Current File : //lib/python3/dist-packages/scipy/stats/_hypotests.py
from collections import namedtuple
from dataclasses import make_dataclass
import numpy as np
import warnings
from itertools import combinations, permutations, product
import scipy.stats
from scipy.optimize import shgo
from . import distributions
from ._common import ConfidenceInterval
from ._continuous_distns import chi2, norm
from scipy.special import gamma, kv, gammaln, comb, factorial
from . import _wilcoxon_data
import scipy.stats._bootstrap as _bootstrap
from scipy._lib._util import check_random_state
from ._hypotests_pythran import _Q, _P, _a_ij_Aij_Dij2
__all__ = ['epps_singleton_2samp', 'cramervonmises', 'somersd',
'barnard_exact', 'boschloo_exact', 'cramervonmises_2samp',
'permutation_test', 'tukey_hsd']
Epps_Singleton_2sampResult = namedtuple('Epps_Singleton_2sampResult',
('statistic', 'pvalue'))
def epps_singleton_2samp(x, y, t=(0.4, 0.8)):
"""Compute the Epps-Singleton (ES) test statistic.
Test the null hypothesis that two samples have the same underlying
probability distribution.
Parameters
----------
x, y : array-like
The two samples of observations to be tested. Input must not have more
than one dimension. Samples can have different lengths.
t : array-like, optional
The points (t1, ..., tn) where the empirical characteristic function is
to be evaluated. It should be positive distinct numbers. The default
value (0.4, 0.8) is proposed in [1]_. Input must not have more than
one dimension.
Returns
-------
statistic : float
The test statistic.
pvalue : float
The associated p-value based on the asymptotic chi2-distribution.
See Also
--------
ks_2samp, anderson_ksamp
Notes
-----
Testing whether two samples are generated by the same underlying
distribution is a classical question in statistics. A widely used test is
the Kolmogorov-Smirnov (KS) test which relies on the empirical
distribution function. Epps and Singleton introduce a test based on the
empirical characteristic function in [1]_.
One advantage of the ES test compared to the KS test is that is does
not assume a continuous distribution. In [1]_, the authors conclude
that the test also has a higher power than the KS test in many
examples. They recommend the use of the ES test for discrete samples as
well as continuous samples with at least 25 observations each, whereas
`anderson_ksamp` is recommended for smaller sample sizes in the
continuous case.
The p-value is computed from the asymptotic distribution of the test
statistic which follows a `chi2` distribution. If the sample size of both
`x` and `y` is below 25, the small sample correction proposed in [1]_ is
applied to the test statistic.
The default values of `t` are determined in [1]_ by considering
various distributions and finding good values that lead to a high power
of the test in general. Table III in [1]_ gives the optimal values for
the distributions tested in that study. The values of `t` are scaled by
the semi-interquartile range in the implementation, see [1]_.
References
----------
.. [1] T. W. Epps and K. J. Singleton, "An omnibus test for the two-sample
problem using the empirical characteristic function", Journal of
Statistical Computation and Simulation 26, p. 177--203, 1986.
.. [2] S. J. Goerg and J. Kaiser, "Nonparametric testing of distributions
- the Epps-Singleton two-sample test using the empirical characteristic
function", The Stata Journal 9(3), p. 454--465, 2009.
"""
x, y, t = np.asarray(x), np.asarray(y), np.asarray(t)
# check if x and y are valid inputs
if x.ndim > 1:
raise ValueError('x must be 1d, but x.ndim equals {}.'.format(x.ndim))
if y.ndim > 1:
raise ValueError('y must be 1d, but y.ndim equals {}.'.format(y.ndim))
nx, ny = len(x), len(y)
if (nx < 5) or (ny < 5):
raise ValueError('x and y should have at least 5 elements, but len(x) '
'= {} and len(y) = {}.'.format(nx, ny))
if not np.isfinite(x).all():
raise ValueError('x must not contain nonfinite values.')
if not np.isfinite(y).all():
raise ValueError('y must not contain nonfinite values.')
n = nx + ny
# check if t is valid
if t.ndim > 1:
raise ValueError('t must be 1d, but t.ndim equals {}.'.format(t.ndim))
if np.less_equal(t, 0).any():
raise ValueError('t must contain positive elements only.')
# rescale t with semi-iqr as proposed in [1]; import iqr here to avoid
# circular import
from scipy.stats import iqr
sigma = iqr(np.hstack((x, y))) / 2
ts = np.reshape(t, (-1, 1)) / sigma
# covariance estimation of ES test
gx = np.vstack((np.cos(ts*x), np.sin(ts*x))).T # shape = (nx, 2*len(t))
gy = np.vstack((np.cos(ts*y), np.sin(ts*y))).T
cov_x = np.cov(gx.T, bias=True) # the test uses biased cov-estimate
cov_y = np.cov(gy.T, bias=True)
est_cov = (n/nx)*cov_x + (n/ny)*cov_y
est_cov_inv = np.linalg.pinv(est_cov)
r = np.linalg.matrix_rank(est_cov_inv)
if r < 2*len(t):
warnings.warn('Estimated covariance matrix does not have full rank. '
'This indicates a bad choice of the input t and the '
'test might not be consistent.') # see p. 183 in [1]_
# compute test statistic w distributed asympt. as chisquare with df=r
g_diff = np.mean(gx, axis=0) - np.mean(gy, axis=0)
w = n*np.dot(g_diff.T, np.dot(est_cov_inv, g_diff))
# apply small-sample correction
if (max(nx, ny) < 25):
corr = 1.0/(1.0 + n**(-0.45) + 10.1*(nx**(-1.7) + ny**(-1.7)))
w = corr * w
p = chi2.sf(w, r)
return Epps_Singleton_2sampResult(w, p)
class CramerVonMisesResult:
def __init__(self, statistic, pvalue):
self.statistic = statistic
self.pvalue = pvalue
def __repr__(self):
return (f"{self.__class__.__name__}(statistic={self.statistic}, "
f"pvalue={self.pvalue})")
def _psi1_mod(x):
"""
psi1 is defined in equation 1.10 in Csorgo, S. and Faraway, J. (1996).
This implements a modified version by excluding the term V(x) / 12
(here: _cdf_cvm_inf(x) / 12) to avoid evaluating _cdf_cvm_inf(x)
twice in _cdf_cvm.
Implementation based on MAPLE code of Julian Faraway and R code of the
function pCvM in the package goftest (v1.1.1), permission granted
by Adrian Baddeley. Main difference in the implementation: the code
here keeps adding terms of the series until the terms are small enough.
"""
def _ed2(y):
z = y**2 / 4
b = kv(1/4, z) + kv(3/4, z)
return np.exp(-z) * (y/2)**(3/2) * b / np.sqrt(np.pi)
def _ed3(y):
z = y**2 / 4
c = np.exp(-z) / np.sqrt(np.pi)
return c * (y/2)**(5/2) * (2*kv(1/4, z) + 3*kv(3/4, z) - kv(5/4, z))
def _Ak(k, x):
m = 2*k + 1
sx = 2 * np.sqrt(x)
y1 = x**(3/4)
y2 = x**(5/4)
e1 = m * gamma(k + 1/2) * _ed2((4 * k + 3)/sx) / (9 * y1)
e2 = gamma(k + 1/2) * _ed3((4 * k + 1) / sx) / (72 * y2)
e3 = 2 * (m + 2) * gamma(k + 3/2) * _ed3((4 * k + 5) / sx) / (12 * y2)
e4 = 7 * m * gamma(k + 1/2) * _ed2((4 * k + 1) / sx) / (144 * y1)
e5 = 7 * m * gamma(k + 1/2) * _ed2((4 * k + 5) / sx) / (144 * y1)
return e1 + e2 + e3 + e4 + e5
x = np.asarray(x)
tot = np.zeros_like(x, dtype='float')
cond = np.ones_like(x, dtype='bool')
k = 0
while np.any(cond):
z = -_Ak(k, x[cond]) / (np.pi * gamma(k + 1))
tot[cond] = tot[cond] + z
cond[cond] = np.abs(z) >= 1e-7
k += 1
return tot
def _cdf_cvm_inf(x):
"""
Calculate the cdf of the Cramér-von Mises statistic (infinite sample size).
See equation 1.2 in Csorgo, S. and Faraway, J. (1996).
Implementation based on MAPLE code of Julian Faraway and R code of the
function pCvM in the package goftest (v1.1.1), permission granted
by Adrian Baddeley. Main difference in the implementation: the code
here keeps adding terms of the series until the terms are small enough.
The function is not expected to be accurate for large values of x, say
x > 4, when the cdf is very close to 1.
"""
x = np.asarray(x)
def term(x, k):
# this expression can be found in [2], second line of (1.3)
u = np.exp(gammaln(k + 0.5) - gammaln(k+1)) / (np.pi**1.5 * np.sqrt(x))
y = 4*k + 1
q = y**2 / (16*x)
b = kv(0.25, q)
return u * np.sqrt(y) * np.exp(-q) * b
tot = np.zeros_like(x, dtype='float')
cond = np.ones_like(x, dtype='bool')
k = 0
while np.any(cond):
z = term(x[cond], k)
tot[cond] = tot[cond] + z
cond[cond] = np.abs(z) >= 1e-7
k += 1
return tot
def _cdf_cvm(x, n=None):
"""
Calculate the cdf of the Cramér-von Mises statistic for a finite sample
size n. If N is None, use the asymptotic cdf (n=inf).
See equation 1.8 in Csorgo, S. and Faraway, J. (1996) for finite samples,
1.2 for the asymptotic cdf.
The function is not expected to be accurate for large values of x, say
x > 2, when the cdf is very close to 1 and it might return values > 1
in that case, e.g. _cdf_cvm(2.0, 12) = 1.0000027556716846.
"""
x = np.asarray(x)
if n is None:
y = _cdf_cvm_inf(x)
else:
# support of the test statistic is [12/n, n/3], see 1.1 in [2]
y = np.zeros_like(x, dtype='float')
sup = (1./(12*n) < x) & (x < n/3.)
# note: _psi1_mod does not include the term _cdf_cvm_inf(x) / 12
# therefore, we need to add it here
y[sup] = _cdf_cvm_inf(x[sup]) * (1 + 1./(12*n)) + _psi1_mod(x[sup]) / n
y[x >= n/3] = 1
if y.ndim == 0:
return y[()]
return y
def cramervonmises(rvs, cdf, args=()):
"""Perform the one-sample Cramér-von Mises test for goodness of fit.
This performs a test of the goodness of fit of a cumulative distribution
function (cdf) :math:`F` compared to the empirical distribution function
:math:`F_n` of observed random variates :math:`X_1, ..., X_n` that are
assumed to be independent and identically distributed ([1]_).
The null hypothesis is that the :math:`X_i` have cumulative distribution
:math:`F`.
Parameters
----------
rvs : array_like
A 1-D array of observed values of the random variables :math:`X_i`.
cdf : str or callable
The cumulative distribution function :math:`F` to test the
observations against. If a string, it should be the name of a
distribution in `scipy.stats`. If a callable, that callable is used
to calculate the cdf: ``cdf(x, *args) -> float``.
args : tuple, optional
Distribution parameters. These are assumed to be known; see Notes.
