'''extra statistical function and helper functions
contains:
* goodness-of-fit tests
- powerdiscrepancy
- gof_chisquare_discrete
- gof_binning_discrete
Author: Josef Perktold
License : BSD-3
changes
-------
2013-02-25 : add chisquare_power, effectsize and "value"
'''
from statsmodels.compat.python import range, lrange
import numpy as np
from scipy import stats
# copied from regression/stats.utils
[docs]def powerdiscrepancy(observed, expected, lambd=0.0, axis=0, ddof=0):
"""Calculates power discrepancy, a class of goodness-of-fit tests
as a measure of discrepancy between observed and expected data.
This contains several goodness-of-fit tests as special cases, see the
describtion of lambd, the exponent of the power discrepancy. The pvalue
is based on the asymptotic chi-square distribution of the test statistic.
freeman_tukey:
D(x|\theta) = \sum_j (\sqrt{x_j} - \sqrt{e_j})^2
Parameters
----------
o : Iterable
Observed values
e : Iterable
Expected values
lambd : float or string
* float : exponent `a` for power discrepancy
* 'loglikeratio': a = 0
* 'freeman_tukey': a = -0.5
* 'pearson': a = 1 (standard chisquare test statistic)
* 'modified_loglikeratio': a = -1
* 'cressie_read': a = 2/3
* 'neyman' : a = -2 (Neyman-modified chisquare, reference from a book?)
axis : int
axis for observations of one series
ddof : int
degrees of freedom correction,
Returns
-------
D_obs : Discrepancy of observed values
pvalue : pvalue
References
----------
Cressie, Noel and Timothy R. C. Read, Multinomial Goodness-of-Fit Tests,
Journal of the Royal Statistical Society. Series B (Methodological),
Vol. 46, No. 3 (1984), pp. 440-464
Campbell B. Read: Freeman-Tukey chi-squared goodness-of-fit statistics,
Statistics & Probability Letters 18 (1993) 271-278
Nobuhiro Taneichi, Yuri Sekiya, Akio Suzukawa, Asymptotic Approximations
for the Distributions of the Multinomial Goodness-of-Fit Statistics
under Local Alternatives, Journal of Multivariate Analysis 81, 335?359 (2002)
Steele, M. 1,2, C. Hurst 3 and J. Chaseling, Simulated Power of Discrete
Goodness-of-Fit Tests for Likert Type Data
Examples
--------
>>> observed = np.array([ 2., 4., 2., 1., 1.])
>>> expected = np.array([ 0.2, 0.2, 0.2, 0.2, 0.2])
for checking correct dimension with multiple series
>>> powerdiscrepancy(np.column_stack((observed,observed)).T, 10*expected, lambd='freeman_tukey',axis=1)
(array([[ 2.745166, 2.745166]]), array([[ 0.6013346, 0.6013346]]))
>>> powerdiscrepancy(np.column_stack((observed,observed)).T, 10*expected,axis=1)
(array([[ 2.77258872, 2.77258872]]), array([[ 0.59657359, 0.59657359]]))
>>> powerdiscrepancy(np.column_stack((observed,observed)).T, 10*expected, lambd=0,axis=1)
(array([[ 2.77258872, 2.77258872]]), array([[ 0.59657359, 0.59657359]]))
>>> powerdiscrepancy(np.column_stack((observed,observed)).T, 10*expected, lambd=1,axis=1)
(array([[ 3., 3.]]), array([[ 0.5578254, 0.5578254]]))
>>> powerdiscrepancy(np.column_stack((observed,observed)).T, 10*expected, lambd=2/3.0,axis=1)
(array([[ 2.89714546, 2.89714546]]), array([[ 0.57518277, 0.57518277]]))
>>> powerdiscrepancy(np.column_stack((observed,observed)).T, expected, lambd=2/3.0,axis=1)
(array([[ 2.89714546, 2.89714546]]), array([[ 0.57518277, 0.57518277]]))
>>> powerdiscrepancy(np.column_stack((observed,observed)), expected, lambd=2/3.0, axis=0)
(array([[ 2.89714546, 2.89714546]]), array([[ 0.57518277, 0.57518277]]))
each random variable can have different total count/sum
>>> powerdiscrepancy(np.column_stack((observed,2*observed)), expected, lambd=2/3.0, axis=0)
(array([[ 2.89714546, 5.79429093]]), array([[ 0.57518277, 0.21504648]]))
