# Source code for statsmodels.stats.nonparametric

"""
Rank based methods for inferential statistics

Created on Sat Aug 15 10:18:53 2020

Author: Josef Perktold

"""

import numpy as np
from scipy import stats
from scipy.stats import rankdata

from statsmodels.stats.base import HolderTuple
from statsmodels.stats.weightstats import (
_tconfint_generic,
_tstat_generic,
_zconfint_generic,
_zstat_generic,
)

[docs]
def rankdata_2samp(x1, x2):
"""Compute midranks for two samples

Parameters
----------
x1, x2 : array_like
Original data for two samples that will be converted to midranks.

Returns
-------
rank1 : ndarray
Midranks of the first sample in the pooled sample.
rank2 : ndarray
Midranks of the second sample in the pooled sample.
ranki1 : ndarray
Internal midranks of the first sample.
ranki2 : ndarray
Internal midranks of the second sample.

"""
x1 = np.asarray(x1)
x2 = np.asarray(x2)

nobs1 = len(x1)
nobs2 = len(x2)
if nobs1 == 0 or nobs2 == 0:
raise ValueError("one sample has zero length")

x_combined = np.concatenate((x1, x2))
if x_combined.ndim > 1:
rank = np.apply_along_axis(rankdata, 0, x_combined)
else:
rank = rankdata(x_combined)  # no axis in older scipy
rank1 = rank[:nobs1]
rank2 = rank[nobs1:]
if x_combined.ndim > 1:
ranki1 = np.apply_along_axis(rankdata, 0, x1)
ranki2 = np.apply_along_axis(rankdata, 0, x2)
else:
ranki1 = rankdata(x1)
ranki2 = rankdata(x2)
return rank1, rank2, ranki1, ranki2

[docs]
class RankCompareResult(HolderTuple):
"""Results for rank comparison

This is a subclass of HolderTuple that includes results from intermediate
computations, as well as methods for hypothesis tests, confidence intervals
and summary.
"""

[docs]
def conf_int(self, value=None, alpha=0.05, alternative="two-sided"):
"""
Confidence interval for probability that sample 1 has larger values

Confidence interval is for the shifted probability

P(x1 > x2) + 0.5 * P(x1 = x2) - value

Parameters
----------
value : float
Value, default 0, shifts the confidence interval,
e.g. value=0.5 centers the confidence interval at zero.
alpha : float
Significance level for the confidence interval, coverage is
1-alpha
alternative : str
The alternative hypothesis, H1, has to be one of the following

* 'two-sided' : H1: prob - value not equal to 0.
* 'larger' :   H1: prob - value > 0
* 'smaller' :  H1: prob - value < 0

Returns
-------
lower : float or ndarray
Lower confidence limit. This is -inf for the one-sided alternative
"smaller".
upper : float or ndarray
Upper confidence limit. This is inf for the one-sided alternative
"larger".

"""

p0 = value
if p0 is None:
p0 = 0
diff = self.prob1 - p0
std_diff = np.sqrt(self.var / self.nobs)

if self.use_t is False:
return _zconfint_generic(diff, std_diff, alpha, alternative)
else:
return _tconfint_generic(diff, std_diff, self.df, alpha,
alternative)

[docs]
def test_prob_superior(self, value=0.5, alternative="two-sided"):
"""test for superiority probability

H0: P(x1 > x2) + 0.5 * P(x1 = x2) = value

The alternative is that the probability is either not equal, larger
or smaller than the null-value depending on the chosen alternative.

Parameters
----------
value : float
Value of the probability under the Null hypothesis.
alternative : str
The alternative hypothesis, H1, has to be one of the following

* 'two-sided' : H1: prob - value not equal to 0.
* 'larger' :   H1: prob - value > 0
* 'smaller' :  H1: prob - value < 0

Returns
-------
res : HolderTuple
HolderTuple instance with the following main attributes

statistic : float
Test statistic for z- or t-test
pvalue : float
Pvalue of the test based on either normal or t distribution.

