494 lines
18 KiB
Python
494 lines
18 KiB
Python
|
import numpy as np
|
||
|
|
from collections import namedtuple
|
||
|
|
from scipy import special
|
||
|
|
from scipy import stats
|
||
|
|
from ._axis_nan_policy import _axis_nan_policy_factory
|
||
|
|
|
||
|
|
|
||
|
|
def _broadcast_concatenate(x, y, axis):
|
||
|
|
'''Broadcast then concatenate arrays, leaving concatenation axis last'''
|
||
|
|
x = np.moveaxis(x, axis, -1)
|
||
|
|
y = np.moveaxis(y, axis, -1)
|
||
|
|
z = np.broadcast(x[..., 0], y[..., 0])
|
||
|
|
x = np.broadcast_to(x, z.shape + (x.shape[-1],))
|
||
|
|
y = np.broadcast_to(y, z.shape + (y.shape[-1],))
|
||
|
|
z = np.concatenate((x, y), axis=-1)
|
||
|
|
return x, y, z
|
||
|
|
|
||
|
|
|
||
|
|
class _MWU:
|
||
|
|
'''Distribution of MWU statistic under the null hypothesis'''
|
||
|
|
# Possible improvement: if m and n are small enough, use integer arithmetic
|
||
|
|
|
||
|
|
def __init__(self):
|
||
|
|
'''Minimal initializer'''
|
||
|
|
self._fmnks = -np.ones((1, 1, 1))
|
||
|
|
self._recursive = None
|
||
|
|
|
||
|
|
def pmf(self, k, m, n):
|
||
|
|
if (self._recursive is None and m <= 500 and n <= 500
|
||
|
|
or self._recursive):
|
||
|
|
return self.pmf_recursive(k, m, n)
|
||
|
|
else:
|
||
|
|
return self.pmf_iterative(k, m, n)
|
||
|
|
|
||
|
|
def pmf_recursive(self, k, m, n):
|
||
|
|
'''Probability mass function, recursive version'''
|
||
|
|
self._resize_fmnks(m, n, np.max(k))
|
||
|
|
# could loop over just the unique elements, but probably not worth
|
||
|
|
# the time to find them
|
||
|
|
for i in np.ravel(k):
|
||
|
|
self._f(m, n, i)
|
||
|
|
return self._fmnks[m, n, k] / special.binom(m + n, m)
|
||
|
|
|
||
|
|
def pmf_iterative(self, k, m, n):
|
||
|
|
'''Probability mass function, iterative version'''
|
||
|
|
fmnks = {}
|
||
|
|
for i in np.ravel(k):
|
||
|
|
fmnks = _mwu_f_iterative(m, n, i, fmnks)
|
||
|
|
return (np.array([fmnks[(m, n, ki)] for ki in k])
|
||
|
|
/ special.binom(m + n, m))
|
||
|
|
|
||
|
|
def cdf(self, k, m, n):
|
||
|
|
'''Cumulative distribution function'''
|
||
|
|
# We could use the fact that the distribution is symmetric to avoid
|
||
|
|
# summing more than m*n/2 terms, but it might not be worth the
|
||
|
|
# overhead. Let's leave that to an improvement.
|
||
|
|
pmfs = self.pmf(np.arange(0, np.max(k) + 1), m, n)
|
||
|
|
cdfs = np.cumsum(pmfs)
|
||
|
|
return cdfs[k]
|
||
|
|
|
||
|
|
def sf(self, k, m, n):
|
||
|
|
'''Survival function'''
