'''
from pystatsmodels mailinglist 20100524
Notes:
- unfinished, unverified, but most parts seem to work in MonteCarlo
- one example taken from lecture notes looks ok
- needs cases with non-monotonic inequality for test to see difference between
one-step, step-up and step-down procedures
- FDR doesn't look really better then Bonferoni in the MC examples that I tried
update:
- now tested against R, stats and multtest,
I have all of their methods for p-value correction
- getting Hommel was impossible until I found reference for pvalue correction
- now, since I have p-values correction, some of the original tests (rej/norej)
implementation is not really needed anymore. I think I keep it for reference.
Test procedure for Hommel in development session log
- I haven't updated other functions and classes in here.
- multtest has some good helper function according to docs
- still need to update references, the real papers
- fdr with estimated true hypothesis still missing
- multiple comparison procedures incomplete or missing
- I will get multiple comparison for now only for independent case, which might
be conservative in correlated case (?).
some References:
Gibbons, Jean Dickinson and Chakraborti Subhabrata, 2003, Nonparametric Statistical
Inference, Fourth Edition, Marcel Dekker
p.363: 10.4 THE KRUSKAL-WALLIS ONE-WAY ANOVA TEST AND MULTIPLE COMPARISONS
p.367: multiple comparison for kruskal formula used in multicomp.kruskal
Sheskin, David J., 2004, Handbook of Parametric and Nonparametric Statistical
Procedures, 3rd ed., Chapman&Hall/CRC
Test 21: The Single-Factor Between-Subjects Analysis of Variance
Test 22: The Kruskal-Wallis One-Way Analysis of Variance by Ranks Test
Zwillinger, Daniel and Stephen Kokoska, 2000, CRC standard probability and
statistics tables and formulae, Chapman&Hall/CRC
14.9 WILCOXON RANKSUM (MANN WHITNEY) TEST
S. Paul Wright, Adjusted P-Values for Simultaneous Inference, Biometrics
Vol. 48, No. 4 (Dec., 1992), pp. 1005-1013, International Biometric Society
Stable URL: http://www.jstor.org/stable/2532694
(p-value correction for Hommel in appendix)
for multicomparison
new book "multiple comparison in R"
Hsu is a good reference but I don't have it.
Author: Josef Pktd and example from H Raja and rewrite from Vincent Davis
TODO
----
* handle exception if empty, shows up only sometimes when running this
- DONE I think
Traceback (most recent call last):
File "C:\Josef\eclipsegworkspace\statsmodels-josef-experimental-gsoc\scikits\statsmodels\sandbox\stats\multicomp.py", line 711, in <module>
print('sh', multipletests(tpval, alpha=0.05, method='sh')
File "C:\Josef\eclipsegworkspace\statsmodels-josef-experimental-gsoc\scikits\statsmodels\sandbox\stats\multicomp.py", line 241, in multipletests
rejectmax = np.max(np.nonzero(reject))
File "C:\Programs\Python25\lib\site-packages\numpy\core\fromnumeric.py", line 1765, in amax
return _wrapit(a, 'max', axis, out)
File "C:\Programs\Python25\lib\site-packages\numpy\core\fromnumeric.py", line 37, in _wrapit
result = getattr(asarray(obj),method)(*args, **kwds)
ValueError: zero-size array to ufunc.reduce without identity
* name of function multipletests, rename to something like pvalue_correction?
'''
#import xlrd
#import xlwt
from __future__ import print_function
from statsmodels.compat.python import lzip, range, lrange, zip
import scipy.stats
import numpy
import numpy as np
import math
import copy
from scipy import stats
from statsmodels.iolib.table import SimpleTable
from numpy.testing import assert_almost_equal, assert_equal
#temporary circular import
from statsmodels.stats.multitest import multipletests, _ecdf as ecdf, fdrcorrection as fdrcorrection0, fdrcorrection_twostage
from statsmodels.graphics import utils
from statsmodels.tools.sm_exceptions import ValueWarning
qcrit = '''
2 3 4 5 6 7 8 9 10
5 3.64 5.70 4.60 6.98 5.22 7.80 5.67 8.42 6.03 8.91 6.33 9.32 6.58 9.67 6.80 9.97 6.99 10.24
6 3.46 5.24 4.34 6.33 4.90 7.03 5.30 7.56 5.63 7.97 5.90 8.32 6.12 8.61 6.32 8.87 6.49 9.10
7 3.34 4.95 4.16 5.92 4.68 6.54 5.06 7.01 5.36 7.37 5.61 7.68 5.82 7.94 6.00 8.17 6.16 8.37
8 3.26 4.75 4.04 5.64 4.53 6.20 4.89 6.62 5.17 6.96 5.40 7.24 5.60 7.47 5.77 7.68 5.92 7.86
9 3.20 4.60 3.95 5.43 4.41 5.96 4.76 6.35 5.02 6.66 5.24 6.91 5.43 7.13 5.59 7.33 5.74 7.49
10 3.15 4.48 3.88 5.27 4.33 5.77 4.65 6.14 4.91 6.43 5.12 6.67 5.30 6.87 5.46 7.05 5.60 7.21
11 3.11 4.39 3.82 5.15 4.26 5.62 4.57 5.97 4.82 6.25 5.03 6.48 5.20 6.67 5.35 6.84 5.49 6.99
12 3.08 4.32 3.77 5.05 4.20 5.50 4.51 5.84 4.75 6.10 4.95 6.32 5.12 6.51 5.27 6.67 5.39 6.81
