'''Ttests and descriptive statistics with weights
Created on 2010-09-18
Author: josef-pktd
License: BSD (3-clause)
This follows in large parts the SPSS manual, which is largely the same as
the SAS manual with different, simpler notation.
Freq, Weight in SAS seems redundant since they always show up as product, SPSS
has only weights.
References
----------
SPSS manual
SAS manual
This has potential problems with ddof, I started to follow numpy with ddof=0
by default and users can change it, but this might still mess up the t-tests,
since the estimates for the standard deviation will be based on the ddof that
the user chooses.
- fixed ddof for the meandiff ttest, now matches scipy.stats.ttest_ind
'''
import numpy as np
from scipy import stats
from statsmodels.tools.decorators import OneTimeProperty
[docs]class DescrStatsW(object):
'''descriptive statistics with weights for simple case
assumes that the data is 1d or 2d with (nobs,nvars) ovservations in rows,
variables in columns, and that the same weight apply to each column.
If degrees of freedom correction is used than weights should add up to the
number of observations. ttest also assumes that the sum of weights
corresponds to the sample size.
This is essentially the same as replicating each observations by it's weight,
if the weights are integers.
Examples
--------
Note: I don't know the seed for the following, so the numbers will
differ
>>> x1_2d = 1.0 + np.random.randn(20, 3)
>>> w1 = np.random.randint(1,4, 20)
>>> d1 = DescrStatsW(x1_2d, weights=w1)
>>> d1.mean
array([ 1.42739844, 1.23174284, 1.083753 ])
>>> d1.var
array([ 0.94855633, 0.52074626, 1.12309325])
>>> d1.std_mean
array([ 0.14682676, 0.10878944, 0.15976497])
>>> tstat, pval, df = d1.ttest_mean(0)
>>> tstat; pval; df
array([ 9.72165021, 11.32226471, 6.78342055])
array([ 1.58414212e-12, 1.26536887e-14, 2.37623126e-08])
44.0
>>> tstat, pval, df = d1.ttest_mean([0, 1, 1])
>>> tstat; pval; df
array([ 9.72165021, 2.13019609, 0.52422632])
array([ 1.58414212e-12, 3.87842808e-02, 6.02752170e-01])
44.0
#if weithts are integers, then asrepeats can be used
>>> x1r = d1.asrepeats()
>>> x1r.shape
...
>>> stats.ttest_1samp(x1r, [0, 1, 1])
...
'''
def __init__(self, data, weights=None, ddof=0):
self.data = np.asarray(data)
if weights is None:
self.weights = np.ones(self.data.shape[0])
else:
self.weights = np.asarray(weights).squeeze().astype(float)
self.ddof = ddof
@OneTimeProperty
def sum_weights(self):
return self.weights.sum(0)
@OneTimeProperty
def nobs(self):
'''alias for number of observations/cases, equal to sum of weights
'''
return self.sum_weights
@OneTimeProperty
def sum(self):
return np.dot(self.data.T, self.weights)
@OneTimeProperty
def mean(self):
return self.sum / self.sum_weights
@OneTimeProperty
def demeaned(self):
return self.data - self.mean
@OneTimeProperty
def sumsquares(self):
return np.dot((self.demeaned**2).T, self.weights)
#need memoize instead of cache decorator
[docs] def var_ddof(self, ddof=0):
return sumsquares(self) / (self.sum_weights - ddof)
[docs] def std_ddof(self, ddof=0):
return np.sqrt(self.var(ddof=ddof))
@OneTimeProperty
def var(self):
'''variance with default degrees of freedom correction
'''
return self.sumsquares / (self.sum_weights - self.ddof)
@OneTimeProperty
def std(self):
return np.sqrt(self.var)
@OneTimeProperty
def cov(self):
'''covariance
'''
return np.dot(self.demeaned.T, self.demeaned) / self.sum_weights
@OneTimeProperty
def corrcoef(self):
'''correlation coefficient with default ddof for standard deviation
'''
return self.cov / self.std() / self.std()[:,None]
@OneTimeProperty
def std_mean(self):
'''standard deviation of mean
'''
return self.std / np.sqrt(self.sum_weights - 1)
[docs] def std_var(self):
pass
[docs] def confint_mean(self, alpha=0.05):
dof = self.sum_weights - 1
tcrit = stats.t.ppf((1+alpha)/2, dof)
lower = self.mean - tcrit * self.std_mean
upper = self.mean + tcrit * self.std_mean
return lower, upper
[docs] def ttest_mean(self, value, alternative='two-sided'):
'''ttest of Null hypothesis that mean is equal to value.