|
||
|
|
# Use the fact that the distribution is symmetric; i.e.
|
||
|
|
# _f(m, n, m*n-k) = _f(m, n, k), and sum from the left
|
||
|
|
k = m*n - k
|
||
|
|
# Note that both CDF and SF include the PMF at k. The p-value is
|
||
|
|
# calculated from the SF and should include the mass at k, so this
|
||
|
|
# is desirable
|
||
|
|
return self.cdf(k, m, n)
|
||
|
|
|
||
|
|
def _resize_fmnks(self, m, n, k):
|
||
|
|
'''If necessary, expand the array that remembers PMF values'''
|
||
|
|
# could probably use `np.pad` but I'm not sure it would save code
|
||
|
|
shape_old = np.array(self._fmnks.shape)
|
||
|
|
shape_new = np.array((m+1, n+1, k+1))
|
||
|
|
if np.any(shape_new > shape_old):
|
||
|
|
shape = np.maximum(shape_old, shape_new)
|
||
|
|
fmnks = -np.ones(shape) # create the new array
|
||
|
|
m0, n0, k0 = shape_old
|
||
|
|
fmnks[:m0, :n0, :k0] = self._fmnks # copy remembered values
|
||
|
|
self._fmnks = fmnks
|
||
|
|
|
||
|
|
def _f(self, m, n, k):
|
||
|
|
'''Recursive implementation of function of [3] Theorem 2.5'''
|
||
|
|
|
||
|
|
# [3] Theorem 2.5 Line 1
|
||
|
|
if k < 0 or m < 0 or n < 0 or k > m*n:
|
||
|
|
return 0
|
||
|
|
|
||
|
|
# if already calculated, return the value
|
||
|
|
if self._fmnks[m, n, k] >= 0:
|
||
|
|
return self._fmnks[m, n, k]
|
||
|
|
|
||
|
|
if k == 0 and m >= 0 and n >= 0: # [3] Theorem 2.5 Line 2
|
||
|
|
fmnk = 1
|
||
|
|
else: # [3] Theorem 2.5 Line 3 / Equation 3
|
||
|
|
fmnk = self._f(m-1, n, k-n) + self._f(m, n-1, k)
|
||
|
|
|
||
|
|
self._fmnks[m, n, k] = fmnk # remember result
|
||
|
|
|
||
|
|
return fmnk
|
||
|
|
|
||
|
|
|
||
|
|
# Maintain state for faster repeat calls to mannwhitneyu w/ method='exact'
|
||
|
|
_mwu_state = _MWU()
|
||
|
|
|
||
|
|
|
||
|
|
def _mwu_f_iterative(m, n, k, fmnks):
|
||
|
|
'''Iterative implementation of function of [3] Theorem 2.5'''
|
||
|
|
|
||
|
|
def _base_case(m, n, k):
|
||
|
|
'''Base cases from recursive version'''
|
||
|
|
|
||
|
|
# if already calculated, return the value
|
||
|
|
if fmnks.get((m, n, k), -1) >= 0:
|
||
|
|
return fmnks[(m, n, k)]
|
||
|
|
|
||
|
|
# [3] Theorem 2.5 Line 1
|
||
|
|
elif k < 0 or m < 0 or n < 0 or k > m*n:
|
||
|
|
return 0
|
||
|
|
|
||
|
|
# [3] Theorem 2.5 Line 2
|
||
|
|
elif k == 0 and m >= 0 and n >= 0:
|
||
|
|
return 1
|
||
|
|
|
||
|
|
return None
|
||
|
|
|
||
|
|
stack = [(m, n, k)]
|
||
|
|
fmnk = None
|
||
|
|
|
||
|
|
while stack:
