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.CondaPkg/env/Lib/site-packages/scipy/stats/_mannwhitneyu.py vendored Normal file
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import numpy as np
from collections import namedtuple
from scipy import special
from scipy import stats
from scipy.stats._stats_py import _rankdata
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'''
def __init__(self, n1, n2):
self._reset(n1, n2)
def set_shapes(self, n1, n2):
n1, n2 = min(n1, n2), max(n1, n2)
if (n1, n2) == (self.n1, self.n2):
return
self.n1 = n1
self.n2 = n2
self.s_array = np.zeros(0, dtype=int)
self.configurations = np.zeros(0, dtype=np.uint64)
def reset(self):
self._reset(self.n1, self.n2)
def _reset(self, n1, n2):
self.n1 = None
self.n2 = None
self.set_shapes(n1, n2)
def pmf(self, k):
# In practice, `pmf` is never called with k > m*n/2.
# If it were, we'd exploit symmetry here:
# k = np.array(k, copy=True)
# k2 = m*n - k
# i = k2 < k
# k[i] = k2[i]
pmfs = self.build_u_freqs_array(np.max(k))
return pmfs[k]
def cdf(self, k):
'''Cumulative distribution function'''
# In practice, `cdf` is never called with k > m*n/2.
# If it were, we'd exploit symmetry here rather than in `sf`
pmfs = self.build_u_freqs_array(np.max(k))
cdfs = np.cumsum(pmfs)
return cdfs[k]
def sf(self, k):
'''Survival function'''
# 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
# Use the fact that the distribution is symmetric and sum from the left
kc = np.asarray(self.n1*self.n2 - k) # complement of k
i = k < kc
if np.any(i):
kc[i] = k[i]
cdfs = np.asarray(self.cdf(kc))
cdfs[i] = 1. - cdfs[i] + self.pmf(kc[i])
else:
cdfs = np.asarray(self.cdf(kc))
return cdfs[()]
# build_sigma_array and build_u_freqs_array adapted from code
# by @toobaz with permission. Thanks to @andreasloe for the suggestion.
# See https://github.com/scipy/scipy/pull/4933#issuecomment-1898082691
def build_sigma_array(self, a):
n1, n2 = self.n1, self.n2
if a + 1 <= self.s_array.size:
return self.s_array[1:a+1]
s_array = np.zeros(a + 1, dtype=int)
for d in np.arange(1, n1 + 1):
# All multiples of d, except 0:
indices = np.arange(d, a + 1, d)
# \epsilon_d = 1:
s_array[indices] += d
for d in np.arange(n2 + 1, n2 + n1 + 1):
# All multiples of d, except 0:
indices = np.arange(d, a + 1, d)
# \epsilon_d = -1:
s_array[indices] -= d
# We don't need 0:
self.s_array = s_array
return s_array[1:]
def build_u_freqs_array(self, maxu):
"""
Build all the array of frequencies for u from 0 to maxu.
Assumptions:
n1 <= n2
maxu <= n1 * n2 / 2
"""
n1, n2 = self.n1, self.n2
total = special.binom(n1 + n2, n1)
if maxu + 1 <= self.configurations.size:
return self.configurations[:maxu + 1] / total
s_array = self.build_sigma_array(maxu)
# Start working with ints, for maximum precision and efficiency:
configurations = np.zeros(maxu + 1, dtype=np.uint64)
configurations_is_uint = True
uint_max = np.iinfo(np.uint64).max
# How many ways to have U=0? 1
configurations[0] = 1
for u in np.arange(1, maxu + 1):
coeffs = s_array[u - 1::-1]
new_val = np.dot(configurations[:u], coeffs) / u
if new_val > uint_max and configurations_is_uint:
# OK, we got into numbers too big for uint64.
# So now we start working with floats.
# By doing this since the beginning, we would have lost precision.
# (And working on python long ints would be unbearably slow)
configurations = configurations.astype(float)
configurations_is_uint = False
configurations[u] = new_val
self.configurations = configurations
return configurations / total
_mwu_state = _MWU(0, 0)
def _get_mwu_z(U, n1, n2, t, axis=0, continuity=True):
'''Standardized MWU statistic'''
# Follows mannwhitneyu [2]
mu = n1 * n2 / 2
n = n1 + n2
# Tie correction according to [2], "Normal approximation and tie correction"
# "A more computationally-efficient form..."
tie_term = (t**3 - t).sum(axis=-1)
s = np.sqrt(n1*n2/12 * ((n + 1) - tie_term/(n*(n-1))))
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.')
if not isinstance(method, stats.PermutationMethod):
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 _mwu_choose_method(n1, n2, ties):
"""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 ties:
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*.
Note that the mathematical expressions in the alternative hypotheses
above describe the CDFs of the underlying distributions. The directions
of the inequalities appear inconsistent with the natural language
description at first glance, but they are not. For example, suppose
*X* and *Y* are random variables that follow distributions with CDFs
*F* and *G*, respectively. If *F(u) > G(u)* for all *u*, samples drawn
from *X* tend to be less than those drawn from *Y*.
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'} or `PermutationMethod` instance, 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 or equal to 8 and there are no ties;
chooses ``'asymptotic'`` otherwise.
* `PermutationMethod` instance. In this case, the p-value
is computed using `permutation_test` with the provided
configuration options and other appropriate settings.
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 algorithm
reported 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. If there are ties and either samples is small (fewer than ~10
observations), consider passing an instance of `PermutationMethod`
as the `method` to perform a permutation test.
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] Andreas Löffler,
"Über eine Partition der nat. Zahlen und ihr Anwendung beim U-Test",
Wiss. Z. Univ. Halle, XXXII'83 pp. 87-89.
.. [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)
TtestResult(statistic=-2.239334696520584,
pvalue=0.030068441095757924,
df=7.0)
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]
# Follows [2]
ranks, t = _rankdata(xy, 'average', return_ties=True) # 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 == "auto":
method = _mwu_choose_method(n1, n2, np.any(t > 1))
if method == "exact":
_mwu_state.set_shapes(n1, n2)
p = _mwu_state.sf(U.astype(int))
elif method == "asymptotic":
z = _get_mwu_z(U, n1, n2, t, continuity=use_continuity)
p = stats.norm.sf(z)
else: # `PermutationMethod` instance (already validated)
def statistic(x, y, axis):
return mannwhitneyu(x, y, use_continuity=use_continuity,
alternative=alternative, axis=axis,
method="asymptotic").statistic
res = stats.permutation_test((x, y), statistic, axis=axis,
**method._asdict(), alternative=alternative)
p = res.pvalue
f = 1
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)