13 3.06 4.26 3.73 4.96 4.15 5.40 4.45 5.73 4.69 5.98 4.88 6.19 5.05 6.37 5.19 6.53 5.32 6.67
14 3.03 4.21 3.70 4.89 4.11 5.32 4.41 5.63 4.64 5.88 4.83 6.08 4.99 6.26 5.13 6.41 5.25 6.54
15 3.01 4.17 3.67 4.84 4.08 5.25 4.37 5.56 4.59 5.80 4.78 5.99 4.94 6.16 5.08 6.31 5.20 6.44
16 3.00 4.13 3.65 4.79 4.05 5.19 4.33 5.49 4.56 5.72 4.74 5.92 4.90 6.08 5.03 6.22 5.15 6.35
17 2.98 4.10 3.63 4.74 4.02 5.14 4.30 5.43 4.52 5.66 4.70 5.85 4.86 6.01 4.99 6.15 5.11 6.27
18 2.97 4.07 3.61 4.70 4.00 5.09 4.28 5.38 4.49 5.60 4.67 5.79 4.82 5.94 4.96 6.08 5.07 6.20
19 2.96 4.05 3.59 4.67 3.98 5.05 4.25 5.33 4.47 5.55 4.65 5.73 4.79 5.89 4.92 6.02 5.04 6.14
20 2.95 4.02 3.58 4.64 3.96 5.02 4.23 5.29 4.45 5.51 4.62 5.69 4.77 5.84 4.90 5.97 5.01 6.09
24 2.92 3.96 3.53 4.55 3.90 4.91 4.17 5.17 4.37 5.37 4.54 5.54 4.68 5.69 4.81 5.81 4.92 5.92
30 2.89 3.89 3.49 4.45 3.85 4.80 4.10 5.05 4.30 5.24 4.46 5.40 4.60 5.54 4.72 5.65 4.82 5.76
40 2.86 3.82 3.44 4.37 3.79 4.70 4.04 4.93 4.23 5.11 4.39 5.26 4.52 5.39 4.63 5.50 4.73 5.60
60 2.83 3.76 3.40 4.28 3.74 4.59 3.98 4.82 4.16 4.99 4.31 5.13 4.44 5.25 4.55 5.36 4.65 5.45
120 2.80 3.70 3.36 4.20 3.68 4.50 3.92 4.71 4.10 4.87 4.24 5.01 4.36 5.12 4.47 5.21 4.56 5.30
infinity 2.77 3.64 3.31 4.12 3.63 4.40 3.86 4.60 4.03 4.76 4.17 4.88 4.29 4.99 4.39 5.08 4.47 5.16
'''
res = [line.split() for line in qcrit.replace('infinity','9999').split('\n')]
c=np.array(res[2:-1]).astype(float)
#c[c==9999] = np.inf
ccols = np.arange(2,11)
crows = c[:,0]
cv005 = c[:, 1::2]
cv001 = c[:, 2::2]
from scipy import interpolate
[docs]def get_tukeyQcrit(k, df, alpha=0.05):
'''
return critical values for Tukey's HSD (Q)
Parameters
----------
k : int in {2, ..., 10}
number of tests
df : int
degrees of freedom of error term
alpha : {0.05, 0.01}
type 1 error, 1-confidence level
not enough error checking for limitations
'''
if alpha == 0.05:
intp = interpolate.interp1d(crows, cv005[:,k-2])
elif alpha == 0.01:
intp = interpolate.interp1d(crows, cv001[:,k-2])
else:
raise ValueError('only implemented for alpha equal to 0.01 and 0.05')
return intp(df)
def get_tukeyQcrit2(k, df, alpha=0.05):
'''
return critical values for Tukey's HSD (Q)
Parameters
----------
k : int in {2, ..., 10}
number of tests
df : int
degrees of freedom of error term
alpha : {0.05, 0.01}
type 1 error, 1-confidence level
not enough error checking for limitations
'''
from statsmodels.stats.libqsturng import qsturng
return qsturng(1-alpha, k, df)
def Tukeythreegene(first,second,third):
#Performing the Tukey HSD post-hoc test for three genes
## qwb = xlrd.open_workbook('F:/Lab/bioinformatics/qcrittable.xls')
## #opening the workbook containing the q crit table
## qwb.sheet_names()
## qcrittable = qwb.sheet_by_name(u'Sheet1')
firstmean = numpy.mean(first) #means of the three arrays
secondmean = numpy.mean(second)
thirdmean = numpy.mean(third)
firststd = numpy.std(first) #standard deviations of the threearrays
secondstd = numpy.std(second)
thirdstd = numpy.std(third)
firsts2 = math.pow(firststd,2) #standard deviation squared of the three arrays
seconds2 = math.pow(secondstd,2)
thirds2 = math.pow(thirdstd,2)
mserrornum = firsts2*2+seconds2*2+thirds2*2 #numerator for mean square error
mserrorden = (len(first)+len(second)+len(third))-3 #denominator for mean square error
mserror = mserrornum/mserrorden #mean square error
standarderror = math.sqrt(mserror/len(first))
#standard error, which is square root of mserror and the number of samples in a group
dftotal = len(first)+len(second)+len(third)-1 #various degrees of freedom
dfgroups = 2
dferror = dftotal-dfgroups
qcrit = 0.5 # fix arbitrary#qcrittable.cell(dftotal, 3).value
qcrit = get_tukeyQcrit(3, dftotal, alpha=0.05)
#getting the q critical value, for degrees of freedom total and 3 groups
qtest3to1 = (math.fabs(thirdmean-firstmean))/standarderror
#calculating q test statistic values
qtest3to2 = (math.fabs(thirdmean-secondmean))/standarderror
qtest2to1 = (math.fabs(secondmean-firstmean))/standarderror
conclusion = []
## print(qcrit
print(qtest3to1)
print(qtest3to2)
print(qtest2to1)
if(qtest3to1>qcrit): #testing all q test statistic values to q critical values
conclusion.append('3to1null')
else:
conclusion.append('3to1alt')
if(qtest3to2>qcrit):
conclusion.append('3to2null')
else:
conclusion.append('3to2alt')
if(qtest2to1>qcrit):
conclusion.append('2to1null')
else:
conclusion.append('2to1alt')
return conclusion
#rewrite by Vincent
def Tukeythreegene2(genes): #Performing the Tukey HSD post-hoc test for three genes
"""gend is a list, ie [first, second, third]"""
# qwb = xlrd.open_workbook('F:/Lab/bioinformatics/qcrittable.xls')
#opening the workbook containing the q crit table
