# -*- coding: utf-8 -*-
"""Tests and Confidence Intervals for Binomial Proportions
Created on Fri Mar 01 00:23:07 2013
Author: Josef Perktold
License: BSD-3
"""
from statsmodels.compat.python import lzip, range
import numpy as np
from scipy import stats, optimize
from statsmodels.stats.base import AllPairsResults
#import statsmodels.stats.multitest as smt
[docs]def proportion_confint(count, nobs, alpha=0.05, method='normal'):
'''confidence interval for a binomial proportion
Parameters
----------
count : int or array
number of successes
nobs : int
total number of trials
alpha : float in (0, 1)
significance level, default 0.05
method : string in ['normal']
method to use for confidence interval,
currently available methods :
- `normal` : asymptotic normal approximation
- `agresti_coull` : Agresti-Coull interval
- `beta` : Clopper-Pearson interval based on Beta distribution
- `wilson` : Wilson Score interval
- `jeffrey` : Jeffrey's Bayesian Interval
- `binom_test` : experimental, inversion of binom_test
Returns
-------
ci_low, ci_upp : float
lower and upper confidence level with coverage (approximately) 1-alpha.
Note: Beta has coverage
coverage is only 1-alpha on average for some other methods.)
Notes
-----
Beta, the Clopper-Pearson interval has coverage at least 1-alpha, but is
in general conservative. Most of the other methods have average coverage
equal to 1-alpha, but will have smaller coverage in some cases.
Method "binom_test" directly inverts the binomial test in scipy.stats.
which has discrete steps.
TODO: binom_test intervals raise an exception in small samples if one
interval bound is close to zero or one.
References
----------
http://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval
Brown, Lawrence D.; Cai, T. Tony; DasGupta, Anirban (2001). "Interval
Estimation for a Binomial Proportion",
Statistical Science 16 (2): 101–133. doi:10.1214/ss/1009213286.
TODO: Is this the correct one ?
'''
q_ = count * 1. / nobs
alpha_2 = 0.5 * alpha
if method == 'normal':
std_ = np.sqrt(q_ * (1 - q_) / nobs)
dist = stats.norm.isf(alpha / 2.) * std_
ci_low = q_ - dist
ci_upp = q_ + dist
elif method == 'binom_test':
# inverting the binomial test
def func(qi):
#return stats.binom_test(qi * nobs, nobs, p=q_) - alpha #/ 2.
return stats.binom_test(q_ * nobs, nobs, p=qi) - alpha
# Note: only approximate, step function at integer values of count
# possible problems if bounds are too narrow
# problem if we hit 0 or 1
# brentq fails ValueError: f(a) and f(b) must have different signs
ci_low = optimize.brentq(func, q_ * 0.1, q_)
#ci_low = stats.binom_test(qi_low * nobs, nobs, p=q_)
#ci_low = np.floor(qi_low * nobs) / nobs
ub = np.minimum(q_ + 2 * (q_ - ci_low), 1)
ci_upp = optimize.brentq(func, q_, ub)
#ci_upp = stats.binom_test(qi_upp * nobs, nobs, p=q_)
#ci_upp = np.ceil(qi_upp * nobs) / nobs
# TODO: check if we should round up or down, or interpolate
elif method == 'beta':
ci_low = stats.beta.ppf(alpha_2 , count, nobs - count + 1)
ci_upp = stats.beta.isf(alpha_2, count + 1, nobs - count)
elif method == 'agresti_coull':
crit = stats.norm.isf(alpha / 2.)
nobs_c = nobs + crit**2
q_c = (count + crit**2 / 2.) / nobs_c
std_c = np.sqrt(q_c * (1. - q_c) / nobs_c)
dist = crit * std_c
ci_low = q_c - dist
ci_upp = q_c + dist
elif method == 'wilson':
crit = stats.norm.isf(alpha / 2.)