>>> powerdiscrepancy(np.column_stack((observed,2*observed)), expected, lambd=2/3.0, axis=0)
(array([[ 2.89714546, 5.79429093]]), array([[ 0.57518277, 0.21504648]]))
>>> powerdiscrepancy(np.column_stack((2*observed,2*observed)), expected, lambd=2/3.0, axis=0)
(array([[ 5.79429093, 5.79429093]]), array([[ 0.21504648, 0.21504648]]))
>>> powerdiscrepancy(np.column_stack((2*observed,2*observed)), 20*expected, lambd=2/3.0, axis=0)
(array([[ 5.79429093, 5.79429093]]), array([[ 0.21504648, 0.21504648]]))
>>> powerdiscrepancy(np.column_stack((observed,2*observed)), np.column_stack((10*expected,20*expected)), lambd=2/3.0, axis=0)
(array([[ 2.89714546, 5.79429093]]), array([[ 0.57518277, 0.21504648]]))
>>> powerdiscrepancy(np.column_stack((observed,2*observed)), np.column_stack((10*expected,20*expected)), lambd=-1, axis=0)
(array([[ 2.77258872, 5.54517744]]), array([[ 0.59657359, 0.2357868 ]]))
"""
o = np.array(observed)
e = np.array(expected)
if np.isfinite(lambd) == True: # check whether lambd is a number
a = lambd
else:
if lambd == 'loglikeratio': a = 0
elif lambd == 'freeman_tukey': a = -0.5
elif lambd == 'pearson': a = 1
elif lambd == 'modified_loglikeratio': a = -1
elif lambd == 'cressie_read': a = 2/3.0
else:
raise ValueError('lambd has to be a number or one of ' + \
'loglikeratio, freeman_tukey, pearson, ' +\
'modified_loglikeratio or cressie_read')
n = np.sum(o, axis=axis)
nt = n
if n.size>1:
n = np.atleast_2d(n)
if axis == 1:
nt = n.T # need both for 2d, n and nt for broadcasting
if e.ndim == 1:
e = np.atleast_2d(e)
if axis == 0:
e = e.T
if np.all(np.sum(e, axis=axis) == n):
p = e/(1.0*nt)
elif np.all(np.sum(e, axis=axis) == 1):
p = e
e = nt * e
else:
raise ValueError('observed and expected need to have the same ' +\
'number of observations, or e needs to add to 1')
k = o.shape[axis]
if e.shape[axis] != k:
raise ValueError('observed and expected need to have the same ' +\
'number of bins')
# Note: taken from formulas, to simplify cancel n
if a == 0: # log likelihood ratio
D_obs = 2*n * np.sum(o/(1.0*nt) * np.log(o/e), axis=axis)
elif a == -1: # modified log likelihood ratio
D_obs = 2*n * np.sum(e/(1.0*nt) * np.log(e/o), axis=axis)
else:
D_obs = 2*n/a/(a+1) * np.sum(o/(1.0*nt) * ((o/e)**a - 1), axis=axis)
return D_obs, stats.chi2.sf(D_obs,k-1-ddof)
#todo: need also binning for continuous distribution
# and separated binning function to be used for powerdiscrepancy
[docs]def gof_chisquare_discrete(distfn, arg, rvs, alpha, msg):
'''perform chisquare test for random sample of a discrete distribution
Parameters
----------
distname : string
name of distribution function
arg : sequence
parameters of distribution
alpha : float
significance level, threshold for p-value
Returns
-------
result : bool
0 if test passes, 1 if test fails
Notes
-----
originally written for scipy.stats test suite,
still needs to be checked for standalone usage, insufficient input checking
may not run yet (after copy/paste)
refactor: maybe a class, check returns, or separate binning from
test results
'''
# define parameters for test
## n=2000
n = len(rvs)
nsupp = 20
wsupp = 1.0/nsupp
## distfn = getattr(stats, distname)
## np.random.seed(9765456)
## rvs = distfn.rvs(size=n,*arg)
# construct intervals with minimum mass 1/nsupp
# intervalls are left-half-open as in a cdf difference
distsupport = lrange(max(distfn.a, -1000), min(distfn.b, 1000) + 1)
last = 0
distsupp = [max(distfn.a, -1000)]
distmass = []
for ii in distsupport:
current = distfn.cdf(ii,*arg)
if current - last >= wsupp-1e-14:
distsupp.append(ii)