The alternative hypothesis H1 is defined by the following
'two-sided': H1: mean different than value
'larger' : H1: mean larger than value
'smaller' : H1: mean smaller than value
'''
tstat = (self.mean - value) / self.std_mean
dof = self.sum_weights - 1
from scipy import stats
if alternative == 'two-sided':
pvalue = stats.t.sf(np.abs(tstat), dof)*2
elif alternative == 'larger':
pvalue = stats.t.sf(tstat, dof)
elif alternative == 'smaller':
pvalue = stats.t.cdf(tstat, dof)
return tstat, pvalue, dof
[docs] def ttest_meandiff(self, other):
pass
[docs] def asrepeats(self):
'''get array that has repeats given by floor(weights)
observations with weight=0 are dropped
'''
w_int = np.floor(self.weights).astype(int)
return np.repeat(self.data, w_int, axis=0)
[docs]def tstat_generic(value, value2, std_diff, dof, alternative):
'''generic ttest to save typing'''
tstat = (value - value2) / std_diff
from scipy import stats
if alternative in ['two-sided', '2-sided', '2']:
pvalue = stats.t.sf(np.abs(tstat), dof)*2
elif alternative in ['larger', 'l']:
pvalue = stats.t.sf(tstat, dof)
elif alternative in ['smaller', 's']:
pvalue = stats.t.cdf(tstat, dof)
return tstat, pvalue
[docs]class CompareMeans(object):
'''temporary just to hold formulas
formulas should also be correct for unweighted means
not sure what happens if we have several variables.
everything should go through vectorized but not checked yet.
extend to any number of groups or write a version that works in that
case, like in SAS and SPSS.
Parameters
----------
'''
def __init__(self, d1, d2):
'''assume d1, d2 hold the relevant attributes
'''
self.d1 = d1
self.d2 = d2
#assume nobs is available
## if not hasattr(self.d1, 'nobs'):
## d1.nobs1 = d1.sum_weights.astype(float) #float just to make sure
## self.nobs2 = d2.sum_weights.astype(float)
@OneTimeProperty
def std_meandiff_separatevar(self):
#note I have too little control so far over ddof since it's an option
#formula assumes var has ddof=0, so we subtract ddof=1 now
d1 = self.d1
d2 = self.d2
return np.sqrt(d1.var / (d1.nobs-1) + d2.var / (d2.nobs-1))
@OneTimeProperty
def std_meandiff_pooledvar(self):
'''
uses d1.ddof, d2.ddof which should be one for the ttest
hardcoding ddof=1 for varpooled
'''
d1 = self.d1
d2 = self.d2
#could make var_pooled into attribute
var_pooled = ((d1.sumsquares + d2.sumsquares) /
#(d1.nobs - d1.ddof + d2.nobs - d2.ddof))
(d1.nobs - 1 + d2.nobs - 1))
return np.sqrt(var_pooled * (1. / d1.nobs + 1. /d2.nobs))
[docs] def ttest_ind(self, alternative='two-sided', usevar='pooled'):
'''ttest for the null hypothesis of identical means
note: I was looking for `usevar` option for the multiple comparison
tests correction
this should also be the same as onewaygls, except for ddof differences
'''
d1 = self.d1
d2 = self.d2
if usevar == 'pooled':
stdm = self.std_meandiff_pooledvar
dof = (d1.nobs - 1 + d2.nobs - 1)
elif usevar == 'separate':
stdm = self.std_meandiff_separatevar
#this follows blindly the SPSS manual
#except I assume var has ddof=0
#I should check d1.ddof, d2.ddof
sem1 = d1.var / (d1.nobs-1)
sem2 = d2.var / (d2.nobs-1)