crit2 = crit**2
denom = 1 + crit2 / nobs
center = (q_ + crit2 / (2 * nobs)) / denom
dist = crit * np.sqrt(q_ * (1. - q_) / nobs + crit2 / (4. * nobs**2))
dist /= denom
ci_low = center - dist
ci_upp = center + dist
elif method == 'jeffrey':
ci_low, ci_upp = stats.beta.interval(1 - alpha, count + 0.5,
nobs - count + 0.5)
else:
raise NotImplementedError('method "%s" is not available' % method)
return ci_low, ci_upp
[docs]def samplesize_confint_proportion(proportion, half_length, alpha=0.05,
method='normal'):
'''find sample size to get desired confidence interval length
Parameters
----------
proportion : float in (0, 1)
proportion or quantile
half_length : float in (0, 1)
desired half length of the confidence interval
alpha : float in (0, 1)
significance level, default 0.05,
coverage of the two-sided interval is (approximately) ``1 - alpha``
method : string in ['normal']
method to use for confidence interval,
currently only normal approximation
Returns
-------
n : float
sample size to get the desired half length of the confidence interval
Notes
-----
this is mainly to store the formula.
possible application: number of replications in bootstrap samples
'''
q_ = proportion
if method == 'normal':
n = q_ * (1 - q_) / (half_length / stats.norm.isf(alpha / 2.))**2
else:
raise NotImplementedError('only "normal" is available')
return n
[docs]def proportion_effectsize(prop1, prop2, method='normal'):
'''effect size for a test comparing two proportions
for use in power function
Parameters
----------
prop1, prop2: float or array_like
Returns
-------
es : float or ndarray
effect size for (transformed) prop1 - prop2
Notes
-----
only method='normal' is implemented to match pwr.p2.test
see http://www.statmethods.net/stats/power.html
Effect size for `normal` is defined as ::
2 * (arcsin(sqrt(prop1)) - arcsin(sqrt(prop2)))
I think other conversions to normality can be used, but I need to check.
Examples
--------
>>> smpr.proportion_effectsize(0.5, 0.4)
0.20135792079033088
>>> smpr.proportion_effectsize([0.3, 0.4, 0.5], 0.4)
array([-0.21015893, 0. , 0.20135792])
'''
if method != 'normal':
raise ValueError('only "normal" is implemented')
es = 2 * (np.arcsin(np.sqrt(prop1)) - np.arcsin(np.sqrt(prop2)))
return es
[docs]def std_prop(prop, nobs):
'''standard error for the estimate of a proportion
This is just ``np.sqrt(p * (1. - p) / nobs)``
Parameters
----------
prop : array_like
proportion
nobs : int, array_like
number of observations
Returns
-------
std : array_like
standard error for a proportion of nobs independent observations
'''
return np.sqrt(prop * (1. - prop) / nobs)
def _power_ztost(mean_low, var_low, mean_upp, var_upp, mean_alt, var_alt,
alpha=0.05, discrete=True, dist='norm', nobs=None,
continuity=0, critval_continuity=0):
'''Generic statistical power function for normal based equivalence test
This includes options to adjust the normal approximation and can use
the binomial to evaluate the probability of the rejection region
see power_ztost_prob for a description of the options
'''
# TODO: refactor structure, separate norm and binom better
if not isinstance(continuity, tuple):
continuity = (continuity, continuity)
crit = stats.norm.isf(alpha)
k_low = mean_low + np.sqrt(var_low) * crit
k_upp = mean_upp - np.sqrt(var_upp) * crit
if discrete or dist == 'binom':
k_low = np.ceil(k_low * nobs + 0.5 * critval_continuity)
k_upp = np.trunc(k_upp * nobs - 0.5 * critval_continuity)
if dist == 'norm':
#need proportion
k_low = (k_low) * 1. / nobs #-1 to match PASS
k_upp = k_upp * 1. / nobs
# else:
# if dist == 'binom':
# #need counts
# k_low *= nobs
# k_upp *= nobs
#print mean_low, np.sqrt(var_low), crit, var_low
#print mean_upp, np.sqrt(var_upp), crit, var_upp
if np.any(k_low > k_upp): #vectorize
import warnings
warnings.warn("no overlap, power is zero", UserWarning)
std_alt = np.sqrt(var_alt)
z_low = (k_low - mean_alt - continuity[0] * 0.5 / nobs) / std_alt
z_upp = (k_upp - mean_alt + continuity[1] * 0.5 / nobs) / std_alt
if dist == 'norm':
power = stats.norm.cdf(z_upp) - stats.norm.cdf(z_low)
elif dist == 'binom':
power = (stats.binom.cdf(k_upp, nobs, mean_alt) -
stats.binom.cdf(k_low-1, nobs, mean_alt))
return power, (k_low, k_upp, z_low, z_upp)
[docs]def binom_tost(count, nobs, low, upp):
'''exact TOST test for one proportion using binomial distribution
Parameters
----------
count : integer or array_like
the number of successes in nobs trials.