# qwb.sheet_names()
# qcrittable = qwb.sheet_by_name(u'Sheet1')
means = []
stds = []
for gene in genes:
means.append(numpy.mean(gene))
std.append(numpy.std(gene))
#firstmean = numpy.mean(first) #means of the three arrays
#secondmean = numpy.mean(second)
#thirdmean = numpy.mean(third)
#firststd = numpy.std(first) #standard deviations of the three arrays
#secondstd = numpy.std(second)
#thirdstd = numpy.std(third)
stds2 = []
for std in stds:
stds2.append(math.pow(std,2))
#firsts2 = math.pow(firststd,2) #standard deviation squared of the three arrays
#seconds2 = math.pow(secondstd,2)
#thirds2 = math.pow(thirdstd,2)
#mserrornum = firsts2*2+seconds2*2+thirds2*2 #numerator for mean square error
mserrornum = sum(stds2)*2
mserrorden = (len(genes[0])+len(genes[1])+len(genes[2]))-3 #denominator for mean square error
mserror = mserrornum/mserrorden #mean square error
[docs]def catstack(args):
x = np.hstack(args)
labels = np.hstack([k*np.ones(len(arr)) for k,arr in enumerate(args)])
return x, labels
[docs]def maxzero(x):
'''find all up zero crossings and return the index of the highest
Not used anymore
>>> np.random.seed(12345)
>>> x = np.random.randn(8)
>>> x
array([-0.20470766, 0.47894334, -0.51943872, -0.5557303 , 1.96578057,
1.39340583, 0.09290788, 0.28174615])
>>> maxzero(x)
(4, array([1, 4]))
no up-zero-crossing at end
>>> np.random.seed(0)
>>> x = np.random.randn(8)
>>> x
array([ 1.76405235, 0.40015721, 0.97873798, 2.2408932 , 1.86755799,
-0.97727788, 0.95008842, -0.15135721])
>>> maxzero(x)
(None, array([6]))
'''
x = np.asarray(x)
cond1 = x[:-1] < 0
cond2 = x[1:] > 0
#allzeros = np.nonzero(np.sign(x[:-1])*np.sign(x[1:]) <= 0)[0] + 1
allzeros = np.nonzero((cond1 & cond2) | (x[1:]==0))[0] + 1
if x[-1] >=0:
maxz = max(allzeros)
else:
maxz = None
return maxz, allzeros
[docs]def maxzerodown(x):
'''find all up zero crossings and return the index of the highest
Not used anymore
>>> np.random.seed(12345)
>>> x = np.random.randn(8)
>>> x
array([-0.20470766, 0.47894334, -0.51943872, -0.5557303 , 1.96578057,
1.39340583, 0.09290788, 0.28174615])
>>> maxzero(x)
(4, array([1, 4]))
no up-zero-crossing at end
>>> np.random.seed(0)
>>> x = np.random.randn(8)
>>> x
array([ 1.76405235, 0.40015721, 0.97873798, 2.2408932 , 1.86755799,
-0.97727788, 0.95008842, -0.15135721])
>>> maxzero(x)
(None, array([6]))
'''
x = np.asarray(x)
cond1 = x[:-1] > 0
cond2 = x[1:] < 0
#allzeros = np.nonzero(np.sign(x[:-1])*np.sign(x[1:]) <= 0)[0] + 1
allzeros = np.nonzero((cond1 & cond2) | (x[1:]==0))[0] + 1
if x[-1] <=0:
maxz = max(allzeros)
else:
maxz = None
return maxz, allzeros
[docs]def rejectionline(n, alpha=0.5):
'''reference line for rejection in multiple tests
Not used anymore
from: section 3.2, page 60
'''
t = np.arange(n)/float(n)
frej = t/( t * (1-alpha) + alpha)
return frej
#I don't remember what I changed or why 2 versions,
#this follows german diss ??? with rline
#this might be useful if the null hypothesis is not "all effects are zero"
#rename to _bak and working again on fdrcorrection0
def fdrcorrection_bak(pvals, alpha=0.05, method='indep'):
'''Reject False discovery rate correction for pvalues
Old version, to be deleted
missing: methods that estimate fraction of true hypotheses
'''
pvals = np.asarray(pvals)
pvals_sortind = np.argsort(pvals)
pvals_sorted = pvals[pvals_sortind]
pecdf = ecdf(pvals_sorted)
if method in ['i', 'indep', 'p', 'poscorr']:
rline = pvals_sorted / alpha
elif method in ['n', 'negcorr']:
cm = np.sum(1./np.arange(1, len(pvals)))
rline = pvals_sorted / alpha * cm
elif method in ['g', 'onegcorr']: #what's this ? german diss
rline = pvals_sorted / (pvals_sorted*(1-alpha) + alpha)
elif method in ['oth', 'o2negcorr']: # other invalid, cut-paste
cm = np.sum(np.arange(len(pvals)))
rline = pvals_sorted / alpha /cm
else:
raise ValueError('method not available')
reject = pecdf >= rline
if reject.any():
rejectmax = max(np.nonzero(reject)[0])
else:
rejectmax = 0
reject[:rejectmax] = True
return reject[pvals_sortind.argsort()]
[docs]def mcfdr(nrepl=100, nobs=50, ntests=10, ntrue=6, mu=0.5, alpha=0.05, rho=0.):
'''MonteCarlo to test fdrcorrection
'''
nfalse = ntests - ntrue
locs = np.array([0.]*ntrue + [mu]*(ntests - ntrue))
results = []
for i in range(nrepl):
#rvs = locs + stats.norm.rvs(size=(nobs, ntests))
rvs = locs + randmvn(rho, size=(nobs, ntests))
tt, tpval = stats.ttest_1samp(rvs, 0)
res = fdrcorrection_bak(np.abs(tpval), alpha=alpha, method='i')
res0 = fdrcorrection0(np.abs(tpval), alpha=alpha)
#res and res0 give the same results
results.append([np.sum(res[:ntrue]), np.sum(res[ntrue:])] +