Returns
-------
res : object with attributes
statistic : float
Cramér-von Mises statistic.
pvalue : float
The p-value.
See Also
--------
kstest, cramervonmises_2samp
Notes
-----
.. versionadded:: 1.6.0
The p-value relies on the approximation given by equation 1.8 in [2]_.
It is important to keep in mind that the p-value is only accurate if
one tests a simple hypothesis, i.e. the parameters of the reference
distribution are known. If the parameters are estimated from the data
(composite hypothesis), the computed p-value is not reliable.
References
----------
.. [1] Cramér-von Mises criterion, Wikipedia,
https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion
.. [2] Csorgo, S. and Faraway, J. (1996). The Exact and Asymptotic
Distribution of Cramér-von Mises Statistics. Journal of the
Royal Statistical Society, pp. 221-234.
Examples
--------
Suppose we wish to test whether data generated by ``scipy.stats.norm.rvs``
were, in fact, drawn from the standard normal distribution. We choose a
significance level of alpha=0.05.
>>> from scipy import stats
>>> rng = np.random.default_rng()
>>> x = stats.norm.rvs(size=500, random_state=rng)
>>> res = stats.cramervonmises(x, 'norm')
>>> res.statistic, res.pvalue
(0.49121480855028343, 0.04189256516661377)
The p-value 0.79 exceeds our chosen significance level, so we do not
reject the null hypothesis that the observed sample is drawn from the
standard normal distribution.
Now suppose we wish to check whether the same samples shifted by 2.1 is
consistent with being drawn from a normal distribution with a mean of 2.
>>> y = x + 2.1
>>> res = stats.cramervonmises(y, 'norm', args=(2,))
>>> res.statistic, res.pvalue
(0.07400330012187435, 0.7274595666160468)
Here we have used the `args` keyword to specify the mean (``loc``)
of the normal distribution to test the data against. This is equivalent
to the following, in which we create a frozen normal distribution with
mean 2.1, then pass its ``cdf`` method as an argument.
>>> frozen_dist = stats.norm(loc=2)
>>> res = stats.cramervonmises(y, frozen_dist.cdf)
>>> res.statistic, res.pvalue
(0.07400330012187435, 0.7274595666160468)
In either case, we would reject the null hypothesis that the observed
sample is drawn from a normal distribution with a mean of 2 (and default
variance of 1) because the p-value 0.04 is less than our chosen
significance level.
"""
if isinstance(cdf, str):
cdf = getattr(distributions, cdf).cdf
vals = np.sort(np.asarray(rvs))
if vals.size <= 1:
raise ValueError('The sample must contain at least two observations.')
if vals.ndim > 1:
raise ValueError('The sample must be one-dimensional.')
n = len(vals)
cdfvals = cdf(vals, *args)
u = (2*np.arange(1, n+1) - 1)/(2*n)
w = 1/(12*n) + np.sum((u - cdfvals)**2)
# avoid small negative values that can occur due to the approximation
p = max(0, 1. - _cdf_cvm(w, n))
return CramerVonMisesResult(statistic=w, pvalue=p)
def _get_wilcoxon_distr(n):
"""
Distribution of counts of the Wilcoxon ranksum statistic r_plus (sum of
ranks of positive differences).
Returns an array with the counts/frequencies of all the possible ranks
r = 0, ..., n*(n+1)/2
"""
cnt = _wilcoxon_data.COUNTS.get(n)
if cnt is None:
raise ValueError("The exact distribution of the Wilcoxon test "
"statistic is not implemented for n={}".format(n))
return np.array(cnt, dtype=int)
def _tau_b(A):
"""Calculate Kendall's tau-b and p-value from contingency table."""
# See [2] 2.2 and 4.2
# contingency table must be truly 2D
if A.shape[0] == 1 or A.shape[1] == 1:
return np.nan, np.nan
NA = A.sum()
PA = _P(A)
QA = _Q(A)
Sri2 = (A.sum(axis=1)**2).sum()
Scj2 = (A.sum(axis=0)**2).sum()
denominator = (NA**2 - Sri2)*(NA**2 - Scj2)
tau = (PA-QA)/(denominator)**0.5
numerator = 4*(_a_ij_Aij_Dij2(A) - (PA - QA)**2 / NA)
s02_tau_b = numerator/denominator
if s02_tau_b == 0: # Avoid divide by zero
return tau, 0
Z = tau/s02_tau_b**0.5
p = 2*norm.sf(abs(Z)) # 2-sided p-value
return tau, p
def _somers_d(A, alternative='two-sided'):
"""Calculate Somers' D and p-value from contingency table."""
# See [3] page 1740
# contingency table must be truly 2D
if A.shape[0] <= 1 or A.shape[1] <= 1:
return np.nan, np.nan
NA = A.sum()
NA2 = NA**2
PA = _P(A)
QA = _Q(A)
Sri2 = (A.sum(axis=1)**2).sum()
d = (PA - QA)/(NA2 - Sri2)
S = _a_ij_Aij_Dij2(A) - (PA-QA)**2/NA
with np.errstate(divide='ignore'):
Z = (PA - QA)/(4*(S))**0.5
_, p = scipy.stats._stats_py._normtest_finish(Z, alternative)
return d, p
SomersDResult = make_dataclass("SomersDResult",
("statistic", "pvalue", "table"))
def somersd(x, y=None, alternative='two-sided'):
r"""Calculates Somers' D, an asymmetric measure of ordinal association.
Like Kendall's :math:`\tau`, Somers' :math:`D` is a measure of the
correspondence between two rankings. Both statistics consider the
difference between the number of concordant and discordant pairs in two
rankings :math:`X` and :math:`Y`, and both are normalized such that values
close to 1 indicate strong agreement and values close to -1 indicate
strong disagreement. They differ in how they are normalized. To show the
relationship, Somers' :math:`D` can be defined in terms of Kendall's
:math:`\tau_a`:
.. math::
D(Y|X) = \frac{\tau_a(X, Y)}{\tau_a(X, X)}
Suppose the first ranking :math:`X` has :math:`r` distinct ranks and the
second ranking :math:`Y` has :math:`s` distinct ranks. These two lists of
:math:`n` rankings can also be viewed as an :math:`r \times s` contingency
table in which element :math:`i, j` is the number of rank pairs with rank
:math:`i` in ranking :math:`X` and rank :math:`j` in ranking :math:`Y`.
Accordingly, `somersd` also allows the input data to be supplied as a
single, 2D contingency table instead of as two separate, 1D rankings.
Note that the definition of Somers' :math:`D` is asymmetric: in general,
:math:`D(Y|X) \neq D(X|Y)`. ``somersd(x, y)`` calculates Somers'
:math:`D(Y|X)`: the "row" variable :math:`X` is treated as an independent
variable, and the "column" variable :math:`Y` is dependent. For Somers'
:math:`D(X|Y)`, swap the input lists or transpose the input table.
Parameters
----------
x: array_like
1D array of rankings, treated as the (row) independent variable.
Alternatively, a 2D contingency table.
y: array_like, optional
If `x` is a 1D array of rankings, `y` is a 1D array of rankings of the
same length, treated as the (column) dependent variable.
If `x` is 2D, `y` is ignored.
alternative : {'two-sided', 'less', 'greater'}, optional
Defines the alternative hypothesis. Default is 'two-sided'.
The following options are available:
* 'two-sided': the rank correlation is nonzero
* 'less': the rank correlation is negative (less than zero)
* 'greater': the rank correlation is positive (greater than zero)
Returns
-------
res : SomersDResult
A `SomersDResult` object with the following fields:
correlation : float
The Somers' :math:`D` statistic.
pvalue : float
The p-value for a hypothesis test whose null
hypothesis is an absence of association, :math:`D=0`.
See notes for more information.
table : 2D array
The contingency table formed from rankings `x` and `y` (or the
provided contingency table, if `x` is a 2D array)
See Also
--------
kendalltau : Calculates Kendall's tau, another correlation measure.
weightedtau : Computes a weighted version of Kendall's tau.
spearmanr : Calculates a Spearman rank-order correlation coefficient.
pearsonr : Calculates a Pearson correlation coefficient.
Notes
-----
This function follows the contingency table approach of [2]_ and
[3]_. *p*-values are computed based on an asymptotic approximation of
the test statistic distribution under the null hypothesis :math:`D=0`.
Theoretically, hypothesis tests based on Kendall's :math:`tau` and Somers'
:math:`D` should be identical.
However, the *p*-values returned by `kendalltau` are based
on the null hypothesis of *independence* between :math:`X` and :math:`Y`
(i.e. the population from which pairs in :math:`X` and :math:`Y` are
sampled contains equal numbers of all possible pairs), which is more
specific than the null hypothesis :math:`D=0` used here. If the null
hypothesis of independence is desired, it is acceptable to use the
*p*-value returned by `kendalltau` with the statistic returned by
`somersd` and vice versa. For more information, see [2]_.
Contingency tables are formatted according to the convention used by
SAS and R: the first ranking supplied (``x``) is the "row" variable, and
the second ranking supplied (``y``) is the "column" variable. This is
opposite the convention of Somers' original paper [1]_.
References
----------
.. [1] Robert H. Somers, "A New Asymmetric Measure of Association for
Ordinal Variables", *American Sociological Review*, Vol. 27, No. 6,
pp. 799--811, 1962.
.. [2] Morton B. Brown and Jacqueline K. Benedetti, "Sampling Behavior of
Tests for Correlation in Two-Way Contingency Tables", *Journal of
the American Statistical Association* Vol. 72, No. 358, pp.
309--315, 1977.
.. [3] SAS Institute, Inc., "The FREQ Procedure (Book Excerpt)",
*SAS/STAT 9.2 User's Guide, Second Edition*, SAS Publishing, 2009.
.. [4] Laerd Statistics, "Somers' d using SPSS Statistics", *SPSS
Statistics Tutorials and Statistical Guides*,
https://statistics.laerd.com/spss-tutorials/somers-d-using-spss-statistics.php,
Accessed July 31, 2020.
Examples
--------
We calculate Somers' D for the example given in [4]_, in which a hotel
chain owner seeks to determine the association between hotel room
cleanliness and customer satisfaction. The independent variable, hotel
room cleanliness, is ranked on an ordinal scale: "below average (1)",
"average (2)", or "above average (3)". The dependent variable, customer
satisfaction, is ranked on a second scale: "very dissatisfied (1)",
"moderately dissatisfied (2)", "neither dissatisfied nor satisfied (3)",
"moderately satisfied (4)", or "very satisfied (5)". 189 customers
respond to the survey, and the results are cast into a contingency table
with the hotel room cleanliness as the "row" variable and customer
satisfaction as the "column" variable.
+-----+-----+-----+-----+-----+-----+
| | (1) | (2) | (3) | (4) | (5) |
+=====+=====+=====+=====+=====+=====+
| (1) | 27 | 25 | 14 | 7 | 0 |
+-----+-----+-----+-----+-----+-----+
| (2) | 7 | 14 | 18 | 35 | 12 |
+-----+-----+-----+-----+-----+-----+
| (3) | 1 | 3 | 2 | 7 | 17 |
+-----+-----+-----+-----+-----+-----+
For example, 27 customers assigned their room a cleanliness ranking of
"below average (1)" and a corresponding satisfaction of "very
dissatisfied (1)". We perform the analysis as follows.