|
||
|
|
# Popping only if necessary would save a tiny bit of time, but NWI.
|
||
|
|
m, n, k = stack.pop()
|
||
|
|
|
||
|
|
# If we're at a base case, continue (stack unwinds)
|
||
|
|
fmnk = _base_case(m, n, k)
|
||
|
|
if fmnk is not None:
|
||
|
|
fmnks[(m, n, k)] = fmnk
|
||
|
|
continue
|
||
|
|
|
||
|
|
# If both terms are base cases, continue (stack unwinds)
|
||
|
|
f1 = _base_case(m-1, n, k-n)
|
||
|
|
f2 = _base_case(m, n-1, k)
|
||
|
|
if f1 is not None and f2 is not None:
|
||
|
|
# [3] Theorem 2.5 Line 3 / Equation 3
|
||
|
|
fmnk = f1 + f2
|
||
|
|
fmnks[(m, n, k)] = fmnk
|
||
|
|
continue
|
||
|
|
|
||
|
|
# recurse deeper
|
||
|
|
stack.append((m, n, k))
|
||
|
|
if f1 is None:
|
||
|
|
stack.append((m-1, n, k-n))
|
||
|
|
if f2 is None:
|
||
|
|
stack.append((m, n-1, k))
|
||
|
|
|
||
|
|
return fmnks
|
||
|
|
|
||
|
|
|
||
|
|
def _tie_term(ranks):
|
||
|
|
"""Tie correction term"""
|
||
|
|
# element i of t is the number of elements sharing rank i
|
||
|
|
_, t = np.unique(ranks, return_counts=True, axis=-1)
|
||
|
|
return (t**3 - t).sum(axis=-1)
|
||
|
|
|
||
|
|
|
||
|
|
def _get_mwu_z(U, n1, n2, ranks, axis=0, continuity=True):
|
||
|
|
'''Standardized MWU statistic'''
|
||
|
|
# Follows mannwhitneyu [2]
|
||
|
|
mu = n1 * n2 / 2
|
||
|
|
n = n1 + n2
|
||
|
|
|
||
|
|
# Tie correction according to [2]
|
||
|
|
tie_term = np.apply_along_axis(_tie_term, -1, ranks)
|
||
|
|
s = np.sqrt(n1*n2/12 * ((n + 1) - tie_term/(n*(n-1))))
|
||
|
|
|
||
|
|
# equivalent to using scipy.stats.tiecorrect
|
||
|
|
# T = np.apply_along_axis(stats.tiecorrect, -1, ranks)
|
||
|
|
# s = np.sqrt(T * n1 * n2 * (n1+n2+1) / 12.0)
|
||
|
|
|
||
|
|
numerator = U - mu
|
||
|
|
|
||
|
|
# Continuity correction.
|
||
|
|
# Because SF is always used to calculate the p-value, we can always
|
||
|
|
# _subtract_ 0.5 for the continuity correction. This always increases the
|
||
|
|
# p-value to account for the rest of the probability mass _at_ q = U.
|
||
|
|
if continuity:
|
||
|
|
numerator -= 0.5
|
||
|
|
|
||
|
|
# no problem evaluating the norm SF at an infinity
|
||
|
|
with np.errstate(divide='ignore', invalid='ignore'):
|
||
|
|
z = numerator / s
|
||
|
|
return z
|
||
|
|
|
||
|
|
|
||
|
|
def _mwu_input_validation(x, y, use_continuity, alternative, axis, method):
|
||
|
|
''' Input validation and standardization for mannwhitneyu '''
|
||
|
|
# Would use np.asarray_chkfinite, but infs are OK
|
||
|
|
x, y = np.atleast_1d(x), np.atleast_1d(y)
|
||
|
|
if np.isnan(x).any() or np.isnan(y).any():
|
||
|
|
raise ValueError('`x` and `y` must not contain NaNs.')
|
||
|
|
if np.size(x) == 0 or np.size(y) == 0:
|
||
|
|
raise ValueError('`x` and `y` must be of nonzero size.')
|
||
|
|
|
||
|
|
bools = {True, False}
|
||
|
|
if use_continuity not in bools:
|
||
|
|
raise ValueError(f'`use_continuity` must be one of {bools}.')
|
||
|
|
|
||
|
|
alternatives = {"two-sided", "less", "greater"}
|
||
|
|
alternative = alternative.lower()
|
||
|
|
if alternative not in alternatives:
|
||
|
|
raise ValueError(f'`alternative` must be one of {alternatives}.')
|
||
|
|
|
||
|
|
axis_int = int(axis)
|
||
|
|
if axis != axis_int:
|
||
|
|
raise ValueError('`axis` must be an integer.')