[np.sum(res0[:ntrue]), np.sum(res0[ntrue:])] +
res.tolist() +
np.sort(tpval).tolist() +
[np.sum(tpval[:ntrue]<alpha),
np.sum(tpval[ntrue:]<alpha)] +
[np.sum(tpval[:ntrue]<alpha/ntests),
np.sum(tpval[ntrue:]<alpha/ntests)])
return np.array(results)
[docs]def randmvn(rho, size=(1, 2), standardize=False):
'''create random draws from equi-correlated multivariate normal distribution
Parameters
----------
rho : float
correlation coefficient
size : tuple of int
size is interpreted (nobs, nvars) where each row
Returns
-------
rvs : ndarray
nobs by nvars where each row is a independent random draw of nvars-
dimensional correlated rvs
'''
nobs, nvars = size
if 0 < rho and rho < 1:
rvs = np.random.randn(nobs, nvars+1)
rvs2 = rvs[:,:-1] * np.sqrt((1-rho)) + rvs[:,-1:] * np.sqrt(rho)
elif rho ==0:
rvs2 = np.random.randn(nobs, nvars)
elif rho < 0:
if rho < -1./(nvars-1):
raise ValueError('rho has to be larger than -1./(nvars-1)')
elif rho == -1./(nvars-1):
rho = -1./(nvars-1+1e-10) #barely positive definite
#use Cholesky
A = rho*np.ones((nvars,nvars))+(1-rho)*np.eye(nvars)
rvs2 = np.dot(np.random.randn(nobs, nvars), np.linalg.cholesky(A).T)
if standardize:
rvs2 = stats.zscore(rvs2)
return rvs2
#============================
#
# Part 2: Multiple comparisons and independent samples tests
#
#============================
[docs]def tiecorrect(xranks):
'''
should be equivalent of scipy.stats.tiecorrect
'''
#casting to int rounds down, but not relevant for this case
rankbincount = np.bincount(np.asarray(xranks,dtype=int))
nties = rankbincount[rankbincount > 1]
ntot = float(len(xranks));
tiecorrection = 1 - (nties**3 - nties).sum()/(ntot**3 - ntot)
return tiecorrection
[docs]class GroupsStats(object):
'''
statistics by groups (another version)
groupstats as a class with lazy evaluation (not yet - decorators are still
missing)
written this time as equivalent of scipy.stats.rankdata
gs = GroupsStats(X, useranks=True)
assert_almost_equal(gs.groupmeanfilter, stats.rankdata(X[:,0]), 15)
TODO: incomplete doc strings
'''
def __init__(self, x, useranks=False, uni=None, intlab=None):
'''descriptive statistics by groups
Parameters
----------
x : array, 2d
first column data, second column group labels
useranks : boolean
if true, then use ranks as data corresponding to the
scipy.stats.rankdata definition (start at 1, ties get mean)
uni, intlab : arrays (optional)
to avoid call to unique, these can be given as inputs
'''
self.x = np.asarray(x)
if intlab is None:
uni, intlab = np.unique(x[:,1], return_inverse=True)
elif uni is None:
uni = np.unique(x[:,1])
self.useranks = useranks
self.uni = uni
self.intlab = intlab
self.groupnobs = groupnobs = np.bincount(intlab)
#temporary until separated and made all lazy
self.runbasic(useranks=useranks)
[docs] def runbasic_old(self, useranks=False):
#check: refactoring screwed up case useranks=True
#groupxsum = np.bincount(intlab, weights=X[:,0])
#groupxmean = groupxsum * 1.0 / groupnobs
x = self.x
if useranks:
self.xx = x[:,1].argsort().argsort() + 1 #rankraw
else:
self.xx = x[:,0]
self.groupsum = groupranksum = np.bincount(self.intlab, weights=self.xx)
#print('groupranksum', groupranksum, groupranksum.shape, self.groupnobs.shape
# start at 1 for stats.rankdata :
self.groupmean = grouprankmean = groupranksum * 1.0 / self.groupnobs # + 1
self.groupmeanfilter = grouprankmean[self.intlab]
#return grouprankmean[intlab]
[docs] def runbasic(self, useranks=False):
#check: refactoring screwed up case useranks=True
#groupxsum = np.bincount(intlab, weights=X[:,0])
#groupxmean = groupxsum * 1.0 / groupnobs
x = self.x
if useranks:
xuni, xintlab = np.unique(x[:,0], return_inverse=True)
ranksraw = x[:,0].argsort().argsort() + 1 #rankraw
self.xx = GroupsStats(np.column_stack([ranksraw, xintlab]),
useranks=False).groupmeanfilter
else:
self.xx = x[:,0]
self.groupsum = groupranksum = np.bincount(self.intlab, weights=self.xx)
#print('groupranksum', groupranksum, groupranksum.shape, self.groupnobs.shape
# start at 1 for stats.rankdata :
self.groupmean = grouprankmean = groupranksum * 1.0 / self.groupnobs # + 1
self.groupmeanfilter = grouprankmean[self.intlab]
#return grouprankmean[intlab]
[docs] def groupdemean(self):
return self.xx - self.groupmeanfilter
[docs] def groupsswithin(self):
xtmp = self.groupdemean()
return np.bincount(self.intlab, weights=xtmp**2)
[docs] def groupvarwithin(self):
return self.groupsswithin()/(self.groupnobs-1) #.sum()
[docs]class TukeyHSDResults(object):
"""Results from Tukey HSD test, with additional plot methods
Can also compute and plot additional post-hoc evaluations using this
results class.