semsum = sem1 + sem2
z1 = (sem1 / semsum)**2 / (d1.nobs - 1)
z2 = (sem2 / semsum)**2 / (d2.nobs - 1)
dof = 1. / (z1 + z2)
tstat, pval = tstat_generic(d1.mean, d2.mean, stdm, dof, alternative)
return tstat, pval, dof
[docs] def test_equal_var():
'''Levene test for independence
'''
d1 = self.d1
d2 = self.d2
#rewrite this, for now just use scipy.stats
return stats.levene(d1.data, d2.data)
def ttest_ind(x1, x2, alternative='two-sided',
usevar='pooled',
weights=(None, None)):
'''ttest independent sample
convenience function that uses the classes and throws away the intermediate
results,
compared to scipy stats: drops axis option, adds alternative, usevar, and
weights option
'''
cm = CompareMeans(DescrStatsW(x1, weights=weights[0], ddof=0),
DescrStatsW(x2, weights=weights[1], ddof=0))
tstat, pval, dof = cm.ttest_ind(alternative=alternative, usevar=usevar)
return tstat, pval, dof
if __name__ == '__main__':
from numpy.testing import assert_almost_equal, assert_equal
n1, n2 = 20,20
m1, m2 = 1, 1.2
x1 = m1 + np.random.randn(n1)
x2 = m2 + np.random.randn(n2)
x1_2d = m1 + np.random.randn(n1, 3)
x2_2d = m2 + np.random.randn(n2, 3)
w1_ = 2. * np.ones(n1)
w2_ = 2. * np.ones(n2)
w1 = np.random.randint(1,4, n1)
w2 = np.random.randint(1,4, n2)
d1 = DescrStatsW(x1)
print ttest_ind(x1, x2)
print ttest_ind(x1, x2, usevar='separate')
#print ttest_ind(x1, x2, usevar='separate')
print stats.ttest_ind(x1, x2)
print ttest_ind(x1, x2, usevar='separate', alternative='larger')
print ttest_ind(x1, x2, usevar='separate', alternative='smaller')
print ttest_ind(x1, x2, usevar='separate', weights=(w1_, w2_))
print stats.ttest_ind(np.r_[x1, x1], np.r_[x2,x2])
assert_almost_equal(ttest_ind(x1, x2, weights=(w1_, w2_))[:2],
stats.ttest_ind(np.r_[x1, x1], np.r_[x2,x2]))
d1w = DescrStatsW(x1, weights=w1)
d2w = DescrStatsW(x2, weights=w2)
x1r = d1w.asrepeats()
x2r = d2w.asrepeats()
print 'random weights'
print ttest_ind(x1, x2, weights=(w1, w2))
print stats.ttest_ind(x1r, x2r)
assert_almost_equal(ttest_ind(x1, x2, weights=(w1, w2))[:2],
stats.ttest_ind(x1r, x2r), 14)
#not the same as new version with random weights/replication
assert x1r.shape[0] == d1w.sum_weights
assert x2r.shape[0] == d2w.sum_weights
assert_almost_equal(x2r.var(), d2w.var, 14)
assert_almost_equal(x2r.std(), d2w.std, 14)
#one-sample tests
print d1.ttest_mean(3)
print stats.ttest_1samp(x1, 3)
print d1w.ttest_mean(3)
print stats.ttest_1samp(x1r, 3)
assert_almost_equal(d1.ttest_mean(3)[:2], stats.ttest_1samp(x1, 3), 11)
assert_almost_equal(d1w.ttest_mean(3)[:2], stats.ttest_1samp(x1r, 3), 11)
d1w_2d = DescrStatsW(x1_2d, weights=w1)
d2w_2d = DescrStatsW(x2_2d, weights=w2)
x1r_2d = d1w_2d.asrepeats()
x2r_2d = d2w_2d.asrepeats()
print d1w_2d.ttest_mean(3)
#scipy.stats.ttest is also vectorized
print stats.ttest_1samp(x1r_2d, 3)
t,p,d = d1w_2d.ttest_mean(3)
assert_almost_equal([t, p], stats.ttest_1samp(x1r_2d, 3), 11)
#print [stats.ttest_1samp(xi, 3) for xi in x1r_2d.T]
ressm = CompareMeans(d1w_2d, d2w_2d).ttest_ind()
resss = stats.ttest_ind(x1r_2d, x2r_2d)
assert_almost_equal(ressm[:2], resss, 14)
print ressm
print resss