"""

p0 = value  # alias
# diff = self.prob1 - p0  # for reporting, not used in computation
# TODO: use var_prob
std_diff = np.sqrt(self.var / self.nobs)

# corresponds to a one-sample test and either p0 or diff could be used
if not self.use_t:
stat, pv = _zstat_generic(self.prob1, p0, std_diff, alternative,
diff=0)
distr = "normal"
else:
stat, pv = _tstat_generic(self.prob1, p0, std_diff, self.df,
alternative, diff=0)
distr = "t"

res = HolderTuple(statistic=stat,
pvalue=pv,
df=self.df,
distribution=distr
)
return res

[docs]
def tost_prob_superior(self, low, upp):
'''test of stochastic (non-)equivalence of p = P(x1 > x2)

Null hypothesis:  p < low or p > upp
Alternative hypothesis:  low < p < upp

where p is the probability that a random draw from the population of
the first sample has a larger value than a random draw from the
population of the second sample, specifically

p = P(x1 > x2) + 0.5 * P(x1 = x2)

If the pvalue is smaller than a threshold, say 0.05, then we reject the
hypothesis that the probability p that distribution 1 is stochastically
superior to distribution 2 is outside of the interval given by
thresholds low and upp.

Parameters
----------
low, upp : float
equivalence interval low < mean < upp

Returns
-------
res : HolderTuple
HolderTuple instance with the following main attributes

pvalue : float
Pvalue of the equivalence test given by the larger pvalue of
the two one-sided tests.
statistic : float
Test statistic of the one-sided test that has the larger
pvalue.
results_larger : HolderTuple
Results instanc with test statistic, pvalue and degrees of
freedom for lower threshold test.
results_smaller : HolderTuple
Results instanc with test statistic, pvalue and degrees of
freedom for upper threshold test.

'''

t1 = self.test_prob_superior(low, alternative='larger')
t2 = self.test_prob_superior(upp, alternative='smaller')

# idx_max = 1 if t1.pvalue < t2.pvalue else 0
idx_max = np.asarray(t1.pvalue < t2.pvalue, int)
title = "Equivalence test for Prob(x1 > x2) + 0.5 Prob(x1 = x2) "
res = HolderTuple(statistic=np.choose(idx_max,
[t1.statistic, t2.statistic]),
# pvalue=[t1.pvalue, t2.pvalue][idx_max], # python
# use np.choose for vectorized selection
pvalue=np.choose(idx_max, [t1.pvalue, t2.pvalue]),
results_larger=t1,
results_smaller=t2,
title=title
)
return res

[docs]
def confint_lintransf(self, const=-1, slope=2, alpha=0.05,
alternative="two-sided"):
"""confidence interval of a linear transformation of prob1

This computes the confidence interval for

d = const + slope * prob1

Default values correspond to Somers' d.

Parameters
----------
const, slope : float
Constant and slope for linear (affine) transformation.
alpha : float
Significance level for the confidence interval, coverage is
1-alpha
alternative : str
The alternative hypothesis, H1, has to be one of the following

* 'two-sided' : H1: prob - value not equal to 0.
* 'larger' :   H1: prob - value > 0
* 'smaller' :  H1: prob - value < 0

Returns
-------
lower : float or ndarray
Lower confidence limit. This is -inf for the one-sided alternative
"smaller".
upper : float or ndarray
Upper confidence limit. This is inf for the one-sided alternative
"larger".

"""

low_p, upp_p = self.conf_int(alpha=alpha, alternative=alternative)
low = const + slope * low_p
upp = const + slope * upp_p
if slope < 0:
low, upp = upp, low
return low, upp

[docs]
def effectsize_normal(self, prob=None):
"""
Cohen's d, standardized mean difference under normality assumption.

This computes the standardized mean difference, Cohen's d, effect size
that is equivalent to the rank based probability p of being
stochastically larger if we assume that the data is normally
distributed, given by

:math: d = F^{-1}(p) * \\sqrt{2}

where :math:F^{-1} is the inverse of the cdf of the normal
distribution.

Parameters
----------
prob : float in (0, 1)
Probability to be converted to Cohen's d effect size.
If prob is None, then the prob1 attribute is used.

Returns
-------
equivalent Cohen's d effect size under normality assumption.

"""
if prob is None:
prob = self.prob1
return stats.norm.ppf(prob) * np.sqrt(2)

[docs]
def summary(self, alpha=0.05, xname=None):
"""summary table for probability that random draw x1 is larger than x2

Parameters
----------
alpha : float
Significance level for confidence intervals. Coverage is 1 - alpha
xname : None or list of str
If None, then each row has a name column with generic names.
If xname is a list of strings, then it will be included as part
of those names.