>>> from scipy.stats import somersd
>>> table = [[27, 25, 14, 7, 0], [7, 14, 18, 35, 12], [1, 3, 2, 7, 17]]
>>> res = somersd(table)
>>> res.statistic
0.6032766111513396
>>> res.pvalue
1.0007091191074533e-27
The value of the Somers' D statistic is approximately 0.6, indicating
a positive correlation between room cleanliness and customer satisfaction
in the sample.
The *p*-value is very small, indicating a very small probability of
observing such an extreme value of the statistic under the null
hypothesis that the statistic of the entire population (from which
our sample of 189 customers is drawn) is zero. This supports the
alternative hypothesis that the true value of Somers' D for the population
is nonzero.
"""
x, y = np.array(x), np.array(y)
if x.ndim == 1:
if x.size != y.size:
raise ValueError("Rankings must be of equal length.")
table = scipy.stats.contingency.crosstab(x, y)[1]
elif x.ndim == 2:
if np.any(x < 0):
raise ValueError("All elements of the contingency table must be "
"non-negative.")
if np.any(x != x.astype(int)):
raise ValueError("All elements of the contingency table must be "
"integer.")
if x.nonzero()[0].size < 2:
raise ValueError("At least two elements of the contingency table "
"must be nonzero.")
table = x
else:
raise ValueError("x must be either a 1D or 2D array")
d, p = _somers_d(table, alternative)
return SomersDResult(d, p, table)
def _all_partitions(nx, ny):
"""
Partition a set of indices into two fixed-length sets in all possible ways
Partition a set of indices 0 ... nx + ny - 1 into two sets of length nx and
ny in all possible ways (ignoring order of elements).
"""
z = np.arange(nx+ny)
for c in combinations(z, nx):
x = np.array(c)
mask = np.ones(nx+ny, bool)
mask[x] = False
y = z[mask]
yield x, y
def _compute_log_combinations(n):
"""Compute all log combination of C(n, k)."""
gammaln_arr = gammaln(np.arange(n + 1) + 1)
return gammaln(n + 1) - gammaln_arr - gammaln_arr[::-1]
BarnardExactResult = make_dataclass(
"BarnardExactResult", [("statistic", float), ("pvalue", float)]
)
def barnard_exact(table, alternative="two-sided", pooled=True, n=32):
r"""Perform a Barnard exact test on a 2x2 contingency table.
Parameters
----------
table : array_like of ints
A 2x2 contingency table. Elements should be non-negative integers.
alternative : {'two-sided', 'less', 'greater'}, optional
Defines the null and alternative hypotheses. Default is 'two-sided'.
Please see explanations in the Notes section below.
pooled : bool, optional
Whether to compute score statistic with pooled variance (as in
Student's t-test, for example) or unpooled variance (as in Welch's
t-test). Default is ``True``.
n : int, optional
Number of sampling points used in the construction of the sampling
method. Note that this argument will automatically be converted to
the next higher power of 2 since `scipy.stats.qmc.Sobol` is used to
select sample points. Default is 32. Must be positive. In most cases,
32 points is enough to reach good precision. More points comes at
performance cost.
Returns
-------
ber : BarnardExactResult
A result object with the following attributes.
statistic : float
The Wald statistic with pooled or unpooled variance, depending
on the user choice of `pooled`.
pvalue : float
P-value, the probability of obtaining a distribution at least as
extreme as the one that was actually observed, assuming that the
null hypothesis is true.
See Also
--------
chi2_contingency : Chi-square test of independence of variables in a
contingency table.
fisher_exact : Fisher exact test on a 2x2 contingency table.
boschloo_exact : Boschloo's exact test on a 2x2 contingency table,
which is an uniformly more powerful alternative to Fisher's exact test.
Notes
-----
Barnard's test is an exact test used in the analysis of contingency
tables. It examines the association of two categorical variables, and
is a more powerful alternative than Fisher's exact test
for 2x2 contingency tables.
Let's define :math:`X_0` a 2x2 matrix representing the observed sample,
where each column stores the binomial experiment, as in the example
below. Let's also define :math:`p_1, p_2` the theoretical binomial
probabilities for :math:`x_{11}` and :math:`x_{12}`. When using
Barnard exact test, we can assert three different null hypotheses :
- :math:`H_0 : p_1 \geq p_2` versus :math:`H_1 : p_1 < p_2`,
with `alternative` = "less"
- :math:`H_0 : p_1 \leq p_2` versus :math:`H_1 : p_1 > p_2`,
with `alternative` = "greater"
- :math:`H_0 : p_1 = p_2` versus :math:`H_1 : p_1 \neq p_2`,
with `alternative` = "two-sided" (default one)
In order to compute Barnard's exact test, we are using the Wald
statistic [3]_ with pooled or unpooled variance.
Under the default assumption that both variances are equal
(``pooled = True``), the statistic is computed as:
.. math::
T(X) = \frac{
\hat{p}_1 - \hat{p}_2
}{
\sqrt{
\hat{p}(1 - \hat{p})
(\frac{1}{c_1} +
\frac{1}{c_2})
}
}
with :math:`\hat{p}_1, \hat{p}_2` and :math:`\hat{p}` the estimator of
:math:`p_1, p_2` and :math:`p`, the latter being the combined probability,
given the assumption that :math:`p_1 = p_2`.
If this assumption is invalid (``pooled = False``), the statistic is:
.. math::
T(X) = \frac{
\hat{p}_1 - \hat{p}_2
}{
\sqrt{
\frac{\hat{p}_1 (1 - \hat{p}_1)}{c_1} +
\frac{\hat{p}_2 (1 - \hat{p}_2)}{c_2}
}
}
The p-value is then computed as:
.. math::
\sum
\binom{c_1}{x_{11}}
\binom{c_2}{x_{12}}
\pi^{x_{11} + x_{12}}
(1 - \pi)^{t - x_{11} - x_{12}}
where the sum is over all 2x2 contingency tables :math:`X` such that:
* :math:`T(X) \leq T(X_0)` when `alternative` = "less",
* :math:`T(X) \geq T(X_0)` when `alternative` = "greater", or
* :math:`T(X) \geq |T(X_0)|` when `alternative` = "two-sided".
Above, :math:`c_1, c_2` are the sum of the columns 1 and 2,
and :math:`t` the total (sum of the 4 sample's element).
The returned p-value is the maximum p-value taken over the nuisance
parameter :math:`\pi`, where :math:`0 \leq \pi \leq 1`.
This function's complexity is :math:`O(n c_1 c_2)`, where `n` is the
number of sample points.
References
----------
.. [1] Barnard, G. A. "Significance Tests for 2x2 Tables". *Biometrika*.
34.1/2 (1947): 123-138. :doi:`dpgkg3`
.. [2] Mehta, Cyrus R., and Pralay Senchaudhuri. "Conditional versus
unconditional exact tests for comparing two binomials."
*Cytel Software Corporation* 675 (2003): 1-5.
.. [3] "Wald Test". *Wikipedia*. https://en.wikipedia.org/wiki/Wald_test
Examples
--------
An example use of Barnard's test is presented in [2]_.
Consider the following example of a vaccine efficacy study
(Chan, 1998). In a randomized clinical trial of 30 subjects, 15 were
inoculated with a recombinant DNA influenza vaccine and the 15 were
inoculated with a placebo. Twelve of the 15 subjects in the placebo
group (80%) eventually became infected with influenza whereas for the
vaccine group, only 7 of the 15 subjects (47%) became infected. The
data are tabulated as a 2 x 2 table::
Vaccine Placebo
Yes 7 12
No 8 3
When working with statistical hypothesis testing, we usually use a
threshold probability or significance level upon which we decide
to reject the null hypothesis :math:`H_0`. Suppose we choose the common
significance level of 5%.
Our alternative hypothesis is that the vaccine will lower the chance of
becoming infected with the virus; that is, the probability :math:`p_1` of
catching the virus with the vaccine will be *less than* the probability
:math:`p_2` of catching the virus without the vaccine. Therefore, we call
`barnard_exact` with the ``alternative="less"`` option:
>>> import scipy.stats as stats
>>> res = stats.barnard_exact([[7, 12], [8, 3]], alternative="less")
>>> res.statistic
-1.894...
>>> res.pvalue
0.03407...
Under the null hypothesis that the vaccine will not lower the chance of
becoming infected, the probability of obtaining test results at least as
extreme as the observed data is approximately 3.4%. Since this p-value is
less than our chosen significance level, we have evidence to reject
:math:`H_0` in favor of the alternative.
Suppose we had used Fisher's exact test instead:
>>> _, pvalue = stats.fisher_exact([[7, 12], [8, 3]], alternative="less")
>>> pvalue
0.0640...
With the same threshold significance of 5%, we would not have been able
to reject the null hypothesis in favor of the alternative. As stated in
[2]_, Barnard's test is uniformly more powerful than Fisher's exact test
because Barnard's test does not condition on any margin. Fisher's test
should only be used when both sets of marginals are fixed.
"""
if n <= 0:
raise ValueError(
"Number of points `n` must be strictly positive, "
f"found {n!r}"
)
table = np.asarray(table, dtype=np.int64)
if not table.shape == (2, 2):
raise ValueError("The input `table` must be of shape (2, 2).")
if np.any(table < 0):
raise ValueError("All values in `table` must be nonnegative.")
if 0 in table.sum(axis=0):
# If both values in column are zero, the p-value is 1 and
# the score's statistic is NaN.
return BarnardExactResult(np.nan, 1.0)
total_col_1, total_col_2 = table.sum(axis=0)
x1 = np.arange(total_col_1 + 1, dtype=np.int64).reshape(-1, 1)
x2 = np.arange(total_col_2 + 1, dtype=np.int64).reshape(1, -1)
# We need to calculate the wald statistics for each combination of x1 and
# x2.
p1, p2 = x1 / total_col_1, x2 / total_col_2
if pooled:
p = (x1 + x2) / (total_col_1 + total_col_2)
variances = p * (1 - p) * (1 / total_col_1 + 1 / total_col_2)
else:
variances = p1 * (1 - p1) / total_col_1 + p2 * (1 - p2) / total_col_2
# To avoid warning when dividing by 0
with np.errstate(divide="ignore", invalid="ignore"):
wald_statistic = np.divide((p1 - p2), np.sqrt(variances))
wald_statistic[p1 == p2] = 0 # Removing NaN values
wald_stat_obs = wald_statistic[table[0, 0], table[0, 1]]
if alternative == "two-sided":
index_arr = np.abs(wald_statistic) >= abs(wald_stat_obs)
elif alternative == "less":
index_arr = wald_statistic <= wald_stat_obs
elif alternative == "greater":
index_arr = wald_statistic >= wald_stat_obs
else:
msg = (
"`alternative` should be one of {'two-sided', 'less', 'greater'},"
f" found {alternative!r}"
)
raise ValueError(msg)
x1_sum_x2 = x1 + x2
x1_log_comb = _compute_log_combinations(total_col_1)
x2_log_comb = _compute_log_combinations(total_col_2)
x1_sum_x2_log_comb = x1_log_comb[x1] + x2_log_comb[x2]
result = shgo(
_get_binomial_log_p_value_with_nuisance_param,
args=(x1_sum_x2, x1_sum_x2_log_comb, index_arr),
bounds=((0, 1),),
n=n,
sampling_method="sobol",
)
# result.fun is the negative log pvalue and therefore needs to be
# changed before return
p_value = np.clip(np.exp(-result.fun), a_min=0, a_max=1)
return BarnardExactResult(wald_stat_obs, p_value)
BoschlooExactResult = make_dataclass(
"BoschlooExactResult", [("statistic", float), ("pvalue", float)]
)
def boschloo_exact(table, alternative="two-sided", n=32):
r"""Perform Boschloo's exact test on a 2x2 contingency table.
