the regression coefficient should be 1.
        y = stats.linregress(X,BIG)
        intercept = y[1]
        r = y[2]
        assert_almost_equal(intercept,99999990)
        assert_almost_equal(r,1.0)

    def test_regressXX(self):
        # W.IV.B.  Regress X on X.
        # The constant should be exactly 0 and the regression coefficient should be 1.
        # This is a perfectly valid regression.  The program should not complain.
        y = stats.linregress(X,X)
        intercept = y[1]
        r = y[2]
        assert_almost_equal(intercept,0.0)
        assert_almost_equal(r,1.0)
#     W.IV.C. Regress X on BIG and LITTLE (two predictors).  The program
#     should tell you that this model is "singular" because BIG and
#     LITTLE are linear combinations of each other.  Cryptic error
#     messages are unacceptable here.  Singularity is the most
#     fundamental regression error.
# Need to figure out how to handle multiple linear regression.  Not obvious

    def test_regressZEROX(self):
        # W.IV.D. Regress ZERO on X.
        # The program should inform you that ZERO has no variance or it should
        # go ahead and compute the regression and report a correlation and
        # total sum of squares of exactly 0.
        y = stats.linregress(X,ZERO)
        intercept = y[1]
        r = y[2]
        assert_almost_equal(intercept,0.0)
        assert_almost_equal(r,0.0)

    def test_regress_simple(self):
        # Regress a line with sinusoidal noise.
        x = np.linspace(0, 100, 100)
        y = 0.2 * np.linspace(0, 100, 100) + 10
        y += np.sin(np.linspace(0, 20, 100))

        res = stats.linregress(x, y)
        assert_almost_equal(res[4], 2.3957814497838803e-3)

    def test_regress_simple_onearg_rows(self):
        # Regress a line w sinusoidal noise, with a single input of shape (2, N).
        x = np.linspace(0, 100, 100)
        y = 0.2 * np.linspace(0, 100, 100) + 10
        y += np.sin(np.linspace(0, 20, 100))
        rows = np.vstack((x, y))

        res = stats.linregress(rows)
        assert_almost_equal(res[4], 2.3957814497838803e-3)

    def test_regress_simple_onearg_cols(self):
        x = np.linspace(0, 100, 100)
        y = 0.2 * np.linspace(0, 100, 100) + 10
        y += np.sin(np.linspace(0, 20, 100))
        cols = np.hstack((np.expand_dims(x, 1), np.expand_dims(y, 1)))

        res = stats.linregress(cols)
        assert_almost_equal(res[4], 2.3957814497838803e-3)

    def test_regress_shape_error(self):
        # Check that a single input argument to linregress with wrong shape
        # results in a ValueError.
        assert_raises(ValueError, stats.linregress, np.ones((3, 3)))

    def test_linregress(self):
        # compared with multivariate ols with pinv
        x = np.arange(11)
        y = np.arange(5,16)
        y[[(1),(-2)]] -= 1
        y[[(0),(-1)]] += 1

        res = (1.0, 5.0, 0.98229948625750, 7.45259691e-008, 0.063564172616372733)
        assert_array_almost_equal(stats.linregress(x,y),res,decimal=14)

    def test_regress_simple_negative_cor(self):
        # If the slope of the regression is negative the factor R tend to -1 not 1.
        # Sometimes rounding errors makes it < -1 leading to stderr being NaN
        a, n = 1e-71, 100000
        x = np.linspace(a, 2 * a, n)
        y = np.linspace(2 * a, a, n)
        stats.linregress(x, y)
        res = stats.linregress(x, y)
        assert_(res[2] >= -1)  # propagated numerical errors were not corrected
        assert_almost_equal(res[2], -1)  # perfect negative correlation case
        assert_(not np.isnan(res[4]))  # stderr should stay finite

    def test_linregress_result_attributes(self):
        # Regress a line with sinusoidal noise.
        x = np.linspace(0, 100, 100)
        y = 0.2 * np.linspace(0, 100, 100) + 10
        y += np.sin(np.linspace(0, 20, 100))

        res = stats.linregress(x, y)
        attributes = ('slope', 'intercept', 'rvalue', 'pvalue', 'stderr')
        check_named_results(res, attributes)

    def test_regress_two_inputs(self):
        # Regress a simple line formed by two points.
        x = np.arange(2)
        y = np.arange(3, 5)

        res = stats.linregress(x, y)
        assert_almost_equal(res[3], 0.0)  # non-horizontal line
        assert_almost_equal(res[4], 0.0)  # zero stderr

    def test_regress_two_inputs_horizontal_line(self):
        # Regress a horizontal line formed by two points.
        x = np.arange(2)
        y = np.ones(2)

        res = stats.linregress(x, y)
        assert_almost_equal(res[3], 1.0)  # horizontal line
        assert_almost_equal(res[4], 0.0)  # zero stderr

    def test_nist_norris(self):
        x = [0.2, 337.4, 118.2, 884.6, 10.1, 226.5, 666.3, 996.3, 448.6, 777.0,
             558.2, 0.4, 0.6, 775.5, 666.9, 338.0, 447.5, 11.6, 556.0, 228.1,
             995.8, 887.6, 120.2, 0.3, 0.3, 556.8, 339.1, 887.2, 999.0, 779.0,
             11.1, 118.3, 229.2, 669.1, 448.9, 0.5]

        y = [0.1, 338.8, 118.1, 888.0, 9.2, 228.1, 668.5, 998.5, 449.1, 778.9,
             559.2, 0.3, 0.1, 778.1, 668.8, 339.3, 448.9, 10.8, 557.7, 228.3,
             998.0, 888.8, 119.6, 0.3, 0.6, 557.6, 339.3, 888.0, 998.5, 778.9,
             10.2, 117.6, 228.9, 668.4, 449.2, 0.2]

        # Expected values
        exp_slope = 1.00211681802045
        exp_intercept = -0.262323073774029
        exp_rvalue = 0.999993745883712

        actual = stats.linregress(x, y)

        assert_almost_equal(actual.slope, exp_slope)
        assert_almost_equal(actual.intercept, exp_intercept)
        assert_almost_equal(actual.rvalue, exp_rvalue, decimal=5)

    def test_empty_input(self):
        assert_raises(ValueError, stats.linregress, [], [])

    def test_nan_input(self):
        x = np.arange(10.)
        x[9] = np.nan

        with np.errstate(invalid="ignore"):
            assert_array_equal(stats.linregress(x, x),
                               (np.nan, np.nan, np.nan, np.nan, np.nan))


def test_theilslopes():
    # Basic slope test.
    slope, intercept, lower, upper = stats.theilslopes([0,1,1])
    assert_almost_equal(slope, 0.5)
    assert_almost_equal(intercept, 0.5)

    # Test of confidence intervals.
    x = [1, 2, 3, 4, 10, 12, 18]
    y = [9, 15, 19, 20, 45, 55, 78]
    slope, intercept, lower, upper = stats.theilslopes(y, x, 0.07)
    assert_almost_equal(slope, 4)
    assert_almost_equal(upper, 4.38, decimal=2)
    assert_almost_equal(lower, 3.71, decimal=2)


def test_cumfreq():
    x = [1, 4, 2, 1, 3, 1]
    cumfreqs, lowlim, binsize, extrapoints = stats.cumfreq(x, numbins=4)
    assert_array_almost_equal(cumfreqs, np.array([3., 4., 5., 6.]))
    cumfreqs, lowlim, binsize, extrapoints = stats.cumfreq(x, numbins=4,
                                                      defaultreallimits=(1.5, 5))
    assert_(extrapoints == 3)

    # test for namedtuple attribute results
    attributes = ('cumcount', 'lowerlimit', 'binsize', 'extrapoints')
    res = stats.cumfreq(x, numbins=4, defaultreallimits=(1.5, 5))
    check_named_results(res, attributes)


def test_relfreq():
    a = np.array([1, 4, 2, 1, 3, 1])
    relfreqs, lowlim, binsize, extrapoints = stats.relfreq(a, numbins=4)
    assert_array_almost_equal(relfreqs,
                              array([0.5, 0.16666667, 0.16666667, 0.16666667]))

    # test for namedtuple attribute results
    attributes = ('frequency', 'lowerlimit', 'binsize', 'extrapoints')
    res = stats.relfreq(a, numbins=4)
    check_named_results(res, attributes)

    # check array_like input is accepted
    relfreqs2, lowlim, binsize, extrapoints = stats.relfreq([1, 4, 2, 1, 3, 1],
                                                            numbins=4)
    assert_array_almost_equal(relfreqs, relfreqs2)


class TestGMean(object):

    def test_1D_list(self):
        a = (1,2,3,4)
        actual = stats.gmean(a)
        desired = power(1*2*3*4,1./4.)
        assert_almost_equal(actual, desired,decimal=14)

        desired1 = stats.gmean(a,axis=-1)
        assert_almost_equal(actual, desired1, decimal=14)

    def test_1D_array(self):
        a = array((1,2,3,4), float32)
        actual = stats.gmean(a)
        desired = power(1*2*3*4,1./4.)
        assert_almost_equal(actual, desired, decimal=7)

        desired1 = stats.gmean(a,axis=-1)
        assert_almost_equal(actual, desired1, decimal=7)

    def test_2D_array_default(self):
        a = array(((1,2,3,4),
                   (1,2,3,4),
                   (1,2,3,4)))
        actual = stats.gmean(a)
        desired = array((1,2,3,4))
        assert_array_almost_equal(actual, desired, decimal=14)

        desired1 = stats.gmean(a,axis=0)
        assert_array_almost_equal(actual, desired1, decimal=14)

    def test_2D_array_dim1(self):
        a = array(((1,2,3,4),
                   (1,2,3,4),
                   (1,2,3,4)))
        actual = stats.gmean(a, axis=1)
        v = power(1*2*3*4,1./4.)
        desired = array((v,v,v))
        assert_array_almost_equal(actual, desired, decimal=14)

    def test_large_values(self):
        a = array([1e100, 1e200, 1e300])
        actual = stats.gmean(a)
        assert_approx_equal(actual, 1e200, significant=13)


class TestHMean(object):
    def test_1D_list(self):
        a = (1,2,3,4)
        actual = stats.hmean(a)
        desired = 4. / (1./1 + 1./2 + 1./3 + 1./4)
        assert_almost_equal(actual, desired, decimal=14)

        desired1 = stats.hmean(array(a),axis=-1)
        assert_almost_equal(actual, desired1, decimal=14)

    def test_1D_array(self):
        a = array((1,2,3,4), float64)
        actual = stats.hmean(a)
        desired = 4. / (1./1 + 1./2 + 1./3 + 1./4)
        assert_almost_equal(actual, desired, decimal=14)

        desired1 = stats.hmean(a,axis=-1)
        assert_almost_equal(actual, desired1, decimal=14)

    def test_2D_array_default(self):
        a = array(((1,2,3,4),
                   (1,2,3,4),
                   (1,2,3,4)))
        actual = stats.hmean(a)
        desired = array((1.,2.,3.,4.))
        assert_array_almost_equal(actual, desired, decimal=14)

        actual1 = stats.hmean(a,axis=0)
        assert_array_almost_equal(actual1, desired, decimal=14)

    def test_2D_array_dim1(self):
        a = array(((1,2,3,4),
                   (1,2,3,4),
                   (1,2,3,4)))

        v = 4. / (1./1 + 1./2 + 1./3 + 1./4)
        desired1 = array((v,v,v))
        actual1 = stats.hmean(a, axis=1)
        assert_array_almost_equal(actual1, desired1, decimal=14)


class TestScoreatpercentile(object):
    def setup_method(self):
        self.a1 = [3, 4, 5, 10, -3, -5, 6]
        self.a2 = [3, -6, -2, 8, 7, 4, 2, 1]
        self.a3 = [3., 4, 5, 10, -3, -5, -6, 7.0]

    def test_basic(self):
        x = arange(8) * 0.5
        assert_equal(stats.scoreatpercentile(x, 0), 0.)
        assert_equal(stats.scoreatpercentile(x, 100), 3.5)
        assert_equal(stats.scoreatpercentile(x, 50), 1.75)

    def test_fraction(self):
        scoreatperc = stats.scoreatpercentile

        # Test defaults
        assert_equal(scoreatperc(list(range(10)), 50), 4.5)
        assert_equal(scoreatperc(list(range(10)), 50, (2,7)), 4.5)
        assert_equal(scoreatperc(list(range(100)), 50, limit=(1, 8)), 4.5)
        assert_equal(scoreatperc(np.array([1, 10,100]), 50, (10,100)), 55)
        assert_equal(scoreatperc(np.array([1, 10,100]), 50, (1,10)), 5.5)

        # explicitly specify interpolation_method 'fraction' (the default)
        assert_equal(scoreatperc(list(range(10)), 50, interpolation_method='fraction'),
                     4.5)
        assert_equal(scoreatperc(list(range(10)), 50, limit=(2, 7),
                                 interpolation_method='fraction'),
                     4.5)
        assert_equal(scoreatperc(list(range(100)), 50, limit=(1, 8),
                                 interpolation_method='fraction'),
                     4.5)
        assert_equal(scoreatperc(np.array([1, 10,100]), 50, (10, 100),
                                 interpolation_method='fraction'),
                     55)
        assert_equal(scoreatperc(np.array([1, 10,100]), 50, (1,10),
                                 interpolation_method='fraction'),
                     5.5)

    def test_lower_higher(self):
        scoreatperc = stats.scoreatpercentile

        # interpolation_method 'lower'/'higher'
        assert_equal(scoreatperc(list(range(10)), 50,
                                 interpolation_method='lower'), 4)
        assert_equal(scoreatperc(list(range(10)), 50,
                                 interpolation_method='higher'), 5)
        assert_equal(scoreatperc(list(range(10)), 50, (2,7),
                                 interpolation_method='lower'), 4)
        assert_equal(scoreatperc(list(range(10)), 50, limit=(2,7),
                                 interpolation_method='higher'), 5)
        assert_equal(scoreatperc(list(range(100)), 50, (1,8),
                                 interpolation_method='lower'), 4)
        assert_equal(scoreatperc(list(range(100)), 50, (1,8),
                                 interpolation_method='higher'), 5)
        assert_equal(scoreatperc(np.array([1, 10, 100]), 50, (10, 100),
                                 interpolation_method='lower'), 10)
        assert_equal(scoreatperc(np.array([1, 10, 100]), 50, limit=(10, 100),
                                 interpolation_method='higher'), 100)
        assert_equal(scoreatperc(np.array([1, 10, 100]), 50, (1, 10),
                                 interpolation_method='lower'), 1)
        assert_equal(scoreatperc(np.array([1, 10, 100]), 50, limit=(1, 10),
                                 interpolation_method='higher'), 10)

    def test_sequence_per(self):
        x = arange(8) * 0.5
        expected = np.array([0, 3.5, 1.75])
        res = stats.scoreatpercentile(x, [0, 100, 50])
        assert_allclose(res, expected)
        assert_(isinstance(res, np.ndarray))
        # Test with ndarray.  Regression test for gh-2861
        assert_allclose(stats.scoreatpercentile(x, np.array([0, 100, 50])),
                        expected)
        # Also test combination of 2-D array, axis not None and array-like per
        res2 = stats.scoreatpercentile(np.arange(12).reshape((3,4)),
                                       np.array([0, 1, 100, 100]), axis=1)
        expected2 = array([[0, 4, 8],
                           [0.03, 4.03, 8.03],
                           [3, 7, 11],
                           [3, 7, 11]])
        assert_allclose(res2, expected2)

    def test_axis(self):
        scoreatperc = stats.scoreatpercentile
        x = arange(12).reshape(3, 4)

        assert_equal(scoreatperc(x, (25, 50, 100)), [2.75, 5.5, 11.0])

        r0 = [[2, 3, 4, 5], [4, 5, 6, 7], [8, 9, 10, 11]]
        assert_equal(scoreatperc(x, (25, 50, 100), axis=0), r0)

        r1 = [[0.75, 4.75, 8.75], [1.5, 5.5, 9.5], [3, 7, 11]]
        assert_equal(scoreatperc(x, (25, 50, 100), axis=1), r1)

        x = array([[1, 1, 1],
                   [1, 1, 1],
                   [4, 4, 3],
                   [1, 1, 1],
                   [1, 1, 1]])
        score = stats.scoreatpercentile(x, 50)
        assert_equal(score.shape, ())
        assert_equal(score, 1.0)
        score = stats.scoreatpercentile(x, 50, axis=0)
        assert_equal(score.shape, (3,))
        assert_equal(score, [1, 1, 1])

    def test_exception(self):
        assert_raises(ValueError, stats.scoreatpercentile, [1, 2], 56,
            interpolation_method='foobar')
        assert_raises(ValueError, stats.scoreatpercentile, [1], 101)
        assert_raises(ValueError, stats.scoreatpercentile, [1], -1)

    def test_empty(self):
        assert_equal(stats.scoreatpercentile([], 50), np.nan)
        assert_equal(stats.scoreatpercentile(np.array([[], []]), 50), np.nan)
        assert_equal(stats.scoreatpercentile([], [50, 99]), [np.nan, np.nan])


class TestItemfreq(object):
    a = [5, 7, 1, 2, 1, 5, 7] * 10
    b = [1, 2, 5, 7]

    def test_numeric_types(self):
        # Check itemfreq works for all dtypes (adapted from np.unique tests)
        def _check_itemfreq(dt):
            a = np.array(self.a, dt)
            with suppress_warnings() as sup:
                sup.filter(DeprecationWarning)
                v = stats.itemfreq(a)
            assert_array_equal(v[:, 0], [1, 2, 5, 7])
            assert_array_equal(v[:, 1], np.array([20, 10, 20, 20], dtype=dt))

        dtypes = [np.int32, np.int64, np.float32, np.float64,
                  np.complex64, np.complex128]
        for dt in dtypes:
            _check_itemfreq(dt)

    def test_object_arrays(self):
        a, b = self.a, self.b
        dt = 'O'
        aa = np.empty(len(a), dt)
        aa[:] = a
        bb = np.empty(len(b), dt)
        bb[:] = b
        with suppress_warnings() as sup:
            sup.filter(DeprecationWarning)
            v = stats.itemfreq(aa)
        assert_array_equal(v[:, 0], bb)

    def test_structured_arrays(self):
        a, b = self.a, self.b
        dt = [('', 'i'), ('', 'i')]
        aa = np.array(list(zip(a, a)), dt)
        bb = np.array(list(zip(b, b)), dt)
        with suppress_warnings() as sup:
            sup.filter(DeprecationWarning)
            v = stats.itemfreq(aa)
        # Arrays don't compare equal because v[:,0] is object array
        assert_equal(tuple(v[2, 0]), tuple(bb[2]))


class TestMode(object):
    def test_empty(self):
        vals, counts = stats.mode([])
        assert_equal(vals, np.array([]))
        assert_equal(counts, np.array([]))

    def test_scalar(self):
        vals, counts = stats.mode(4.)
        assert_equal(vals, np.array([4.]))
        assert_equal(counts, np.array([1]))

    def test_basic(self):
        data1 = [3, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6]
        vals = stats.mode(data1)
        assert_equal(vals[0][0], 6)
        assert_equal(vals[1][0], 3)

    def test_axes(self):
        data1 = [10, 10, 30, 40]
        data2 = [10, 10, 10, 10]
        data3 = [20, 10, 20, 20]
        data4 = [30, 30, 30, 30]
        data5 = [40, 30, 30, 30]
        arr = np.array([data1, data2, data3, data4, data5])

        vals = stats.mode(arr, axis=None)
        assert_equal(vals[0], np.array([30]))
        assert_equal(vals[1], np.array([8]))

        vals = stats.mode(arr, axis=0)
        assert_equal(vals[0], np.array([[10, 10, 30, 30]]))
        assert_equal(vals[1], np.array([[2, 3, 3, 2]]))

        vals = stats.mode(arr, axis=1)
        assert_equal(vals[0], np.array([[10], [10], [20], [30], [30]]))
        assert_equal(vals[1], np.array([[2], [4], [3], [4], [3]]))

    def test_strings(self):
        data1 = ['rain', 'showers', 'showers']

        with suppress_warnings() as sup:
            r = sup.record(RuntimeWarning, ".*checked for nan values")
            vals = stats.mode(data1)
            assert_equal(len(r), 1)

        assert_equal(vals[0][0], 'showers')
        assert_equal(vals[1][0], 2)

    def test_mixed_objects(self):
        objects = [10, True, np.nan, 'hello', 10]
        arr = np.empty((5,), dtype=object)
        arr[:] = objects
        with suppress_warnings() as sup:
            r = sup.record(RuntimeWarning, ".*checked for nan values")
            vals = stats.mode(arr)
            assert_equal(len(r), 1)
        assert_equal(vals[0][0], 10)
        assert_equal(vals[1][0], 2)

    def test_objects(self):
        # Python objects must be sortable (le + eq) and have ne defined
        # for np.unique to work. hash is for set.
        class Point(object):
            def __init__(self, x):
                self.x = x

            def __eq__(self, other):
                return self.x == other.x

            def __ne__(self, other):
                return self.x != other.x

            def __lt__(self, other):
                return self.x < other.x

            def __hash__(self):
                return hash(self.x)

        points = [Point(x) for x in [1, 2, 3, 4, 3, 2, 2, 2]]
        arr = np.empty((8,), dtype=object)
        arr[:] = points
        assert_(len(set(points)) == 4)
        assert_equal(np.unique(arr).shape, (4,))
        with suppress_warnings() as sup:
            r = sup.record(RuntimeWarning, ".*checked for nan values")
            vals = stats.mode(arr)
            assert_equal(len(r), 1)

        assert_equal(vals[0][0], Point(2))
        assert_equal(vals[1][0], 4)

    def test_mode_result_attributes(self):
        data1 = [3, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6]
        data2 = []
        actual = stats.mode(data1)
        attributes = ('mode', 'count')
        check_named_results(actual, attributes)
        actual2 = stats.mode(data2)
        check_named_results(actual2, attributes)

    def test_mode_nan(self):
        data1 = [3, np.nan, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6]
        actual = stats.mode(data1)
        assert_equal(actual, (6, 3))

        actual = stats.mode(data1, nan_policy='omit')
        assert_equal(actual, (6, 3))
        assert_raises(ValueError, stats.mode, data1, nan_policy='raise')
        assert_raises(ValueError, stats.mode, data1, nan_policy='foobar')