Returns
-------
SimpleTable instance with methods to convert to different output
formats.
"""

yname = "None"
effect = np.atleast_1d(self.prob1)
if self.pvalue is None:
statistic, pvalue = self.test_prob_superior()
else:
pvalue = self.pvalue
statistic = self.statistic
pvalues = np.atleast_1d(pvalue)
ci = np.atleast_2d(self.conf_int(alpha=alpha))
if ci.shape[0] > 1:
ci = ci.T
use_t = self.use_t
sd = np.atleast_1d(np.sqrt(self.var_prob))
statistic = np.atleast_1d(statistic)
if xname is None:
xname = ['c%d' % ii for ii in range(len(effect))]

xname2 = ['prob(x1>x2) %s' % ii for ii in xname]

title = "Probability sample 1 is stochastically larger"
from statsmodels.iolib.summary import summary_params

summ = summary_params((self, effect, sd, statistic,
pvalues, ci),
yname=yname, xname=xname2, use_t=use_t,
title=title, alpha=alpha)
return summ

[docs]
def rank_compare_2indep(x1, x2, use_t=True):
"""
Statistics and tests for the probability that x1 has larger values than x2.

p is the probability that a random draw from the population of
the first sample has a larger value than a random draw from the
population of the second sample, specifically

p = P(x1 > x2) + 0.5 * P(x1 = x2)

This is a measure underlying Wilcoxon-Mann-Whitney's U test,
Fligner-Policello test and Brunner-Munzel test, and
Inference is based on the asymptotic distribution of the Brunner-Munzel
test. The half probability for ties corresponds to the use of midranks
and make it valid for discrete variables.

The Null hypothesis for stochastic equality is p = 0.5, which corresponds
to the Brunner-Munzel test.

Parameters
----------
x1, x2 : array_like
Array of samples, should be one-dimensional.
use_t : boolean
If use_t is true, the t distribution with Welch-Satterthwaite type
degrees of freedom is used for p-value and confidence interval.
If use_t is false, then the normal distribution is used.

Returns
-------
res : RankCompareResult
The results instance contains the results for the Brunner-Munzel test
and has methods for hypothesis tests, confidence intervals and summary.

statistic : float
The Brunner-Munzel W statistic.
pvalue : float
p-value assuming an t distribution. One-sided or
two-sided, depending on the choice of alternative and use_t.

--------
RankCompareResult
scipy.stats.brunnermunzel : Brunner-Munzel test for stochastic equality
scipy.stats.mannwhitneyu : Mann-Whitney rank test on two samples.

Notes
-----
Wilcoxon-Mann-Whitney assumes equal variance or equal distribution under
the Null hypothesis. Fligner-Policello test allows for unequal variances
but assumes continuous distribution, i.e. no ties.
Brunner-Munzel extend the test to allow for unequal variance and discrete
or ordered categorical random variables.

Brunner and Munzel recommended to estimate the p-value by t-distribution
when the size of data is 50 or less. If the size is lower than 10, it would
be better to use permuted Brunner Munzel test (see [2]_) for the test
of stochastic equality.

This measure has been introduced in the literature under many different
names relying on a variety of assumptions.
In psychology, McGraw and Wong (1992) introduced it as Common Language
effect size for the continuous, normal distribution case,
Vargha and Delaney (2000) [3]_ extended it to the nonparametric
continuous distribution case as in Fligner-Policello.

WMW and related tests can only be interpreted as test of medians or tests
of central location only under very restrictive additional assumptions
such as both distribution are identical under the equality null hypothesis
(assumed by Mann-Whitney) or both distributions are symmetric (shown by
Fligner-Policello). If the distribution of the two samples can differ in
an arbitrary way, then the equality Null hypothesis corresponds to p=0.5
against an alternative p != 0.5.  see for example Conroy (2012) [4]_ and
Divine et al (2018) [5]_ .

Note: Brunner-Munzel and related literature define the probability that x1
is stochastically smaller than x2, while here we use stochastically larger.
This equivalent to switching x1 and x2 in the two sample case.