Attributes
----------
reject : array of boolean, True if we reject Null for group pair
meandiffs : pairwise mean differences
confint : confidence interval for pairwise mean differences
std_pairs : standard deviation of pairwise mean differences
q_crit : critical value of studentized range statistic at given alpha
halfwidths : half widths of simultaneous confidence interval
Notes
-----
halfwidths is only available after call to `plot_simultaneous`.
Other attributes contain information about the data from the
MultiComparison instance: data, df_total, groups, groupsunique, variance.
"""
def __init__(self, mc_object, results_table, q_crit, reject=None,
meandiffs=None, std_pairs=None, confint=None, df_total=None,
reject2=None, variance=None):
self._multicomp = mc_object
self._results_table = results_table
self.q_crit = q_crit
self.reject = reject
self.meandiffs = meandiffs
self.std_pairs = std_pairs
self.confint = confint
self.df_total = df_total
self.reject2 = reject2
self.variance = variance
# Taken out of _multicomp for ease of access for unknowledgeable users
self.data = self._multicomp.data
self.groups =self._multicomp.groups
self.groupsunique = self._multicomp.groupsunique
def __str__(self):
return str(self._results_table)
[docs] def summary(self):
'''Summary table that can be printed
'''
return self._results_table
def _simultaneous_ci(self):
"""Compute simultaneous confidence intervals for comparison of means.
"""
self.halfwidths = simultaneous_ci(self.q_crit, self.variance,
self._multicomp.groupstats.groupnobs,
self._multicomp.pairindices)
[docs] def plot_simultaneous(self, comparison_name=None, ax=None, figsize=(10,6),
xlabel=None, ylabel=None):
"""Plot a universal confidence interval of each group mean
Visiualize significant differences in a plot with one confidence
interval per group instead of all pairwise confidence intervals.
Parameters
----------
comparison_name : string, optional
if provided, plot_intervals will color code all groups that are
significantly different from the comparison_name red, and will
color code insignificant groups gray. Otherwise, all intervals will
just be plotted in black.
ax : matplotlib axis, optional
An axis handle on which to attach the plot.
figsize : tuple, optional
tuple for the size of the figure generated
xlabel : string, optional
Name to be displayed on x axis
ylabel : string, optional
Name to be displayed on y axis
Returns
-------
fig : Matplotlib Figure object
handle to figure object containing interval plots
Notes
-----
Multiple comparison tests are nice, but lack a good way to be
visualized. If you have, say, 6 groups, showing a graph of the means
between each group will require 15 confidence intervals.
Instead, we can visualize inter-group differences with a single
interval for each group mean. Hochberg et al. [1] first proposed this
idea and used Tukey's Q critical value to compute the interval widths.
Unlike plotting the differences in the means and their respective
confidence intervals, any two pairs can be compared for significance
by looking for overlap.
References
----------
.. [*] Hochberg, Y., and A. C. Tamhane. Multiple Comparison Procedures.
Hoboken, NJ: John Wiley & Sons, 1987.
Examples
--------
>>> from statsmodels.examples.try_tukey_hsd import cylinders, cyl_labels
>>> from statsmodels.stats.multicomp import MultiComparison
>>> cardata = MultiComparison(cylinders, cyl_labels)
>>> results = cardata.tukeyhsd()
>>> results.plot_simultaneous()
<matplotlib.figure.Figure at 0x...>
This example shows an example plot comparing significant differences
in group means. Significant differences at the alpha=0.05 level can be
identified by intervals that do not overlap (i.e. USA vs Japan,
USA vs Germany).
>>> results.plot_simultaneous(comparison_name="USA")
<matplotlib.figure.Figure at 0x...>
Optionally provide one of the group names to color code the plot to
highlight group means different from comparison_name.
"""
fig, ax1 = utils.create_mpl_ax(ax)
if figsize is not None:
fig.set_size_inches(figsize)
if getattr(self, 'halfwidths', None) is None:
self._simultaneous_ci()
means = self._multicomp.groupstats.groupmean
sigidx = []
nsigidx = []
minrange = [means[i] - self.halfwidths[i] for i in range(len(means))]
maxrange = [means[i] + self.halfwidths[i] for i in range(len(means))]
if comparison_name is None:
ax1.errorbar(means, lrange(len(means)), xerr=self.halfwidths,
marker='o', linestyle='None', color='k', ecolor='k')
else:
if comparison_name not in self.groupsunique:
raise ValueError('comparison_name not found in group names.')
midx = np.where(self.groupsunique==comparison_name)[0][0]
for i in range(len(means)):
if self.groupsunique[i] == comparison_name:
continue
if (min(maxrange[i], maxrange[midx]) -
max(minrange[i], minrange[midx]) < 0):
sigidx.append(i)
else:
nsigidx.append(i)
#Plot the master comparison
ax1.errorbar(means[midx], midx, xerr=self.halfwidths[midx],
marker='o', linestyle='None', color='b', ecolor='b')
ax1.plot([minrange[midx]]*2, [-1, self._multicomp.ngroups],
linestyle='--', color='0.7')
ax1.plot([maxrange[midx]]*2, [-1, self._multicomp.ngroups],
linestyle='--', color='0.7')
#Plot those that are significantly different
if len(sigidx) > 0:
ax1.errorbar(means[sigidx], sigidx,
xerr=self.halfwidths[sigidx], marker='o',
linestyle='None', color='r', ecolor='r')
#Plot those that are not significantly different
if len(nsigidx) > 0:
ax1.errorbar(means[nsigidx], nsigidx,
xerr=self.halfwidths[nsigidx], marker='o',
linestyle='None', color='0.5', ecolor='0.5')
ax1.set_title('Multiple Comparisons Between All Pairs (Tukey)')
r = np.max(maxrange) - np.min(minrange)
ax1.set_ylim([-1, self._multicomp.ngroups])
ax1.set_xlim([np.min(minrange) - r / 10., np.max(maxrange) + r / 10.])
ax1.set_yticklabels(np.insert(self.groupsunique.astype(str), 0, ''))
ax1.set_yticks(np.arange(-1, len(means)+1))
ax1.set_xlabel(xlabel if xlabel is not None else '')
ax1.set_ylabel(ylabel if ylabel is not None else '')
return fig
[docs]class MultiComparison(object):
'''Tests for multiple comparisons
Parameters
----------
data : array
independent data samples
groups : array
group labels corresponding to each data point
group_order : list of strings, optional
the desired order for the group mean results to be reported in. If
not specified, results are reported in increasing order.
