Source code for statsmodels.stats.nonparametric
"""
Rank based methods for inferential statistics
Created on Sat Aug 15 10:18:53 2020
Author: Josef Perktold
License: BSD-3
"""
import numpy as np
from scipy import stats
from scipy.stats import rankdata
from statsmodels.tools.testing import Holder
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`.
See Also
--------
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.
See Also
--------
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))
def _compute_rank_placements(x1, x2) -> Holder:
"""
Compute ranks and placements for two samples.
This helper is used by `samplesize_rank_compare_onetail`
to calculate rank-based statistics for two input samples.
It assumes that the input data has been validated beforehand.
Parameters
----------
x1, x2 : array_like
Data samples used to compute ranks and placements.
Returns
-------
res : Holder
An instance of Holder containing the following attributes:
n_1 : int
Number of observations in the first sample.
n_2 : int
Number of observations in the second sample.
overall_ranks_pooled : ndarray
Ranks of the pooled sample.
overall_ranks_1 : ndarray
Ranks of the first sample in the pooled sample.
overall_ranks_2 : ndarray
Ranks of the second sample in the pooled sample.
within_group_ranks_1 : ndarray
Internal ranks of the first sample.
within_group_ranks_2 : ndarray
Internal ranks of the second sample.
placements_1 : ndarray
Placements of the first sample in the pooled sample.
placements_2 : ndarray
Placements of the second sample in the pooled sample.
Notes
-----
* The overall rank for each observation is determined
by ranking all data points from both samples combined
(`x1` and `x2`) in ascending order, with ties averaged.
* The within-group rank for each observation is determined
by ranking the data points within each sample separately,
* The placement of each observation is calculated by
taking the difference between the overall rank and the
within-group rank of the observation. Placements can be
thought of as measuress of the degree of overlap or
separation between two samples.
"""
n_1 = len(x1)
n_2 = len(x2)
# Overall ranks for each obs among combined sample
overall_ranks_pooled = rankdata(
np.r_[x1, x2], method="average"
)
overall_ranks_1 = overall_ranks_pooled[:n_1]
overall_ranks_2 = overall_ranks_pooled[n_1:]
# Within group ranks for each obs
within_group_ranks_1 = rankdata(x1, method="average")
within_group_ranks_2 = rankdata(x2, method="average")
placements_1 = overall_ranks_1 - within_group_ranks_1
placements_2 = overall_ranks_2 - within_group_ranks_2
return Holder(
n_1=n_1,
n_2=n_2,
overall_ranks_pooled=overall_ranks_pooled,
overall_ranks_1=overall_ranks_1,
overall_ranks_2=overall_ranks_2,
within_group_ranks_1=within_group_ranks_1,
within_group_ranks_2=within_group_ranks_2,
placements_1=placements_1,
placements_2=placements_2,
)
[docs]
def samplesize_rank_compare_onetail(
synthetic_sample,
reference_sample,
alpha,
power,
nobs_ratio=1,
alternative="two-sided",
) -> Holder:
"""
Compute sample size for the non-parametric Mann-Whitney U test.
This function implements the method of Happ et al (2019).
Parameters
----------
synthetic_sample : array_like
Generated `synthetic` data representing the treatment
group under the research hypothesis.
reference_sample : array_like
Advance information for the reference group.
alpha : float
The type I error rate for the test (two-sided).
power : float
The desired power of the test.
nobs_ratio : float, optional
Sample size ratio, `nobs_ref` = `nobs_ratio` *
`nobs_treat`. This is the ratio of the reference
group sample size to the treatment group sample
size, by default 1 (balanced design). See Notes.
alternative : str, ‘two-sided’ (default), ‘larger’, or ‘smaller’
Extra argument to choose whether the sample size is
calculated for a two-sided (default) or one-sided test.
See Notes.
Returns
-------
res : Holder
An instance of Holder containing the following attributes:
nobs_total : float
The total sample size required for the experiment.
nobs_treat : float
Sample size for the treatment group.
nobs_ref : float
Sample size for the reference group.
relative_effect : float
The estimated relative effect size.
power : float
The desired power for the test.
alpha : float
The type I error rate for the test.
Notes
-----
In the context of the two-sample Wilcoxon Mann-Whitney
U test, the `reference_sample` typically represents data
from the control group or previous studies. The
`synthetic_sample` is generated based on this reference
data and a prespecified relative effect size that is
meaningful for the research question. This effect size
is often determined in collaboration with subject matter
experts to reflect a significant difference worth detecting.
By comparing the reference and synthetic samples, this
function estimates the sample size needed to acheve the
desired power at the specified Type-I error rate.
Choosing between `one-sided` and `two-sided` tests has
important implications for sample size planning. A
`two-sided` test is more conservative and requires a
larger sample size but covers effects in both directions.
In contrast, a `larger` (`relative_effect > 0.5`) or `smaller`
(`relative_effect < 0.5`) one-sided test assumes the effect
occurs only in one direction, leading to a smaller required
sample size. However, if the true effect is in the opposite
direction, the `one-sided` test have virtually no power to
detect it. Additionally, if a two-sided test ends up being
used instead of the planned one-sided test, the original
sample size may be insufficient, resulting in an underpowered
study. It is important to carefully consider these trade-offs
when planning a study.
For `nobs_ratio > 1`, `nobs_ratio = 1`, or `nobs_ratio < 1`,
the reference group sample size is larger, equal to, or smaller
than the treatment group sample size, respectively.
Example
-------
The data for the placebo group of a clinical trial published in
Thall and Vail [2] is shown below. A relevant effect for the treatment
under investigation is considered to be a 50% reduction in the number
of seizures. To compute the required sample size with a power of 0.8
and holding the type I error rate at 0.05, we generate synthetic data
for the treatment group under the alternative assuming this reduction.