References
----------
.. [1] Brunner, E. and Munzel, U. "The nonparametric Benhrens-Fisher
problem: Asymptotic theory and a small-sample approximation".
Biometrical Journal. Vol. 42(2000): 17-25.
.. [2] Neubert, K. and Brunner, E. "A studentized permutation test for the
non-parametric Behrens-Fisher problem". Computational Statistics and
Data Analysis. Vol. 51(2007): 5192-5204.
.. [3] Vargha, András, and Harold D. Delaney. 2000. “A Critique and
Improvement of the CL Common Language Effect Size Statistics of
McGraw and Wong.” Journal of Educational and Behavioral Statistics
25 (2): 101–32. https://doi.org/10.3102/10769986025002101.
.. [4] Conroy, Ronán M. 2012. “What Hypotheses Do ‘Nonparametric’ Two-Group
Tests Actually Test?” The Stata Journal: Promoting Communications on
Statistics and Stata 12 (2): 182–90.
https://doi.org/10.1177/1536867X1201200202.
.. [5] Divine, George W., H. James Norton, Anna E. Barón, and Elizabeth
Juarez-Colunga. 2018. “The Wilcoxon–Mann–Whitney Procedure Fails as
a Test of Medians.” The American Statistician 72 (3): 278–86.
https://doi.org/10.1080/00031305.2017.1305291.

"""
x1 = np.asarray(x1)
x2 = np.asarray(x2)

nobs1 = len(x1)
nobs2 = len(x2)
nobs = nobs1 + nobs2
if nobs1 == 0 or nobs2 == 0:
raise ValueError("one sample has zero length")

rank1, rank2, ranki1, ranki2 = rankdata_2samp(x1, x2)

meanr1 = np.mean(rank1, axis=0)
meanr2 = np.mean(rank2, axis=0)
meanri1 = np.mean(ranki1, axis=0)
meanri2 = np.mean(ranki2, axis=0)

S1 = np.sum(np.power(rank1 - ranki1 - meanr1 + meanri1, 2.0), axis=0)
S1 /= nobs1 - 1
S2 = np.sum(np.power(rank2 - ranki2 - meanr2 + meanri2, 2.0), axis=0)
S2 /= nobs2 - 1

wbfn = nobs1 * nobs2 * (meanr1 - meanr2)
wbfn /= (nobs1 + nobs2) * np.sqrt(nobs1 * S1 + nobs2 * S2)

# Here we only use alternative == "two-sided"
if use_t:
df_numer = np.power(nobs1 * S1 + nobs2 * S2, 2.0)
df_denom = np.power(nobs1 * S1, 2.0) / (nobs1 - 1)
df_denom += np.power(nobs2 * S2, 2.0) / (nobs2 - 1)
df = df_numer / df_denom
pvalue = 2 * stats.t.sf(np.abs(wbfn), df)
else:
pvalue = 2 * stats.norm.sf(np.abs(wbfn))
df = None

# other info
var1 = S1 / (nobs - nobs1)**2
var2 = S2 / (nobs - nobs2)**2
var_prob = (var1 / nobs1 + var2 / nobs2)
var = nobs * (var1 / nobs1 + var2 / nobs2)
prob1 = (meanr1 - (nobs1 + 1) / 2) / nobs2
prob2 = (meanr2 - (nobs2 + 1) / 2) / nobs1

return RankCompareResult(statistic=wbfn, pvalue=pvalue, s1=S1, s2=S2,
var1=var1, var2=var2, var=var,
var_prob=var_prob,
nobs1=nobs1, nobs2=nobs2, nobs=nobs,
mean1=meanr1, mean2=meanr2,
prob1=prob1, prob2=prob2,
somersd1=prob1 * 2 - 1, somersd2=prob2 * 2 - 1,
df=df, use_t=use_t
)

[docs]
def rank_compare_2ordinal(count1, count2, ddof=1, use_t=True):
"""
Stochastically larger probability for 2 independent ordinal samples.

This is a special case of rank_compare_2indep when the data are given as
counts of two independent ordinal, i.e. ordered multinomial, samples.