    @pytest.mark.parametrize("data", [
        [3, 5, 1, 1, 3],
        [3, np.nan, 5, 1, 1, 3],
        [3, 5, 1],
        [3, np.nan, 5, 1],
    ])
    def test_smallest_equal(self, data):
        result = stats.mode(data, nan_policy='omit')
        assert_equal(result[0][0], 1)


class TestVariability(object):

    testcase = [1,2,3,4]
    scalar_testcase = 4.

    def test_sem(self):
        # This is not in R, so used:
        #     sqrt(var(testcase)*3/4)/sqrt(3)

        # y = stats.sem(self.shoes[0])
        # assert_approx_equal(y,0.775177399)
        with suppress_warnings() as sup, np.errstate(invalid="ignore"):
            sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
            y = stats.sem(self.scalar_testcase)
        assert_(np.isnan(y))

        y = stats.sem(self.testcase)
        assert_approx_equal(y, 0.6454972244)
        n = len(self.testcase)
        assert_allclose(stats.sem(self.testcase, ddof=0) * np.sqrt(n/(n-2)),
                        stats.sem(self.testcase, ddof=2))

        x = np.arange(10.)
        x[9] = np.nan
        assert_equal(stats.sem(x), np.nan)
        assert_equal(stats.sem(x, nan_policy='omit'), 0.9128709291752769)
        assert_raises(ValueError, stats.sem, x, nan_policy='raise')
        assert_raises(ValueError, stats.sem, x, nan_policy='foobar')

    def test_zmap(self):
        # not in R, so tested by using:
        #     (testcase[i] - mean(testcase, axis=0)) / sqrt(var(testcase) * 3/4)
        y = stats.zmap(self.testcase,self.testcase)
        desired = ([-1.3416407864999, -0.44721359549996, 0.44721359549996, 1.3416407864999])
        assert_array_almost_equal(desired,y,decimal=12)

    def test_zmap_axis(self):
        # Test use of 'axis' keyword in zmap.
        x = np.array([[0.0, 0.0, 1.0, 1.0],
                      [1.0, 1.0, 1.0, 2.0],
                      [2.0, 0.0, 2.0, 0.0]])

        t1 = 1.0/np.sqrt(2.0/3)
        t2 = np.sqrt(3.)/3
        t3 = np.sqrt(2.)

        z0 = stats.zmap(x, x, axis=0)
        z1 = stats.zmap(x, x, axis=1)

        z0_expected = [[-t1, -t3/2, -t3/2, 0.0],
                       [0.0, t3, -t3/2, t1],
                       [t1, -t3/2, t3, -t1]]
        z1_expected = [[-1.0, -1.0, 1.0, 1.0],
                       [-t2, -t2, -t2, np.sqrt(3.)],
                       [1.0, -1.0, 1.0, -1.0]]

        assert_array_almost_equal(z0, z0_expected)
        assert_array_almost_equal(z1, z1_expected)

    def test_zmap_ddof(self):
        # Test use of 'ddof' keyword in zmap.
        x = np.array([[0.0, 0.0, 1.0, 1.0],
                      [0.0, 1.0, 2.0, 3.0]])

        z = stats.zmap(x, x, axis=1, ddof=1)

        z0_expected = np.array([-0.5, -0.5, 0.5, 0.5])/(1.0/np.sqrt(3))
        z1_expected = np.array([-1.5, -0.5, 0.5, 1.5])/(np.sqrt(5./3))
        assert_array_almost_equal(z[0], z0_expected)
        assert_array_almost_equal(z[1], z1_expected)

    def test_zscore(self):
        # not in R, so tested by using:
        #    (testcase[i] - mean(testcase, axis=0)) / sqrt(var(testcase) * 3/4)
        y = stats.zscore(self.testcase)
        desired = ([-1.3416407864999, -0.44721359549996, 0.44721359549996, 1.3416407864999])
        assert_array_almost_equal(desired,y,decimal=12)

    def test_zscore_axis(self):
        # Test use of 'axis' keyword in zscore.
        x = np.array([[0.0, 0.0, 1.0, 1.0],
                      [1.0, 1.0, 1.0, 2.0],
                      [2.0, 0.0, 2.0, 0.0]])

        t1 = 1.0/np.sqrt(2.0/3)
        t2 = np.sqrt(3.)/3
        t3 = np.sqrt(2.)

        z0 = stats.zscore(x, axis=0)
        z1 = stats.zscore(x, axis=1)

        z0_expected = [[-t1, -t3/2, -t3/2, 0.0],
                       [0.0, t3, -t3/2, t1],
                       [t1, -t3/2, t3, -t1]]
        z1_expected = [[-1.0, -1.0, 1.0, 1.0],
                       [-t2, -t2, -t2, np.sqrt(3.)],
                       [1.0, -1.0, 1.0, -1.0]]

        assert_array_almost_equal(z0, z0_expected)
        assert_array_almost_equal(z1, z1_expected)

    def test_zscore_ddof(self):
        # Test use of 'ddof' keyword in zscore.
        x = np.array([[0.0, 0.0, 1.0, 1.0],
                      [0.0, 1.0, 2.0, 3.0]])

        z = stats.zscore(x, axis=1, ddof=1)

        z0_expected = np.array([-0.5, -0.5, 0.5, 0.5])/(1.0/np.sqrt(3))
        z1_expected = np.array([-1.5, -0.5, 0.5, 1.5])/(np.sqrt(5./3))
        assert_array_almost_equal(z[0], z0_expected)
        assert_array_almost_equal(z[1], z1_expected)


class _numpy_version_warn_context_mgr(object):
    """
    A simple context maneger class to avoid retyping the same code for
    different versions of numpy when the only difference is that older
    versions raise warnings.

    This manager does not apply for cases where the old code returns
    different values.
    """
    def __init__(self, min_numpy_version, warning_type, num_warnings):
        if NumpyVersion(np.__version__) < min_numpy_version:
            self.numpy_is_old = True
            self.warning_type = warning_type
            self.num_warnings = num_warnings
            self.delegate = warnings.catch_warnings(record = True)
        else:
            self.numpy_is_old = False

    def __enter__(self):
        if self.numpy_is_old:
            self.warn_list = self.delegate.__enter__()
            warnings.simplefilter("always")
        return None

    def __exit__(self, exc_type, exc_value, traceback):
        if self.numpy_is_old:
            self.delegate.__exit__(exc_type, exc_value, traceback)
            _check_warnings(self.warn_list, self.warning_type, self.num_warnings)


def _check_warnings(warn_list, expected_type, expected_len):
    """
    Checks that all of the warnings from a list returned by
    `warnings.catch_all(record=True)` are of the required type and that the list
    contains expected number of warnings.
    """
    assert_equal(len(warn_list), expected_len, "number of warnings")
    for warn_ in warn_list:
        assert_(warn_.category is expected_type)


class TestIQR(object):

    def test_basic(self):
        x = np.arange(8) * 0.5
        np.random.shuffle(x)
        assert_equal(stats.iqr(x), 1.75)

    def test_api(self):
        d = np.ones((5, 5))
        stats.iqr(d)
        stats.iqr(d, None)
        stats.iqr(d, 1)
        stats.iqr(d, (0, 1))
        stats.iqr(d, None, (10, 90))
        stats.iqr(d, None, (30, 20), 'raw')
        stats.iqr(d, None, (25, 75), 1.5, 'propagate')
        if NumpyVersion(np.__version__) >= '1.9.0a':
            stats.iqr(d, None, (50, 50), 'normal', 'raise', 'linear')
            stats.iqr(d, None, (25, 75), -0.4, 'omit', 'lower', True)

    def test_empty(self):
        assert_equal(stats.iqr([]), np.nan)
        assert_equal(stats.iqr(np.arange(0)), np.nan)

    def test_constant(self):
        # Constant array always gives 0
        x = np.ones((7, 4))
        assert_equal(stats.iqr(x), 0.0)
        assert_array_equal(stats.iqr(x, axis=0), np.zeros(4))
        assert_array_equal(stats.iqr(x, axis=1), np.zeros(7))
        # Even for older versions, 'linear' does not raise a warning
        with _numpy_version_warn_context_mgr('1.9.0a', RuntimeWarning, 4):
            assert_equal(stats.iqr(x, interpolation='linear'), 0.0)
            assert_equal(stats.iqr(x, interpolation='midpoint'), 0.0)
            assert_equal(stats.iqr(x, interpolation='nearest'), 0.0)
            assert_equal(stats.iqr(x, interpolation='lower'), 0.0)
            assert_equal(stats.iqr(x, interpolation='higher'), 0.0)

        # 0 only along constant dimensions
        # This also tests much of `axis`
        y = np.ones((4, 5, 6)) * np.arange(6)
        assert_array_equal(stats.iqr(y, axis=0), np.zeros((5, 6)))
        assert_array_equal(stats.iqr(y, axis=1), np.zeros((4, 6)))
        assert_array_equal(stats.iqr(y, axis=2), 2.5 * np.ones((4, 5)))
        assert_array_equal(stats.iqr(y, axis=(0, 1)), np.zeros(6))
        assert_array_equal(stats.iqr(y, axis=(0, 2)), 3. * np.ones(5))
        assert_array_equal(stats.iqr(y, axis=(1, 2)), 3. * np.ones(4))

    def test_scalarlike(self):
        x = np.arange(1) + 7.0
        assert_equal(stats.iqr(x[0]), 0.0)
        assert_equal(stats.iqr(x), 0.0)
        if NumpyVersion(np.__version__) >= '1.9.0a':
            assert_array_equal(stats.iqr(x, keepdims=True), [0.0])
        else:
            with warnings.catch_warnings(record=True) as w:
                warnings.simplefilter("always")
                assert_array_equal(stats.iqr(x, keepdims=True), 0.0)
                _check_warnings(w, RuntimeWarning, 1)

    def test_2D(self):
        x = np.arange(15).reshape((3, 5))
        assert_equal(stats.iqr(x), 7.0)
        assert_array_equal(stats.iqr(x, axis=0), 5. * np.ones(5))
        assert_array_equal(stats.iqr(x, axis=1), 2. * np.ones(3))
        assert_array_equal(stats.iqr(x, axis=(0, 1)), 7.0)
        assert_array_equal(stats.iqr(x, axis=(1, 0)), 7.0)

    def test_axis(self):
        # The `axis` keyword is also put through its paces in `test_keepdims`.
        o = np.random.normal(size=(71, 23))
        x = np.dstack([o] * 10)                 # x.shape = (71, 23, 10)
        q = stats.iqr(o)

        assert_equal(stats.iqr(x, axis=(0, 1)), q)
        x = np.rollaxis(x, -1, 0)               # x.shape = (10, 71, 23)
        assert_equal(stats.iqr(x, axis=(2, 1)), q)
        x = x.swapaxes(0, 1)                    # x.shape = (71, 10, 23)
        assert_equal(stats.iqr(x, axis=(0, 2)), q)
        x = x.swapaxes(0, 1)                    # x.shape = (10, 71, 23)

        assert_equal(stats.iqr(x, axis=(0, 1, 2)),
                     stats.iqr(x, axis=None))
        assert_equal(stats.iqr(x, axis=(0,)),
                     stats.iqr(x, axis=0))

        d = np.arange(3 * 5 * 7 * 11)
        # Older versions of numpy only shuffle along axis=0.
        # Not sure about newer, don't care.
        np.random.shuffle(d)
        d = d.reshape((3, 5, 7, 11))
        assert_equal(stats.iqr(d, axis=(0, 1, 2))[0],
                     stats.iqr(d[:,:,:, 0].ravel()))
        assert_equal(stats.iqr(d, axis=(0, 1, 3))[1],
                     stats.iqr(d[:,:, 1,:].ravel()))
        assert_equal(stats.iqr(d, axis=(3, 1, -4))[2],
                     stats.iqr(d[:,:, 2,:].ravel()))
        assert_equal(stats.iqr(d, axis=(3, 1, 2))[2],
                     stats.iqr(d[2,:,:,:].ravel()))
        assert_equal(stats.iqr(d, axis=(3, 2))[2, 1],
                     stats.iqr(d[2, 1,:,:].ravel()))
        assert_equal(stats.iqr(d, axis=(1, -2))[2, 1],
                     stats.iqr(d[2, :, :, 1].ravel()))
        assert_equal(stats.iqr(d, axis=(1, 3))[2, 2],
                     stats.iqr(d[2, :, 2,:].ravel()))

        if NumpyVersion(np.__version__) >= '1.9.0a':
            assert_raises(IndexError, stats.iqr, d, axis=4)
        else:
            assert_raises(ValueError, stats.iqr, d, axis=4)
        assert_raises(ValueError, stats.iqr, d, axis=(0, 0))

    def test_rng(self):
        x = np.arange(5)
        assert_equal(stats.iqr(x), 2)
        assert_equal(stats.iqr(x, rng=(25, 87.5)), 2.5)
        assert_equal(stats.iqr(x, rng=(12.5, 75)), 2.5)
        assert_almost_equal(stats.iqr(x, rng=(10, 50)), 1.6)  # 3-1.4

        assert_raises(ValueError, stats.iqr, x, rng=(0, 101))
        assert_raises(ValueError, stats.iqr, x, rng=(np.nan, 25))
        assert_raises(TypeError, stats.iqr, x, rng=(0, 50, 60))

    def test_interpolation(self):
        x = np.arange(5)
        y = np.arange(4)
        # Default
        assert_equal(stats.iqr(x), 2)
        assert_equal(stats.iqr(y), 1.5)
        if NumpyVersion(np.__version__) >= '1.9.0a':
            # Linear
            assert_equal(stats.iqr(x, interpolation='linear'), 2)
            assert_equal(stats.iqr(y, interpolation='linear'), 1.5)
            # Higher
            assert_equal(stats.iqr(x, interpolation='higher'), 2)
            assert_equal(stats.iqr(x, rng=(25, 80), interpolation='higher'), 3)
            assert_equal(stats.iqr(y, interpolation='higher'), 2)
            # Lower (will generally, but not always be the same as higher)
            assert_equal(stats.iqr(x, interpolation='lower'), 2)
            assert_equal(stats.iqr(x, rng=(25, 80), interpolation='lower'), 2)
            assert_equal(stats.iqr(y, interpolation='lower'), 2)
            # Nearest
            assert_equal(stats.iqr(x, interpolation='nearest'), 2)
            assert_equal(stats.iqr(y, interpolation='nearest'), 1)
            # Midpoint
            if NumpyVersion(np.__version__) >= '1.11.0a':
                assert_equal(stats.iqr(x, interpolation='midpoint'), 2)
                assert_equal(stats.iqr(x, rng=(25, 80), interpolation='midpoint'), 2.5)
                assert_equal(stats.iqr(y, interpolation='midpoint'), 2)
            else:
                # midpoint did not work correctly before numpy 1.11.0
                assert_equal(stats.iqr(x, interpolation='midpoint'), 2)
                assert_equal(stats.iqr(x, rng=(25, 80), interpolation='midpoint'), 2)
                assert_equal(stats.iqr(y, interpolation='midpoint'), 2)
        else:
            with warnings.catch_warnings(record=True) as w:
                warnings.simplefilter("always")
                # Linear
                assert_equal(stats.iqr(x, interpolation='linear'), 2)
                assert_equal(stats.iqr(y, interpolation='linear'), 1.5)
                # Higher
                assert_equal(stats.iqr(x, interpolation='higher'), 2)
                assert_almost_equal(stats.iqr(x, rng=(25, 80), interpolation='higher'), 2.2)
                assert_equal(stats.iqr(y, interpolation='higher'), 1.5)
                # Lower
                assert_equal(stats.iqr(x, interpolation='lower'), 2)
                assert_almost_equal(stats.iqr(x, rng=(25, 80), interpolation='lower'), 2.2)
                assert_equal(stats.iqr(y, interpolation='lower'), 1.5)
                # Nearest
                assert_equal(stats.iqr(x, interpolation='nearest'), 2)
                assert_equal(stats.iqr(y, interpolation='nearest'), 1.5)
                # Midpoint
                assert_equal(stats.iqr(x, interpolation='midpoint'), 2)
                assert_almost_equal(stats.iqr(x, rng=(25, 80), interpolation='midpoint'), 2.2)
                assert_equal(stats.iqr(y, interpolation='midpoint'), 1.5)
                _check_warnings(w, RuntimeWarning, 11)

        if NumpyVersion(np.__version__) >= '1.9.0a':
            assert_raises(ValueError, stats.iqr, x, interpolation='foobar')
        else:
            with warnings.catch_warnings(record=True) as w:
                warnings.simplefilter("always")
                assert_equal(stats.iqr(x, interpolation='foobar'), 2)
                _check_warnings(w, RuntimeWarning, 1)

    def test_keepdims(self):
        numpy_version = NumpyVersion(np.__version__)