nobs : integer
the number of trials or observations.
low, upp : floats
lower and upper limit of equivalence region
Returns
-------
pvalue : float
p-value of equivalence test
pval_low, pval_upp : floats
p-values of lower and upper one-sided tests
'''
# binom_test_stat only returns pval
tt1 = binom_test(count, nobs, alternative='larger', prop=low)
tt2 = binom_test(count, nobs, alternative='smaller', prop=upp)
return np.maximum(tt1, tt2), tt1, tt2,
[docs]def binom_tost_reject_interval(low, upp, nobs, alpha=0.05):
'''rejection region for binomial TOST
The interval includes the end points,
`reject` if and only if `r_low <= x <= r_upp`.
The interval might be empty with `r_upp < r_low`.
Parameters
----------
low, upp : floats
lower and upper limit of equivalence region
nobs : integer
the number of trials or observations.
Returns
-------
x_low, x_upp : float
lower and upper bound of rejection region
'''
x_low = stats.binom.isf(alpha, nobs, low) + 1
x_upp = stats.binom.ppf(alpha, nobs, upp) - 1
return x_low, x_upp
[docs]def binom_test_reject_interval(value, nobs, alpha=0.05, alternative='two-sided'):
'''rejection region for binomial test for one sample proportion
The interval includes the end points of the rejection region.
Parameters
----------
value : float
proportion under the Null hypothesis
nobs : integer
the number of trials or observations.
Returns
-------
x_low, x_upp : float
lower and upper bound of rejection region
'''
if alternative in ['2s', 'two-sided']:
alternative = '2s' # normalize alternative name
alpha = alpha / 2
if alternative in ['2s', 'smaller']:
x_low = stats.binom.ppf(alpha, nobs, value) - 1
else:
x_low = 0
if alternative in ['2s', 'larger']:
x_upp = stats.binom.isf(alpha, nobs, value) + 1
else :
x_upp = nobs
return x_low, x_upp
[docs]def binom_test(count, nobs, prop=0.5, alternative='two-sided'):
'''Perform a test that the probability of success is p.
This is an exact, two-sided test of the null hypothesis
that the probability of success in a Bernoulli experiment
is `p`.
Parameters
----------
count : integer or array_like
the number of successes in nobs trials.
nobs : integer
the number of trials or observations.
prop : float, optional
The probability of success under the null hypothesis,
`0 <= prop <= 1`. The default value is `prop = 0.5`
alternative : string in ['two-sided', 'smaller', 'larger']
alternative hypothesis, which can be two-sided or either one of the
one-sided tests.
Returns
-------
p-value : float
The p-value of the hypothesis test
Notes
-----
This uses scipy.stats.binom_test for the two-sided alternative.