Parameters
----------
table : array_like of ints
A 2x2 contingency table. Elements should be non-negative integers.
alternative : {'two-sided', 'less', 'greater'}, optional
Defines the null and alternative hypotheses. Default is 'two-sided'.
Please see explanations in the Notes section below.
n : int, optional
Number of sampling points used in the construction of the sampling
method. Note that this argument will automatically be converted to
the next higher power of 2 since `scipy.stats.qmc.Sobol` is used to
select sample points. Default is 32. Must be positive. In most cases,
32 points is enough to reach good precision. More points comes at
performance cost.
Returns
-------
ber : BoschlooExactResult
A result object with the following attributes.
statistic : float
The statistic used in Boschloo's test; that is, the p-value
from Fisher's exact test.
pvalue : float
P-value, the probability of obtaining a distribution at least as
extreme as the one that was actually observed, assuming that the
null hypothesis is true.
See Also
--------
chi2_contingency : Chi-square test of independence of variables in a
contingency table.
fisher_exact : Fisher exact test on a 2x2 contingency table.
barnard_exact : Barnard's exact test, which is a more powerful alternative
than Fisher's exact test for 2x2 contingency tables.
Notes
-----
Boschloo's test is an exact test used in the analysis of contingency
tables. It examines the association of two categorical variables, and
is a uniformly more powerful alternative to Fisher's exact test
for 2x2 contingency tables.
Let's define :math:`X_0` a 2x2 matrix representing the observed sample,
where each column stores the binomial experiment, as in the example
below. Let's also define :math:`p_1, p_2` the theoretical binomial
probabilities for :math:`x_{11}` and :math:`x_{12}`. When using
Boschloo exact test, we can assert three different null hypotheses :
- :math:`H_0 : p_1=p_2` versus :math:`H_1 : p_1 < p_2`,
with `alternative` = "less"
- :math:`H_0 : p_1=p_2` versus :math:`H_1 : p_1 > p_2`,
with `alternative` = "greater"
- :math:`H_0 : p_1=p_2` versus :math:`H_1 : p_1 \neq p_2`,
with `alternative` = "two-sided" (default one)
Boschloo's exact test uses the p-value of Fisher's exact test as a
statistic, and Boschloo's p-value is the probability under the null
hypothesis of observing such an extreme value of this statistic.
Boschloo's and Barnard's are both more powerful than Fisher's exact
test.
.. versionadded:: 1.7.0
References
----------
.. [1] R.D. Boschloo. "Raised conditional level of significance for the
2 x 2-table when testing the equality of two probabilities",
Statistica Neerlandica, 24(1), 1970
.. [2] "Boschloo's test", Wikipedia,
https://en.wikipedia.org/wiki/Boschloo%27s_test
.. [3] Lise M. Saari et al. "Employee attitudes and job satisfaction",
Human Resource Management, 43(4), 395-407, 2004,
:doi:`10.1002/hrm.20032`.
Examples
--------
In the following example, we consider the article "Employee
attitudes and job satisfaction" [3]_
which reports the results of a survey from 63 scientists and 117 college
professors. Of the 63 scientists, 31 said they were very satisfied with
their jobs, whereas 74 of the college professors were very satisfied
with their work. Is this significant evidence that college
professors are happier with their work than scientists?
The following table summarizes the data mentioned above::
college professors scientists
Very Satisfied 74 31
Dissatisfied 43 32
When working with statistical hypothesis testing, we usually use a
threshold probability or significance level upon which we decide
to reject the null hypothesis :math:`H_0`. Suppose we choose the common
significance level of 5%.
Our alternative hypothesis is that college professors are truly more
satisfied with their work than scientists. Therefore, we expect
:math:`p_1` the proportion of very satisfied college professors to be
greater than :math:`p_2`, the proportion of very satisfied scientists.
We thus call `boschloo_exact` with the ``alternative="greater"`` option:
>>> import scipy.stats as stats
>>> res = stats.boschloo_exact([[74, 31], [43, 32]], alternative="greater")
>>> res.statistic
0.0483...
>>> res.pvalue
0.0355...
Under the null hypothesis that scientists are happier in their work than
college professors, the probability of obtaining test
results at least as extreme as the observed data is approximately 3.55%.
Since this p-value is less than our chosen significance level, we have
evidence to reject :math:`H_0` in favor of the alternative hypothesis.
"""
hypergeom = distributions.hypergeom
if n <= 0:
raise ValueError(
"Number of points `n` must be strictly positive,"
f" found {n!r}"
)
table = np.asarray(table, dtype=np.int64)
if not table.shape == (2, 2):
raise ValueError("The input `table` must be of shape (2, 2).")
if np.any(table < 0):
raise ValueError("All values in `table` must be nonnegative.")
if 0 in table.sum(axis=0):
# If both values in column are zero, the p-value is 1 and
# the score's statistic is NaN.
return BoschlooExactResult(np.nan, np.nan)
total_col_1, total_col_2 = table.sum(axis=0)
total = total_col_1 + total_col_2
x1 = np.arange(total_col_1 + 1, dtype=np.int64).reshape(1, -1)
x2 = np.arange(total_col_2 + 1, dtype=np.int64).reshape(-1, 1)
x1_sum_x2 = x1 + x2
if alternative == 'less':
pvalues = hypergeom.cdf(x1, total, x1_sum_x2, total_col_1).T
elif alternative == 'greater':
# Same formula as the 'less' case, but with the second column.
pvalues = hypergeom.cdf(x2, total, x1_sum_x2, total_col_2).T
elif alternative == 'two-sided':
boschloo_less = boschloo_exact(table, alternative="less", n=n)
boschloo_greater = boschloo_exact(table, alternative="greater", n=n)
res = (
boschloo_less if boschloo_less.pvalue < boschloo_greater.pvalue
else boschloo_greater
)
# Two-sided p-value is defined as twice the minimum of the one-sided
# p-values
pvalue = 2 * res.pvalue
return BoschlooExactResult(res.statistic, pvalue)
else:
msg = (
f"`alternative` should be one of {'two-sided', 'less', 'greater'},"
f" found {alternative!r}"
)
raise ValueError(msg)
fisher_stat = pvalues[table[0, 0], table[0, 1]]
# fisher_stat * (1+1e-13) guards us from small numerical error. It is
# equivalent to np.isclose with relative tol of 1e-13 and absolute tol of 0
# For more throughout explanations, see gh-14178
index_arr = pvalues <= fisher_stat * (1+1e-13)
x1, x2, x1_sum_x2 = x1.T, x2.T, x1_sum_x2.T
x1_log_comb = _compute_log_combinations(total_col_1)
x2_log_comb = _compute_log_combinations(total_col_2)
x1_sum_x2_log_comb = x1_log_comb[x1] + x2_log_comb[x2]
result = shgo(
_get_binomial_log_p_value_with_nuisance_param,
args=(x1_sum_x2, x1_sum_x2_log_comb, index_arr),
bounds=((0, 1),),
n=n,
sampling_method="sobol",
)
# result.fun is the negative log pvalue and therefore needs to be
# changed before return
p_value = np.clip(np.exp(-result.fun), a_min=0, a_max=1)
return BoschlooExactResult(fisher_stat, p_value)
def _get_binomial_log_p_value_with_nuisance_param(
nuisance_param, x1_sum_x2, x1_sum_x2_log_comb, index_arr
):
r"""
Compute the log pvalue in respect of a nuisance parameter considering
a 2x2 sample space.
Parameters
----------
nuisance_param : float
nuisance parameter used in the computation of the maximisation of
the p-value. Must be between 0 and 1
x1_sum_x2 : ndarray
Sum of x1 and x2 inside barnard_exact
x1_sum_x2_log_comb : ndarray
sum of the log combination of x1 and x2
index_arr : ndarray of boolean
Returns
-------
p_value : float
Return the maximum p-value considering every nuisance paramater
between 0 and 1
Notes
-----
Both Barnard's test and Boschloo's test iterate over a nuisance parameter
:math:`\pi \in [0, 1]` to find the maximum p-value. To search this
maxima, this function return the negative log pvalue with respect to the
nuisance parameter passed in params. This negative log p-value is then
used in `shgo` to find the minimum negative pvalue which is our maximum
pvalue.
Also, to compute the different combination used in the
p-values' computation formula, this function uses `gammaln` which is
more tolerant for large value than `scipy.special.comb`. `gammaln` gives
a log combination. For the little precision loss, performances are
improved a lot.
"""
t1, t2 = x1_sum_x2.shape
n = t1 + t2 - 2
with np.errstate(divide="ignore", invalid="ignore"):
log_nuisance = np.log(
nuisance_param,
out=np.zeros_like(nuisance_param),
where=nuisance_param >= 0,
)
log_1_minus_nuisance = np.log(
1 - nuisance_param,
out=np.zeros_like(nuisance_param),
where=1 - nuisance_param >= 0,
)
nuisance_power_x1_x2 = log_nuisance * x1_sum_x2
nuisance_power_x1_x2[(x1_sum_x2 == 0)[:, :]] = 0
nuisance_power_n_minus_x1_x2 = log_1_minus_nuisance * (n - x1_sum_x2)
nuisance_power_n_minus_x1_x2[(x1_sum_x2 == n)[:, :]] = 0
tmp_log_values_arr = (
x1_sum_x2_log_comb
+ nuisance_power_x1_x2
+ nuisance_power_n_minus_x1_x2
)
tmp_values_from_index = tmp_log_values_arr[index_arr]
# To avoid dividing by zero in log function and getting inf value,
# values are centered according to the max
max_value = tmp_values_from_index.max()
# To have better result's precision, the log pvalue is taken here.
# Indeed, pvalue is included inside [0, 1] interval. Passing the
# pvalue to log makes the interval a lot bigger ([-inf, 0]), and thus
# help us to achieve better precision
with np.errstate(divide="ignore", invalid="ignore"):
log_probs = np.exp(tmp_values_from_index - max_value).sum()
log_pvalue = max_value + np.log(
log_probs,
out=np.full_like(log_probs, -np.inf),
where=log_probs > 0,
)
# Since shgo find the minima, minus log pvalue is returned
return -log_pvalue
def _pval_cvm_2samp_exact(s, nx, ny):
"""
Compute the exact p-value of the Cramer-von Mises two-sample test
for a given value s (float) of the test statistic by enumerating
all possible combinations. nx and ny are the sizes of the samples.