distmass.append(current - last)
last = current
if current > (1-wsupp):
break
if distsupp[-1] < distfn.b:
distsupp.append(distfn.b)
distmass.append(1-last)
distsupp = np.array(distsupp)
distmass = np.array(distmass)
# convert intervals to right-half-open as required by histogram
histsupp = distsupp+1e-8
histsupp[0] = distfn.a
# find sample frequencies and perform chisquare test
#TODO: move to compatibility.py
if np.__version__ < '1.5':
freq,hsupp = np.histogram(rvs, histsupp, new=True)
else:
freq,hsupp = np.histogram(rvs,histsupp)
cdfs = distfn.cdf(distsupp,*arg)
(chis,pval) = stats.chisquare(np.array(freq),n*distmass)
return chis, pval, (pval > alpha), 'chisquare - test for %s' \
'at arg = %s with pval = %s' % (msg,str(arg),str(pval))
# copy/paste, remove code duplication when it works
[docs]def gof_binning_discrete(rvs, distfn, arg, nsupp=20):
'''get bins for chisquare type gof tests for a discrete distribution
Parameters
----------
rvs : array
sample data
distname : string
name of distribution function
arg : sequence
parameters of distribution
nsupp : integer
number of bins. The algorithm tries to find bins with equal weights.
depending on the distribution, the actual number of bins can be smaller.
Returns
-------
freq : array
empirical frequencies for sample; not normalized, adds up to sample size
expfreq : array
theoretical frequencies according to distribution
histsupp : array
bin boundaries for histogram, (added 1e-8 for numerical robustness)
Notes
-----
The results can be used for a chisquare test ::
(chis,pval) = stats.chisquare(freq, expfreq)
originally written for scipy.stats test suite,
still needs to be checked for standalone usage, insufficient input checking
may not run yet (after copy/paste)
refactor: maybe a class, check returns, or separate binning from
test results
todo :
optimal number of bins ? (check easyfit),
recommendation in literature at least 5 expected observations in each bin
'''
# define parameters for test
## n=2000
n = len(rvs)
wsupp = 1.0/nsupp
## distfn = getattr(stats, distname)
## np.random.seed(9765456)
## rvs = distfn.rvs(size=n,*arg)
# construct intervals with minimum mass 1/nsupp
# intervalls are left-half-open as in a cdf difference
distsupport = lrange(max(distfn.a, -1000), min(distfn.b, 1000) + 1)
last = 0
distsupp = [max(distfn.a, -1000)]
distmass = []
for ii in distsupport:
current = distfn.cdf(ii,*arg)
if current - last >= wsupp-1e-14:
distsupp.append(ii)
distmass.append(current - last)
last = current
if current > (1-wsupp):
break
if distsupp[-1] < distfn.b:
distsupp.append(distfn.b)
distmass.append(1-last)
distsupp = np.array(distsupp)
distmass = np.array(distmass)
# convert intervals to right-half-open as required by histogram
histsupp = distsupp+1e-8
histsupp[0] = distfn.a
# find sample frequencies and perform chisquare test
if np.__version__ < '1.5':
freq,hsupp = np.histogram(rvs, histsupp, new=True)
else:
freq,hsupp = np.histogram(rvs,histsupp)
#freq,hsupp = np.histogram(rvs,histsupp,new=True)
cdfs = distfn.cdf(distsupp,*arg)
return np.array(freq), n*distmass, histsupp
# -*- coding: utf-8 -*-
"""Extension to chisquare goodness-of-fit test
Created on Mon Feb 25 13:46:53 2013
Author: Josef Perktold
License: BSD-3
"""
def chisquare(f_obs, f_exp=None, value=0, ddof=0, return_basic=True):
'''chisquare goodness-of-fit test
The null hypothesis is that the distance between the expected distribution
and the observed frequencies is ``value``. The alternative hypothesis is
that the distance is larger than ``value``. ``value`` is normalized in
terms of effect size.