        # Also tests most of `axis`
        x = np.ones((3, 5, 7, 11))
        assert_equal(stats.iqr(x, axis=None, keepdims=False).shape, ())
        assert_equal(stats.iqr(x, axis=2, keepdims=False).shape, (3, 5, 11))
        assert_equal(stats.iqr(x, axis=(0, 1), keepdims=False).shape, (7, 11))
        assert_equal(stats.iqr(x, axis=(0, 3), keepdims=False).shape, (5, 7))
        assert_equal(stats.iqr(x, axis=(1,), keepdims=False).shape, (3, 7, 11))
        assert_equal(stats.iqr(x, (0, 1, 2, 3), keepdims=False).shape, ())
        assert_equal(stats.iqr(x, axis=(0, 1, 3), keepdims=False).shape, (7,))

        if numpy_version >= '1.9.0a':
            assert_equal(stats.iqr(x, axis=None, keepdims=True).shape, (1, 1, 1, 1))
            assert_equal(stats.iqr(x, axis=2, keepdims=True).shape, (3, 5, 1, 11))
            assert_equal(stats.iqr(x, axis=(0, 1), keepdims=True).shape, (1, 1, 7, 11))
            assert_equal(stats.iqr(x, axis=(0, 3), keepdims=True).shape, (1, 5, 7, 1))
            assert_equal(stats.iqr(x, axis=(1,), keepdims=True).shape, (3, 1, 7, 11))
            assert_equal(stats.iqr(x, (0, 1, 2, 3), keepdims=True).shape, (1, 1, 1, 1))
            assert_equal(stats.iqr(x, axis=(0, 1, 3), keepdims=True).shape, (1, 1, 7, 1))
        else:
            with warnings.catch_warnings(record=True) as w:
                warnings.simplefilter("always")
                assert_equal(stats.iqr(x, axis=None, keepdims=True).shape, ())
                assert_equal(stats.iqr(x, axis=2, keepdims=True).shape, (3, 5, 11))
                assert_equal(stats.iqr(x, axis=(0, 1), keepdims=True).shape, (7, 11))
                assert_equal(stats.iqr(x, axis=(0, 3), keepdims=True).shape, (5, 7))
                assert_equal(stats.iqr(x, axis=(1,), keepdims=True).shape, (3, 7, 11))
                assert_equal(stats.iqr(x, (0, 1, 2, 3), keepdims=True).shape, ())
                assert_equal(stats.iqr(x, axis=(0, 1, 3), keepdims=True).shape, (7,))
                _check_warnings(w, RuntimeWarning, 7)

    def test_nanpolicy(self):
        numpy_version = NumpyVersion(np.__version__)
        x = np.arange(15.0).reshape((3, 5))

        # No NaNs
        assert_equal(stats.iqr(x, nan_policy='propagate'), 7)
        assert_equal(stats.iqr(x, nan_policy='omit'), 7)
        assert_equal(stats.iqr(x, nan_policy='raise'), 7)

        # Yes NaNs
        x[1, 2] = np.nan
        with warnings.catch_warnings(record=True) as w:
            warnings.simplefilter("always")
            if numpy_version < '1.10.0a':
                # Fails over to mishmash of omit/propagate, but mostly omit
                # The first case showcases the "incorrect" behavior of np.percentile
                assert_equal(stats.iqr(x, nan_policy='propagate'), 8)
                assert_equal(stats.iqr(x, axis=0, nan_policy='propagate'), [5, 5, np.nan, 5, 5])
                if numpy_version < '1.9.0a':
                    assert_equal(stats.iqr(x, axis=1, nan_policy='propagate'), [2, 3, 2])
                else:
                    # some fixes to percentile nan handling in 1.9
                    assert_equal(stats.iqr(x, axis=1, nan_policy='propagate'), [2, np.nan, 2])
                _check_warnings(w, RuntimeWarning, 3)
            else:
                assert_equal(stats.iqr(x, nan_policy='propagate'), np.nan)
                assert_equal(stats.iqr(x, axis=0, nan_policy='propagate'), [5, 5, np.nan, 5, 5])
                assert_equal(stats.iqr(x, axis=1, nan_policy='propagate'), [2, np.nan, 2])

        with warnings.catch_warnings(record=True) as w:
            warnings.simplefilter("always")
            if numpy_version < '1.9.0a':
                # Fails over to mishmash of omit/propagate, but mostly omit
                assert_equal(stats.iqr(x, nan_policy='omit'), 8)
                assert_equal(stats.iqr(x, axis=0, nan_policy='omit'), [5, 5, np.nan, 5, 5])
                assert_equal(stats.iqr(x, axis=1, nan_policy='omit'), [2, 3, 2])
                _check_warnings(w, RuntimeWarning, 3)
            else:
                assert_equal(stats.iqr(x, nan_policy='omit'), 7.5)
                assert_equal(stats.iqr(x, axis=0, nan_policy='omit'), 5 * np.ones(5))
                assert_equal(stats.iqr(x, axis=1, nan_policy='omit'), [2, 2.5, 2])

        assert_raises(ValueError, stats.iqr, x, nan_policy='raise')
        assert_raises(ValueError, stats.iqr, x, axis=0, nan_policy='raise')
        assert_raises(ValueError, stats.iqr, x, axis=1, nan_policy='raise')

        # Bad policy
        assert_raises(ValueError, stats.iqr, x, nan_policy='barfood')

    def test_scale(self):
        numpy_version = NumpyVersion(np.__version__)
        x = np.arange(15.0).reshape((3, 5))

        # No NaNs
        assert_equal(stats.iqr(x, scale='raw'), 7)
        assert_almost_equal(stats.iqr(x, scale='normal'), 7 / 1.3489795)
        assert_equal(stats.iqr(x, scale=2.0), 3.5)

        # Yes NaNs
        x[1, 2] = np.nan
        with warnings.catch_warnings(record=True) as w:
            warnings.simplefilter("always")
            if numpy_version < '1.10.0a':
                # Fails over to mishmash of omit/propagate, but mostly omit
                assert_equal(stats.iqr(x, scale='raw', nan_policy='propagate'), 8)
                assert_almost_equal(stats.iqr(x, scale='normal',
                                              nan_policy='propagate'),
                                    8 / 1.3489795)
                assert_equal(stats.iqr(x, scale=2.0, nan_policy='propagate'), 4)
                # axis=1 chosen to show behavior with both nans and without
                if numpy_version < '1.9.0a':
                    assert_equal(stats.iqr(x, axis=1, nan_policy='propagate'), [2, 3, 2])
                    assert_almost_equal(stats.iqr(x, axis=1, scale='normal',
                                                  nan_policy='propagate'),
                                        np.array([2, 3, 2]) / 1.3489795)
                    assert_equal(stats.iqr(x, axis=1, scale=2.0,
                                           nan_policy='propagate'), [1, 1.5, 1])
                else:
                    # some fixes to percentile nan handling in 1.9
                    assert_equal(stats.iqr(x, axis=1, nan_policy='propagate'), [2, np.nan, 2])
                    assert_almost_equal(stats.iqr(x, axis=1, scale='normal',
                                                  nan_policy='propagate'),
                                        np.array([2, np.nan, 2]) / 1.3489795)
                    assert_equal(stats.iqr(x, axis=1, scale=2.0,
                                           nan_policy='propagate'), [1, np.nan, 1])
                _check_warnings(w, RuntimeWarning, 6)
            else:
                assert_equal(stats.iqr(x, scale='raw', nan_policy='propagate'), np.nan)
                assert_equal(stats.iqr(x, scale='normal', nan_policy='propagate'), np.nan)
                assert_equal(stats.iqr(x, scale=2.0, nan_policy='propagate'), np.nan)
                # axis=1 chosen to show behavior with both nans and without
                assert_equal(stats.iqr(x, axis=1, scale='raw',
                                       nan_policy='propagate'), [2, np.nan, 2])
                assert_almost_equal(stats.iqr(x, axis=1, scale='normal',
                                              nan_policy='propagate'),
                                    np.array([2, np.nan, 2]) / 1.3489795)
                assert_equal(stats.iqr(x, axis=1, scale=2.0, nan_policy='propagate'),
                             [1, np.nan, 1])
                _check_warnings(w, RuntimeWarning, 6)

        if numpy_version < '1.9.0a':
            with warnings.catch_warnings(record=True) as w:
                warnings.simplefilter("always")
                # Fails over to mishmash of omit/propagate, but mostly omit
                assert_equal(stats.iqr(x, scale='raw', nan_policy='omit'), 8)
                assert_almost_equal(stats.iqr(x, scale='normal', nan_policy='omit'),
                                    8 / 1.3489795)
                assert_equal(stats.iqr(x, scale=2.0, nan_policy='omit'), 4)
                _check_warnings(w, RuntimeWarning, 3)
        else:
            assert_equal(stats.iqr(x, scale='raw', nan_policy='omit'), 7.5)
            assert_almost_equal(stats.iqr(x, scale='normal', nan_policy='omit'),
                                7.5 / 1.3489795)
            assert_equal(stats.iqr(x, scale=2.0, nan_policy='omit'), 3.75)

        # Bad scale
        assert_raises(ValueError, stats.iqr, x, scale='foobar')


class TestMoments(object):
    """
        Comparison numbers are found using R v.1.5.1
        note that length(testcase) = 4
        testmathworks comes from documentation for the
        Statistics Toolbox for Matlab and can be found at both
        https://www.mathworks.com/help/stats/kurtosis.html
        https://www.mathworks.com/help/stats/skewness.html
        Note that both test cases came from here.
    """
    testcase = [1,2,3,4]
    scalar_testcase = 4.
    np.random.seed(1234)
    testcase_moment_accuracy = np.random.rand(42)
    testmathworks = [1.165, 0.6268, 0.0751, 0.3516, -0.6965]

    def test_moment(self):
        # mean((testcase-mean(testcase))**power,axis=0),axis=0))**power))
        y = stats.moment(self.scalar_testcase)
        assert_approx_equal(y, 0.0)
        y = stats.moment(self.testcase, 0)
        assert_approx_equal(y, 1.0)
        y = stats.moment(self.testcase, 1)
        assert_approx_equal(y, 0.0, 10)
        y = stats.moment(self.testcase, 2)
        assert_approx_equal(y, 1.25)
        y = stats.moment(self.testcase, 3)
        assert_approx_equal(y, 0.0)
        y = stats.moment(self.testcase, 4)
        assert_approx_equal(y, 2.5625)

        # check array_like input for moment
        y = stats.moment(self.testcase, [1, 2, 3, 4])
        assert_allclose(y, [0, 1.25, 0, 2.5625])

        # check moment input consists only of integers
        y = stats.moment(self.testcase, 0.0)
        assert_approx_equal(y, 1.0)
        assert_raises(ValueError, stats.moment, self.testcase, 1.2)
        y = stats.moment(self.testcase, [1.0, 2, 3, 4.0])
        assert_allclose(y, [0, 1.25, 0, 2.5625])

        # test empty input
        y = stats.moment([])
        assert_equal(y, np.nan)

        x = np.arange(10.)
        x[9] = np.nan
        assert_equal(stats.moment(x, 2), np.nan)
        assert_almost_equal(stats.moment(x, nan_policy='omit'), 0.0)
        assert_raises(ValueError, stats.moment, x, nan_policy='raise')
        assert_raises(ValueError, stats.moment, x, nan_policy='foobar')

    def test_moment_propagate_nan(self):
        # Check that the shape of the result is the same for inputs
        # with and without nans, cf gh-5817
        a = np.arange(8).reshape(2, -1).astype(float)
        a[1, 0] = np.nan
        mm = stats.moment(a, 2, axis=1, nan_policy="propagate")
        np.testing.assert_allclose(mm, [1.25, np.nan], atol=1e-15)

    def test_variation(self):
        # variation = samplestd / mean
        y = stats.variation(self.scalar_testcase)
        assert_approx_equal(y, 0.0)
        y = stats.variation(self.testcase)
        assert_approx_equal(y, 0.44721359549996, 10)

        x = np.arange(10.)
        x[9] = np.nan
        assert_equal(stats.variation(x), np.nan)
        assert_almost_equal(stats.variation(x, nan_policy='omit'),
                            0.6454972243679028)
        assert_raises(ValueError, stats.variation, x, nan_policy='raise')
        assert_raises(ValueError, stats.variation, x, nan_policy='foobar')

    def test_variation_propagate_nan(self):
        # Check that the shape of the result is the same for inputs
        # with and without nans, cf gh-5817
        a = np.arange(8).reshape(2, -1).astype(float)
        a[1, 0] = np.nan
        vv = stats.variation(a, axis=1, nan_policy="propagate")
        np.testing.assert_allclose(vv, [0.7453559924999299, np.nan], atol=1e-15)

    def test_skewness(self):
        # Scalar test case
        y = stats.skew(self.scalar_testcase)
        assert_approx_equal(y, 0.0)
        # sum((testmathworks-mean(testmathworks,axis=0))**3,axis=0) /
        #     ((sqrt(var(testmathworks)*4/5))**3)/5
        y = stats.skew(self.testmathworks)
        assert_approx_equal(y, -0.29322304336607, 10)
        y = stats.skew(self.testmathworks, bias=0)
        assert_approx_equal(y, -0.437111105023940, 10)
        y = stats.skew(self.testcase)
        assert_approx_equal(y, 0.0, 10)

        x = np.arange(10.)
        x[9] = np.nan
        with np.errstate(invalid='ignore'):
            assert_equal(stats.skew(x), np.nan)
        assert_equal(stats.skew(x, nan_policy='omit'), 0.)
        assert_raises(ValueError, stats.skew, x, nan_policy='raise')
        assert_raises(ValueError, stats.skew, x, nan_policy='foobar')

    def test_skewness_scalar(self):
        # `skew` must return a scalar for 1-dim input
        assert_equal(stats.skew(arange(10)), 0.0)

    def test_skew_propagate_nan(self):
        # Check that the shape of the result is the same for inputs
        # with and without nans, cf gh-5817
        a = np.arange(8).reshape(2, -1).astype(float)
        a[1, 0] = np.nan
        with np.errstate(invalid='ignore'):
            s = stats.skew(a, axis=1, nan_policy="propagate")
        np.testing.assert_allclose(s, [0, np.nan], atol=1e-15)

    def test_kurtosis(self):
        # Scalar test case
        y = stats.kurtosis(self.scalar_testcase)
        assert_approx_equal(y, -3.0)
        #   sum((testcase-mean(testcase,axis=0))**4,axis=0)/((sqrt(var(testcase)*3/4))**4)/4
        #   sum((test2-mean(testmathworks,axis=0))**4,axis=0)/((sqrt(var(testmathworks)*4/5))**4)/5
        #   Set flags for axis = 0 and
        #   fisher=0 (Pearson's defn of kurtosis for compatibility with Matlab)
        y = stats.kurtosis(self.testmathworks, 0, fisher=0, bias=1)
        assert_approx_equal(y, 2.1658856802973, 10)

        # Note that MATLAB has confusing docs for the following case
        #  kurtosis(x,0) gives an unbiased estimate of Pearson's skewness
        #  kurtosis(x)  gives a biased estimate of Fisher's skewness (Pearson-3)
        #  The MATLAB docs imply that both should give Fisher's
        y = stats.kurtosis(self.testmathworks, fisher=0, bias=0)
        assert_approx_equal(y, 3.663542721189047, 10)
        y = stats.kurtosis(self.testcase, 0, 0)
        assert_approx_equal(y, 1.64)

        x = np.arange(10.)
        x[9] = np.nan
        assert_equal(stats.kurtosis(x), np.nan)
        assert_almost_equal(stats.kurtosis(x, nan_policy='omit'), -1.230000)
        assert_raises(ValueError, stats.kurtosis, x, nan_policy='raise')
        assert_raises(ValueError, stats.kurtosis, x, nan_policy='foobar')

    def test_kurtosis_array_scalar(self):
        assert_equal(type(stats.kurtosis([1,2,3])), float)

    def test_kurtosis_propagate_nan(self):
        # Check that the shape of the result is the same for inputs
        # with and without nans, cf gh-5817
        a = np.arange(8).reshape(2, -1).astype(float)
        a[1, 0] = np.nan
        k = stats.kurtosis(a, axis=1, nan_policy="propagate")
        np.testing.assert_allclose(k, [-1.36, np.nan], atol=1e-15)

    def test_moment_accuracy(self):
        # 'moment' must have a small enough error compared to the slower
        #  but very accurate numpy.power() implementation.
        tc_no_mean = self.testcase_moment_accuracy - \
                     np.mean(self.testcase_moment_accuracy)
        assert_allclose(np.power(tc_no_mean, 42).mean(),
                            stats.moment(self.testcase_moment_accuracy, 42))


class TestStudentTest(object):
    X1 = np.array([-1, 0, 1])
    X2 = np.array([0, 1, 2])
    T1_0 = 0
    P1_0 = 1
    T1_1 = -1.732051
    P1_1 = 0.2254033
    T1_2 = -3.464102
    P1_2 = 0.0741799
    T2_0 = 1.732051
    P2_0 = 0.2254033

    def test_onesample(self):
        with suppress_warnings() as sup, np.errstate(invalid="ignore"):
            sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
            t, p = stats.ttest_1samp(4., 3.)
        assert_(np.isnan(t))
        assert_(np.isnan(p))

        t, p = stats.ttest_1samp(self.X1, 0)

        assert_array_almost_equal(t, self.T1_0)
        assert_array_almost_equal(p, self.P1_0)

        res = stats.ttest_1samp(self.X1, 0)
        attributes = ('statistic', 'pvalue')
        check_named_results(res, attributes)

        t, p = stats.ttest_1samp(self.X2, 0)

        assert_array_almost_equal(t, self.T2_0)
        assert_array_almost_equal(p, self.P2_0)

        t, p = stats.ttest_1samp(self.X1, 1)

        assert_array_almost_equal(t, self.T1_1)
        assert_array_almost_equal(p, self.P1_1)

        t, p = stats.ttest_1samp(self.X1, 2)

        assert_array_almost_equal(t, self.T1_2)
        assert_array_almost_equal(p, self.P1_2)

        # check nan policy
        np.random.seed(7654567)
        x = stats.norm.rvs(loc=5, scale=10, size=51)
        x[50] = np.nan
        with np.errstate(invalid="ignore"):
            assert_array_equal(stats.ttest_1samp(x, 5.0), (np.nan, np.nan))

            assert_array_almost_equal(stats.ttest_1samp(x, 5.0, nan_policy='omit'),
                                      (-1.6412624074367159, 0.107147027334048005))
            assert_raises(ValueError, stats.ttest_1samp, x, 5.0, nan_policy='raise')
            assert_raises(ValueError, stats.ttest_1samp, x, 5.0,
                          nan_policy='foobar')


def test_percentileofscore():
    pcos = stats.percentileofscore

    assert_equal(pcos([1,2,3,4,5,6,7,8,9,10],4), 40.0)

    for (kind, result) in [('mean', 35.0),
                           ('strict', 30.0),
                           ('weak', 40.0)]:
        assert_equal(pcos(np.arange(10) + 1, 4, kind=kind), result)

    # multiple - 2
    for (kind, result) in [('rank', 45.0),
                           ('strict', 30.0),
                           ('weak', 50.0),
                           ('mean', 40.0)]:
        assert_equal(pcos([1,2,3,4,4,5,6,7,8,9], 4, kind=kind), result)

    # multiple - 3
    assert_equal(pcos([1,2,3,4,4,4,5,6,7,8], 4), 50.0)
    for (kind, result) in [('rank', 50.0),
                           ('mean', 45.0),
                           ('strict', 30.0),
                           ('weak', 60.0)]:

        assert_equal(pcos([1,2,3,4,4,4,5,6,7,8], 4, kind=kind), result)