"""
rangex = np.arange(nx)
rangey = np.arange(ny)
us = []
# x and y are all possible partitions of ranks from 0 to nx + ny - 1
# into two sets of length nx and ny
# Here, ranks are from 0 to nx + ny - 1 instead of 1 to nx + ny, but
# this does not change the value of the statistic.
for x, y in _all_partitions(nx, ny):
# compute the statistic
u = nx * np.sum((x - rangex)**2)
u += ny * np.sum((y - rangey)**2)
us.append(u)
# compute the values of u and the frequencies
u, cnt = np.unique(us, return_counts=True)
return np.sum(cnt[u >= s]) / np.sum(cnt)
def cramervonmises_2samp(x, y, method='auto'):
"""Perform the two-sample Cramér-von Mises test for goodness of fit.
This is the two-sample version of the Cramér-von Mises test ([1]_):
for two independent samples :math:`X_1, ..., X_n` and
:math:`Y_1, ..., Y_m`, the null hypothesis is that the samples
come from the same (unspecified) continuous distribution.
Parameters
----------
x : array_like
A 1-D array of observed values of the random variables :math:`X_i`.
y : array_like
A 1-D array of observed values of the random variables :math:`Y_i`.
method : {'auto', 'asymptotic', 'exact'}, optional
The method used to compute the p-value, see Notes for details.
The default is 'auto'.
Returns
-------
res : object with attributes
statistic : float
Cramér-von Mises statistic.
pvalue : float
The p-value.
See Also
--------
cramervonmises, anderson_ksamp, epps_singleton_2samp, ks_2samp
Notes
-----
.. versionadded:: 1.7.0
The statistic is computed according to equation 9 in [2]_. The
calculation of the p-value depends on the keyword `method`:
- ``asymptotic``: The p-value is approximated by using the limiting
distribution of the test statistic.
- ``exact``: The exact p-value is computed by enumerating all
possible combinations of the test statistic, see [2]_.
The exact calculation will be very slow even for moderate sample
sizes as the number of combinations increases rapidly with the
size of the samples. If ``method=='auto'``, the exact approach
is used if both samples contain less than 10 observations,
otherwise the asymptotic distribution is used.
If the underlying distribution is not continuous, the p-value is likely to
be conservative (Section 6.2 in [3]_). When ranking the data to compute
the test statistic, midranks are used if there are ties.
References
----------
.. [1] https://en.wikipedia.org/wiki/Cramer-von_Mises_criterion
.. [2] Anderson, T.W. (1962). On the distribution of the two-sample
Cramer-von-Mises criterion. The Annals of Mathematical
Statistics, pp. 1148-1159.
.. [3] Conover, W.J., Practical Nonparametric Statistics, 1971.
Examples
--------
Suppose we wish to test whether two samples generated by
``scipy.stats.norm.rvs`` have the same distribution. We choose a
significance level of alpha=0.05.
>>> from scipy import stats
>>> rng = np.random.default_rng()
>>> x = stats.norm.rvs(size=100, random_state=rng)
>>> y = stats.norm.rvs(size=70, random_state=rng)
>>> res = stats.cramervonmises_2samp(x, y)
>>> res.statistic, res.pvalue
(0.29376470588235293, 0.1412873014573014)
The p-value exceeds our chosen significance level, so we do not
reject the null hypothesis that the observed samples are drawn from the
same distribution.
For small sample sizes, one can compute the exact p-values:
>>> x = stats.norm.rvs(size=7, random_state=rng)
>>> y = stats.t.rvs(df=2, size=6, random_state=rng)
>>> res = stats.cramervonmises_2samp(x, y, method='exact')
>>> res.statistic, res.pvalue
(0.197802197802198, 0.31643356643356646)
The p-value based on the asymptotic distribution is a good approximation
even though the sample size is small.
>>> res = stats.cramervonmises_2samp(x, y, method='asymptotic')
>>> res.statistic, res.pvalue
(0.197802197802198, 0.2966041181527128)
Independent of the method, one would not reject the null hypothesis at the
chosen significance level in this example.
"""
xa = np.sort(np.asarray(x))
ya = np.sort(np.asarray(y))
if xa.size <= 1 or ya.size <= 1:
raise ValueError('x and y must contain at least two observations.')
if xa.ndim > 1 or ya.ndim > 1:
raise ValueError('The samples must be one-dimensional.')
if method not in ['auto', 'exact', 'asymptotic']:
raise ValueError('method must be either auto, exact or asymptotic.')
nx = len(xa)
ny = len(ya)
if method == 'auto':
if max(nx, ny) > 10:
method = 'asymptotic'
else:
method = 'exact'
# get ranks of x and y in the pooled sample
z = np.concatenate([xa, ya])
# in case of ties, use midrank (see [1])
r = scipy.stats.rankdata(z, method='average')
rx = r[:nx]
ry = r[nx:]
# compute U (eq. 10 in [2])
u = nx * np.sum((rx - np.arange(1, nx+1))**2)
u += ny * np.sum((ry - np.arange(1, ny+1))**2)
# compute T (eq. 9 in [2])
k, N = nx*ny, nx + ny
t = u / (k*N) - (4*k - 1)/(6*N)
if method == 'exact':
p = _pval_cvm_2samp_exact(u, nx, ny)
else:
# compute expected value and variance of T (eq. 11 and 14 in [2])
et = (1 + 1/N)/6
vt = (N+1) * (4*k*N - 3*(nx**2 + ny**2) - 2*k)
vt = vt / (45 * N**2 * 4 * k)
# computed the normalized statistic (eq. 15 in [2])
tn = 1/6 + (t - et) / np.sqrt(45 * vt)
# approximate distribution of tn with limiting distribution
# of the one-sample test statistic
# if tn < 0.003, the _cdf_cvm_inf(tn) < 1.28*1e-18, return 1.0 directly
if tn < 0.003:
p = 1.0
else:
p = max(0, 1. - _cdf_cvm_inf(tn))
return CramerVonMisesResult(statistic=t, pvalue=p)
attributes = ('statistic', 'pvalue', 'null_distribution')
PermutationTestResult = make_dataclass('PermutationTestResult', attributes)
def _broadcast_arrays(arrays, axis=None):
"""
Broadcast shapes of arrays, ignoring incompatibility of specified axes
"""
new_shapes = _broadcast_array_shapes(arrays, axis=axis)
if axis is None:
new_shapes = [new_shapes]*len(arrays)
return [np.broadcast_to(array, new_shape)
for array, new_shape in zip(arrays, new_shapes)]
def _broadcast_array_shapes(arrays, axis=None):
"""
Broadcast shapes of arrays, ignoring incompatibility of specified axes
"""
shapes = [np.asarray(arr).shape for arr in arrays]
return _broadcast_shapes(shapes, axis)
def _broadcast_shapes(shapes, axis=None):
"""
Broadcast shapes, ignoring incompatibility of specified axes
"""
# First, ensure all shapes have same number of dimensions by prepending 1s.
n_dims = max([len(shape) for shape in shapes])
new_shapes = np.ones((len(shapes), n_dims), dtype=int)
for row, shape in zip(new_shapes, shapes):
row[len(row)-len(shape):] = shape # can't use negative indices (-0:)
# Remove the shape elements of the axes to be ignored, but remember them.
if axis is not None:
axis = np.atleast_1d(axis)
axis[axis < 0] = n_dims + axis[axis < 0]
removed_shapes = new_shapes[:, axis]
new_shapes = np.delete(new_shapes, axis, axis=1)
# If arrays are broadcastable, shape elements that are 1 may be replaced
# with a corresponding non-1 shape element. Assuming arrays are
# broadcastable, that final shape element can be found with:
new_shape = np.max(new_shapes, axis=0)
# except in case of an empty array:
new_shape *= new_shapes.all(axis=0)
# Among all arrays, there can only be one unique non-1 shape element.
# Therefore, if any non-1 shape element does not match what we found
# above, the arrays must not be broadcastable after all.
if np.any(~((new_shapes == 1) | (new_shapes == new_shape))):
raise ValueError("Array shapes are incompatible for broadcasting.")
if axis is not None:
# Add back the shape elements that were ignored
new_axis = axis - np.arange(len(axis))
new_shapes = [tuple(np.insert(new_shape, new_axis, removed_shape))
for removed_shape in removed_shapes]
return new_shapes
else:
return tuple(new_shape)
def _all_partitions_concatenated(ns):
"""
Generate all partitions of indices of groups of given sizes, concatenated
`ns` is an iterable of ints.
"""
def all_partitions(z, n):
for c in combinations(z, n):
x0 = set(c)
x1 = z - x0
yield [x0, x1]
def all_partitions_n(z, ns):
if len(ns) == 0:
yield [z]
return
for c in all_partitions(z, ns[0]):
for d in all_partitions_n(c[1], ns[1:]):
yield c[0:1] + d
z = set(range(np.sum(ns)))
for partitioning in all_partitions_n(z, ns[:]):
x = np.concatenate([list(partition)
for partition in partitioning]).astype(int)
yield x
def _batch_generator(iterable, batch):
"""A generator that yields batches of elements from an iterable"""
iterator = iter(iterable)
if batch <= 0:
raise ValueError("`batch` must be positive.")
while True:
z = []
try:
# get elements from iterator `batch` at a time
for i in range(batch):
z.append(next(iterator))
yield z
except StopIteration:
# when there are no more elements, yield the final batch and stop
if z:
yield z
break
def _calculate_null_both(data, statistic, n_permutations, batch,
random_state=None):
"""
Calculate null distribution for independent sample tests.
"""
n_samples = len(data)
# compute number of permutations
# (distinct partitions of data into samples of these sizes)
n_obs_i = [sample.shape[-1] for sample in data] # observations per sample
n_obs_ic = np.cumsum(n_obs_i)
n_obs = n_obs_ic[-1] # total number of observations
n_max = np.prod([comb(n_obs_ic[i], n_obs_ic[i-1])
for i in range(n_samples-1, 0, -1)])
# perm_generator is an iterator that produces permutations of indices
# from 0 to n_obs. We'll concatenate the samples, use these indices to
# permute the data, then split the samples apart again.
if n_permutations >= n_max:
exact_test = True
n_permutations = n_max
perm_generator = _all_partitions_concatenated(n_obs_i)
else:
exact_test = False
# Neither RandomState.permutation nor Generator.permutation
# can permute axis-slices independently. If this feature is
# added in the future, batches of the desired size should be
# generated in a single call.
perm_generator = (random_state.permutation(n_obs)
for i in range(n_permutations))
batch = batch or int(n_permutations)
null_distribution = []
# First, concatenate all the samples. In batches, permute samples with
# indices produced by the `perm_generator`, split them into new samples of
# the original sizes, compute the statistic for each batch, and add these
# statistic values to the null distribution.
data = np.concatenate(data, axis=-1)
for indices in _batch_generator(perm_generator, batch=batch):
indices = np.array(indices)
# `indices` is 2D: each row is a permutation of the indices.
# We use it to index `data` along its last axis, which corresponds
# with observations.
# After indexing, the second to last axis of `data_batch` corresponds
# with permutations, and the last axis corresponds with observations.
data_batch = data[..., indices]
# Move the permutation axis to the front: we'll concatenate a list
# of batched statistic values along this zeroth axis to form the
# null distribution.
data_batch = np.moveaxis(data_batch, -2, 0)
data_batch = np.split(data_batch, n_obs_ic[:-1], axis=-1)
null_distribution.append(statistic(*data_batch, axis=-1))
null_distribution = np.concatenate(null_distribution, axis=0)
return null_distribution, n_permutations, exact_test
def _calculate_null_pairings(data, statistic, n_permutations, batch,
random_state=None):
"""
Calculate null distribution for association tests.