The standard chisquare test has the null hypothesis that ``value=0``, that
is the distributions are the same.
Notes
-----
The case with value greater than zero is similar to an equivalence test,
that the exact null hypothesis is replaced by an approximate hypothesis.
However, TOST "reverses" null and alternative hypothesis, while here the
alternative hypothesis is that the distance (divergence) is larger than a
threshold.
References
----------
McLaren, ...
Drost,...
See Also
--------
powerdiscrepancy
scipy.stats.chisquare
'''
f_obs = np.asarray(f_obs)
n_bins = len(f_obs)
nobs = f_obs.sum(0)
if f_exp is None:
# uniform distribution
f_exp = np.empty(n_bins, float)
f_exp.fill(nobs / float(n_bins))
f_exp = np.asarray(f_exp, float)
chisq = ((f_obs - f_exp)**2 / f_exp).sum(0)
if value == 0:
pvalue = stats.chi2.sf(chisq, n_bins - 1 - ddof)
else:
pvalue = stats.ncx2.sf(chisq, n_bins - 1 - ddof, value**2 * nobs)
if return_basic:
return chisq, pvalue
else:
return chisq, pvalue #TODO: replace with TestResults
def chisquare_power(effect_size, nobs, n_bins, alpha=0.05, ddof=0):
'''power of chisquare goodness of fit test
effect size is sqrt of chisquare statistic divided by nobs
Parameters
----------
effect_size : float
This is the deviation from the Null of the normalized chi_square
statistic. This follows Cohen's definition (sqrt).
nobs : int or float
number of observations
n_bins : int (or float)
number of bins, or points in the discrete distribution
alpha : float in (0,1)
significance level of the test, default alpha=0.05
Returns
-------
power : float
power of the test at given significance level at effect size
Notes
-----
This function also works vectorized if all arguments broadcast.
This can also be used to calculate the power for power divergence test.
However, for the range of more extreme values of the power divergence
parameter, this power is not a very good approximation for samples of
small to medium size (Drost et al. 1989)
References
----------
Drost, ...
See Also
--------
chisquare_effectsize
statsmodels.stats.GofChisquarePower
'''
crit = stats.chi2.isf(alpha, n_bins - 1 - ddof)
power = stats.ncx2.sf(crit, n_bins - 1 - ddof, effect_size**2 * nobs)
return power
[docs]def chisquare_effectsize(probs0, probs1, correction=None, cohen=True, axis=0):
'''effect size for a chisquare goodness-of-fit test
Parameters
----------
probs0 : array_like
probabilities or cell frequencies under the Null hypothesis
probs1 : array_like
probabilities or cell frequencies under the Alternative hypothesis
probs0 and probs1 need to have the same length in the ``axis`` dimension.
and broadcast in the other dimensions
Both probs0 and probs1 are normalized to add to one (in the ``axis``
dimension).
correction : None or tuple (nobs, df)
If None, then the effect size is the chisquare statistic divide by
the number of observations.
If the correction is a tuple (nobs, df), then the effectsize is
corrected to have less bias and a smaller variance. However, the
correction can make the effectsize negative. In that case, the
effectsize is set to zero.
Pederson and Johnson (1990) as referenced in McLaren et all. (1994)
cohen : bool
If True, then the square root is returned as in the definition of the
effect size by Cohen (1977), If False, then the original effect size
is returned.
axis : int
If the probability arrays broadcast to more than 1 dimension, then
this is the axis over which the sums are taken.
Returns
-------
effectsize : float
effect size of chisquare test
'''
probs0 = np.asarray(probs0, float)
probs1 = np.asarray(probs1, float)
probs0 = probs0 / probs0.sum(axis)
probs1 = probs1 / probs1.sum(axis)
d2 = ((probs1 - probs0)**2 / probs0).sum(axis)
if correction is not None:
nobs, df = correction
diff = ((probs1 - probs0) / probs0).sum(axis)
d2 = np.maximum((d2 * nobs - diff - df) / (nobs - 1.), 0)
if cohen:
return np.sqrt(d2)
else:
return d2