If group_order does not contain all labels that are in groups, then
only those observations are kept that have a label in group_order.
'''
def __init__(self, data, groups, group_order=None):
if len(data) != len(groups):
raise ValueError('data has %d elements and groups has %d' % (len(data), len(groups)))
self.data = np.asarray(data)
self.groups = groups = np.asarray(groups)
# Allow for user-provided sorting of groups
if group_order is None:
self.groupsunique, self.groupintlab = np.unique(groups,
return_inverse=True)
else:
#check if group_order has any names not in groups
for grp in group_order:
if grp not in groups:
raise ValueError(
"group_order value '%s' not found in groups"%grp)
self.groupsunique = np.array(group_order)
self.groupintlab = np.empty(len(data), int)
self.groupintlab.fill(-999) # instead of a nan
count = 0
for name in self.groupsunique:
idx = np.where(self.groups == name)[0]
count += len(idx)
self.groupintlab[idx] = np.where(self.groupsunique == name)[0]
if count != data.shape[0]:
#raise ValueError('group_order does not contain all groups')
# warn and keep only observations with label in group_order
import warnings
warnings.warn('group_order does not contain all groups:' +
' dropping observations', ValueWarning)
mask_keep = self.groupintlab != -999
self.groupintlab = self.groupintlab[mask_keep]
self.data = self.data[mask_keep]
self.groups = self.groups[mask_keep]
if len(self.groupsunique) < 2:
raise ValueError('2 or more groups required for multiple comparisons')
self.datali = [self.data[self.groups == k] for k in self.groupsunique]
self.pairindices = np.triu_indices(len(self.groupsunique), 1) #tuple
self.nobs = self.data.shape[0]
self.ngroups = len(self.groupsunique)
[docs] def getranks(self):
'''convert data to rankdata and attach
This creates rankdata as it is used for non-parametric tests, where
in the case of ties the average rank is assigned.
'''
#bug: the next should use self.groupintlab instead of self.groups
#update: looks fixed
#self.ranks = GroupsStats(np.column_stack([self.data, self.groups]),
self.ranks = GroupsStats(np.column_stack([self.data, self.groupintlab]),
useranks=True)
self.rankdata = self.ranks.groupmeanfilter
[docs] def kruskal(self, pairs=None, multimethod='T'):
'''
pairwise comparison for kruskal-wallis test
This is just a reimplementation of scipy.stats.kruskal and does
not yet use a multiple comparison correction.
'''
self.getranks()
tot = self.nobs
meanranks = self.ranks.groupmean
groupnobs = self.ranks.groupnobs
# simultaneous/separate treatment of multiple tests
f=(tot * (tot + 1.) / 12.) / stats.tiecorrect(self.rankdata) #(xranks)
print('MultiComparison.kruskal')
for i,j in zip(*self.pairindices):
#pdiff = np.abs(mrs[i] - mrs[j])
pdiff = np.abs(meanranks[i] - meanranks[j])
se = np.sqrt(f * np.sum(1. / groupnobs[[i,j]] )) #np.array([8,8]))) #Fixme groupnobs[[i,j]] ))
Q = pdiff / se
# TODO : print(statments, fix
print(i,j, pdiff, se, pdiff / se, pdiff / se > 2.6310)
print(stats.norm.sf(Q) * 2)
return stats.norm.sf(Q) * 2
[docs] def allpairtest(self, testfunc, alpha=0.05, method='bonf', pvalidx=1):
'''run a pairwise test on all pairs with multiple test correction
The statistical test given in testfunc is calculated for all pairs
and the p-values are adjusted by methods in multipletests. The p-value
correction is generic and based only on the p-values, and does not
take any special structure of the hypotheses into account.
Parameters
----------
testfunc : function
A test function for two (independent) samples. It is assumed that
the return value on position pvalidx is the p-value.
alpha : float
familywise error rate
method : string
This specifies the method for the p-value correction. Any method
of multipletests is possible.
pvalidx : int (default: 1)
position of the p-value in the return of testfunc
Returns
-------
sumtab : SimpleTable instance
summary table for printing
errors: TODO: check if this is still wrong, I think it's fixed.
results from multipletests are in different order
pval_corrected can be larger than 1 ???
'''
res = []
for i,j in zip(*self.pairindices):
res.append(testfunc(self.datali[i], self.datali[j]))
res = np.array(res)
reject, pvals_corrected, alphacSidak, alphacBonf = \
multipletests(res[:, pvalidx], alpha=0.05, method=method)
#print(np.column_stack([res[:,0],res[:,1], reject, pvals_corrected])
i1, i2 = self.pairindices
if pvals_corrected is None:
resarr = np.array(lzip(self.groupsunique[i1], self.groupsunique[i2],
np.round(res[:,0],4),
np.round(res[:,1],4),
reject),
dtype=[('group1', object),
('group2', object),
('stat',float),
('pval',float),
('reject', np.bool8)])
else:
resarr = np.array(lzip(self.groupsunique[i1], self.groupsunique[i2],
np.round(res[:,0],4),
np.round(res[:,1],4),
np.round(pvals_corrected,4),
reject),
dtype=[('group1', object),
('group2', object),
('stat',float),
('pval',float),
('pval_corr',float),
('reject', np.bool8)])
from statsmodels.iolib.table import SimpleTable
results_table = SimpleTable(resarr, headers=resarr.dtype.names)
results_table.title = (
'Test Multiple Comparison %s \n%s%4.2f method=%s'
% (testfunc.__name__, 'FWER=', alpha, method) +
'\nalphacSidak=%4.2f, alphacBonf=%5.3f'
% (alphacSidak, alphacBonf))
return results_table, (res, reject, pvals_corrected,
alphacSidak, alphacBonf), resarr
[docs] def tukeyhsd(self, alpha=0.05):
"""Tukey's range test to compare means of all pairs of groups
Parameters
----------
alpha : float, optional
Value of FWER at which to calculate HSD.