    # missing
    for kind in ('rank', 'mean', 'strict', 'weak'):
        assert_equal(pcos([1,2,3,5,6,7,8,9,10,11], 4, kind=kind), 30)

    # larger numbers
    for (kind, result) in [('mean', 35.0),
                           ('strict', 30.0),
                           ('weak', 40.0)]:
        assert_equal(
              pcos([10, 20, 30, 40, 50, 60, 70, 80, 90, 100], 40,
                   kind=kind), result)

    for (kind, result) in [('mean', 45.0),
                           ('strict', 30.0),
                           ('weak', 60.0)]:
        assert_equal(
              pcos([10, 20, 30, 40, 40, 40, 50, 60, 70, 80],
                   40, kind=kind), result)

    for kind in ('rank', 'mean', 'strict', 'weak'):
        assert_equal(
              pcos([10, 20, 30, 50, 60, 70, 80, 90, 100, 110],
                   40, kind=kind), 30.0)

    # boundaries
    for (kind, result) in [('rank', 10.0),
                           ('mean', 5.0),
                           ('strict', 0.0),
                           ('weak', 10.0)]:
        assert_equal(
              pcos([10, 20, 30, 50, 60, 70, 80, 90, 100, 110],
                   10, kind=kind), result)

    for (kind, result) in [('rank', 100.0),
                           ('mean', 95.0),
                           ('strict', 90.0),
                           ('weak', 100.0)]:
        assert_equal(
              pcos([10, 20, 30, 50, 60, 70, 80, 90, 100, 110],
                   110, kind=kind), result)

    # out of bounds
    for (kind, score, result) in [('rank', 200, 100.0),
                                  ('mean', 200, 100.0),
                                  ('mean', 0, 0.0)]:
        assert_equal(
              pcos([10, 20, 30, 50, 60, 70, 80, 90, 100, 110],
                   score, kind=kind), result)

    assert_raises(ValueError, pcos, [1, 2, 3, 3, 4], 3, kind='unrecognized')


PowerDivCase = namedtuple('Case', ['f_obs', 'f_exp', 'ddof', 'axis',
                                   'chi2',     # Pearson's
                                   'log',      # G-test (log-likelihood)
                                   'mod_log',  # Modified log-likelihood
                                   'cr',       # Cressie-Read (lambda=2/3)
                                   ])

# The details of the first two elements in power_div_1d_cases are used
# in a test in TestPowerDivergence.  Check that code before making
# any changes here.
power_div_1d_cases = [
    # Use the default f_exp.
    PowerDivCase(f_obs=[4, 8, 12, 8], f_exp=None, ddof=0, axis=None,
                 chi2=4,
                 log=2*(4*np.log(4/8) + 12*np.log(12/8)),
                 mod_log=2*(8*np.log(8/4) + 8*np.log(8/12)),
                 cr=(4*((4/8)**(2/3) - 1) + 12*((12/8)**(2/3) - 1))/(5/9)),
    # Give a non-uniform f_exp.
    PowerDivCase(f_obs=[4, 8, 12, 8], f_exp=[2, 16, 12, 2], ddof=0, axis=None,
                 chi2=24,
                 log=2*(4*np.log(4/2) + 8*np.log(8/16) + 8*np.log(8/2)),
                 mod_log=2*(2*np.log(2/4) + 16*np.log(16/8) + 2*np.log(2/8)),
                 cr=(4*((4/2)**(2/3) - 1) + 8*((8/16)**(2/3) - 1) +
                     8*((8/2)**(2/3) - 1))/(5/9)),
    # f_exp is a scalar.
    PowerDivCase(f_obs=[4, 8, 12, 8], f_exp=8, ddof=0, axis=None,
                 chi2=4,
                 log=2*(4*np.log(4/8) + 12*np.log(12/8)),
                 mod_log=2*(8*np.log(8/4) + 8*np.log(8/12)),
                 cr=(4*((4/8)**(2/3) - 1) + 12*((12/8)**(2/3) - 1))/(5/9)),
    # f_exp equal to f_obs.
    PowerDivCase(f_obs=[3, 5, 7, 9], f_exp=[3, 5, 7, 9], ddof=0, axis=0,
                 chi2=0, log=0, mod_log=0, cr=0),
]


power_div_empty_cases = [
    # Shape is (0,)--a data set with length 0.  The computed
    # test statistic should be 0.
    PowerDivCase(f_obs=[],
                 f_exp=None, ddof=0, axis=0,
                 chi2=0, log=0, mod_log=0, cr=0),
    # Shape is (0, 3).  This is 3 data sets, but each data set has
    # length 0, so the computed test statistic should be [0, 0, 0].
    PowerDivCase(f_obs=np.array([[],[],[]]).T,
                 f_exp=None, ddof=0, axis=0,
                 chi2=[0, 0, 0],
                 log=[0, 0, 0],
                 mod_log=[0, 0, 0],
                 cr=[0, 0, 0]),
    # Shape is (3, 0).  This represents an empty collection of
    # data sets in which each data set has length 3.  The test
    # statistic should be an empty array.
    PowerDivCase(f_obs=np.array([[],[],[]]),
                 f_exp=None, ddof=0, axis=0,
                 chi2=[],
                 log=[],
                 mod_log=[],
                 cr=[]),
]


class TestPowerDivergence(object):

    def check_power_divergence(self, f_obs, f_exp, ddof, axis, lambda_,
                               expected_stat):
        f_obs = np.asarray(f_obs)
        if axis is None:
            num_obs = f_obs.size
        else:
            b = np.broadcast(f_obs, f_exp)
            num_obs = b.shape[axis]

        with suppress_warnings() as sup:
            sup.filter(RuntimeWarning, "Mean of empty slice")
            stat, p = stats.power_divergence(
                                f_obs=f_obs, f_exp=f_exp, ddof=ddof,
                                axis=axis, lambda_=lambda_)
            assert_allclose(stat, expected_stat)

            if lambda_ == 1 or lambda_ == "pearson":
                # Also test stats.chisquare.
                stat, p = stats.chisquare(f_obs=f_obs, f_exp=f_exp, ddof=ddof,
                                          axis=axis)
                assert_allclose(stat, expected_stat)

        ddof = np.asarray(ddof)
        expected_p = stats.distributions.chi2.sf(expected_stat,
                                                 num_obs - 1 - ddof)
        assert_allclose(p, expected_p)

    def test_basic(self):
        for case in power_div_1d_cases:
            self.check_power_divergence(
                   case.f_obs, case.f_exp, case.ddof, case.axis,
                   None, case.chi2)
            self.check_power_divergence(
                   case.f_obs, case.f_exp, case.ddof, case.axis,
                   "pearson", case.chi2)
            self.check_power_divergence(
                   case.f_obs, case.f_exp, case.ddof, case.axis,
                   1, case.chi2)
            self.check_power_divergence(
                   case.f_obs, case.f_exp, case.ddof, case.axis,
                   "log-likelihood", case.log)
            self.check_power_divergence(
                   case.f_obs, case.f_exp, case.ddof, case.axis,
                   "mod-log-likelihood", case.mod_log)
            self.check_power_divergence(
                   case.f_obs, case.f_exp, case.ddof, case.axis,
                   "cressie-read", case.cr)
            self.check_power_divergence(
                   case.f_obs, case.f_exp, case.ddof, case.axis,
                   2/3, case.cr)

    def test_basic_masked(self):
        for case in power_div_1d_cases:
            mobs = np.ma.array(case.f_obs)
            self.check_power_divergence(
                   mobs, case.f_exp, case.ddof, case.axis,
                   None, case.chi2)
            self.check_power_divergence(
                   mobs, case.f_exp, case.ddof, case.axis,
                   "pearson", case.chi2)
            self.check_power_divergence(
                   mobs, case.f_exp, case.ddof, case.axis,
                   1, case.chi2)
            self.check_power_divergence(
                   mobs, case.f_exp, case.ddof, case.axis,
                   "log-likelihood", case.log)
            self.check_power_divergence(
                   mobs, case.f_exp, case.ddof, case.axis,
                   "mod-log-likelihood", case.mod_log)
            self.check_power_divergence(
                   mobs, case.f_exp, case.ddof, case.axis,
                   "cressie-read", case.cr)
            self.check_power_divergence(
                   mobs, case.f_exp, case.ddof, case.axis,
                   2/3, case.cr)

    def test_axis(self):
        case0 = power_div_1d_cases[0]
        case1 = power_div_1d_cases[1]
        f_obs = np.vstack((case0.f_obs, case1.f_obs))
        f_exp = np.vstack((np.ones_like(case0.f_obs)*np.mean(case0.f_obs),
                           case1.f_exp))
        # Check the four computational code paths in power_divergence
        # using a 2D array with axis=1.
        self.check_power_divergence(
               f_obs, f_exp, 0, 1,
               "pearson", [case0.chi2, case1.chi2])
        self.check_power_divergence(
               f_obs, f_exp, 0, 1,
               "log-likelihood", [case0.log, case1.log])
        self.check_power_divergence(
               f_obs, f_exp, 0, 1,
               "mod-log-likelihood", [case0.mod_log, case1.mod_log])
        self.check_power_divergence(
               f_obs, f_exp, 0, 1,
               "cressie-read", [case0.cr, case1.cr])
        # Reshape case0.f_obs to shape (2,2), and use axis=None.
        # The result should be the same.
        self.check_power_divergence(
               np.array(case0.f_obs).reshape(2, 2), None, 0, None,
               "pearson", case0.chi2)

    def test_ddof_broadcasting(self):
        # Test that ddof broadcasts correctly.
        # ddof does not affect the test statistic.  It is broadcast
        # with the computed test statistic for the computation of
        # the p value.

        case0 = power_div_1d_cases[0]
        case1 = power_div_1d_cases[1]
        # Create 4x2 arrays of observed and expected frequencies.
        f_obs = np.vstack((case0.f_obs, case1.f_obs)).T
        f_exp = np.vstack((np.ones_like(case0.f_obs)*np.mean(case0.f_obs),
                           case1.f_exp)).T

        expected_chi2 = [case0.chi2, case1.chi2]

        # ddof has shape (2, 1).  This is broadcast with the computed
        # statistic, so p will have shape (2,2).
        ddof = np.array([[0], [1]])

        stat, p = stats.power_divergence(f_obs, f_exp, ddof=ddof)
        assert_allclose(stat, expected_chi2)

        # Compute the p values separately, passing in scalars for ddof.
        stat0, p0 = stats.power_divergence(f_obs, f_exp, ddof=ddof[0,0])
        stat1, p1 = stats.power_divergence(f_obs, f_exp, ddof=ddof[1,0])

        assert_array_equal(p, np.vstack((p0, p1)))

    def test_empty_cases(self):
        with warnings.catch_warnings():
            for case in power_div_empty_cases:
                self.check_power_divergence(
                       case.f_obs, case.f_exp, case.ddof, case.axis,
                       "pearson", case.chi2)
                self.check_power_divergence(
                       case.f_obs, case.f_exp, case.ddof, case.axis,
                       "log-likelihood", case.log)
                self.check_power_divergence(
                       case.f_obs, case.f_exp, case.ddof, case.axis,
                       "mod-log-likelihood", case.mod_log)
                self.check_power_divergence(
                       case.f_obs, case.f_exp, case.ddof, case.axis,
                       "cressie-read", case.cr)

    def test_power_divergence_result_attributes(self):
        f_obs = power_div_1d_cases[0].f_obs
        f_exp = power_div_1d_cases[0].f_exp
        ddof = power_div_1d_cases[0].ddof
        axis = power_div_1d_cases[0].axis

        res = stats.power_divergence(f_obs=f_obs, f_exp=f_exp, ddof=ddof,
                                     axis=axis, lambda_="pearson")
        attributes = ('statistic', 'pvalue')
        check_named_results(res, attributes)


def test_chisquare_masked_arrays():
    # Test masked arrays.
    obs = np.array([[8, 8, 16, 32, -1], [-1, -1, 3, 4, 5]]).T
    mask = np.array([[0, 0, 0, 0, 1], [1, 1, 0, 0, 0]]).T
    mobs = np.ma.masked_array(obs, mask)
    expected_chisq = np.array([24.0, 0.5])
    expected_g = np.array([2*(2*8*np.log(0.5) + 32*np.log(2.0)),
                           2*(3*np.log(0.75) + 5*np.log(1.25))])

    chi2 = stats.distributions.chi2

    chisq, p = stats.chisquare(mobs)
    mat.assert_array_equal(chisq, expected_chisq)
    mat.assert_array_almost_equal(p, chi2.sf(expected_chisq,
                                             mobs.count(axis=0) - 1))

    g, p = stats.power_divergence(mobs, lambda_='log-likelihood')
    mat.assert_array_almost_equal(g, expected_g, decimal=15)
    mat.assert_array_almost_equal(p, chi2.sf(expected_g,
                                             mobs.count(axis=0) - 1))

    chisq, p = stats.chisquare(mobs.T, axis=1)
    mat.assert_array_equal(chisq, expected_chisq)
    mat.assert_array_almost_equal(p, chi2.sf(expected_chisq,
                                             mobs.T.count(axis=1) - 1))
    g, p = stats.power_divergence(mobs.T, axis=1, lambda_="log-likelihood")
    mat.assert_array_almost_equal(g, expected_g, decimal=15)
    mat.assert_array_almost_equal(p, chi2.sf(expected_g,
                                             mobs.count(axis=0) - 1))

    obs1 = np.ma.array([3, 5, 6, 99, 10], mask=[0, 0, 0, 1, 0])
    exp1 = np.ma.array([2, 4, 8, 10, 99], mask=[0, 0, 0, 0, 1])
    chi2, p = stats.chisquare(obs1, f_exp=exp1)
    # Because of the mask at index 3 of obs1 and at index 4 of exp1,
    # only the first three elements are included in the calculation
    # of the statistic.
    mat.assert_array_equal(chi2, 1/2 + 1/4 + 4/8)

    # When axis=None, the two values should have type np.float64.
    chisq, p = stats.chisquare(np.ma.array([1,2,3]), axis=None)
    assert_(isinstance(chisq, np.float64))
    assert_(isinstance(p, np.float64))
    assert_equal(chisq, 1.0)
    assert_almost_equal(p, stats.distributions.chi2.sf(1.0, 2))

    # Empty arrays:
    # A data set with length 0 returns a masked scalar.
    with np.errstate(invalid='ignore'):
        with suppress_warnings() as sup:
            sup.filter(RuntimeWarning, "Mean of empty slice")
            chisq, p = stats.chisquare(np.ma.array([]))
    assert_(isinstance(chisq, np.ma.MaskedArray))
    assert_equal(chisq.shape, ())
    assert_(chisq.mask)

    empty3 = np.ma.array([[],[],[]])

    # empty3 is a collection of 0 data sets (whose lengths would be 3, if
    # there were any), so the return value is an array with length 0.
    chisq, p = stats.chisquare(empty3)
    assert_(isinstance(chisq, np.ma.MaskedArray))
    mat.assert_array_equal(chisq, [])

    # empty3.T is an array containing 3 data sets, each with length 0,
    # so an array of size (3,) is returned, with all values masked.
    with np.errstate(invalid='ignore'):
        with suppress_warnings() as sup:
            sup.filter(RuntimeWarning, "Mean of empty slice")
            chisq, p = stats.chisquare(empty3.T)

    assert_(isinstance(chisq, np.ma.MaskedArray))
    assert_equal(chisq.shape, (3,))
    assert_(np.all(chisq.mask))


def test_power_divergence_against_cressie_read_data():
    # Test stats.power_divergence against tables 4 and 5 from
    # Cressie and Read, "Multimonial Goodness-of-Fit Tests",
    # J. R. Statist. Soc. B (1984), Vol 46, No. 3, pp. 440-464.
    # This tests the calculation for several values of lambda.

    # `table4` holds just the second and third columns from Table 4.
    table4 = np.array([
        # observed, expected,
        15, 15.171,
        11, 13.952,
        14, 12.831,
        17, 11.800,
        5, 10.852,
        11, 9.9796,
        10, 9.1777,
        4, 8.4402,
        8, 7.7620,
        10, 7.1383,
        7, 6.5647,
        9, 6.0371,
        11, 5.5520,
        3, 5.1059,
        6, 4.6956,
        1, 4.3183,
        1, 3.9713,
        4, 3.6522,
        ]).reshape(-1, 2)
    table5 = np.array([
        # lambda, statistic
        -10.0, 72.2e3,
        -5.0, 28.9e1,
        -3.0, 65.6,
        -2.0, 40.6,
        -1.5, 34.0,
        -1.0, 29.5,
        -0.5, 26.5,
        0.0, 24.6,
        0.5, 23.4,
        0.67, 23.1,
        1.0, 22.7,
        1.5, 22.6,
        2.0, 22.9,
        3.0, 24.8,
        5.0, 35.5,
        10.0, 21.4e1,
        ]).reshape(-1, 2)

    for lambda_, expected_stat in table5:
        stat, p = stats.power_divergence(table4[:,0], table4[:,1],
                                         lambda_=lambda_)
        assert_allclose(stat, expected_stat, rtol=5e-3)


def test_friedmanchisquare():
    # see ticket:113
    # verified with matlab and R
    # From Demsar "Statistical Comparisons of Classifiers over Multiple Data Sets"
    # 2006, Xf=9.28 (no tie handling, tie corrected Xf >=9.28)
    x1 = [array([0.763, 0.599, 0.954, 0.628, 0.882, 0.936, 0.661, 0.583,
                 0.775, 1.0, 0.94, 0.619, 0.972, 0.957]),
          array([0.768, 0.591, 0.971, 0.661, 0.888, 0.931, 0.668, 0.583,
                 0.838, 1.0, 0.962, 0.666, 0.981, 0.978]),
          array([0.771, 0.590, 0.968, 0.654, 0.886, 0.916, 0.609, 0.563,
                 0.866, 1.0, 0.965, 0.614, 0.9751, 0.946]),
          array([0.798, 0.569, 0.967, 0.657, 0.898, 0.931, 0.685, 0.625,
                 0.875, 1.0, 0.962, 0.669, 0.975, 0.970])]

    # From "Bioestadistica para las ciencias de la salud" Xf=18.95 p<0.001:
    x2 = [array([4,3,5,3,5,3,2,5,4,4,4,3]),
          array([2,2,1,2,3,1,2,3,2,1,1,3]),
          array([2,4,3,3,4,3,3,4,4,1,2,1]),
          array([3,5,4,3,4,4,3,3,3,4,4,4])]

    # From Jerrorl H. Zar, "Biostatistical Analysis"(example 12.6), Xf=10.68, 0.005 < p < 0.01:
    # Probability from this example is inexact using Chisquare approximation of Friedman Chisquare.
    x3 = [array([7.0,9.9,8.5,5.1,10.3]),
          array([5.3,5.7,4.7,3.5,7.7]),
          array([4.9,7.6,5.5,2.8,8.4]),
          array([8.8,8.9,8.1,3.3,9.1])]

    assert_array_almost_equal(stats.friedmanchisquare(x1[0],x1[1],x1[2],x1[3]),
                              (10.2283464566929, 0.0167215803284414))
    assert_array_almost_equal(stats.friedmanchisquare(x2[0],x2[1],x2[2],x2[3]),
                              (18.9428571428571, 0.000280938375189499))
    assert_array_almost_equal(stats.friedmanchisquare(x3[0],x3[1],x3[2],x3[3]),
                              (10.68, 0.0135882729582176))
    assert_raises(ValueError, stats.friedmanchisquare,x3[0],x3[1])