|
||
|
|
|
||
|
|
methods = {"asymptotic", "exact", "auto"}
|
||
|
|
method = method.lower()
|
||
|
|
if method not in methods:
|
||
|
|
raise ValueError(f'`method` must be one of {methods}.')
|
||
|
|
|
||
|
|
return x, y, use_continuity, alternative, axis_int, method
|
||
|
|
|
||
|
|
|
||
|
|
def _tie_check(xy):
|
||
|
|
"""Find any ties in data"""
|
||
|
|
_, t = np.unique(xy, return_counts=True, axis=-1)
|
||
|
|
return np.any(t != 1)
|
||
|
|
|
||
|
|
|
||
|
|
def _mwu_choose_method(n1, n2, xy, method):
|
||
|
|
"""Choose method 'asymptotic' or 'exact' depending on input size, ties"""
|
||
|
|
|
||
|
|
# if both inputs are large, asymptotic is OK
|
||
|
|
if n1 > 8 and n2 > 8:
|
||
|
|
return "asymptotic"
|
||
|
|
|
||
|
|
# if there are any ties, asymptotic is preferred
|
||
|
|
if np.apply_along_axis(_tie_check, -1, xy).any():
|
||
|
|
return "asymptotic"
|
||
|
|
|
||
|
|
return "exact"
|
||
|
|
|
||
|
|
|
||
|
|
MannwhitneyuResult = namedtuple('MannwhitneyuResult', ('statistic', 'pvalue'))
|
||
|
|
|
||
|
|
|
||
|
|
@_axis_nan_policy_factory(MannwhitneyuResult, n_samples=2)
|
||
|
|
def mannwhitneyu(x, y, use_continuity=True, alternative="two-sided",
|
||
|
|
axis=0, method="auto"):
|
||
|
|
r'''Perform the Mann-Whitney U rank test on two independent samples.
|
||
|
|
|
||
|
|
The Mann-Whitney U test is a nonparametric test of the null hypothesis
|
||
|
|
that the distribution underlying sample `x` is the same as the
|
||
|
|
distribution underlying sample `y`. It is often used as a test of
|
||
|
|
difference in location between distributions.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x, y : array-like
|
||
|
|
N-d arrays of samples. The arrays must be broadcastable except along
|
||
|
|
the dimension given by `axis`.
|
||
|
|
use_continuity : bool, optional
|
||
|
|
Whether a continuity correction (1/2) should be applied.
|
||
|
|
Default is True when `method` is ``'asymptotic'``; has no effect
|
||
|
|
otherwise.
|
||
|
|
alternative : {'two-sided', 'less', 'greater'}, optional
|
||
|
|
Defines the alternative hypothesis. Default is 'two-sided'.
|
||
|
|
Let *F(u)* and *G(u)* be the cumulative distribution functions of the
|
||
|
|
distributions underlying `x` and `y`, respectively. Then the following
|
||
|
|
alternative hypotheses are available:
|
||
|
|
|
||
|
|
* 'two-sided': the distributions are not equal, i.e. *F(u) ≠ G(u)* for
|
||
|
|
at least one *u*.
|
||
|
|
* 'less': the distribution underlying `x` is stochastically less
|
||
|
|
than the distribution underlying `y`, i.e. *F(u) > G(u)* for all *u*.
|
||
|
|
* 'greater': the distribution underlying `x` is stochastically greater
|
||
|
|
than the distribution underlying `y`, i.e. *F(u) < G(u)* for all *u*.
|
||
|
|
|
||
|
|
Under a more restrictive set of assumptions, the alternative hypotheses
|
||
|
|
can be expressed in terms of the locations of the distributions;
|
||
|
|
see [5] section 5.1.
|
||
|
|
axis : int, optional
|
||
|
|
Axis along which to perform the test. Default is 0.
|
||
|
|
method : {'auto', 'asymptotic', 'exact'}, optional
|
||
|
|
Selects the method used to calculate the *p*-value.