"""
n_samples = len(data)
# compute number of permutations (factorial(n) permutations of each sample)
n_obs_sample = data[0].shape[-1] # observations per sample; same for each
n_max = factorial(n_obs_sample)**n_samples
# `perm_generator` is an iterator that produces a list of permutations of
# indices from 0 to n_obs_sample, one for each sample.
if n_permutations >= n_max:
exact_test = True
n_permutations = n_max
# cartesian product of the sets of all permutations of indices
perm_generator = product(*(permutations(range(n_obs_sample))
for i in range(n_samples)))
else:
exact_test = False
# Separate random permutations of indices for each sample.
# Again, it would be nice if RandomState/Generator.permutation
# could permute each axis-slice separately.
perm_generator = ([random_state.permutation(n_obs_sample)
for i in range(n_samples)]
for j in range(n_permutations))
batch = batch or int(n_permutations)
null_distribution = []
for indices in _batch_generator(perm_generator, batch=batch):
indices = np.array(indices)
# `indices` is 3D: the zeroth axis is for permutations, the next is
# for samples, and the last is for observations. Swap the first two
# to make the zeroth axis correspond with samples, as it does for
# `data`.
indices = np.swapaxes(indices, 0, 1)
# When we're done, `data_batch` will be a list of length `n_samples`.
# Each element will be a batch of random permutations of one sample.
# The zeroth axis of each batch will correspond with permutations,
# and the last will correspond with observations. (This makes it
# easy to pass into `statistic`.)
data_batch = [None]*n_samples
for i in range(n_samples):
data_batch[i] = data[i][..., indices[i]]
data_batch[i] = np.moveaxis(data_batch[i], -2, 0)
null_distribution.append(statistic(*data_batch, axis=-1))
null_distribution = np.concatenate(null_distribution, axis=0)
return null_distribution, n_permutations, exact_test
def _calculate_null_samples(data, statistic, n_permutations, batch,
random_state=None):
"""
Calculate null distribution for paired-sample tests.
"""
n_samples = len(data)
# By convention, the meaning of the "samples" permutations type for
# data with only one sample is to flip the sign of the observations.
# Achieve this by adding a second sample - the negative of the original.
if n_samples == 1:
data = [data[0], -data[0]]
# The "samples" permutation strategy is the same as the "pairings"
# strategy except the roles of samples and observations are flipped.
# So swap these axes, then we'll use the function for the "pairings"
# strategy to do all the work!
data = np.swapaxes(data, 0, -1)
# (Of course, the user's statistic doesn't know what we've done here,
# so we need to pass it what it's expecting.)
def statistic_wrapped(*data, axis):
data = np.swapaxes(data, 0, -1)
if n_samples == 1:
data = data[0:1]
return statistic(*data, axis=axis)
return _calculate_null_pairings(data, statistic_wrapped, n_permutations,
batch, random_state)
def _permutation_test_iv(data, statistic, permutation_type, vectorized,
n_resamples, batch, alternative, axis, random_state):
"""Input validation for `permutation_test`."""
axis_int = int(axis)
if axis != axis_int:
raise ValueError("`axis` must be an integer.")
permutation_types = {'samples', 'pairings', 'independent'}
permutation_type = permutation_type.lower()
if permutation_type not in permutation_types:
raise ValueError(f"`permutation_type` must be in {permutation_types}.")
if vectorized not in {True, False}:
raise ValueError("`vectorized` must be `True` or `False`.")
if not vectorized:
statistic = _bootstrap._vectorize_statistic(statistic)
message = "`data` must be a tuple containing at least two samples"
try:
if len(data) < 2 and permutation_type == 'independent':
raise ValueError(message)
except TypeError:
raise TypeError(message)
data = _broadcast_arrays(data, axis)
data_iv = []
for sample in data:
sample = np.atleast_1d(sample)
if sample.shape[axis] <= 1:
raise ValueError("each sample in `data` must contain two or more "
"observations along `axis`.")
sample = np.moveaxis(sample, axis_int, -1)
data_iv.append(sample)
n_resamples_int = (int(n_resamples) if not np.isinf(n_resamples)
else np.inf)
if n_resamples != n_resamples_int or n_resamples_int <= 0:
raise ValueError("`n_resamples` must be a positive integer.")
if batch is None:
batch_iv = batch
else:
batch_iv = int(batch)
if batch != batch_iv or batch_iv <= 0:
raise ValueError("`batch` must be a positive integer or None.")
alternatives = {'two-sided', 'greater', 'less'}
alternative = alternative.lower()
if alternative not in alternatives:
raise ValueError(f"`alternative` must be in {alternatives}")
random_state = check_random_state(random_state)
return (data_iv, statistic, permutation_type, vectorized, n_resamples_int,
batch_iv, alternative, axis_int, random_state)
def permutation_test(data, statistic, *, permutation_type='independent',
vectorized=False, n_resamples=9999, batch=None,
alternative="two-sided", axis=0, random_state=None):
r"""
Performs a permutation test of a given statistic on provided data.
For independent sample statistics, the null hypothesis is that the data are
randomly sampled from the same distribution.
For paired sample statistics, two null hypothesis can be tested:
that the data are paired at random or that the data are assigned to samples
at random.
Parameters
----------
data : iterable of array-like
Contains the samples, each of which is an array of observations.
Dimensions of sample arrays must be compatible for broadcasting except
along `axis`.
statistic : callable
Statistic for which the p-value of the hypothesis test is to be
calculated. `statistic` must be a callable that accepts samples
as separate arguments (e.g. ``statistic(*data)``) and returns the
resulting statistic.
If `vectorized` is set ``True``, `statistic` must also accept a keyword
argument `axis` and be vectorized to compute the statistic along the
provided `axis` of the sample arrays.
permutation_type : {'independent', 'samples', 'pairings'}, optional
The type of permutations to be performed, in accordance with the
null hypothesis. The first two permutation types are for paired sample
statistics, in which all samples contain the same number of
observations and observations with corresponding indices along `axis`
are considered to be paired; the third is for independent sample
statistics.
- ``'samples'`` : observations are assigned to different samples
but remain paired with the same observations from other samples.
This permutation type is appropriate for paired sample hypothesis
tests such as the Wilcoxon signed-rank test and the paired t-test.
- ``'pairings'`` : observations are paired with different observations,
but they remain within the same sample. This permutation type is
appropriate for association/correlation tests with statistics such
as Spearman's :math:`\rho`, Kendall's :math:`\tau`, and Pearson's
:math:`r`.
- ``'independent'`` (default) : observations are assigned to different
samples. Samples may contain different numbers of observations. This
permutation type is appropriate for independent sample hypothesis
tests such as the Mann-Whitney :math:`U` test and the independent
sample t-test.
Please see the Notes section below for more detailed descriptions
of the permutation types.
vectorized : bool, default: ``False``
By default, `statistic` is assumed to calculate the statistic only for
1D arrays contained in `data`. If `vectorized` is set ``True``,
`statistic` must also accept a keyword argument `axis` and be
vectorized to compute the statistic along the provided `axis` of the ND
arrays in `data`. Use of a vectorized statistic can reduce computation
time.
n_resamples : int or np.inf, default: 9999
Number of random permutations (resamples) used to approximate the null
distribution. If greater than or equal to the number of distinct
permutations, the exact null distribution will be computed.
Note that the number of distinct permutations grows very rapidly with
the sizes of samples, so exact tests are feasible only for very small
data sets.
batch : int, optional
The number of permutations to process in each call to `statistic`.
Memory usage is O(`batch`*``n``), where ``n`` is the total size
of all samples, regardless of the value of `vectorized`. Default is
``None``, in which case ``batch`` is the number of permutations.
alternative : {'two-sided', 'less', 'greater'}, optional
The alternative hypothesis for which the p-value is calculated.
For each alternative, the p-value is defined for exact tests as
follows.
- ``'greater'`` : the percentage of the null distribution that is
greater than or equal to the observed value of the test statistic.
- ``'less'`` : the percentage of the null distribution that is
less than or equal to the observed value of the test statistic.
- ``'two-sided'`` (default) : twice the smaller of the p-values above.
Note that p-values for randomized tests are calculated according to the
conservative (over-estimated) approximation suggested in [2]_ and [3]_
rather than the unbiased estimator suggested in [4]_. That is, when
calculating the proportion of the randomized null distribution that is
as extreme as the observed value of the test statistic, the values in
the numerator and denominator are both increased by one. An
interpretation of this adjustment is that the observed value of the
test statistic is always included as an element of the randomized
null distribution.
The convention used for two-sided p-values is not universal;
the observed test statistic and null distribution are returned in
case a different definition is preferred.
axis : int, default: 0
The axis of the (broadcasted) samples over which to calculate the
statistic. If samples have a different number of dimensions,
singleton dimensions are prepended to samples with fewer dimensions
before `axis` is considered.
random_state : {None, int, `numpy.random.Generator`,
`numpy.random.RandomState`}, optional
Pseudorandom number generator state used to generate permutations.
If `random_state` is ``None`` (default), the
`numpy.random.RandomState` singleton is used.
If `random_state` is an int, a new ``RandomState`` instance is used,
seeded with `random_state`.
If `random_state` is already a ``Generator`` or ``RandomState``
instance then that instance is used.
Returns
-------
statistic : float or ndarray
The observed test statistic of the data.
pvalue : float or ndarray
The p-value for the given alternative.
null_distribution : ndarray
The values of the test statistic generated under the null hypothesis.
Notes
-----
The three types of permutation tests supported by this function are
described below.
**Unpaired statistics** (``permutation_type='independent'``):
The null hypothesis associated with this permutation type is that all
observations are sampled from the same underlying distribution and that
they have been assigned to one of the samples at random.
Suppose ``data`` contains two samples; e.g. ``a, b = data``.
When ``1 < n_resamples < binom(n, k)``, where
* ``k`` is the number of observations in ``a``,
* ``n`` is the total number of observations in ``a`` and ``b``, and
* ``binom(n, k)`` is the binomial coefficient (``n`` choose ``k``),
the data are pooled (concatenated), randomly assigned to either the first
or second sample, and the statistic is calculated. This process is
performed repeatedly, `permutation` times, generating a distribution of the
statistic under the null hypothesis. The statistic of the original
data is compared to this distribution to determine the p-value.