    # test for namedtuple attribute results
    attributes = ('statistic', 'pvalue')
    res = stats.friedmanchisquare(*x1)
    check_named_results(res, attributes)

    # test using mstats
    assert_array_almost_equal(mstats.friedmanchisquare(x1[0], x1[1],
                                                       x1[2], x1[3]),
                              (10.2283464566929, 0.0167215803284414))
    # the following fails
    # assert_array_almost_equal(mstats.friedmanchisquare(x2[0],x2[1],x2[2],x2[3]),
    #                           (18.9428571428571, 0.000280938375189499))
    assert_array_almost_equal(mstats.friedmanchisquare(x3[0], x3[1],
                                                       x3[2], x3[3]),
                              (10.68, 0.0135882729582176))
    assert_raises(ValueError, mstats.friedmanchisquare,x3[0],x3[1])


def test_kstest():
    # from numpy.testing import assert_almost_equal

    # comparing with values from R
    x = np.linspace(-1,1,9)
    D,p = stats.kstest(x,'norm')
    assert_almost_equal(D, 0.15865525393145705, 12)
    assert_almost_equal(p, 0.95164069201518386, 1)

    x = np.linspace(-15,15,9)
    D,p = stats.kstest(x,'norm')
    assert_almost_equal(D, 0.44435602715924361, 15)
    assert_almost_equal(p, 0.038850140086788665, 8)

    # test for namedtuple attribute results
    attributes = ('statistic', 'pvalue')
    res = stats.kstest(x, 'norm')
    check_named_results(res, attributes)

    # the following tests rely on deterministicaly replicated rvs
    np.random.seed(987654321)
    x = stats.norm.rvs(loc=0.2, size=100)
    D,p = stats.kstest(x, 'norm', mode='asymp')
    assert_almost_equal(D, 0.12464329735846891, 15)
    assert_almost_equal(p, 0.089444888711820769, 15)
    assert_almost_equal(np.array(stats.kstest(x, 'norm', mode='asymp')),
                np.array((0.12464329735846891, 0.089444888711820769)), 15)
    assert_almost_equal(np.array(stats.kstest(x,'norm', alternative='less')),
                np.array((0.12464329735846891, 0.040989164077641749)), 15)
    # this 'greater' test fails with precision of decimal=14
    assert_almost_equal(np.array(stats.kstest(x,'norm', alternative='greater')),
                np.array((0.0072115233216310994, 0.98531158590396228)), 12)

    # missing: no test that uses *args


def test_ks_2samp():
    # exact small sample solution
    data1 = np.array([1.0,2.0])
    data2 = np.array([1.0,2.0,3.0])
    assert_almost_equal(np.array(stats.ks_2samp(data1+0.01,data2)),
                np.array((0.33333333333333337, 0.99062316386915694)))
    assert_almost_equal(np.array(stats.ks_2samp(data1-0.01,data2)),
                np.array((0.66666666666666674, 0.42490954988801982)))
    # these can also be verified graphically
    assert_almost_equal(
        np.array(stats.ks_2samp(np.linspace(1,100,100),
                              np.linspace(1,100,100)+2+0.1)),
        np.array((0.030000000000000027, 0.99999999996005062)))
    assert_almost_equal(
        np.array(stats.ks_2samp(np.linspace(1,100,100),
                              np.linspace(1,100,100)+2-0.1)),
        np.array((0.020000000000000018, 0.99999999999999933)))
    # these are just regression tests
    assert_almost_equal(
        np.array(stats.ks_2samp(np.linspace(1,100,100),
                              np.linspace(1,100,110)+20.1)),
        np.array((0.21090909090909091, 0.015880386730710221)))
    assert_almost_equal(
        np.array(stats.ks_2samp(np.linspace(1,100,100),
                              np.linspace(1,100,110)+20-0.1)),
        np.array((0.20818181818181825, 0.017981441789762638)))

    # test for namedtuple attribute results
    attributes = ('statistic', 'pvalue')
    res = stats.ks_2samp(data1 - 0.01, data2)
    check_named_results(res, attributes)


def test_ttest_rel():
    # regression test
    tr,pr = 0.81248591389165692, 0.41846234511362157
    tpr = ([tr,-tr],[pr,pr])

    rvs1 = np.linspace(1,100,100)
    rvs2 = np.linspace(1.01,99.989,100)
    rvs1_2D = np.array([np.linspace(1,100,100), np.linspace(1.01,99.989,100)])
    rvs2_2D = np.array([np.linspace(1.01,99.989,100), np.linspace(1,100,100)])

    t,p = stats.ttest_rel(rvs1, rvs2, axis=0)
    assert_array_almost_equal([t,p],(tr,pr))
    t,p = stats.ttest_rel(rvs1_2D.T, rvs2_2D.T, axis=0)
    assert_array_almost_equal([t,p],tpr)
    t,p = stats.ttest_rel(rvs1_2D, rvs2_2D, axis=1)
    assert_array_almost_equal([t,p],tpr)

    # test scalars
    with suppress_warnings() as sup, np.errstate(invalid="ignore"):
        sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
        t, p = stats.ttest_rel(4., 3.)
    assert_(np.isnan(t))
    assert_(np.isnan(p))

    # test for namedtuple attribute results
    attributes = ('statistic', 'pvalue')
    res = stats.ttest_rel(rvs1, rvs2, axis=0)
    check_named_results(res, attributes)

    # test on 3 dimensions
    rvs1_3D = np.dstack([rvs1_2D,rvs1_2D,rvs1_2D])
    rvs2_3D = np.dstack([rvs2_2D,rvs2_2D,rvs2_2D])
    t,p = stats.ttest_rel(rvs1_3D, rvs2_3D, axis=1)
    assert_array_almost_equal(np.abs(t), tr)
    assert_array_almost_equal(np.abs(p), pr)
    assert_equal(t.shape, (2, 3))

    t,p = stats.ttest_rel(np.rollaxis(rvs1_3D,2), np.rollaxis(rvs2_3D,2), axis=2)
    assert_array_almost_equal(np.abs(t), tr)
    assert_array_almost_equal(np.abs(p), pr)
    assert_equal(t.shape, (3, 2))

    # check nan policy
    np.random.seed(12345678)
    x = stats.norm.rvs(loc=5, scale=10, size=501)
    x[500] = np.nan
    y = (stats.norm.rvs(loc=5, scale=10, size=501) +
         stats.norm.rvs(scale=0.2, size=501))
    y[500] = np.nan

    with np.errstate(invalid="ignore"):
        assert_array_equal(stats.ttest_rel(x, x), (np.nan, np.nan))

    assert_array_almost_equal(stats.ttest_rel(x, y, nan_policy='omit'),
                              (0.25299925303978066, 0.8003729814201519))
    assert_raises(ValueError, stats.ttest_rel, x, y, nan_policy='raise')
    assert_raises(ValueError, stats.ttest_rel, x, y, nan_policy='foobar')

    # test zero division problem
    t, p = stats.ttest_rel([0, 0, 0], [1, 1, 1])
    assert_equal((np.abs(t), p), (np.inf, 0))
    with np.errstate(invalid="ignore"):
        assert_equal(stats.ttest_rel([0, 0, 0], [0, 0, 0]), (np.nan, np.nan))

        # check that nan in input array result in nan output
        anan = np.array([[1, np.nan], [-1, 1]])
        assert_equal(stats.ttest_rel(anan, np.zeros((2, 2))),
                     ([0, np.nan], [1, np.nan]))

    # test incorrect input shape raise an error
    x = np.arange(24)
    assert_raises(ValueError, stats.ttest_rel, x.reshape((8, 3)),
                  x.reshape((2, 3, 4)))


def test_ttest_rel_nan_2nd_arg():
    # regression test for gh-6134: nans in the second arg were not handled
    x = [np.nan, 2.0, 3.0, 4.0]
    y = [1.0, 2.0, 1.0, 2.0]

    r1 = stats.ttest_rel(x, y, nan_policy='omit')
    r2 = stats.ttest_rel(y, x, nan_policy='omit')
    assert_allclose(r2.statistic, -r1.statistic, atol=1e-15)
    assert_allclose(r2.pvalue, r1.pvalue, atol=1e-15)

    # NB: arguments are paired when NaNs are dropped
    r3 = stats.ttest_rel(y[1:], x[1:])
    assert_allclose(r2, r3, atol=1e-15)

    # .. and this is consistent with R. R code:
    # x = c(NA, 2.0, 3.0, 4.0)
    # y = c(1.0, 2.0, 1.0, 2.0)
    # t.test(x, y, paired=TRUE)
    assert_allclose(r2, (-2, 0.1835), atol=1e-4)


def _desc_stats(x1, x2, axis=0):
    def _stats(x, axis=0):
        x = np.asarray(x)
        mu = np.mean(x, axis=axis)
        std = np.std(x, axis=axis, ddof=1)
        nobs = x.shape[axis]
        return mu, std, nobs
    return _stats(x1, axis) + _stats(x2, axis)


def test_ttest_ind():
    # regression test
    tr = 1.0912746897927283
    pr = 0.27647818616351882
    tpr = ([tr,-tr],[pr,pr])

    rvs2 = np.linspace(1,100,100)
    rvs1 = np.linspace(5,105,100)
    rvs1_2D = np.array([rvs1, rvs2])
    rvs2_2D = np.array([rvs2, rvs1])

    t,p = stats.ttest_ind(rvs1, rvs2, axis=0)
    assert_array_almost_equal([t,p],(tr,pr))
    # test from_stats API
    assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(rvs1,
                                                                      rvs2)),
                              [t, p])
    t,p = stats.ttest_ind(rvs1_2D.T, rvs2_2D.T, axis=0)
    assert_array_almost_equal([t,p],tpr)
    args = _desc_stats(rvs1_2D.T, rvs2_2D.T)
    assert_array_almost_equal(stats.ttest_ind_from_stats(*args),
                              [t, p])
    t,p = stats.ttest_ind(rvs1_2D, rvs2_2D, axis=1)
    assert_array_almost_equal([t,p],tpr)
    args = _desc_stats(rvs1_2D, rvs2_2D, axis=1)
    assert_array_almost_equal(stats.ttest_ind_from_stats(*args),
                              [t, p])

    # test scalars
    with suppress_warnings() as sup, np.errstate(invalid="ignore"):
        sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
        t, p = stats.ttest_ind(4., 3.)
    assert_(np.isnan(t))
    assert_(np.isnan(p))

    # test on 3 dimensions
    rvs1_3D = np.dstack([rvs1_2D,rvs1_2D,rvs1_2D])
    rvs2_3D = np.dstack([rvs2_2D,rvs2_2D,rvs2_2D])
    t,p = stats.ttest_ind(rvs1_3D, rvs2_3D, axis=1)
    assert_almost_equal(np.abs(t), np.abs(tr))
    assert_array_almost_equal(np.abs(p), pr)
    assert_equal(t.shape, (2, 3))

    t,p = stats.ttest_ind(np.rollaxis(rvs1_3D,2), np.rollaxis(rvs2_3D,2), axis=2)
    assert_array_almost_equal(np.abs(t), np.abs(tr))
    assert_array_almost_equal(np.abs(p), pr)
    assert_equal(t.shape, (3, 2))

    # check nan policy
    np.random.seed(12345678)
    x = stats.norm.rvs(loc=5, scale=10, size=501)
    x[500] = np.nan
    y = stats.norm.rvs(loc=5, scale=10, size=500)

    with np.errstate(invalid="ignore"):
        assert_array_equal(stats.ttest_ind(x, y), (np.nan, np.nan))

    assert_array_almost_equal(stats.ttest_ind(x, y, nan_policy='omit'),
                              (0.24779670949091914, 0.80434267337517906))
    assert_raises(ValueError, stats.ttest_ind, x, y, nan_policy='raise')
    assert_raises(ValueError, stats.ttest_ind, x, y, nan_policy='foobar')

    # test zero division problem
    t, p = stats.ttest_ind([0, 0, 0], [1, 1, 1])
    assert_equal((np.abs(t), p), (np.inf, 0))

    with np.errstate(invalid="ignore"):
        assert_equal(stats.ttest_ind([0, 0, 0], [0, 0, 0]), (np.nan, np.nan))

        # check that nan in input array result in nan output
        anan = np.array([[1, np.nan], [-1, 1]])
        assert_equal(stats.ttest_ind(anan, np.zeros((2, 2))),
                     ([0, np.nan], [1, np.nan]))


def test_ttest_ind_with_uneq_var():
    # check vs. R
    a = (1, 2, 3)
    b = (1.1, 2.9, 4.2)
    pr = 0.53619490753126731
    tr = -0.68649512735572582
    t, p = stats.ttest_ind(a, b, equal_var=False)
    assert_array_almost_equal([t,p], [tr, pr])
    # test from desc stats API
    assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(a, b),
                                                         equal_var=False),
                              [t, p])

    a = (1, 2, 3, 4)
    pr = 0.84354139131608286
    tr = -0.2108663315950719
    t, p = stats.ttest_ind(a, b, equal_var=False)
    assert_array_almost_equal([t,p], [tr, pr])
    assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(a, b),
                                                         equal_var=False),
                              [t, p])

    # regression test
    tr = 1.0912746897927283
    tr_uneq_n = 0.66745638708050492
    pr = 0.27647831993021388
    pr_uneq_n = 0.50873585065616544
    tpr = ([tr,-tr],[pr,pr])

    rvs3 = np.linspace(1,100, 25)
    rvs2 = np.linspace(1,100,100)
    rvs1 = np.linspace(5,105,100)
    rvs1_2D = np.array([rvs1, rvs2])
    rvs2_2D = np.array([rvs2, rvs1])

    t,p = stats.ttest_ind(rvs1, rvs2, axis=0, equal_var=False)
    assert_array_almost_equal([t,p],(tr,pr))
    assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(rvs1,
                                                                      rvs2),
                                                         equal_var=False),
                              (t, p))

    t,p = stats.ttest_ind(rvs1, rvs3, axis=0, equal_var=False)
    assert_array_almost_equal([t,p], (tr_uneq_n, pr_uneq_n))
    assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(rvs1,
                                                                      rvs3),
                                                         equal_var=False),
                              (t, p))

    t,p = stats.ttest_ind(rvs1_2D.T, rvs2_2D.T, axis=0, equal_var=False)
    assert_array_almost_equal([t,p],tpr)
    args = _desc_stats(rvs1_2D.T, rvs2_2D.T)
    assert_array_almost_equal(stats.ttest_ind_from_stats(*args,
                                                         equal_var=False),
                              (t, p))

    t,p = stats.ttest_ind(rvs1_2D, rvs2_2D, axis=1, equal_var=False)
    assert_array_almost_equal([t,p],tpr)
    args = _desc_stats(rvs1_2D, rvs2_2D, axis=1)
    assert_array_almost_equal(stats.ttest_ind_from_stats(*args,
                                                         equal_var=False),
                              (t, p))

    # test for namedtuple attribute results
    attributes = ('statistic', 'pvalue')
    res = stats.ttest_ind(rvs1, rvs2, axis=0, equal_var=False)
    check_named_results(res, attributes)

    # test on 3 dimensions
    rvs1_3D = np.dstack([rvs1_2D,rvs1_2D,rvs1_2D])
    rvs2_3D = np.dstack([rvs2_2D,rvs2_2D,rvs2_2D])
    t,p = stats.ttest_ind(rvs1_3D, rvs2_3D, axis=1, equal_var=False)
    assert_almost_equal(np.abs(t), np.abs(tr))
    assert_array_almost_equal(np.abs(p), pr)
    assert_equal(t.shape, (2, 3))
    args = _desc_stats(rvs1_3D, rvs2_3D, axis=1)
    t, p = stats.ttest_ind_from_stats(*args, equal_var=False)
    assert_almost_equal(np.abs(t), np.abs(tr))
    assert_array_almost_equal(np.abs(p), pr)
    assert_equal(t.shape, (2, 3))

    t,p = stats.ttest_ind(np.rollaxis(rvs1_3D,2), np.rollaxis(rvs2_3D,2),
                                   axis=2, equal_var=False)
    assert_array_almost_equal(np.abs(t), np.abs(tr))
    assert_array_almost_equal(np.abs(p), pr)
    assert_equal(t.shape, (3, 2))
    args = _desc_stats(np.rollaxis(rvs1_3D, 2),
                       np.rollaxis(rvs2_3D, 2), axis=2)
    t, p = stats.ttest_ind_from_stats(*args, equal_var=False)
    assert_array_almost_equal(np.abs(t), np.abs(tr))
    assert_array_almost_equal(np.abs(p), pr)
    assert_equal(t.shape, (3, 2))

    # test zero division problem
    t, p = stats.ttest_ind([0, 0, 0], [1, 1, 1], equal_var=False)
    assert_equal((np.abs(t), p), (np.inf, 0))
    with np.errstate(all='ignore'):
        assert_equal(stats.ttest_ind([0, 0, 0], [0, 0, 0], equal_var=False),
                     (np.nan, np.nan))

        # check that nan in input array result in nan output
        anan = np.array([[1, np.nan], [-1, 1]])
        assert_equal(stats.ttest_ind(anan, np.zeros((2, 2)), equal_var=False),
                     ([0, np.nan], [1, np.nan]))


def test_ttest_ind_nan_2nd_arg():
    # regression test for gh-6134: nans in the second arg were not handled
    x = [np.nan, 2.0, 3.0, 4.0]
    y = [1.0, 2.0, 1.0, 2.0]

    r1 = stats.ttest_ind(x, y, nan_policy='omit')
    r2 = stats.ttest_ind(y, x, nan_policy='omit')
    assert_allclose(r2.statistic, -r1.statistic, atol=1e-15)
    assert_allclose(r2.pvalue, r1.pvalue, atol=1e-15)

    # NB: arguments are not paired when NaNs are dropped
    r3 = stats.ttest_ind(y, x[1:])
    assert_allclose(r2, r3, atol=1e-15)

    # .. and this is consistent with R. R code:
    # x = c(NA, 2.0, 3.0, 4.0)
    # y = c(1.0, 2.0, 1.0, 2.0)
    # t.test(x, y, var.equal=TRUE)
    assert_allclose(r2, (-2.5354627641855498, 0.052181400457057901), atol=1e-15)


def test_gh5686():
    mean1, mean2 = np.array([1, 2]), np.array([3, 4])
    std1, std2 = np.array([5, 3]), np.array([4, 5])
    nobs1, nobs2 = np.array([130, 140]), np.array([100, 150])
    # This will raise a TypeError unless gh-5686 is fixed.
    stats.ttest_ind_from_stats(mean1, std1, nobs1, mean2, std2, nobs2)


def test_ttest_1samp_new():
    n1, n2, n3 = (10,15,20)
    rvn1 = stats.norm.rvs(loc=5,scale=10,size=(n1,n2,n3))

