495 lines
19 KiB
Python
495 lines
19 KiB
Python
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)
|