When ``n_resamples >= binom(n, k)``, an exact test is performed: the data
are *partitioned* between the samples in each distinct way exactly once,
and the exact null distribution is formed.
Note that for a given partitioning of the data between the samples,
only one ordering/permutation of the data *within* each sample is
considered. For statistics that do not depend on the order of the data
within samples, this dramatically reduces computational cost without
affecting the shape of the null distribution (because the frequency/count
of each value is affected by the same factor).
For ``a = [a1, a2, a3, a4]`` and ``b = [b1, b2, b3]``, an example of this
permutation type is ``x = [b3, a1, a2, b2]`` and ``y = [a4, b1, a3]``.
Because only one ordering/permutation of the data *within* each sample
is considered in an exact test, a resampling like ``x = [b3, a1, b2, a2]``
and ``y = [a4, a3, b1]`` would *not* be considered distinct from the
example above.
``permutation_type='independent'`` does not support one-sample statistics,
but it can be applied to statistics with more than two samples. In this
case, if ``n`` is an array of the number of observations within each
sample, the number of distinct partitions is::
np.product([binom(sum(n[i:]), sum(n[i+1:])) for i in range(len(n)-1)])
**Paired statistics, permute pairings** (``permutation_type='pairings'``):
The null hypothesis associated with this permutation type is that
observations within each sample are drawn from the same underlying
distribution and that pairings with elements of other samples are
assigned at random.
Suppose ``data`` contains only one sample; e.g. ``a, = data``, and we
wish to consider all possible pairings of elements of ``a`` with elements
of a second sample, ``b``. Let ``n`` be the number of observations in
``a``, which must also equal the number of observations in ``b``.
When ``1 < n_resamples < factorial(n)``, the elements of ``a`` are
randomly permuted. The user-supplied statistic accepts one data argument,
say ``a_perm``, and calculates the statistic considering ``a_perm`` and
``b``. This process is performed repeatedly, `permutation` times,
generating a distribution of the statistic under the null hypothesis.
The statistic of the original data is compared to this distribution to
determine the p-value.
When ``n_resamples >= factorial(n)``, an exact test is performed:
``a`` is permuted in each distinct way exactly once. Therefore, the
`statistic` is computed for each unique pairing of samples between ``a``
and ``b`` exactly once.
For ``a = [a1, a2, a3]`` and ``b = [b1, b2, b3]``, an example of this
permutation type is ``a_perm = [a3, a1, a2]`` while ``b`` is left
in its original order.
``permutation_type='pairings'`` supports ``data`` containing any number
of samples, each of which must contain the same number of observations.
All samples provided in ``data`` are permuted *independently*. Therefore,
if ``m`` is the number of samples and ``n`` is the number of observations
within each sample, then the number of permutations in an exact test is::
factorial(n)**m
Note that if a two-sample statistic, for example, does not inherently
depend on the order in which observations are provided - only on the
*pairings* of observations - then only one of the two samples should be
provided in ``data``. This dramatically reduces computational cost without
affecting the shape of the null distribution (because the frequency/count
of each value is affected by the same factor).
**Paired statistics, permute samples** (``permutation_type='samples'``):
The null hypothesis associated with this permutation type is that
observations within each pair are drawn from the same underlying
distribution and that the sample to which they are assigned is random.
Suppose ``data`` contains two samples; e.g. ``a, b = data``.
Let ``n`` be the number of observations in ``a``, which must also equal
the number of observations in ``b``.
When ``1 < n_resamples < 2**n``, the elements of ``a`` are ``b`` are
randomly swapped between samples (maintaining their pairings) and the
statistic is calculated. This process is performed repeatedly,
`permutation` times, generating a distribution of the statistic under the
null hypothesis. The statistic of the original data is compared to this
distribution to determine the p-value.
When ``n_resamples >= 2**n``, an exact test is performed: the observations
are assigned to the two samples in each distinct way (while maintaining
pairings) exactly once.
For ``a = [a1, a2, a3]`` and ``b = [b1, b2, b3]``, an example of this
permutation type is ``x = [b1, a2, b3]`` and ``y = [a1, b2, a3]``.
``permutation_type='samples'`` supports ``data`` containing any number
of samples, each of which must contain the same number of observations.
If ``data`` contains more than one sample, paired observations within
``data`` are exchanged between samples *independently*. Therefore, if ``m``
is the number of samples and ``n`` is the number of observations within
each sample, then the number of permutations in an exact test is::
factorial(m)**n
Several paired-sample statistical tests, such as the Wilcoxon signed rank
test and paired-sample t-test, can be performed considering only the
*difference* between two paired elements. Accordingly, if ``data`` contains
only one sample, then the null distribution is formed by independently
changing the *sign* of each observation.
References
----------
.. [1] R. A. Fisher. The Design of Experiments, 6th Ed (1951).
.. [2] B. Phipson and G. K. Smyth. "Permutation P-values Should Never Be
Zero: Calculating Exact P-values When Permutations Are Randomly Drawn."
Statistical Applications in Genetics and Molecular Biology 9.1 (2010).
.. [3] M. D. Ernst. "Permutation Methods: A Basis for Exact Inference".
Statistical Science (2004).
.. [4] B. Efron and R. J. Tibshirani. An Introduction to the Bootstrap
(1993).
Examples
--------
Suppose we wish to test whether two samples are drawn from the same
distribution. Assume that the underlying distributions are unknown to us,
and that before observing the data, we hypothesized that the mean of the
first sample would be less than that of the second sample. We decide that
we will use the difference between the sample means as a test statistic,
and we will consider a p-value of 0.05 to be statistically significant.
For efficiency, we write the function defining the test statistic in a
vectorized fashion: the samples ``x`` and ``y`` can be ND arrays, and the
statistic will be calculated for each axis-slice along `axis`.
>>> def statistic(x, y, axis):
... return np.mean(x, axis=axis) - np.mean(y, axis=axis)
After collecting our data, we calculate the observed value of the test
statistic.
>>> from scipy.stats import norm
>>> rng = np.random.default_rng()
>>> x = norm.rvs(size=5, random_state=rng)
>>> y = norm.rvs(size=6, loc = 3, random_state=rng)
>>> statistic(x, y, 0)
-3.5411688580987266
Indeed, the test statistic is negative, suggesting that the true mean of
the distribution underlying ``x`` is less than that of the distribution
underlying ``y``. To determine the probability of this occuring by chance
if the two samples were drawn from the same distribution, we perform
a permutation test.
>>> from scipy.stats import permutation_test
>>> # because our statistic is vectorized, we pass `vectorized=True`
>>> # `n_resamples=np.inf` indicates that an exact test is to be performed
>>> res = permutation_test((x, y), statistic, vectorized=True,
... n_resamples=np.inf, alternative='less')
>>> print(res.statistic)
-3.5411688580987266
>>> print(res.pvalue)
0.004329004329004329
The probability of obtaining a test statistic less than or equal to the
observed value under the null hypothesis is 0.4329%. This is less than our
chosen threshold of 5%, so we consider this to to be significant evidence
against the null hypothesis in favor of the alternative.
Because the size of the samples above was small, `permutation_test` could
perform an exact test. For larger samples, we resort to a randomized
permutation test.
>>> x = norm.rvs(size=100, random_state=rng)
>>> y = norm.rvs(size=120, loc=0.3, random_state=rng)
>>> res = permutation_test((x, y), statistic, n_resamples=100000,
... vectorized=True, alternative='less',
... random_state=rng)
>>> print(res.statistic)
-0.5230459671240913
>>> print(res.pvalue)
0.00016999830001699983
The approximate probability of obtaining a test statistic less than or
equal to the observed value under the null hypothesis is 0.0225%. This is
again less than our chosen threshold of 5%, so again we have significant
evidence to reject the null hypothesis in favor of the alternative.
For large samples and number of permutations, the result is comparable to
that of the corresponding asymptotic test, the independent sample t-test.
>>> from scipy.stats import ttest_ind
>>> res_asymptotic = ttest_ind(x, y, alternative='less')
>>> print(res_asymptotic.pvalue)
0.00012688101537979522
The permutation distribution of the test statistic is provided for
further investigation.
>>> import matplotlib.pyplot as plt
>>> plt.hist(res.null_distribution, bins=50)
>>> plt.title("Permutation distribution of test statistic")
>>> plt.xlabel("Value of Statistic")
>>> plt.ylabel("Frequency")
"""
args = _permutation_test_iv(data, statistic, permutation_type, vectorized,
n_resamples, batch, alternative, axis,
random_state)
(data, statistic, permutation_type, vectorized, n_resamples, batch,
alternative, axis, random_state) = args
observed = statistic(*data, axis=-1)
null_calculators = {"pairings": _calculate_null_pairings,
"samples": _calculate_null_samples,
"independent": _calculate_null_both}
null_calculator_args = (data, statistic, n_resamples,
batch, random_state)
calculate_null = null_calculators[permutation_type]
null_distribution, n_resamples, exact_test = (
calculate_null(*null_calculator_args))
# See References [2] and [3]
adjustment = 0 if exact_test else 1
def less(null_distribution, observed):
cmps = null_distribution <= observed
pvalues = (cmps.sum(axis=0) + adjustment) / (n_resamples + adjustment)
return pvalues
def greater(null_distribution, observed):
cmps = null_distribution >= observed
pvalues = (cmps.sum(axis=0) + adjustment) / (n_resamples + adjustment)
return pvalues
def two_sided(null_distribution, observed):
pvalues_less = less(null_distribution, observed)
pvalues_greater = greater(null_distribution, observed)
pvalues = np.minimum(pvalues_less, pvalues_greater) * 2
return pvalues
compare = {"less": less,
"greater": greater,
"two-sided": two_sided}
pvalues = compare[alternative](null_distribution, observed)
pvalues = np.clip(pvalues, 0, 1)
return PermutationTestResult(observed, pvalues, null_distribution)
class TukeyHSDResult:
"""Result of `scipy.stats.tukey_hsd`.
Attributes
----------
statistic : float ndarray
The computed statistic of the test for each comparison. The element
at index ``(i, j)`` is the statistic for the comparison between groups
``i`` and ``j``.
pvalue : float ndarray
The associated p-value from the studentized range distribution. The
element at index ``(i, j)`` is the p-value for the comparison
between groups ``i`` and ``j``.
Notes
-----
The string representation of this object displays the most recently
calculated confidence interval, and if none have been previously
calculated, it will evaluate ``confidence_interval()``.