    # check multidimensional array and correct axis handling
    # deterministic rvn1 and rvn2 would be better as in test_ttest_rel
    t1,p1 = stats.ttest_1samp(rvn1[:,:,:], np.ones((n2,n3)),axis=0)
    t2,p2 = stats.ttest_1samp(rvn1[:,:,:], 1,axis=0)
    t3,p3 = stats.ttest_1samp(rvn1[:,0,0], 1)
    assert_array_almost_equal(t1,t2, decimal=14)
    assert_almost_equal(t1[0,0],t3, decimal=14)
    assert_equal(t1.shape, (n2,n3))

    t1,p1 = stats.ttest_1samp(rvn1[:,:,:], np.ones((n1,n3)),axis=1)
    t2,p2 = stats.ttest_1samp(rvn1[:,:,:], 1,axis=1)
    t3,p3 = stats.ttest_1samp(rvn1[0,:,0], 1)
    assert_array_almost_equal(t1,t2, decimal=14)
    assert_almost_equal(t1[0,0],t3, decimal=14)
    assert_equal(t1.shape, (n1,n3))

    t1,p1 = stats.ttest_1samp(rvn1[:,:,:], np.ones((n1,n2)),axis=2)
    t2,p2 = stats.ttest_1samp(rvn1[:,:,:], 1,axis=2)
    t3,p3 = stats.ttest_1samp(rvn1[0,0,:], 1)
    assert_array_almost_equal(t1,t2, decimal=14)
    assert_almost_equal(t1[0,0],t3, decimal=14)
    assert_equal(t1.shape, (n1,n2))

    # test zero division problem
    t, p = stats.ttest_1samp([0, 0, 0], 1)
    assert_equal((np.abs(t), p), (np.inf, 0))

    with np.errstate(all='ignore'):
        assert_equal(stats.ttest_1samp([0, 0, 0], 0), (np.nan, np.nan))

        # check that nan in input array result in nan output
        anan = np.array([[1, np.nan],[-1, 1]])
        assert_equal(stats.ttest_1samp(anan, 0), ([0, np.nan], [1, np.nan]))


class TestDescribe(object):
    def test_describe_scalar(self):
        with suppress_warnings() as sup, np.errstate(invalid="ignore"):
            sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
            n, mm, m, v, sk, kurt = stats.describe(4.)
        assert_equal(n, 1)
        assert_equal(mm, (4.0, 4.0))
        assert_equal(m, 4.0)
        assert_(np.isnan(v))
        assert_array_almost_equal(sk, 0.0, decimal=13)
        assert_array_almost_equal(kurt, -3.0, decimal=13)

    def test_describe_numbers(self):
        x = np.vstack((np.ones((3,4)), 2 * np.ones((2,4))))
        nc, mmc = (5, ([1., 1., 1., 1.], [2., 2., 2., 2.]))
        mc = np.array([1.4, 1.4, 1.4, 1.4])
        vc = np.array([0.3, 0.3, 0.3, 0.3])
        skc = [0.40824829046386357] * 4
        kurtc = [-1.833333333333333] * 4
        n, mm, m, v, sk, kurt = stats.describe(x)
        assert_equal(n, nc)
        assert_equal(mm, mmc)
        assert_equal(m, mc)
        assert_equal(v, vc)
        assert_array_almost_equal(sk, skc, decimal=13)
        assert_array_almost_equal(kurt, kurtc, decimal=13)
        n, mm, m, v, sk, kurt = stats.describe(x.T, axis=1)
        assert_equal(n, nc)
        assert_equal(mm, mmc)
        assert_equal(m, mc)
        assert_equal(v, vc)
        assert_array_almost_equal(sk, skc, decimal=13)
        assert_array_almost_equal(kurt, kurtc, decimal=13)

        x = np.arange(10.)
        x[9] = np.nan

        nc, mmc = (9, (0.0, 8.0))
        mc = 4.0
        vc = 7.5
        skc = 0.0
        kurtc = -1.2300000000000002
        n, mm, m, v, sk, kurt = stats.describe(x, nan_policy='omit')
        assert_equal(n, nc)
        assert_equal(mm, mmc)
        assert_equal(m, mc)
        assert_equal(v, vc)
        assert_array_almost_equal(sk, skc)
        assert_array_almost_equal(kurt, kurtc, decimal=13)

        assert_raises(ValueError, stats.describe, x, nan_policy='raise')
        assert_raises(ValueError, stats.describe, x, nan_policy='foobar')

    def test_describe_result_attributes(self):
        actual = stats.describe(np.arange(5))
        attributes = ('nobs', 'minmax', 'mean', 'variance', 'skewness',
                      'kurtosis')
        check_named_results(actual, attributes)

    def test_describe_ddof(self):
        x = np.vstack((np.ones((3, 4)), 2 * np.ones((2, 4))))
        nc, mmc = (5, ([1., 1., 1., 1.], [2., 2., 2., 2.]))
        mc = np.array([1.4, 1.4, 1.4, 1.4])
        vc = np.array([0.24, 0.24, 0.24, 0.24])
        skc = [0.40824829046386357] * 4
        kurtc = [-1.833333333333333] * 4
        n, mm, m, v, sk, kurt = stats.describe(x, ddof=0)
        assert_equal(n, nc)
        assert_allclose(mm, mmc, rtol=1e-15)
        assert_allclose(m, mc, rtol=1e-15)
        assert_allclose(v, vc, rtol=1e-15)
        assert_array_almost_equal(sk, skc, decimal=13)
        assert_array_almost_equal(kurt, kurtc, decimal=13)

    def test_describe_axis_none(self):
        x = np.vstack((np.ones((3, 4)), 2 * np.ones((2, 4))))

        # expected values
        e_nobs, e_minmax = (20, (1.0, 2.0))
        e_mean = 1.3999999999999999
        e_var = 0.25263157894736848
        e_skew = 0.4082482904638634
        e_kurt = -1.8333333333333333

        # actual values
        a = stats.describe(x, axis=None)

        assert_equal(a.nobs, e_nobs)
        assert_almost_equal(a.minmax, e_minmax)
        assert_almost_equal(a.mean, e_mean)
        assert_almost_equal(a.variance, e_var)
        assert_array_almost_equal(a.skewness, e_skew, decimal=13)
        assert_array_almost_equal(a.kurtosis, e_kurt, decimal=13)

    def test_describe_empty(self):
        assert_raises(ValueError, stats.describe, [])


def test_normalitytests():
    assert_raises(ValueError, stats.skewtest, 4.)
    assert_raises(ValueError, stats.kurtosistest, 4.)
    assert_raises(ValueError, stats.normaltest, 4.)

    # numbers verified with R: dagoTest in package fBasics
    st_normal, st_skew, st_kurt = (3.92371918, 1.98078826, -0.01403734)
    pv_normal, pv_skew, pv_kurt = (0.14059673, 0.04761502, 0.98880019)
    x = np.array((-2, -1, 0, 1, 2, 3)*4)**2
    attributes = ('statistic', 'pvalue')

    assert_array_almost_equal(stats.normaltest(x), (st_normal, pv_normal))
    check_named_results(stats.normaltest(x), attributes)
    assert_array_almost_equal(stats.skewtest(x), (st_skew, pv_skew))
    check_named_results(stats.skewtest(x), attributes)
    assert_array_almost_equal(stats.kurtosistest(x), (st_kurt, pv_kurt))
    check_named_results(stats.kurtosistest(x), attributes)

    # Test axis=None (equal to axis=0 for 1-D input)
    assert_array_almost_equal(stats.normaltest(x, axis=None),
           (st_normal, pv_normal))
    assert_array_almost_equal(stats.skewtest(x, axis=None),
           (st_skew, pv_skew))
    assert_array_almost_equal(stats.kurtosistest(x, axis=None),
           (st_kurt, pv_kurt))

    x = np.arange(10.)
    x[9] = np.nan
    with np.errstate(invalid="ignore"):
        assert_array_equal(stats.skewtest(x), (np.nan, np.nan))

    expected = (1.0184643553962129, 0.30845733195153502)
    assert_array_almost_equal(stats.skewtest(x, nan_policy='omit'), expected)

    with np.errstate(all='ignore'):
        assert_raises(ValueError, stats.skewtest, x, nan_policy='raise')
    assert_raises(ValueError, stats.skewtest, x, nan_policy='foobar')

    x = np.arange(30.)
    x[29] = np.nan
    with np.errstate(all='ignore'):
        assert_array_equal(stats.kurtosistest(x), (np.nan, np.nan))

    expected = (-2.2683547379505273, 0.023307594135872967)
    assert_array_almost_equal(stats.kurtosistest(x, nan_policy='omit'),
                              expected)

    assert_raises(ValueError, stats.kurtosistest, x, nan_policy='raise')
    assert_raises(ValueError, stats.kurtosistest, x, nan_policy='foobar')

    with np.errstate(all='ignore'):
        assert_array_equal(stats.normaltest(x), (np.nan, np.nan))

    expected = (6.2260409514287449, 0.04446644248650191)
    assert_array_almost_equal(stats.normaltest(x, nan_policy='omit'), expected)

    assert_raises(ValueError, stats.normaltest, x, nan_policy='raise')
    assert_raises(ValueError, stats.normaltest, x, nan_policy='foobar')

    # regression test for issue gh-9033: x cleary non-normal but power of
    # negtative denom needs to be handled correctly to reject normality
    counts = [128, 0, 58, 7, 0, 41, 16, 0, 0, 167]
    x = np.hstack([np.full(c, i) for i, c in enumerate(counts)])
    assert_equal(stats.kurtosistest(x)[1] < 0.01, True)


class TestRankSums(object):
    def test_ranksums_result_attributes(self):
        res = stats.ranksums(np.arange(5), np.arange(25))
        attributes = ('statistic', 'pvalue')
        check_named_results(res, attributes)


class TestJarqueBera(object):
    def test_jarque_bera_stats(self):
        np.random.seed(987654321)
        x = np.random.normal(0, 1, 100000)
        y = np.random.chisquare(10000, 100000)
        z = np.random.rayleigh(1, 100000)

        assert_(stats.jarque_bera(x)[1] > stats.jarque_bera(y)[1])
        assert_(stats.jarque_bera(x)[1] > stats.jarque_bera(z)[1])
        assert_(stats.jarque_bera(y)[1] > stats.jarque_bera(z)[1])

    def test_jarque_bera_array_like(self):
        np.random.seed(987654321)
        x = np.random.normal(0, 1, 100000)

        JB1, p1 = stats.jarque_bera(list(x))
        JB2, p2 = stats.jarque_bera(tuple(x))
        JB3, p3 = stats.jarque_bera(x.reshape(2, 50000))

        assert_(JB1 == JB2 == JB3)
        assert_(p1 == p2 == p3)

    def test_jarque_bera_size(self):
        assert_raises(ValueError, stats.jarque_bera, [])


def test_skewtest_too_few_samples():
    # Regression test for ticket #1492.
    # skewtest requires at least 8 samples; 7 should raise a ValueError.
    x = np.arange(7.0)
    assert_raises(ValueError, stats.skewtest, x)


def test_kurtosistest_too_few_samples():
    # Regression test for ticket #1425.
    # kurtosistest requires at least 5 samples; 4 should raise a ValueError.
    x = np.arange(4.0)
    assert_raises(ValueError, stats.kurtosistest, x)


class TestMannWhitneyU(object):
    X = [19.8958398126694, 19.5452691647182, 19.0577309166425, 21.716543054589,
         20.3269502208702, 20.0009273294025, 19.3440043632957, 20.4216806548105,
         19.0649894736528, 18.7808043120398, 19.3680942943298, 19.4848044069953,
         20.7514611265663, 19.0894948874598, 19.4975522356628, 18.9971170734274,
         20.3239606288208, 20.6921298083835, 19.0724259532507, 18.9825187935021,
         19.5144462609601, 19.8256857844223, 20.5174677102032, 21.1122407995892,
         17.9490854922535, 18.2847521114727, 20.1072217648826, 18.6439891962179,
         20.4970638083542, 19.5567594734914]

    Y = [19.2790668029091, 16.993808441865, 18.5416338448258, 17.2634018833575,
         19.1577183624616, 18.5119655377495, 18.6068455037221, 18.8358343362655,
         19.0366413269742, 18.1135025515417, 19.2201873866958, 17.8344909022841,
         18.2894380745856, 18.6661374133922, 19.9688601693252, 16.0672254617636,
         19.00596360572, 19.201561539032, 19.0487501090183, 19.0847908674356]

    significant = 14

    def test_mannwhitneyu_one_sided(self):
        u1, p1 = stats.mannwhitneyu(self.X, self.Y, alternative='less')
        u2, p2 = stats.mannwhitneyu(self.Y, self.X, alternative='greater')
        u3, p3 = stats.mannwhitneyu(self.X, self.Y, alternative='greater')
        u4, p4 = stats.mannwhitneyu(self.Y, self.X, alternative='less')

        assert_equal(p1, p2)
        assert_equal(p3, p4)
        assert_(p1 != p3)
        assert_equal(u1, 498)
        assert_equal(u2, 102)
        assert_equal(u3, 498)
        assert_equal(u4, 102)
        assert_approx_equal(p1, 0.999957683256589, significant=self.significant)
        assert_approx_equal(p3, 4.5941632666275e-05, significant=self.significant)

    def test_mannwhitneyu_two_sided(self):
        u1, p1 = stats.mannwhitneyu(self.X, self.Y, alternative='two-sided')
        u2, p2 = stats.mannwhitneyu(self.Y, self.X, alternative='two-sided')

        assert_equal(p1, p2)
        assert_equal(u1, 498)
        assert_equal(u2, 102)
        assert_approx_equal(p1, 9.188326533255e-05,
                            significant=self.significant)

    def test_mannwhitneyu_default(self):
        # The default value for alternative is None
        with suppress_warnings() as sup:
            sup.filter(DeprecationWarning,
                       "Calling `mannwhitneyu` without .*`alternative`")
            u1, p1 = stats.mannwhitneyu(self.X, self.Y)
            u2, p2 = stats.mannwhitneyu(self.Y, self.X)
            u3, p3 = stats.mannwhitneyu(self.X, self.Y, alternative=None)

        assert_equal(p1, p2)
        assert_equal(p1, p3)
        assert_equal(u1, 102)
        assert_equal(u2, 102)
        assert_equal(u3, 102)
        assert_approx_equal(p1, 4.5941632666275e-05,
                            significant=self.significant)

    def test_mannwhitneyu_no_correct_one_sided(self):
        u1, p1 = stats.mannwhitneyu(self.X, self.Y, False,
                                    alternative='less')
        u2, p2 = stats.mannwhitneyu(self.Y, self.X, False,
                                    alternative='greater')
        u3, p3 = stats.mannwhitneyu(self.X, self.Y, False,
                                    alternative='greater')
        u4, p4 = stats.mannwhitneyu(self.Y, self.X, False,
                                    alternative='less')

        assert_equal(p1, p2)
        assert_equal(p3, p4)
        assert_(p1 != p3)
        assert_equal(u1, 498)
        assert_equal(u2, 102)
        assert_equal(u3, 498)
        assert_equal(u4, 102)
        assert_approx_equal(p1, 0.999955905990004, significant=self.significant)
        assert_approx_equal(p3, 4.40940099958089e-05, significant=self.significant)

    def test_mannwhitneyu_no_correct_two_sided(self):
        u1, p1 = stats.mannwhitneyu(self.X, self.Y, False,
                                    alternative='two-sided')
        u2, p2 = stats.mannwhitneyu(self.Y, self.X, False,
                                    alternative='two-sided')

        assert_equal(p1, p2)
        assert_equal(u1, 498)
        assert_equal(u2, 102)
        assert_approx_equal(p1, 8.81880199916178e-05,
                            significant=self.significant)

    def test_mannwhitneyu_no_correct_default(self):
        # The default value for alternative is None
        with suppress_warnings() as sup:
            sup.filter(DeprecationWarning,
                       "Calling `mannwhitneyu` without .*`alternative`")
            u1, p1 = stats.mannwhitneyu(self.X, self.Y, False)
            u2, p2 = stats.mannwhitneyu(self.Y, self.X, False)
            u3, p3 = stats.mannwhitneyu(self.X, self.Y, False,
                                        alternative=None)

        assert_equal(p1, p2)
        assert_equal(p1, p3)
        assert_equal(u1, 102)
        assert_equal(u2, 102)
        assert_equal(u3, 102)
        assert_approx_equal(p1, 4.40940099958089e-05,
                            significant=self.significant)

    def test_mannwhitneyu_ones(self):
        x = np.array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2.,
                      1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                      1., 1., 2., 1., 1., 1., 1., 2., 1., 1., 2., 1., 1., 2.,
                      1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2., 1.,
                      1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 2., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 1., 3., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1.])

        y = np.array([1., 1., 1., 1., 1., 1., 1., 2., 1., 2., 1., 1., 1., 1.,
                      2., 1., 1., 1., 2., 1., 1., 1., 1., 1., 2., 1., 1., 3.,
                      1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2., 1., 2., 1.,
                      1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 2.,
                      2., 1., 1., 2., 1., 1., 2., 1., 2., 1., 1., 1., 1., 2.,
                      2., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                      1., 2., 1., 1., 1., 1., 1., 2., 2., 2., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                      2., 1., 1., 2., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1.,
                      1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 2., 1., 1.,
                      1., 1., 1., 1.])

        # p-value verified with matlab and R to 5 significant digits
        assert_array_almost_equal(stats.stats.mannwhitneyu(x, y,
                                                           alternative='less'),
                                  (16980.5, 2.8214327656317373e-005),
                                  decimal=12)

    def test_mannwhitneyu_result_attributes(self):
        # test for namedtuple attribute results
        attributes = ('statistic', 'pvalue')
        res = stats.mannwhitneyu(self.X, self.Y, alternative="less")
        check_named_results(res, attributes)


def test_pointbiserial():
    # same as mstats test except for the nan
    # Test data: https://web.archive.org/web/20060504220742/https://support.sas.com/ctx/samples/index.jsp?sid=490&tab=output
    x = [1,0,1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0,0,1,0,
         0,0,0,0,1]
    y = [14.8,13.8,12.4,10.1,7.1,6.1,5.8,4.6,4.3,3.5,3.3,3.2,3.0,
         2.8,2.8,2.5,2.4,2.3,2.1,1.7,1.7,1.5,1.3,1.3,1.2,1.2,1.1,
         0.8,0.7,0.6,0.5,0.2,0.2,0.1]
    assert_almost_equal(stats.pointbiserialr(x, y)[0], 0.36149, 5)

    # test for namedtuple attribute results
    attributes = ('correlation', 'pvalue')
    res = stats.pointbiserialr(x, y)
    check_named_results(res, attributes)


def test_obrientransform():
    # A couple tests calculated by hand.
    x1 = np.array([0, 2, 4])
    t1 = stats.obrientransform(x1)
    expected = [7, -2, 7]
    assert_allclose(t1[0], expected)

    x2 = np.array([0, 3, 6, 9])
    t2 = stats.obrientransform(x2)
    expected = np.array([30, 0, 0, 30])
    assert_allclose(t2[0], expected)

    # Test two arguments.
    a, b = stats.obrientransform(x1, x2)
    assert_equal(a, t1[0])
    assert_equal(b, t2[0])

    # Test three arguments.
    a, b, c = stats.obrientransform(x1, x2, x1)
    assert_equal(a, t1[0])
    assert_equal(b, t2[0])
    assert_equal(c, t1[0])

    # This is a regression test to check np.var replacement.
    # The author of this test didn't separately verify the numbers.
    x1 = np.arange(5)
    result = np.array(
      [[5.41666667, 1.04166667, -0.41666667, 1.04166667, 5.41666667],
       [21.66666667, 4.16666667, -1.66666667, 4.16666667, 21.66666667]])
    assert_array_almost_equal(stats.obrientransform(x1, 2*x1), result, decimal=8)