|
||
|
|
Default is 'auto'. The following options are available.
|
||
|
|
|
||
|
|
* ``'asymptotic'``: compares the standardized test statistic
|
||
|
|
against the normal distribution, correcting for ties.
|
||
|
|
* ``'exact'``: computes the exact *p*-value by comparing the observed
|
||
|
|
:math:`U` statistic against the exact distribution of the :math:`U`
|
||
|
|
statistic under the null hypothesis. No correction is made for ties.
|
||
|
|
* ``'auto'``: chooses ``'exact'`` when the size of one of the samples
|
||
|
|
is less than 8 and there are no ties; chooses ``'asymptotic'``
|
||
|
|
otherwise.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
res : MannwhitneyuResult
|
||
|
|
An object containing attributes:
|
||
|
|
|
||
|
|
statistic : float
|
||
|
|
The Mann-Whitney U statistic corresponding with sample `x`. See
|
||
|
|
Notes for the test statistic corresponding with sample `y`.
|
||
|
|
pvalue : float
|
||
|
|
The associated *p*-value for the chosen `alternative`.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
If ``U1`` is the statistic corresponding with sample `x`, then the
|
||
|
|
statistic corresponding with sample `y` is
|
||
|
|
`U2 = `x.shape[axis] * y.shape[axis] - U1``.
|
||
|
|
|
||
|
|
`mannwhitneyu` is for independent samples. For related / paired samples,
|
||
|
|
consider `scipy.stats.wilcoxon`.
|
||
|
|
|
||
|
|
`method` ``'exact'`` is recommended when there are no ties and when either
|
||
|
|
sample size is less than 8 [1]_. The implementation follows the recurrence
|
||
|
|
relation originally proposed in [1]_ as it is described in [3]_.
|
||
|
|
Note that the exact method is *not* corrected for ties, but
|
||
|
|
`mannwhitneyu` will not raise errors or warnings if there are ties in the
|
||
|
|
data.
|
||
|
|
|
||
|
|
The Mann-Whitney U test is a non-parametric version of the t-test for
|
||
|
|
independent samples. When the means of samples from the populations
|
||
|
|
are normally distributed, consider `scipy.stats.ttest_ind`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
scipy.stats.wilcoxon, scipy.stats.ranksums, scipy.stats.ttest_ind
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] H.B. Mann and D.R. Whitney, "On a test of whether one of two random
|
||
|
|
variables is stochastically larger than the other", The Annals of
|
||
|
|
Mathematical Statistics, Vol. 18, pp. 50-60, 1947.
|
||
|
|
.. [2] Mann-Whitney U Test, Wikipedia,
|
||
|
|
http://en.wikipedia.org/wiki/Mann-Whitney_U_test
|
||
|
|
.. [3] A. Di Bucchianico, "Combinatorics, computer algebra, and the
|
||
|
|
Wilcoxon-Mann-Whitney test", Journal of Statistical Planning and
|
||
|
|
Inference, Vol. 79, pp. 349-364, 1999.
|
||
|
|
.. [4] Rosie Shier, "Statistics: 2.3 The Mann-Whitney U Test", Mathematics
|
||
|
|
Learning Support Centre, 2004.
|
||
|
|
.. [5] Michael P. Fay and Michael A. Proschan. "Wilcoxon-Mann-Whitney
|
||
|
|
or t-test? On assumptions for hypothesis tests and multiple \
|
||
|
|
interpretations of decision rules." Statistics surveys, Vol. 4, pp.
|
||
|
|
1-39, 2010. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2857732/
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
We follow the example from [4]_: nine randomly sampled young adults were
|
||
|
|
diagnosed with type II diabetes at the ages below.
|
||
|
|
|
||
|
|
>>> males = [19, 22, 16, 29, 24]
|
||
|
|
>>> females = [20, 11, 17, 12]
|
||
|
|
|
||
|
|
We use the Mann-Whitney U test to assess whether there is a statistically
|
||
|
|
significant difference in the diagnosis age of males and females.