References
----------
.. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "7.4.7.1. Tukey's
Method."
https://www.itl.nist.gov/div898/handbook/prc/section4/prc471.htm,
28 November 2020.
"""
def __init__(self, statistic, pvalue, _nobs, _ntreatments, _stand_err):
self.statistic = statistic
self.pvalue = pvalue
self._ntreatments = _ntreatments
self._nobs = _nobs
self._stand_err = _stand_err
self._ci = None
self._ci_cl = None
def __str__(self):
# Note: `__str__` prints the confidence intervals from the most
# recent call to `confidence_interval`. If it has not been called,
# it will be called with the default CL of .95.
if self._ci is None:
self.confidence_interval(confidence_level=.95)
s = ("Tukey's HSD Pairwise Group Comparisons"
f" ({self._ci_cl*100:.1f}% Confidence Interval)\n")
s += "Comparison Statistic p-value Lower CI Upper CI\n"
for i in range(self.pvalue.shape[0]):
for j in range(self.pvalue.shape[0]):
if i != j:
s += (f" ({i} - {j}) {self.statistic[i, j]:>10.3f}"
f"{self.pvalue[i, j]:>10.3f}"
f"{self._ci.low[i, j]:>10.3f}"
f"{self._ci.high[i, j]:>10.3f}\n")
return s
def confidence_interval(self, confidence_level=.95):
"""Compute the confidence interval for the specified confidence level.
Parameters
----------
confidence_level : float, optional
Confidence level for the computed confidence interval
of the estimated proportion. Default is .95.
Returns
-------
ci : ``ConfidenceInterval`` object
The object has attributes ``low`` and ``high`` that hold the
lower and upper bounds of the confidence intervals for each
comparison. The high and low values are accessible for each
comparison at index ``(i, j)`` between groups ``i`` and ``j``.
References
----------
.. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "7.4.7.1.
Tukey's Method."
https://www.itl.nist.gov/div898/handbook/prc/section4/prc471.htm,
28 November 2020.
Examples
--------
>>> from scipy.stats import tukey_hsd
>>> group0 = [24.5, 23.5, 26.4, 27.1, 29.9]
>>> group1 = [28.4, 34.2, 29.5, 32.2, 30.1]
>>> group2 = [26.1, 28.3, 24.3, 26.2, 27.8]
>>> result = tukey_hsd(group0, group1, group2)
>>> ci = result.confidence_interval()
>>> ci.low
array([[-3.649159, -8.249159, -3.909159],
[ 0.950841, -3.649159, 0.690841],
[-3.389159, -7.989159, -3.649159]])
>>> ci.high
array([[ 3.649159, -0.950841, 3.389159],
[ 8.249159, 3.649159, 7.989159],
[ 3.909159, -0.690841, 3.649159]])
"""
# check to see if the supplied confidence level matches that of the
# previously computed CI.
if (self._ci is not None and self._ci_cl is not None and
confidence_level == self._ci_cl):
return self._ci
if not 0 < confidence_level < 1:
raise ValueError("Confidence level must be between 0 and 1.")
# determine the critical value of the studentized range using the
# appropriate confidence level, number of treatments, and degrees
# of freedom as determined by the number of data less the number of
# treatments. ("Confidence limits for Tukey's method")[1]. Note that
# in the cases of unequal sample sizes there will be a criterion for
# each group comparison.
params = (confidence_level, self._nobs, self._ntreatments - self._nobs)
srd = distributions.studentized_range.ppf(*params)
# also called maximum critical value, the Tukey criterion is the
# studentized range critical value * the square root of mean square
# error over the sample size.
tukey_criterion = srd * self._stand_err
# the confidence levels are determined by the
# `mean_differences` +- `tukey_criterion`
upper_conf = self.statistic + tukey_criterion
lower_conf = self.statistic - tukey_criterion
self._ci = ConfidenceInterval(low=lower_conf, high=upper_conf)
self._ci_cl = confidence_level
return self._ci
def _tukey_hsd_iv(args):
if (len(args)) < 2:
raise ValueError("There must be more than 1 treatment.")
args = [np.asarray(arg) for arg in args]
for arg in args:
if arg.ndim != 1:
raise ValueError("Input samples must be one-dimensional.")
if arg.size <= 1:
raise ValueError("Input sample size must be greater than one.")
if np.isinf(arg).any():
raise ValueError("Input samples must be finite.")
return args
def tukey_hsd(*args):
"""Perform Tukey's HSD test for equality of means over multiple treatments.
Tukey's honestly significant difference (HSD) test performs pairwise
comparison of means for a set of samples. Whereas ANOVA (e.g. `f_oneway`)
assesses whether the true means underlying each sample are identical,
Tukey's HSD is a post hoc test used to compare the mean of each sample
to the mean of each other sample.
The null hypothesis is that the distributions underlying the samples all
have the same mean. The test statistic, which is computed for every
possible pairing of samples, is simply the difference between the sample
means. For each pair, the p-value is the probability under the null
hypothesis (and other assumptions; see notes) of observing such an extreme
value of the statistic, considering that many pairwise comparisons are
being performed. Confidence intervals for the difference between each pair
of means are also available.
Parameters
----------
sample1, sample2, ... : array_like
The sample measurements for each group. There must be at least
two arguments.
Returns
-------
result : `~scipy.stats._result_classes.TukeyHSDResult` instance
The return value is an object with the following attributes:
statistic : float ndarray
The computed statistic of the test for each comparison. The element
at index ``(i, j)`` is the statistic for the comparison between
groups ``i`` and ``j``.
pvalue : float ndarray
The computed p-value of the test for each comparison. The element
at index ``(i, j)`` is the p-value for the comparison between
groups ``i`` and ``j``.
The object has the following methods:
confidence_interval(confidence_level=0.95):
Compute the confidence interval for the specified confidence level.
Notes
-----
The use of this test relies on several assumptions.
1. The observations are independent within and among groups.
2. The observations within each group are normally distributed.
3. The distributions from which the samples are drawn have the same finite
variance.
The original formulation of the test was for samples of equal size [6]_.
In case of unequal sample sizes, the test uses the Tukey-Kramer method
[4]_.
References
----------
.. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "7.4.7.1. Tukey's
Method."
https://www.itl.nist.gov/div898/handbook/prc/section4/prc471.htm,
28 November 2020.
.. [2] Abdi, Herve & Williams, Lynne. (2021). "Tukey's Honestly Significant
Difference (HSD) Test."
https://personal.utdallas.edu/~herve/abdi-HSD2010-pretty.pdf
.. [3] "One-Way ANOVA Using SAS PROC ANOVA & PROC GLM." SAS
Tutorials, 2007, www.stattutorials.com/SAS/TUTORIAL-PROC-GLM.htm.
.. [4] Kramer, Clyde Young. "Extension of Multiple Range Tests to Group
Means with Unequal Numbers of Replications." Biometrics, vol. 12,
no. 3, 1956, pp. 307-310. JSTOR, www.jstor.org/stable/3001469.
Accessed 25 May 2021.
.. [5] NIST/SEMATECH e-Handbook of Statistical Methods, "7.4.3.3.
The ANOVA table and tests of hypotheses about means"
https://www.itl.nist.gov/div898/handbook/prc/section4/prc433.htm,
2 June 2021.
.. [6] Tukey, John W. "Comparing Individual Means in the Analysis of
Variance." Biometrics, vol. 5, no. 2, 1949, pp. 99-114. JSTOR,
www.jstor.org/stable/3001913. Accessed 14 June 2021.
Examples
--------
Here are some data comparing the time to relief of three brands of
headache medicine, reported in minutes. Data adapted from [3]_.
>>> from scipy.stats import tukey_hsd
>>> group0 = [24.5, 23.5, 26.4, 27.1, 29.9]
>>> group1 = [28.4, 34.2, 29.5, 32.2, 30.1]
>>> group2 = [26.1, 28.3, 24.3, 26.2, 27.8]
We would like to see if the means between any of the groups are
significantly different. First, visually examine a box and whisker plot.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
>>> ax.boxplot([group0, group1, group2])
>>> ax.set_xticklabels(["group0", "group1", "group2"]) # doctest: +SKIP
>>> ax.set_ylabel("mean") # doctest: +SKIP
>>> plt.show()
From the box and whisker plot, we can see overlap in the interquartile
ranges group 1 to group 2 and group 3, but we can apply the ``tukey_hsd``
test to determine if the difference between means is significant. We
set a significance level of .05 to reject the null hypothesis.
>>> res = tukey_hsd(group0, group1, group2)
>>> print(res)
Tukey's HSD Pairwise Group Comparisons (95.0% Confidence Interval)
Comparison Statistic p-value Lower CI Upper CI
(0 - 1) -4.600 0.014 -8.249 -0.951
(0 - 2) -0.260 0.980 -3.909 3.389
(1 - 0) 4.600 0.014 0.951 8.249
(1 - 2) 4.340 0.020 0.691 7.989
(2 - 0) 0.260 0.980 -3.389 3.909
(2 - 1) -4.340 0.020 -7.989 -0.691
The null hypothesis is that each group has the same mean. The p-value for
comparisons between ``group0`` and ``group1`` as well as ``group1`` and
``group2`` do not exceed .05, so we reject the null hypothesis that they
have the same means. The p-value of the comparison between ``group0``
and ``group2`` exceeds .05, so we accept the null hypothesis that there
is not a significant difference between their means.
We can also compute the confidence interval associated with our chosen
confidence level.
>>> group0 = [24.5, 23.5, 26.4, 27.1, 29.9]
>>> group1 = [28.4, 34.2, 29.5, 32.2, 30.1]
>>> group2 = [26.1, 28.3, 24.3, 26.2, 27.8]
>>> result = tukey_hsd(group0, group1, group2)
>>> conf = res.confidence_interval(confidence_level=.99)
>>> for ((i, j), l) in np.ndenumerate(conf.low):
... # filter out self comparisons
... if i != j:
... h = conf.high[i,j]
... print(f"({i} - {j}) {l:>6.3f} {h:>6.3f}")
(0 - 1) -9.480 0.280
(0 - 2) -5.140 4.620
(1 - 0) -0.280 9.480
(1 - 2) -0.540 9.220
(2 - 0) -4.620 5.140
(2 - 1) -9.220 0.540
"""
args = _tukey_hsd_iv(args)
ntreatments = len(args)
means = np.asarray([np.mean(arg) for arg in args])
nsamples_treatments = np.asarray([a.size for a in args])
nobs = np.sum(nsamples_treatments)
# determine mean square error [5]. Note that this is sometimes called
# mean square error within.
mse = (np.sum([np.var(arg, ddof=1) for arg in args] *
(nsamples_treatments - 1)) / (nobs - ntreatments))
# The calculation of the standard error differs when treatments differ in
# size. See ("Unequal sample sizes")[1].
if np.unique(nsamples_treatments).size == 1:
# all input groups are the same length, so only one value needs to be
# calculated [1].
normalize = 2 / nsamples_treatments[0]
else:
# to compare groups of differing sizes, we must compute a variance
# value for each individual comparison. Use broadcasting to get the
# resulting matrix. [3], verified against [4] (page 308).
normalize = 1 / nsamples_treatments + 1 / nsamples_treatments[None].T
# the standard error is used in the computation of the tukey criterion and
# finding the p-values.
stand_err = np.sqrt(normalize * mse / 2)
# the mean difference is the test statistic.
mean_differences = means[None].T - means
# Calculate the t-statistic to use within the survival function of the
# studentized range to get the p-value.
t_stat = np.abs(mean_differences) / stand_err
params = t_stat, ntreatments, nobs - ntreatments
pvalues = distributions.studentized_range.sf(*params)
return TukeyHSDResult(mean_differences, pvalues, ntreatments,
nobs, stand_err)
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