The statistic of interest is the probability that a random draw from the
population of the first sample has a larger value than a random draw from
the population of the second sample, specifically

p = P(x1 > x2) + 0.5 * P(x1 = x2)

Parameters
----------
count1 : array_like
Counts of the first sample, categories are assumed to be ordered.
count2 : array_like
Counts of the second sample, number of categories and ordering needs
to be the same as for sample 1.
ddof : scalar
Degrees of freedom correction for variance estimation. The default
ddof=1 corresponds to rank_compare_2indep.
use_t : bool
If use_t is true, the t distribution with Welch-Satterthwaite type
degrees of freedom is used for p-value and confidence interval.
If use_t is false, then the normal distribution is used.

Returns
-------
res : RankCompareResult
This includes methods for hypothesis tests and confidence intervals
for the probability that sample 1 is stochastically larger than
sample 2.

--------
rank_compare_2indep
RankCompareResult

Notes
-----
The implementation is based on the appendix of Munzel and Hauschke (2003)
with the addition of ddof so that the results match the general
function rank_compare_2indep.

"""

count1 = np.asarray(count1)
count2 = np.asarray(count2)
nobs1, nobs2 = count1.sum(), count2.sum()
freq1 = count1 / nobs1
freq2 = count2 / nobs2
cdf1 = np.concatenate(([0], freq1)).cumsum(axis=0)
cdf2 = np.concatenate(([0], freq2)).cumsum(axis=0)

# mid rank cdf
cdfm1 = (cdf1[1:] + cdf1[:-1]) / 2
cdfm2 = (cdf2[1:] + cdf2[:-1]) / 2
prob1 = (cdfm2 * freq1).sum()
prob2 = (cdfm1 * freq2).sum()

var1 = (cdfm2**2 * freq1).sum() - prob1**2
var2 = (cdfm1**2 * freq2).sum() - prob2**2

var_prob = (var1 / (nobs1 - ddof) + var2 / (nobs2 - ddof))
nobs = nobs1 + nobs2
var = nobs * var_prob
vn1 = var1 * nobs2 * nobs1 / (nobs1 - ddof)
vn2 = var2 * nobs1 * nobs2 / (nobs2 - ddof)
df = (vn1 + vn2)**2 / (vn1**2 / (nobs1 - 1) + vn2**2 / (nobs2 - 1))
res = RankCompareResult(statistic=None, pvalue=None, s1=None, s2=None,
var1=var1, var2=var2, var=var,
var_prob=var_prob,
nobs1=nobs1, nobs2=nobs2, nobs=nobs,
mean1=None, mean2=None,
prob1=prob1, prob2=prob2,
somersd1=prob1 * 2 - 1, somersd2=prob2 * 2 - 1,
df=df, use_t=use_t
)

return res

[docs]
def prob_larger_continuous(distr1, distr2):
"""
Probability indicating that distr1 is stochastically larger than distr2.

This computes

p = P(x1 > x2)

for two continuous distributions, where distr1 and distr2 are the
distributions of random variables x1 and x2 respectively.

Parameters
----------
distr1, distr2 : distributions
Two instances of scipy.stats.distributions. The required methods are
cdf of the second distribution and expect of the first distribution.

Returns
-------
p : probability x1 is larger than x2

Notes
-----
This is a one-liner that is added mainly as reference.

Examples
--------
>>> from scipy import stats
>>> prob_larger_continuous(stats.norm, stats.t(5))
0.4999999999999999

# which is the same as
>>> stats.norm.expect(stats.t(5).cdf)
0.4999999999999999

# distribution 1 with smaller mean (loc) than distribution 2
>>> prob_larger_continuous(stats.norm, stats.norm(loc=1))
0.23975006109347669

"""

return distr1.expect(distr2.cdf)

[docs]
def cohensd2problarger(d):
"""
Convert Cohen's d effect size to stochastically-larger-probability.

This assumes observations are normally distributed.

Computed as

p = Prob(x1 > x2) = F(d / sqrt(2))

where F is cdf of normal distribution. Cohen's d is defined as

d = (mean1 - mean2) / std

where std is the pooled within standard deviation.

Parameters
----------
d : float or array_like
Cohen's d effect size for difference mean1 - mean2.

Returns
-------
prob : float or ndarray
Prob(x1 > x2)
"""

return stats.norm.cdf(d / np.sqrt(2))



Last update: Sep 16, 2024