|
||
|
|
The null hypothesis is that the distribution of male diagnosis ages is
|
||
|
|
the same as the distribution of female diagnosis ages. We decide
|
||
|
|
that a confidence level of 95% is required to reject the null hypothesis
|
||
|
|
in favor of the alternative that the distributions are different.
|
||
|
|
Since the number of samples is very small and there are no ties in the
|
||
|
|
data, we can compare the observed test statistic against the *exact*
|
||
|
|
distribution of the test statistic under the null hypothesis.
|
||
|
|
|
||
|
|
>>> from scipy.stats import mannwhitneyu
|
||
|
|
>>> U1, p = mannwhitneyu(males, females, method="exact")
|
||
|
|
>>> print(U1)
|
||
|
|
17.0
|
||
|
|
|
||
|
|
`mannwhitneyu` always reports the statistic associated with the first
|
||
|
|
sample, which, in this case, is males. This agrees with :math:`U_M = 17`
|
||
|
|
reported in [4]_. The statistic associated with the second statistic
|
||
|
|
can be calculated:
|
||
|
|
|
||
|
|
>>> nx, ny = len(males), len(females)
|
||
|
|
>>> U2 = nx*ny - U1
|
||
|
|
>>> print(U2)
|
||
|
|
3.0
|
||
|
|
|
||
|
|
This agrees with :math:`U_F = 3` reported in [4]_. The two-sided
|
||
|
|
*p*-value can be calculated from either statistic, and the value produced
|
||
|
|
by `mannwhitneyu` agrees with :math:`p = 0.11` reported in [4]_.
|
||
|
|
|
||
|
|
>>> print(p)
|
||
|
|
0.1111111111111111
|
||
|
|
|
||
|
|
The exact distribution of the test statistic is asymptotically normal, so
|
||
|
|
the example continues by comparing the exact *p*-value against the
|
||
|
|
*p*-value produced using the normal approximation.
|
||
|
|
|
||
|
|
>>> _, pnorm = mannwhitneyu(males, females, method="asymptotic")
|
||
|
|
>>> print(pnorm)
|
||
|
|
0.11134688653314041
|
||
|
|
|
||
|
|
Here `mannwhitneyu`'s reported *p*-value appears to conflict with the
|
||
|
|
value :math:`p = 0.09` given in [4]_. The reason is that [4]_
|
||
|
|
does not apply the continuity correction performed by `mannwhitneyu`;
|
||
|
|
`mannwhitneyu` reduces the distance between the test statistic and the
|
||
|
|
mean :math:`\mu = n_x n_y / 2` by 0.5 to correct for the fact that the
|
||
|
|
discrete statistic is being compared against a continuous distribution.
|
||
|
|
Here, the :math:`U` statistic used is less than the mean, so we reduce
|
||
|
|
the distance by adding 0.5 in the numerator.
|
||
|
|
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from scipy.stats import norm
|
||
|
|
>>> U = min(U1, U2)
|
||
|
|
>>> N = nx + ny
|
||
|
|
>>> z = (U - nx*ny/2 + 0.5) / np.sqrt(nx*ny * (N + 1)/ 12)
|
||
|
|
>>> p = 2 * norm.cdf(z) # use CDF to get p-value from smaller statistic
|
||
|
|
>>> print(p)
|
||
|
|
0.11134688653314041
|
||
|
|
|
||
|
|
If desired, we can disable the continuity correction to get a result
|
||
|
|
that agrees with that reported in [4]_.
|
||
|
|
|
||
|
|
>>> _, pnorm = mannwhitneyu(males, females, use_continuity=False,
|
||
|
|
... method="asymptotic")
|
||
|
|
>>> print(pnorm)
|
||
|
|
0.0864107329737
|
||
|
|
|
||
|
|
Regardless of whether we perform an exact or asymptotic test, the
|
||
|
|
probability of the test statistic being as extreme or more extreme by
|
||
|
|
chance exceeds 5%, so we do not consider the results statistically
|
||
|
|
significant.