    # Example from "O'Brien Test for Homogeneity of Variance"
    # by Herve Abdi.
    values = range(5, 11)
    reps = np.array([5, 11, 9, 3, 2, 2])
    data = np.repeat(values, reps)
    transformed_values = np.array([3.1828, 0.5591, 0.0344,
                                   1.6086, 5.2817, 11.0538])
    expected = np.repeat(transformed_values, reps)
    result = stats.obrientransform(data)
    assert_array_almost_equal(result[0], expected, decimal=4)


class HarMeanTestCase:
    def test_1dlist(self):
        #  Test a 1d list
        a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
        b = 34.1417152147
        self.do(a, b)

    def test_1darray(self):
        #  Test a 1d array
        a = np.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100])
        b = 34.1417152147
        self.do(a, b)

    def test_1dma(self):
        #  Test a 1d masked array
        a = np.ma.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100])
        b = 34.1417152147
        self.do(a, b)

    def test_1dmavalue(self):
        #  Test a 1d masked array with a masked value
        a = np.ma.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100],
                      mask=[0,0,0,0,0,0,0,0,0,1])
        b = 31.8137186141
        self.do(a, b)

    # Note the next tests use axis=None as default, not axis=0
    def test_2dlist(self):
        #  Test a 2d list
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = 38.6696271841
        self.do(a, b)

    def test_2darray(self):
        #  Test a 2d array
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = 38.6696271841
        self.do(np.array(a), b)

    def test_2dma(self):
        #  Test a 2d masked array
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = 38.6696271841
        self.do(np.ma.array(a), b)

    def test_2daxis0(self):
        #  Test a 2d list with axis=0
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = np.array([22.88135593, 39.13043478, 52.90076336, 65.45454545])
        self.do(a, b, axis=0)

    def test_2daxis1(self):
        #  Test a 2d list with axis=1
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = np.array([19.2, 63.03939962, 103.80078637])
        self.do(a, b, axis=1)

    def test_2dmatrixdaxis0(self):
        #  Test a 2d list with axis=0
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = np.matrix([[22.88135593, 39.13043478, 52.90076336, 65.45454545]])
        self.do(np.matrix(a), b, axis=0)

    def test_2dmatrixaxis1(self):
        #  Test a 2d list with axis=1
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = np.matrix([[19.2, 63.03939962, 103.80078637]]).T
        self.do(np.matrix(a), b, axis=1)


class TestHarMean(HarMeanTestCase):
    def do(self, a, b, axis=None, dtype=None):
        x = stats.hmean(a, axis=axis, dtype=dtype)
        assert_almost_equal(b, x)
        assert_equal(x.dtype, dtype)


class GeoMeanTestCase:
    def test_1dlist(self):
        #  Test a 1d list
        a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
        b = 45.2872868812
        self.do(a, b)

    def test_1darray(self):
        #  Test a 1d array
        a = np.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100])
        b = 45.2872868812
        self.do(a, b)

    def test_1dma(self):
        #  Test a 1d masked array
        a = np.ma.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100])
        b = 45.2872868812
        self.do(a, b)

    def test_1dmavalue(self):
        #  Test a 1d masked array with a masked value
        a = np.ma.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100], mask=[0,0,0,0,0,0,0,0,0,1])
        b = 41.4716627439
        self.do(a, b)

    # Note the next tests use axis=None as default, not axis=0
    def test_2dlist(self):
        #  Test a 2d list
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = 52.8885199
        self.do(a, b)

    def test_2darray(self):
        #  Test a 2d array
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = 52.8885199
        self.do(np.array(a), b)

    def test_2dma(self):
        #  Test a 2d masked array
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = 52.8885199
        self.do(np.ma.array(a), b)

    def test_2daxis0(self):
        #  Test a 2d list with axis=0
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = np.array([35.56893304, 49.32424149, 61.3579244, 72.68482371])
        self.do(a, b, axis=0)

    def test_2daxis1(self):
        #  Test a 2d list with axis=1
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = np.array([22.13363839, 64.02171746, 104.40086817])
        self.do(a, b, axis=1)

    def test_2dmatrixdaxis0(self):
        #  Test a 2d list with axis=0
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = np.matrix([[35.56893304, 49.32424149, 61.3579244, 72.68482371]])
        self.do(np.matrix(a), b, axis=0)

    def test_2dmatrixaxis1(self):
        #  Test a 2d list with axis=1
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        b = np.matrix([[22.13363839, 64.02171746, 104.40086817]]).T
        self.do(np.matrix(a), b, axis=1)

    def test_1dlist0(self):
        #  Test a 1d list with zero element
        a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 0]
        b = 0.0  # due to exp(-inf)=0
        olderr = np.seterr(all='ignore')
        try:
            self.do(a, b)
        finally:
            np.seterr(**olderr)

    def test_1darray0(self):
        #  Test a 1d array with zero element
        a = np.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 0])
        b = 0.0  # due to exp(-inf)=0
        olderr = np.seterr(all='ignore')
        try:
            self.do(a, b)
        finally:
            np.seterr(**olderr)

    def test_1dma0(self):
        #  Test a 1d masked array with zero element
        a = np.ma.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 0])
        b = 41.4716627439
        olderr = np.seterr(all='ignore')
        try:
            self.do(a, b)
        finally:
            np.seterr(**olderr)

    def test_1dmainf(self):
        #  Test a 1d masked array with negative element
        a = np.ma.array([10, 20, 30, 40, 50, 60, 70, 80, 90, -1])
        b = 41.4716627439
        olderr = np.seterr(all='ignore')
        try:
            self.do(a, b)
        finally:
            np.seterr(**olderr)


class TestGeoMean(GeoMeanTestCase):
    def do(self, a, b, axis=None, dtype=None):
        # Note this doesn't test when axis is not specified
        x = stats.gmean(a, axis=axis, dtype=dtype)
        assert_almost_equal(b, x)
        assert_equal(x.dtype, dtype)


def test_binomtest():
    # precision tests compared to R for ticket:986
    pp = np.concatenate((np.linspace(0.1,0.2,5), np.linspace(0.45,0.65,5),
                          np.linspace(0.85,0.95,5)))
    n = 501
    x = 450
    results = [0.0, 0.0, 1.0159969301994141e-304,
    2.9752418572150531e-275, 7.7668382922535275e-250,
    2.3381250925167094e-099, 7.8284591587323951e-081,
    9.9155947819961383e-065, 2.8729390725176308e-050,
    1.7175066298388421e-037, 0.0021070691951093692,
    0.12044570587262322, 0.88154763174802508, 0.027120993063129286,
    2.6102587134694721e-006]

    for p, res in zip(pp,results):
        assert_approx_equal(stats.binom_test(x, n, p), res,
                            significant=12, err_msg='fail forp=%f' % p)

    assert_approx_equal(stats.binom_test(50,100,0.1), 5.8320387857343647e-024,
                            significant=12, err_msg='fail forp=%f' % p)


def test_binomtest2():
    # test added for issue #2384
    res2 = [
    [1.0, 1.0],
    [0.5,1.0,0.5],
    [0.25,1.00,1.00,0.25],
    [0.125,0.625,1.000,0.625,0.125],
    [0.0625,0.3750,1.0000,1.0000,0.3750,0.0625],
    [0.03125,0.21875,0.68750,1.00000,0.68750,0.21875,0.03125],
    [0.015625,0.125000,0.453125,1.000000,1.000000,0.453125,0.125000,0.015625],
    [0.0078125,0.0703125,0.2890625,0.7265625,1.0000000,0.7265625,0.2890625,
     0.0703125,0.0078125],
    [0.00390625,0.03906250,0.17968750,0.50781250,1.00000000,1.00000000,
     0.50781250,0.17968750,0.03906250,0.00390625],
    [0.001953125,0.021484375,0.109375000,0.343750000,0.753906250,1.000000000,
     0.753906250,0.343750000,0.109375000,0.021484375,0.001953125]
    ]

    for k in range(1, 11):
        res1 = [stats.binom_test(v, k, 0.5) for v in range(k + 1)]
        assert_almost_equal(res1, res2[k-1], decimal=10)


def test_binomtest3():
    # test added for issue #2384
    # test when x == n*p and neighbors
    res3 = [stats.binom_test(v, v*k, 1./k) for v in range(1, 11)
                                           for k in range(2, 11)]
    assert_equal(res3, np.ones(len(res3), int))

    #> bt=c()
    #> for(i in as.single(1:10)){for(k in as.single(2:10)){bt = c(bt, binom.test(i-1, k*i,(1/k))$p.value); print(c(i+1, k*i,(1/k)))}}
    binom_testm1 = np.array([
         0.5, 0.5555555555555556, 0.578125, 0.5904000000000003,
         0.5981224279835393, 0.603430543396034, 0.607304096221924,
         0.610255656871054, 0.612579511000001, 0.625, 0.670781893004115,
         0.68853759765625, 0.6980101120000006, 0.703906431368616,
         0.70793209416498, 0.7108561134173507, 0.713076544331419,
         0.714820192935702, 0.6875, 0.7268709038256367, 0.7418963909149174,
         0.74986110468096, 0.7548015520398076, 0.7581671424768577,
         0.760607984787832, 0.762459425024199, 0.7639120677676575, 0.7265625,
         0.761553963657302, 0.774800934828818, 0.7818005980538996,
         0.78613491480358, 0.789084353140195, 0.7912217659828884,
         0.79284214559524, 0.794112956558801, 0.75390625, 0.7856929451142176,
         0.7976688481430754, 0.8039848974727624, 0.807891868948366,
         0.8105487660137676, 0.812473307174702, 0.8139318233591120,
         0.815075399104785, 0.7744140625, 0.8037322594985427,
         0.814742863657656, 0.8205425178645808, 0.8241275984172285,
         0.8265645374416, 0.8283292196088257, 0.829666291102775,
         0.8307144686362666, 0.7905273437499996, 0.8178712053954738,
         0.828116983756619, 0.833508948940494, 0.8368403871552892,
         0.839104213210105, 0.840743186196171, 0.84198481438049,
         0.8429580531563676, 0.803619384765625, 0.829338573944648,
         0.8389591907548646, 0.84401876783902, 0.84714369697889,
         0.8492667010581667, 0.850803474598719, 0.851967542858308,
         0.8528799045949524, 0.8145294189453126, 0.838881732845347,
         0.847979024541911, 0.852760894015685, 0.8557134656773457,
         0.8577190131799202, 0.85917058278431, 0.860270010472127,
         0.861131648404582, 0.823802947998047, 0.846984756807511,
         0.855635653643743, 0.860180994825685, 0.86298688573253,
         0.864892525675245, 0.866271647085603, 0.867316125625004,
         0.8681346531755114
        ])

    # > bt=c()
    # > for(i in as.single(1:10)){for(k in as.single(2:10)){bt = c(bt, binom.test(i+1, k*i,(1/k))$p.value); print(c(i+1, k*i,(1/k)))}}

    binom_testp1 = np.array([
         0.5, 0.259259259259259, 0.26171875, 0.26272, 0.2632244513031551,
         0.2635138663069203, 0.2636951804161073, 0.2638162407564354,
         0.2639010709000002, 0.625, 0.4074074074074074, 0.42156982421875,
         0.4295746560000003, 0.43473045988554, 0.4383309503172684,
         0.4409884859402103, 0.4430309389962837, 0.444649849401104, 0.6875,
         0.4927602499618962, 0.5096031427383425, 0.5189636628480,
         0.5249280070771274, 0.5290623300865124, 0.5320974248125793,
         0.5344204730474308, 0.536255847400756, 0.7265625, 0.5496019313526808,
         0.5669248746708034, 0.576436455045805, 0.5824538812831795,
         0.5866053321547824, 0.589642781414643, 0.5919618019300193,
         0.593790427805202, 0.75390625, 0.590868349763505, 0.607983393277209,
         0.617303847446822, 0.623172512167948, 0.627208862156123,
         0.6301556891501057, 0.632401894928977, 0.6341708982290303,
         0.7744140625, 0.622562037497196, 0.639236102912278, 0.648263335014579,
         0.65392850011132, 0.657816519817211, 0.660650782947676,
         0.662808780346311, 0.6645068560246006, 0.7905273437499996,
         0.6478843304312477, 0.6640468318879372, 0.6727589686071775,
         0.6782129857784873, 0.681950188903695, 0.684671508668418,
         0.686741824999918, 0.688369886732168, 0.803619384765625,
         0.668716055304315, 0.684360013879534, 0.6927642396829181,
         0.6980155964704895, 0.701609591890657, 0.7042244320992127,
         0.7062125081341817, 0.707775152962577, 0.8145294189453126,
         0.686243374488305, 0.7013873696358975, 0.709501223328243,
         0.714563595144314, 0.718024953392931, 0.7205416252126137,
         0.722454130389843, 0.723956813292035, 0.823802947998047,
         0.701255953767043, 0.715928221686075, 0.723772209289768,
         0.7286603031173616, 0.7319999279787631, 0.7344267920995765,
         0.736270323773157, 0.737718376096348
        ])

    res4_p1 = [stats.binom_test(v+1, v*k, 1./k) for v in range(1, 11)
                                                for k in range(2, 11)]
    res4_m1 = [stats.binom_test(v-1, v*k, 1./k) for v in range(1, 11)
                                                for k in range(2, 11)]

    assert_almost_equal(res4_p1, binom_testp1, decimal=13)
    assert_almost_equal(res4_m1, binom_testm1, decimal=13)


class TestTrim(object):
    # test trim functions
    def test_trim1(self):
        a = np.arange(11)
        assert_equal(np.sort(stats.trim1(a, 0.1)), np.arange(10))
        assert_equal(np.sort(stats.trim1(a, 0.2)), np.arange(9))
        assert_equal(np.sort(stats.trim1(a, 0.2, tail='left')),
                     np.arange(2, 11))
        assert_equal(np.sort(stats.trim1(a, 3/11., tail='left')),
                     np.arange(3, 11))
        assert_equal(stats.trim1(a, 1.0), [])
        assert_equal(stats.trim1(a, 1.0, tail='left'), [])

        # empty input
        assert_equal(stats.trim1([], 0.1), [])
        assert_equal(stats.trim1([], 3/11., tail='left'), [])
        assert_equal(stats.trim1([], 4/6.), [])

    def test_trimboth(self):
        a = np.arange(11)
        assert_equal(np.sort(stats.trimboth(a, 3/11.)), np.arange(3, 8))
        assert_equal(np.sort(stats.trimboth(a, 0.2)),
                     np.array([2, 3, 4, 5, 6, 7, 8]))
        assert_equal(np.sort(stats.trimboth(np.arange(24).reshape(6, 4), 0.2)),
                     np.arange(4, 20).reshape(4, 4))
        assert_equal(np.sort(stats.trimboth(np.arange(24).reshape(4, 6).T,
                                            2/6.)),
                     np.array([[2, 8, 14, 20], [3, 9, 15, 21]]))
        assert_raises(ValueError, stats.trimboth,
                      np.arange(24).reshape(4, 6).T, 4/6.)

        # empty input
        assert_equal(stats.trimboth([], 0.1), [])
        assert_equal(stats.trimboth([], 3/11.), [])
        assert_equal(stats.trimboth([], 4/6.), [])

    def test_trim_mean(self):
        # don't use pre-sorted arrays
        a = np.array([4, 8, 2, 0, 9, 5, 10, 1, 7, 3, 6])
        idx = np.array([3, 5, 0, 1, 2, 4])
        a2 = np.arange(24).reshape(6, 4)[idx, :]
        a3 = np.arange(24).reshape(6, 4, order='F')[idx, :]
        assert_equal(stats.trim_mean(a3, 2/6.),
                     np.array([2.5, 8.5, 14.5, 20.5]))
        assert_equal(stats.trim_mean(a2, 2/6.),
                     np.array([10., 11., 12., 13.]))
        idx4 = np.array([1, 0, 3, 2])
        a4 = np.arange(24).reshape(4, 6)[idx4, :]
        assert_equal(stats.trim_mean(a4, 2/6.),
                     np.array([9., 10., 11., 12., 13., 14.]))
        # shuffled arange(24) as array_like
        a = [7, 11, 12, 21, 16, 6, 22, 1, 5, 0, 18, 10, 17, 9, 19, 15, 23,
             20, 2, 14, 4, 13, 8, 3]
        assert_equal(stats.trim_mean(a, 2/6.), 11.5)
        assert_equal(stats.trim_mean([5,4,3,1,2,0], 2/6.), 2.5)

        # check axis argument
        np.random.seed(1234)
        a = np.random.randint(20, size=(5, 6, 4, 7))
        for axis in [0, 1, 2, 3, -1]:
            res1 = stats.trim_mean(a, 2/6., axis=axis)
            res2 = stats.trim_mean(np.rollaxis(a, axis), 2/6.)
            assert_equal(res1, res2)

        res1 = stats.trim_mean(a, 2/6., axis=None)
        res2 = stats.trim_mean(a.ravel(), 2/6.)
        assert_equal(res1, res2)

        assert_raises(ValueError, stats.trim_mean, a, 0.6)

        # empty input
        assert_equal(stats.trim_mean([], 0.0), np.nan)
        assert_equal(stats.trim_mean([], 0.6), np.nan)


class TestSigmaClip(object):
    def test_sigmaclip1(self):
        a = np.concatenate((np.linspace(9.5, 10.5, 31), np.linspace(0, 20, 5)))
        fact = 4  # default
        c, low, upp = stats.sigmaclip(a)
        assert_(c.min() > low)
        assert_(c.max() < upp)
        assert_equal(low, c.mean() - fact*c.std())
        assert_equal(upp, c.mean() + fact*c.std())
        assert_equal(c.size, a.size)

    def test_sigmaclip2(self):
        a = np.concatenate((np.linspace(9.5, 10.5, 31), np.linspace(0, 20, 5)))
        fact = 1.5
        c, low, upp = stats.sigmaclip(a, fact, fact)
        assert_(c.min() > low)
        assert_(c.max() < upp)
        assert_equal(low, c.mean() - fact*c.std())
        assert_equal(upp, c.mean() + fact*c.std())
        assert_equal(c.size, 4)
        assert_equal(a.size, 36)  # check original array unchanged

    def test_sigmaclip3(self):
        a = np.concatenate((np.linspace(9.5, 10.5, 11),
                            np.linspace(-100, -50, 3)))
        fact = 1.8
        c, low, upp = stats.sigmaclip(a, fact, fact)
        assert_(c.min() > low)
        assert_(c.max() < upp)
        assert_equal(low, c.mean() - fact*c.std())
        assert_equal(upp, c.mean() + fact*c.std())
        assert_equal(c, np.linspace(9.5, 10.5, 11))

    def test_sigmaclip_result_attributes(self):
        a = np.concatenate((np.linspace(9.5, 10.5, 11),
                            np.linspace(-100, -50, 3)))
        fact = 1.8
        res = stats.sigmaclip(a, fact, fact)
        attributes = ('clipped', 'lower', 'upper')
        check_named_results(res, attributes)

    def test_std_zero(self):
        # regression test #8632
        x = np.ones(10)
        assert_equal(stats.sigmaclip(x)[0], x)


class TestFOneWay(object):
    def test_trivial(self):
        # A trivial test of stats.f_oneway, with F=0.
        F, p = stats.f_oneway([0,2], [0,2])
        assert_equal(F, 0.0)

    def test_basic(self):
        # Despite being a floating point calculation, this data should
        # result in F being exactly 2.0.
        F, p = stats.f_oneway([0,2], [2,4])
        assert_equal(F, 2.0)

    def test_large_integer_array(self):
        a = np.array([655, 788], dtype=np.uint16)
        b = np.array([789, 772], dtype=np.uint16)
        F, p = stats.f_oneway(a, b)
        assert_almost_equal(F, 0.77450216931805538)

    def test_result_attributes(self):
        a = np.array([655, 788], dtype=np.uint16)
        b = np.array([789, 772], dtype=np.uint16)
        res = stats.f_oneway(a, b)
        attributes = ('statistic', 'pvalue')
        check_named_results(res, attributes)

    def test_nist(self):
        # These are the nist ANOVA files. They can be found at:
        # https://www.itl.nist.gov/div898/strd/anova/anova.html
        filenames = ['SiRstv.dat', 'SmLs01.dat', 'SmLs02.dat', 'SmLs03.dat',
                     'AtmWtAg.dat', 'SmLs04.dat', 'SmLs05.dat', 'SmLs06.dat',
                     'SmLs07.dat', 'SmLs08.dat', 'SmLs09.dat']

        for test_case in filenames:
            rtol = 1e-7
            fname = os.path.abspath(os.path.join(os.path.dirname(__file__),
                                                 'data/nist_anova', test_case))
            with open(fname, 'r') as f:
                content = f.read().split('\n')
            certified = [line.split() for line in content[40:48]
                         if line.strip()]
            dataf = np.loadtxt(fname, skiprows=60)
            y, x = dataf.T
            y = y.astype(int)
            caty = np.unique(y)
            f = float(certified[0][-1])

            xlist = [x[y == i] for i in caty]
            res = stats.f_oneway(*xlist)