'''
if np.any(prop > 1.0) or np.any(prop < 0.0):
raise ValueError("p must be in range [0,1]")
if alternative in ['2s', 'two-sided']:
pval = stats.binom_test(count, n=nobs, p=prop)
elif alternative in ['l', 'larger']:
pval = stats.binom.sf(count-1, nobs, prop)
elif alternative in ['s', 'smaller']:
pval = stats.binom.cdf(count, nobs, prop)
else:
raise ValueError('alternative not recognized\n'
'should be two-sided, larger or smaller')
return pval
[docs]def power_binom_tost(low, upp, nobs, p_alt=None, alpha=0.05):
if p_alt is None:
p_alt = 0.5 * (low + upp)
x_low, x_upp = binom_tost_reject_interval(low, upp, nobs, alpha=alpha)
power = (stats.binom.cdf(x_upp, nobs, p_alt) -
stats.binom.cdf(x_low-1, nobs, p_alt))
return power
[docs]def power_ztost_prop(low, upp, nobs, p_alt, alpha=0.05, dist='norm',
variance_prop=None, discrete=True, continuity=0,
critval_continuity=0):
'''Power of proportions equivalence test based on normal distribution
Parameters
----------
low, upp : floats
lower and upper limit of equivalence region
nobs : int
number of observations
p_alt : float in (0,1)
proportion under the alternative
alpha : float in (0,1)
significance level of the test
dist : string in ['norm', 'binom']
This defines the distribution to evalute the power of the test. The
critical values of the TOST test are always based on the normal
approximation, but the distribution for the power can be either the
normal (default) or the binomial (exact) distribution.
variance_prop : None or float in (0,1)
If this is None, then the variances for the two one sided tests are
based on the proportions equal to the equivalence limits.
If variance_prop is given, then it is used to calculate the variance
for the TOST statistics. If this is based on an sample, then the
estimated proportion can be used.
discrete : bool
If true, then the critical values of the rejection region are converted
to integers. If dist is "binom", this is automatically assumed.
If discrete is false, then the TOST critical values are used as
floating point numbers, and the power is calculated based on the
rejection region that is not discretized.
continuity : bool or float
adjust the rejection region for the normal power probability. This has
and effect only if ``dist='norm'``
critval_continuity : bool or float
If this is non-zero, then the critical values of the tost rejection
region are adjusted before converting to integers. This affects both
distributions, ``dist='norm'`` and ``dist='binom'``.
Returns
-------
power : float
statistical power of the equivalence test.
(k_low, k_upp, z_low, z_upp) : tuple of floats
critical limits in intermediate steps
temporary return, will be changed
Notes
-----
In small samples the power for the ``discrete`` version, has a sawtooth
pattern as a function of the number of observations. As a consequence,
small changes in the number of observations or in the normal approximation
can have a large effect on the power.
``continuity`` and ``critval_continuity`` are added to match some results
of PASS, and are mainly to investigate the sensitivity of the ztost power
to small changes in the rejection region. From my interpretation of the
equations in the SAS manual, both are zero in SAS.
works vectorized
**verification:**
The ``dist='binom'`` results match PASS,
The ``dist='norm'`` results look reasonable, but no benchmark is available.
References
----------
SAS Manual: Chapter 68: The Power Procedure, Computational Resources
PASS Chapter 110: Equivalence Tests for One Proportion.
'''
mean_low = low
var_low = std_prop(low, nobs)**2
mean_upp = upp
var_upp = std_prop(upp, nobs)**2
mean_alt = p_alt
var_alt = std_prop(p_alt, nobs)**2
if variance_prop is not None:
var_low = var_upp = std_prop(variance_prop, nobs)**2
power = _power_ztost(mean_low, var_low, mean_upp, var_upp, mean_alt, var_alt,
alpha=alpha, discrete=discrete, dist=dist, nobs=nobs,
continuity=continuity, critval_continuity=critval_continuity)
return np.maximum(power[0], 0), power[1:]
def _table_proportion(count, nobs):
'''create a k by 2 contingency table for proportion
helper function for proportions_chisquare
Parameters
----------
count : integer or array_like
the number of successes in nobs trials.
nobs : integer
the number of trials or observations.