Returns
-------
results : TukeyHSDResults instance
A results class containing relevant data and some post-hoc
calculations
"""
self.groupstats = GroupsStats(
np.column_stack([self.data, self.groupintlab]),
useranks=False)
gmeans = self.groupstats.groupmean
gnobs = self.groupstats.groupnobs #var_ = self.groupstats.groupvarwithin() #possibly an error in varcorrection in this case
var_ = np.var(self.groupstats.groupdemean(), ddof=len(gmeans))
#res contains: 0:(idx1, idx2), 1:reject, 2:meandiffs, 3: std_pairs, 4:confint, 5:q_crit,
#6:df_total, 7:reject2
res = tukeyhsd(gmeans, gnobs, var_, df=None, alpha=alpha, q_crit=None)
resarr = np.array(lzip(self.groupsunique[res[0][0]], self.groupsunique[res[0][1]],
np.round(res[2],4),
np.round(res[4][:, 0],4),
np.round(res[4][:, 1],4),
res[1]),
dtype=[('group1', object),
('group2', object),
('meandiff',float),
('lower',float),
('upper',float),
('reject', np.bool8)])
results_table = SimpleTable(resarr, headers=resarr.dtype.names)
results_table.title = 'Multiple Comparison of Means - Tukey HSD,' + \
'FWER=%4.2f' % alpha
return TukeyHSDResults(self, results_table, res[5], res[1], res[2],
res[3], res[4], res[6], res[7], var_)
[docs]def rankdata(x):
'''rankdata, equivalent to scipy.stats.rankdata
just a different implementation, I have not yet compared speed
'''
uni, intlab = np.unique(x[:,0], return_inverse=True)
groupnobs = np.bincount(intlab)
groupxsum = np.bincount(intlab, weights=X[:,0])
groupxmean = groupxsum * 1.0 / groupnobs
rankraw = x[:,0].argsort().argsort()
groupranksum = np.bincount(intlab, weights=rankraw)
# start at 1 for stats.rankdata :
grouprankmean = groupranksum * 1.0 / groupnobs + 1
return grouprankmean[intlab]
#new
[docs]def compare_ordered(vals, alpha):
'''simple ordered sequential comparison of means
vals : array_like
means or rankmeans for independent groups
incomplete, no return, not used yet
'''
vals = np.asarray(vals)
alphaf = alpha # Notation ?
sortind = np.argsort(vals)
pvals = vals[sortind]
sortrevind = sortind.argsort()
ntests = len(vals)
#alphacSidak = 1 - np.power((1. - alphaf), 1./ntests)
#alphacBonf = alphaf / float(ntests)
v1, v2 = np.triu_indices(ntests, 1)
#v1,v2 have wrong sequence
for i in range(4):
for j in range(4,i, -1):
print(i,j)
[docs]def varcorrection_unbalanced(nobs_all, srange=False):
'''correction factor for variance with unequal sample sizes
this is just a harmonic mean
Parameters
----------
nobs_all : array_like
The number of observations for each sample
srange : bool
if true, then the correction is divided by the number of samples
for the variance of the studentized range statistic
Returns
-------
correction : float
Correction factor for variance.
Notes
-----
variance correction factor is
1/k * sum_i 1/n_i
where k is the number of samples and summation is over i=0,...,k-1.
If all n_i are the same, then the correction factor is 1.
This needs to be multiplied by the joint variance estimate, means square
error, MSE. To obtain the correction factor for the standard deviation,
square root needs to be taken.
'''
nobs_all = np.asarray(nobs_all)
if not srange:
return (1./nobs_all).sum()
else:
return (1./nobs_all).sum()/len(nobs_all)
[docs]def varcorrection_pairs_unbalanced(nobs_all, srange=False):
'''correction factor for variance with unequal sample sizes for all pairs
this is just a harmonic mean
Parameters
----------
nobs_all : array_like
The number of observations for each sample
srange : bool
if true, then the correction is divided by 2 for the variance of
the studentized range statistic
Returns
-------
correction : array
Correction factor for variance.
Notes
-----
variance correction factor is
1/k * sum_i 1/n_i
where k is the number of samples and summation is over i=0,...,k-1.
If all n_i are the same, then the correction factor is 1.
This needs to be multiplies by the joint variance estimate, means square
error, MSE. To obtain the correction factor for the standard deviation,
square root needs to be taken.
For the studentized range statistic, the resulting factor has to be
divided by 2.
'''
#TODO: test and replace with broadcasting
n1, n2 = np.meshgrid(nobs_all, nobs_all)
if not srange:
return (1./n1 + 1./n2)
else:
return (1./n1 + 1./n2) / 2.
[docs]def varcorrection_unequal(var_all, nobs_all, df_all):
'''return joint variance from samples with unequal variances and unequal
sample sizes
something is wrong
Parameters
----------
var_all : array_like
The variance for each sample
nobs_all : array_like
The number of observations for each sample
df_all : array_like
degrees of freedom for each sample
Returns
-------
varjoint : float
joint variance.
dfjoint : float
joint Satterthwait's degrees of freedom
Notes
-----
(copy, paste not correct)
variance is
1/k * sum_i 1/n_i
where k is the number of samples and summation is over i=0,...,k-1.