|
||
|
|
|
||
|
|
Suppose that, before seeing the data, we had hypothesized that females
|
||
|
|
would tend to be diagnosed at a younger age than males.
|
||
|
|
In that case, it would be natural to provide the female ages as the
|
||
|
|
first input, and we would have performed a one-sided test using
|
||
|
|
``alternative = 'less'``: females are diagnosed at an age that is
|
||
|
|
stochastically less than that of males.
|
||
|
|
|
||
|
|
>>> res = mannwhitneyu(females, males, alternative="less", method="exact")
|
||
|
|
>>> print(res)
|
||
|
|
MannwhitneyuResult(statistic=3.0, pvalue=0.05555555555555555)
|
||
|
|
|
||
|
|
Again, the probability of getting a sufficiently low value of the
|
||
|
|
test statistic by chance under the null hypothesis is greater than 5%,
|
||
|
|
so we do not reject the null hypothesis in favor of our alternative.
|
||
|
|
|
||
|
|
If it is reasonable to assume that the means of samples from the
|
||
|
|
populations are normally distributed, we could have used a t-test to
|
||
|
|
perform the analysis.
|
||
|
|
|
||
|
|
>>> from scipy.stats import ttest_ind
|
||
|
|
>>> res = ttest_ind(females, males, alternative="less")
|
||
|
|
>>> print(res)
|
||
|
|
Ttest_indResult(statistic=-2.239334696520584, pvalue=0.030068441095757924)
|
||
|
|
|
||
|
|
Under this assumption, the *p*-value would be low enough to reject the
|
||
|
|
null hypothesis in favor of the alternative.
|
||
|
|
|
||
|
|
'''
|
||
|
|
|
||
|
|
x, y, use_continuity, alternative, axis_int, method = (
|
||
|
|
_mwu_input_validation(x, y, use_continuity, alternative, axis, method))
|
||
|
|
|
||
|
|
x, y, xy = _broadcast_concatenate(x, y, axis)
|
||
|
|
|
||
|
|
n1, n2 = x.shape[-1], y.shape[-1]
|
||
|
|
|
||
|
|
if method == "auto":
|
||
|
|
method = _mwu_choose_method(n1, n2, xy, method)
|
||
|
|
|
||
|
|
# Follows [2]
|
||
|
|
ranks = stats.rankdata(xy, axis=-1) # method 2, step 1
|
||
|
|
R1 = ranks[..., :n1].sum(axis=-1) # method 2, step 2
|
||
|
|
U1 = R1 - n1*(n1+1)/2 # method 2, step 3
|
||
|
|
U2 = n1 * n2 - U1 # as U1 + U2 = n1 * n2
|
||
|
|
|
||
|
|
if alternative == "greater":
|
||
|
|
U, f = U1, 1 # U is the statistic to use for p-value, f is a factor
|
||
|
|
elif alternative == "less":
|
||
|
|
U, f = U2, 1 # Due to symmetry, use SF of U2 rather than CDF of U1
|
||
|
|
else:
|
||
|
|
U, f = np.maximum(U1, U2), 2 # multiply SF by two for two-sided test
|
||
|
|
|
||
|
|
if method == "exact":
|
||
|
|
p = _mwu_state.sf(U.astype(int), n1, n2)
|
||
|
|
elif method == "asymptotic":
|
||
|
|
z = _get_mwu_z(U, n1, n2, ranks, continuity=use_continuity)
|
||
|
|
p = stats.norm.sf(z)
|
||
|
|
p *= f
|
||
|
|
|
||
|
|
# Ensure that test statistic is not greater than 1
|
||
|
|
# This could happen for exact test when U = m*n/2
|
||
|
|
p = np.clip(p, 0, 1)
|
||
|
|
|
||
|
|
return MannwhitneyuResult(U1, p)
|