Returns
-------
table : ndarray
(k, 2) contingency table
Notes
-----
recent scipy has more elaborate contingency table functions
'''
table = np.column_stack((count, nobs - count))
expected = table.sum(0) * table.sum(1)[:,None] * 1. / table.sum()
n_rows = table.shape[0]
return table, expected, n_rows
[docs]def proportions_ztest(count, nobs, value=None, alternative='two-sided',
prop_var=False):
'''test for proportions based on normal (z) test
Parameters
----------
count : integer or array_like
the number of successes in nobs trials. If this is array_like, then
the assumption is that this represents the number of successes for
each independent sample
nobs : integer
the number of trials or observations, with the same length as
count.
value : None or float or array_like
This is the value of the null hypothesis equal to the proportion in the
case of a one sample test. In the case of a two-sample test, the
null hypothesis is that prop[0] - prop[1] = value, where prop is the
proportion in the two samples
alternative : string in ['two-sided', 'smaller', 'larger']
The alternative hypothesis can be either two-sided or one of the one-
sided tests, smaller means that the alternative hypothesis is
``prop < value` and larger means ``prop > value``, or the corresponding
inequality for the two sample test.
prop_var : False or float in (0, 1)
If prop_var is false, then the variance of the proportion estimate is
calculated based on the sample proportion. Alternatively, a proportion
can be specified to calculate this variance. Common use case is to
use the proportion under the Null hypothesis to specify the variance
of the proportion estimate.
TODO: change options similar to propotion_ztost ?
Returns
-------
zstat : float
test statistic for the z-test
p-value : float
p-value for the z-test
Notes
-----
This uses a simple normal test for proportions. It should be the same as
running the mean z-test on the data encoded 1 for event and 0 for no event,
so that the sum corresponds to count.
In the one and two sample cases with two-sided alternative, this test
produces the same p-value as ``proportions_chisquare``, since the
chisquare is the distribution of the square of a standard normal
distribution.
(TODO: verify that this really holds)
TODO: add continuity correction or other improvements for small samples.
'''
prop = count * 1. / nobs
k_sample = np.size(prop)
if k_sample == 1:
diff = prop - value
elif k_sample == 2:
diff = prop[0] - prop[1] - value
else:
msg = 'more than two samples are not implemented yet'
raise NotImplementedError(msg)
p_pooled = np.sum(count) * 1. / np.sum(nobs)
nobs_fact = np.sum(1. / nobs)
if prop_var:
p_pooled = prop_var
var_ = p_pooled * (1 - p_pooled) * nobs_fact
std_diff = np.sqrt(var_)
from statsmodels.stats.weightstats import _zstat_generic2
return _zstat_generic2(diff, std_diff, alternative)
[docs]def proportions_ztost(count, nobs, low, upp, prop_var='sample'):
'''Equivalence test based on normal distribution
Parameters
----------
count : integer or array_like
the number of successes in nobs trials. If this is array_like, then
the assumption is that this represents the number of successes for
each independent sample
nobs : integer
the number of trials or observations, with the same length as
count.
low, upp : float
equivalence interval low < prop1 - prop2 < upp
prop_var : string or float in (0, 1)
prop_var determines which proportion is used for the calculation
of the standard deviation of the proportion estimate
The available options for string are 'sample' (default), 'null' and
'limits'. If prop_var is a float, then it is used directly.
Returns
-------
pvalue : float
pvalue of the non-equivalence test
t1, pv1 : tuple of floats
test statistic and pvalue for lower threshold test
t2, pv2 : tuple of floats
test statistic and pvalue for upper threshold test
Notes
-----
checked only for 1 sample case
'''
if prop_var == 'limits':
prop_var_low = low
prop_var_upp = upp
elif prop_var == 'sample':
prop_var_low = prop_var_upp = False #ztest uses sample
elif prop_var == 'null':
prop_var_low = prop_var_upp = 0.5 * (low + upp)
elif np.isreal(prop_var):
prop_var_low = prop_var_upp = prop_var
tt1 = proportions_ztest(count, nobs, alternative='larger',
prop_var=prop_var_low, value=low)
tt2 = proportions_ztest(count, nobs, alternative='smaller',
prop_var=prop_var_upp, value=upp)
return np.maximum(tt1[1], tt2[1]), tt1, tt2,
[docs]def proportions_chisquare(count, nobs, value=None):
'''test for proportions based on chisquare test
Parameters
----------
count : integer or array_like
the number of successes in nobs trials. If this is array_like, then
the assumption is that this represents the number of successes for
each independent sample
nobs : integer
the number of trials or observations, with the same length as
count.
value : None or float or array_like
Returns
-------
chi2stat : float
test statistic for the chisquare test
p-value : float
p-value for the chisquare test
(table, expected)
table is a (k, 2) contingency table, ``expected`` is the corresponding
table of counts that are expected under independence with given
margins
Notes
-----
Recent version of scipy.stats have a chisquare test for independence in
contingency tables.