>>> from statsmodels.stats.nonparametric import samplesize_rank_compare_onetail
>>> import numpy as np
>>> reference_sample = np.array([3, 3, 5, 4, 21, 7, 2, 12, 5, 0, 22, 4, 2, 12,
... 9, 5, 3, 29, 5, 7, 4, 4, 5, 8, 25, 1, 2, 12])
>>> # Apply 50% reduction in seizure counts and floor operation
>>> synthetic_sample = np.floor(reference_sample / 2)
>>> result = samplesize_rank_compare_onetail(
... synthetic_sample=synthetic_sample,
... reference_sample=reference_sample,
... alpha=0.05, power=0.8
... )
>>> print(f"Total sample size: {result.nobs_total}, "
... f"Treatment group: {result.nobs_treat}, "
... f"Reference group: {result.nobs_ref}")
References
----------
.. [1] Happ, M., Bathke, A. C., and Brunner, E. "Optimal sample size
planning for the Wilcoxon-Mann-Whitney test". Statistics in Medicine.
Vol. 38(2019): 363-375. https://doi.org/10.1002/sim.7983.
.. [2] Thall, P. F., and Vail, S. C. "Some covariance models for longitudinal
count data with overdispersion". Biometrics, pp. 657-671, 1990.
"""
synthetic_sample = np.asarray(synthetic_sample)
reference_sample = np.asarray(reference_sample)
if not (len(synthetic_sample) > 0 and len(reference_sample) > 0):
raise ValueError(
"Both `synthetic_sample` and `reference_sample`"
" must have at least one element."
)
if not (
np.all(np.isfinite(reference_sample))
and np.all(np.isfinite(synthetic_sample))
):
raise ValueError(
"All elements of `synthetic_sample` and `reference_sample`"
" must be finite; check for missing values."
)
if not (0 < alpha < 1):
raise ValueError("Alpha must be between 0 and 1 non-inclusive.")
if not (0 < power < 1):
raise ValueError("Power must be between 0 and 1 non-inclusive.")
if not (0 < nobs_ratio):
raise ValueError(
"Ratio of reference group to treatment group must be"
" strictly positive."
)
if alternative not in ("two-sided", "larger", "smaller"):
raise ValueError(
"Alternative must be one of `two-sided`, `larger`, or `smaller`."
)
# Group 1 is the treatment group, Group 2 is the reference group
rank_place = _compute_rank_placements(
synthetic_sample,
reference_sample,
)
# Extra few bytes of name binding for explicitness & readability
n_syn = rank_place.n_1
n_ref = rank_place.n_2
overall_ranks_pooled = rank_place.overall_ranks_pooled
placements_syn = rank_place.placements_1
placements_ref = rank_place.placements_2
relative_effect = (
np.mean(placements_syn) - np.mean(placements_ref)
) / (n_syn + n_ref) + 0.5
# Values [0.499, 0.501] considered 'practically' = 0.5 (0.1% atol)
if np.isclose(relative_effect, 0.5, atol=1e-3):
raise ValueError(
"Estimated relative effect is effectively 0.5, i.e."
" stochastic equality between `synthetic_sample` and"
" `reference_sample`. Given null hypothesis is true,"
" sample size cannot be calculated. Please review data"
" samples to ensure they reflect appropriate relative"
" effect size assumptions."
)
if relative_effect < 0.5 and alternative == "larger":
raise ValueError(
"Estimated relative effect is smaller than 0.5;"
" `synthetic_sample` is stochastically smaller than"
" `reference_sample`. No sample size can be calculated"
" for `alternative == 'larger'`. Please review data"
" samples to ensure they reflect appropriate relative"
" effect size assumptions."
)
if relative_effect > 0.5 and alternative == "smaller":
raise ValueError(
"Estimated relative effect is larger than 0.5;"
" `synthetic_sample` is stochastically larger than"
" `reference_sample`. No sample size can be calculated"
" for `alternative == 'smaller'`. Please review data"
" samples to ensure they reflect appropriate relative"
" effect size assumptions."
)
sd_overall = np.sqrt(
np.sum(
(overall_ranks_pooled - (n_syn + n_ref + 1) / 2) ** 2
)
/ (n_syn + n_ref) ** 3
)
var_ref = (
np.sum(
(placements_ref - np.mean(placements_ref)) ** 2
) / (n_ref * (n_syn ** 2))
)
var_syn = (
np.sum(
(placements_syn - np.mean(placements_syn)) ** 2
) / ((n_ref ** 2) * n_syn)
)
quantile_prob = (1 - alpha / 2) if alternative == "two-sided" else (1 - alpha)
quantile_alpha = stats.norm.ppf(quantile_prob, loc=0, scale=1)
quantile_power = stats.norm.ppf(power, loc=0, scale=1)
# Convert `nobs_ratio` to proportion of total allocated to reference group
prop_treatment = 1 / (1 + nobs_ratio)
prop_reference = 1 - prop_treatment
var_terms = np.sqrt(
prop_reference * var_syn + (1 - prop_reference) * var_ref
)
quantiles_terms = sd_overall * quantile_alpha + quantile_power * var_terms
# Add a small epsilon to avoid division by zero when there is no
# treatment effect, i.e. p_hat = 0.5
nobs_total = (quantiles_terms**2) / (
prop_reference
* (1 - prop_reference)
* (relative_effect - 0.5 + 1e-12) ** 2
)
nobs_treat = nobs_total * (1 - prop_reference)
nobs_ref = nobs_total * prop_reference
return Holder(
nobs_total=nobs_total.item(),
nobs_treat=nobs_treat.item(),
nobs_ref=nobs_ref.item(),
relative_effect=relative_effect.item(),
power=power,
alpha=alpha,
)
Last update:
Nov 14, 2024