            # With the hard test cases we relax the tolerance a bit.
            hard_tc = ('SmLs07.dat', 'SmLs08.dat', 'SmLs09.dat')
            if test_case in hard_tc:
                rtol = 1e-4

            assert_allclose(res[0], f, rtol=rtol,
                            err_msg='Failing testcase: %s' % test_case)


class TestKruskal(object):
    def test_simple(self):
        x = [1]
        y = [2]
        h, p = stats.kruskal(x, y)
        assert_equal(h, 1.0)
        assert_approx_equal(p, stats.distributions.chi2.sf(h, 1))
        h, p = stats.kruskal(np.array(x), np.array(y))
        assert_equal(h, 1.0)
        assert_approx_equal(p, stats.distributions.chi2.sf(h, 1))

    def test_basic(self):
        x = [1, 3, 5, 7, 9]
        y = [2, 4, 6, 8, 10]
        h, p = stats.kruskal(x, y)
        assert_approx_equal(h, 3./11, significant=10)
        assert_approx_equal(p, stats.distributions.chi2.sf(3./11, 1))
        h, p = stats.kruskal(np.array(x), np.array(y))
        assert_approx_equal(h, 3./11, significant=10)
        assert_approx_equal(p, stats.distributions.chi2.sf(3./11, 1))

    def test_simple_tie(self):
        x = [1]
        y = [1, 2]
        h_uncorr = 1.5**2 + 2*2.25**2 - 12
        corr = 0.75
        expected = h_uncorr / corr   # 0.5
        h, p = stats.kruskal(x, y)
        # Since the expression is simple and the exact answer is 0.5, it
        # should be safe to use assert_equal().
        assert_equal(h, expected)

    def test_another_tie(self):
        x = [1, 1, 1, 2]
        y = [2, 2, 2, 2]
        h_uncorr = (12. / 8. / 9.) * 4 * (3**2 + 6**2) - 3 * 9
        corr = 1 - float(3**3 - 3 + 5**3 - 5) / (8**3 - 8)
        expected = h_uncorr / corr
        h, p = stats.kruskal(x, y)
        assert_approx_equal(h, expected)

    def test_three_groups(self):
        # A test of stats.kruskal with three groups, with ties.
        x = [1, 1, 1]
        y = [2, 2, 2]
        z = [2, 2]
        h_uncorr = (12. / 8. / 9.) * (3*2**2 + 3*6**2 + 2*6**2) - 3 * 9  # 5.0
        corr = 1 - float(3**3 - 3 + 5**3 - 5) / (8**3 - 8)
        expected = h_uncorr / corr  # 7.0
        h, p = stats.kruskal(x, y, z)
        assert_approx_equal(h, expected)
        assert_approx_equal(p, stats.distributions.chi2.sf(h, 2))

    def test_empty(self):
        # A test of stats.kruskal with three groups, with ties.
        x = [1, 1, 1]
        y = [2, 2, 2]
        z = []
        assert_equal(stats.kruskal(x, y, z), (np.nan, np.nan))

    def test_kruskal_result_attributes(self):
        x = [1, 3, 5, 7, 9]
        y = [2, 4, 6, 8, 10]
        res = stats.kruskal(x, y)
        attributes = ('statistic', 'pvalue')
        check_named_results(res, attributes)

    def test_nan_policy(self):
        x = np.arange(10.)
        x[9] = np.nan
        assert_equal(stats.kruskal(x, x), (np.nan, np.nan))
        assert_almost_equal(stats.kruskal(x, x, nan_policy='omit'), (0.0, 1.0))
        assert_raises(ValueError, stats.kruskal, x, x, nan_policy='raise')
        assert_raises(ValueError, stats.kruskal, x, x, nan_policy='foobar')


class TestCombinePvalues(object):

    def test_fisher(self):
        # Example taken from https://en.wikipedia.org/wiki/Fisher%27s_exact_test#Example
        xsq, p = stats.combine_pvalues([.01, .2, .3], method='fisher')
        assert_approx_equal(p, 0.02156, significant=4)

    def test_stouffer(self):
        Z, p = stats.combine_pvalues([.01, .2, .3], method='stouffer')
        assert_approx_equal(p, 0.01651, significant=4)

    def test_stouffer2(self):
        Z, p = stats.combine_pvalues([.5, .5, .5], method='stouffer')
        assert_approx_equal(p, 0.5, significant=4)

    def test_weighted_stouffer(self):
        Z, p = stats.combine_pvalues([.01, .2, .3], method='stouffer',
                                     weights=np.ones(3))
        assert_approx_equal(p, 0.01651, significant=4)

    def test_weighted_stouffer2(self):
        Z, p = stats.combine_pvalues([.01, .2, .3], method='stouffer',
                                     weights=np.array((1, 4, 9)))
        assert_approx_equal(p, 0.1464, significant=4)


class TestCdfDistanceValidation(object):
    """
    Test that _cdf_distance() (via wasserstein_distance()) raises ValueErrors
    for bad inputs.
    """

    def test_distinct_value_and_weight_lengths(self):
        # When the number of weights does not match the number of values,
        # a ValueError should be raised.
        assert_raises(ValueError, stats.wasserstein_distance,
                      [1], [2], [4], [3, 1])
        assert_raises(ValueError, stats.wasserstein_distance, [1], [2], [1, 0])

    def test_zero_weight(self):
        # When a distribution is given zero weight, a ValueError should be
        # raised.
        assert_raises(ValueError, stats.wasserstein_distance,
                      [0, 1], [2], [0, 0])
        assert_raises(ValueError, stats.wasserstein_distance,
                      [0, 1], [2], [3, 1], [0])

    def test_negative_weights(self):
        # A ValueError should be raised if there are any negative weights.
        assert_raises(ValueError, stats.wasserstein_distance,
                      [0, 1], [2, 2], [1, 1], [3, -1])

    def test_empty_distribution(self):
        # A ValueError should be raised when trying to measure the distance
        # between something and nothing.
        assert_raises(ValueError, stats.wasserstein_distance, [], [2, 2])
        assert_raises(ValueError, stats.wasserstein_distance, [1], [])

    def test_inf_weight(self):
        # An inf weight is not valid.
        assert_raises(ValueError, stats.wasserstein_distance,
                      [1, 2, 1], [1, 1], [1, np.inf, 1], [1, 1])


class TestWassersteinDistance(object):
    """ Tests for wasserstein_distance() output values.
    """

    def test_simple(self):
        # For basic distributions, the value of the Wasserstein distance is
        # straightforward.
        assert_almost_equal(
            stats.wasserstein_distance([0, 1], [0], [1, 1], [1]),
            .5)
        assert_almost_equal(stats.wasserstein_distance(
            [0, 1], [0], [3, 1], [1]),
            .25)
        assert_almost_equal(stats.wasserstein_distance(
            [0, 2], [0], [1, 1], [1]),
            1)
        assert_almost_equal(stats.wasserstein_distance(
            [0, 1, 2], [1, 2, 3]),
            1)

    def test_same_distribution(self):
        # Any distribution moved to itself should have a Wasserstein distance of
        # zero.
        assert_equal(stats.wasserstein_distance([1, 2, 3], [2, 1, 3]), 0)
        assert_equal(
            stats.wasserstein_distance([1, 1, 1, 4], [4, 1],
                                       [1, 1, 1, 1], [1, 3]),
            0)

    def test_shift(self):
        # If the whole distribution is shifted by x, then the Wasserstein
        # distance should be x.
        assert_almost_equal(stats.wasserstein_distance([0], [1]), 1)
        assert_almost_equal(stats.wasserstein_distance([-5], [5]), 10)
        assert_almost_equal(
            stats.wasserstein_distance([1, 2, 3, 4, 5], [11, 12, 13, 14, 15]),
            10)
        assert_almost_equal(
            stats.wasserstein_distance([4.5, 6.7, 2.1], [4.6, 7, 9.2],
                                       [3, 1, 1], [1, 3, 1]),
            2.5)

    def test_combine_weights(self):
        # Assigning a weight w to a value is equivalent to including that value
        # w times in the value array with weight of 1.
        assert_almost_equal(
            stats.wasserstein_distance(
                [0, 0, 1, 1, 1, 1, 5], [0, 3, 3, 3, 3, 4, 4],
                [1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]),
            stats.wasserstein_distance([5, 0, 1], [0, 4, 3],
                                       [1, 2, 4], [1, 2, 4]))

    def test_collapse(self):
        # Collapsing a distribution to a point distribution at zero is
        # equivalent to taking the average of the absolute values of the values.
        u = np.arange(-10, 30, 0.3)
        v = np.zeros_like(u)
        assert_almost_equal(
            stats.wasserstein_distance(u, v),
            np.mean(np.abs(u)))

        u_weights = np.arange(len(u))
        v_weights = u_weights[::-1]
        assert_almost_equal(
            stats.wasserstein_distance(u, v, u_weights, v_weights),
            np.average(np.abs(u), weights=u_weights))

    def test_zero_weight(self):
        # Values with zero weight have no impact on the Wasserstein distance.
        assert_almost_equal(
            stats.wasserstein_distance([1, 2, 100000], [1, 1],
                                       [1, 1, 0], [1, 1]),
            stats.wasserstein_distance([1, 2], [1, 1], [1, 1], [1, 1]))

    def test_inf_values(self):
        # Inf values can lead to an inf distance or trigger a RuntimeWarning
        # (and return NaN) if the distance is undefined.
        assert_equal(
            stats.wasserstein_distance([1, 2, np.inf], [1, 1]),
            np.inf)
        assert_equal(
            stats.wasserstein_distance([1, 2, np.inf], [-np.inf, 1]),
            np.inf)
        assert_equal(
            stats.wasserstein_distance([1, -np.inf, np.inf], [1, 1]),
            np.inf)
        with suppress_warnings() as sup:
            r = sup.record(RuntimeWarning, "invalid value*")
            assert_equal(
                stats.wasserstein_distance([1, 2, np.inf], [np.inf, 1]),
                np.nan)


class TestEnergyDistance(object):
    """ Tests for energy_distance() output values.
    """

    def test_simple(self):
        # For basic distributions, the value of the energy distance is
        # straightforward.
        assert_almost_equal(
            stats.energy_distance([0, 1], [0], [1, 1], [1]),
            np.sqrt(2) * .5)
        assert_almost_equal(stats.energy_distance(
            [0, 1], [0], [3, 1], [1]),
            np.sqrt(2) * .25)
        assert_almost_equal(stats.energy_distance(
            [0, 2], [0], [1, 1], [1]),
            2 * .5)
        assert_almost_equal(
            stats.energy_distance([0, 1, 2], [1, 2, 3]),
            np.sqrt(2) * (3*(1./3**2))**.5)

    def test_same_distribution(self):
        # Any distribution moved to itself should have a energy distance of
        # zero.
        assert_equal(stats.energy_distance([1, 2, 3], [2, 1, 3]), 0)
        assert_equal(
            stats.energy_distance([1, 1, 1, 4], [4, 1], [1, 1, 1, 1], [1, 3]),
            0)

    def test_shift(self):
        # If a single-point distribution is shifted by x, then the energy
        # distance should be sqrt(2) * sqrt(x).
        assert_almost_equal(stats.energy_distance([0], [1]), np.sqrt(2))
        assert_almost_equal(
            stats.energy_distance([-5], [5]),
            np.sqrt(2) * 10**.5)

    def test_combine_weights(self):
        # Assigning a weight w to a value is equivalent to including that value
        # w times in the value array with weight of 1.
        assert_almost_equal(
            stats.energy_distance([0, 0, 1, 1, 1, 1, 5], [0, 3, 3, 3, 3, 4, 4],
                                  [1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]),
            stats.energy_distance([5, 0, 1], [0, 4, 3], [1, 2, 4], [1, 2, 4]))

    def test_zero_weight(self):
        # Values with zero weight have no impact on the energy distance.
        assert_almost_equal(
            stats.energy_distance([1, 2, 100000], [1, 1], [1, 1, 0], [1, 1]),
            stats.energy_distance([1, 2], [1, 1], [1, 1], [1, 1]))

    def test_inf_values(self):
        # Inf values can lead to an inf distance or trigger a RuntimeWarning
        # (and return NaN) if the distance is undefined.
        assert_equal(stats.energy_distance([1, 2, np.inf], [1, 1]), np.inf)
        assert_equal(
            stats.energy_distance([1, 2, np.inf], [-np.inf, 1]),
            np.inf)
        assert_equal(
            stats.energy_distance([1, -np.inf, np.inf], [1, 1]),
            np.inf)
        with suppress_warnings() as sup:
            r = sup.record(RuntimeWarning, "invalid value*")
            assert_equal(
                stats.energy_distance([1, 2, np.inf], [np.inf, 1]),
                np.nan)


class TestBrunnerMunzel(object):
    # Data from (Lumley, 1996)
    X = [1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1]
    Y = [3, 3, 4, 3, 1, 2, 3, 1, 1, 5, 4]
    significant = 13

    def test_brunnermunzel_one_sided(self):
        # Results are compared with R's lawstat package.
        u1, p1 = stats.brunnermunzel(self.X, self.Y, alternative='less')
        u2, p2 = stats.brunnermunzel(self.Y, self.X, alternative='greater')
        u3, p3 = stats.brunnermunzel(self.X, self.Y, alternative='greater')
        u4, p4 = stats.brunnermunzel(self.Y, self.X, alternative='less')

        assert_approx_equal(p1, p2, significant=self.significant)
        assert_approx_equal(p3, p4, significant=self.significant)
        assert_(p1 != p3)
        assert_approx_equal(u1, 3.1374674823029505,
                            significant=self.significant)
        assert_approx_equal(u2, -3.1374674823029505,
                            significant=self.significant)
        assert_approx_equal(u3, 3.1374674823029505,
                            significant=self.significant)
        assert_approx_equal(u4, -3.1374674823029505,
                            significant=self.significant)
        assert_approx_equal(p1, 0.0028931043330757342,
                            significant=self.significant)
        assert_approx_equal(p3, 0.99710689566692423,
                            significant=self.significant)

    def test_brunnermunzel_two_sided(self):
        # Results are compared with R's lawstat package.
        u1, p1 = stats.brunnermunzel(self.X, self.Y, alternative='two-sided')
        u2, p2 = stats.brunnermunzel(self.Y, self.X, alternative='two-sided')

        assert_approx_equal(p1, p2, significant=self.significant)
        assert_approx_equal(u1, 3.1374674823029505,
                            significant=self.significant)
        assert_approx_equal(u2, -3.1374674823029505,
                            significant=self.significant)
        assert_approx_equal(p1, 0.0057862086661515377,
                            significant=self.significant)

    def test_brunnermunzel_default(self):
        # The default value for alternative is two-sided
        u1, p1 = stats.brunnermunzel(self.X, self.Y)
        u2, p2 = stats.brunnermunzel(self.Y, self.X)

        assert_approx_equal(p1, p2, significant=self.significant)
        assert_approx_equal(u1, 3.1374674823029505,
                            significant=self.significant)
        assert_approx_equal(u2, -3.1374674823029505,
                            significant=self.significant)
        assert_approx_equal(p1, 0.0057862086661515377,
                            significant=self.significant)

    def test_brunnermunzel_alternative_error(self):
        alternative = "error"
        distribution = "t"
        nan_policy = "propagate"
        assert_(alternative not in ["two-sided", "greater", "less"])
        assert_raises(ValueError,
                      stats.brunnermunzel,
                      self.X,
                      self.Y,
                      alternative,
                      distribution,
                      nan_policy)

    def test_brunnermunzel_distribution_norm(self):
        u1, p1 = stats.brunnermunzel(self.X, self.Y, distribution="normal")
        u2, p2 = stats.brunnermunzel(self.Y, self.X, distribution="normal")
        assert_approx_equal(p1, p2, significant=self.significant)
        assert_approx_equal(u1, 3.1374674823029505,
                            significant=self.significant)
        assert_approx_equal(u2, -3.1374674823029505,
                            significant=self.significant)
        assert_approx_equal(p1, 0.0017041417600383024,
                            significant=self.significant)

    def test_brunnermunzel_distribution_error(self):
        alternative = "two-sided"
        distribution = "error"
        nan_policy = "propagate"
        assert_(alternative not in ["t", "normal"])
        assert_raises(ValueError,
                      stats.brunnermunzel,
                      self.X,
                      self.Y,
                      alternative,
                      distribution,
                      nan_policy)

    def test_brunnermunzel_empty_imput(self):
        u1, p1 = stats.brunnermunzel(self.X, [])
        u2, p2 = stats.brunnermunzel([], self.Y)
        u3, p3 = stats.brunnermunzel([], [])

        assert_equal(u1, np.nan)
        assert_equal(p1, np.nan)
        assert_equal(u2, np.nan)
        assert_equal(p2, np.nan)
        assert_equal(u3, np.nan)
        assert_equal(p3, np.nan)

    def test_brunnermunzel_nan_input_propagate(self):
        X = [1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, np.nan]
        Y = [3, 3, 4, 3, 1, 2, 3, 1, 1, 5, 4]
        u1, p1 = stats.brunnermunzel(X, Y, nan_policy="propagate")
        u2, p2 = stats.brunnermunzel(Y, X, nan_policy="propagate")

        assert_equal(u1, np.nan)
        assert_equal(p1, np.nan)
        assert_equal(u2, np.nan)
        assert_equal(p2, np.nan)

    def test_brunnermunzel_nan_input_raise(self):
        X = [1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, np.nan]
        Y = [3, 3, 4, 3, 1, 2, 3, 1, 1, 5, 4]
        alternative = "two-sided"
        distribution = "t"
        nan_policy = "raise"

        assert_raises(ValueError,
                      stats.brunnermunzel,
                      X,
                      Y,
                      alternative,
                      distribution,
                      nan_policy)
        assert_raises(ValueError,
                      stats.brunnermunzel,
                      Y,
                      X,
                      alternative,
                      distribution,
                      nan_policy)

    def test_brunnermunzel_nan_input_omit(self):
        X = [1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, np.nan]
        Y = [3, 3, 4, 3, 1, 2, 3, 1, 1, 5, 4]
        u1, p1 = stats.brunnermunzel(X, Y, nan_policy="omit")
        u2, p2 = stats.brunnermunzel(Y, X, nan_policy="omit")

        assert_approx_equal(p1, p2, significant=self.significant)
        assert_approx_equal(u1, 3.1374674823029505,
                            significant=self.significant)
        assert_approx_equal(u2, -3.1374674823029505,
                            significant=self.significant)
        assert_approx_equal(p1, 0.0057862086661515377,
                            significant=self.significant)