This function provides a similar interface to chisquare tests as
``prop.test`` in R, however without the option for Yates continuity
correction.
count can be the count for the number of events for a single proportion,
or the counts for several independent proportions. If value is given, then
all proportions are jointly tested against this value. If value is not
given and count and nobs are not scalar, then the null hypothesis is
that all samples have the same proportion.
'''
nobs = np.atleast_1d(nobs)
table, expected, n_rows = _table_proportion(count, nobs)
if value is not None:
expected = np.column_stack((nobs * value, nobs * (1 - value)))
ddof = n_rows - 1
else:
ddof = n_rows
#print table, expected
chi2stat, pval = stats.chisquare(table.ravel(), expected.ravel(),
ddof=ddof)
return chi2stat, pval, (table, expected)
[docs]def proportions_chisquare_allpairs(count, nobs, multitest_method='hs'):
'''chisquare test of proportions for all pairs of k samples
Performs a chisquare test for proportions for all pairwise comparisons.
The alternative is two-sided
Parameters
----------
count : integer or array_like
the number of successes in nobs trials.
nobs : integer
the number of trials or observations.
prop : float, optional
The probability of success under the null hypothesis,
`0 <= prop <= 1`. The default value is `prop = 0.5`
multitest_method : string
This chooses the method for the multiple testing p-value correction,
that is used as default in the results.
It can be any method that is available in ``multipletesting``.
The default is Holm-Sidak 'hs'.
Returns
-------
result : AllPairsResults instance
The returned results instance has several statistics, such as p-values,
attached, and additional methods for using a non-default
``multitest_method``.
Notes
-----
Yates continuity correction is not available.
'''
#all_pairs = lmap(list, lzip(*np.triu_indices(4, 1)))
all_pairs = lzip(*np.triu_indices(len(count), 1))
pvals = [proportions_chisquare(count[list(pair)], nobs[list(pair)])[1]
for pair in all_pairs]
return AllPairsResults(pvals, all_pairs, multitest_method=multitest_method)
[docs]def proportions_chisquare_pairscontrol(count, nobs, value=None,
multitest_method='hs', alternative='two-sided'):
'''chisquare test of proportions for pairs of k samples compared to control
Performs a chisquare test for proportions for pairwise comparisons with a
control (Dunnet's test). The control is assumed to be the first element
of ``count`` and ``nobs``. The alternative is two-sided, larger or
smaller.
Parameters
----------
count : integer or array_like
the number of successes in nobs trials.
nobs : integer
the number of trials or observations.
prop : float, optional
The probability of success under the null hypothesis,
`0 <= prop <= 1`. The default value is `prop = 0.5`
multitest_method : string
This chooses the method for the multiple testing p-value correction,
that is used as default in the results.
It can be any method that is available in ``multipletesting``.
The default is Holm-Sidak 'hs'.
alternative : string in ['two-sided', 'smaller', 'larger']
alternative hypothesis, which can be two-sided or either one of the
one-sided tests.
Returns
-------
result : AllPairsResults instance
The returned results instance has several statistics, such as p-values,
attached, and additional methods for using a non-default
``multitest_method``.
Notes
-----
Yates continuity correction is not available.
``value`` and ``alternative`` options are not yet implemented.
'''
if (value is not None) or (not alternative in ['two-sided', '2s']):
raise NotImplementedError
#all_pairs = lmap(list, lzip(*np.triu_indices(4, 1)))
all_pairs = [(0, k) for k in range(1, len(count))]
pvals = [proportions_chisquare(count[list(pair)], nobs[list(pair)],
#alternative=alternative)[1]
)[1]
for pair in all_pairs]
return AllPairsResults(pvals, all_pairs, multitest_method=multitest_method)