If all n_i are the same, then the correction factor is 1/n.
This needs to be multiplies by the joint variance estimate, means square
error, MSE. To obtain the correction factor for the standard deviation,
square root needs to be taken.
This is for variance of mean difference not of studentized range.
'''
var_all = np.asarray(var_all)
var_over_n = var_all *1./ nobs_all #avoid integer division
varjoint = var_over_n.sum()
dfjoint = varjoint**2 / (var_over_n**2 * df_all).sum()
return varjoint, dfjoint
[docs]def varcorrection_pairs_unequal(var_all, nobs_all, df_all):
'''return joint variance from samples with unequal variances and unequal
sample sizes for all pairs
something is wrong
Parameters
----------
var_all : array_like
The variance for each sample
nobs_all : array_like
The number of observations for each sample
df_all : array_like
degrees of freedom for each sample
Returns
-------
varjoint : array
joint variance.
dfjoint : array
joint Satterthwait's degrees of freedom
Notes
-----
(copy, paste not correct)
variance is
1/k * sum_i 1/n_i
where k is the number of samples and summation is over i=0,...,k-1.
If all n_i are the same, then the correction factor is 1.
This needs to be multiplies by the joint variance estimate, means square
error, MSE. To obtain the correction factor for the standard deviation,
square root needs to be taken.
TODO: something looks wrong with dfjoint, is formula from SPSS
'''
#TODO: test and replace with broadcasting
v1, v2 = np.meshgrid(var_all, var_all)
n1, n2 = np.meshgrid(nobs_all, nobs_all)
df1, df2 = np.meshgrid(df_all, df_all)
varjoint = v1/n1 + v2/n2
dfjoint = varjoint**2 / (df1 * (v1/n1)**2 + df2 * (v2/n2)**2)
return varjoint, dfjoint
def tukeyhsd(mean_all, nobs_all, var_all, df=None, alpha=0.05, q_crit=None):
'''simultaneous Tukey HSD
check: instead of sorting, I use absolute value of pairwise differences
in means. That's irrelevant for the test, but maybe reporting actual
differences would be better.
CHANGED: meandiffs are with sign, studentized range uses abs
q_crit added for testing
TODO: error in variance calculation when nobs_all is scalar, missing 1/n
'''
mean_all = np.asarray(mean_all)
#check if or when other ones need to be arrays
n_means = len(mean_all)
if df is None:
df = nobs_all - 1
if np.size(df) == 1: # assumes balanced samples with df = n - 1, n_i = n
df_total = n_means * df
df = np.ones(n_means) * df
else:
df_total = np.sum(df)
if (np.size(nobs_all) == 1) and (np.size(var_all) == 1):
#balanced sample sizes and homogenous variance
var_pairs = 1. * var_all / nobs_all * np.ones((n_means, n_means))
elif np.size(var_all) == 1:
#unequal sample sizes and homogenous variance
var_pairs = var_all * varcorrection_pairs_unbalanced(nobs_all,
srange=True)
elif np.size(var_all) > 1:
var_pairs, df_sum = varcorrection_pairs_unequal(nobs_all, var_all, df)
var_pairs /= 2.
#check division by two for studentized range
else:
raise ValueError('not supposed to be here')
#meandiffs_ = mean_all[:,None] - mean_all
meandiffs_ = mean_all - mean_all[:,None] #reverse sign, check with R example
std_pairs_ = np.sqrt(var_pairs)
#select all pairs from upper triangle of matrix
idx1, idx2 = np.triu_indices(n_means, 1)
meandiffs = meandiffs_[idx1, idx2]
std_pairs = std_pairs_[idx1, idx2]
st_range = np.abs(meandiffs) / std_pairs #studentized range statistic
df_total_ = max(df_total, 5) #TODO: smallest df in table
if q_crit is None:
q_crit = get_tukeyQcrit2(n_means, df_total, alpha=alpha)
reject = st_range > q_crit
crit_int = std_pairs * q_crit
reject2 = np.abs(meandiffs) > crit_int
confint = np.column_stack((meandiffs - crit_int, meandiffs + crit_int))
return (idx1, idx2), reject, meandiffs, std_pairs, confint, q_crit, \
df_total, reject2
def simultaneous_ci(q_crit, var, groupnobs, pairindices=None):
"""Compute simultaneous confidence intervals for comparison of means.
q_crit value is generated from tukey hsd test. Variance is considered
across all groups. Returned halfwidths can be thought of as uncertainty
intervals around each group mean. They allow for simultaneous
comparison of pairwise significance among any pairs (by checking for
overlap)
Parameters
----------
q_crit : float
The Q critical value studentized range statistic from Tukey's HSD
var : float
The group variance
groupnobs : array-like object
Number of observations contained in each group.
pairindices : tuple of lists, optional
Indices corresponding to the upper triangle of matrix. Computed
here if not supplied
Returns
-------
halfwidths : ndarray
Half the width of each confidence interval for each group given in
groupnobs
See Also
--------
MultiComparison : statistics class providing significance tests
tukeyhsd : among other things, computes q_crit value
References
----------
.. [*] Hochberg, Y., and A. C. Tamhane. Multiple Comparison Procedures.
Hoboken, NJ: John Wiley & Sons, 1987.)
"""
# Set initial variables
ng = len(groupnobs)
if pairindices is None:
pairindices = np.triu_indices(ng, 1)
# Compute dij for all pairwise comparisons ala hochberg p. 95
gvar = var / groupnobs
d12 = np.sqrt(gvar[pairindices[0]] + gvar[pairindices[1]])
# Create the full d matrix given all known dij vals
d = np.zeros((ng, ng))
d[pairindices] = d12
d = d + d.conj().T
# Compute the two global sums from hochberg eq 3.32
sum1 = np.sum(d12)
sum2 = np.sum(d, axis=0)
if (ng > 2):
w = ((ng-1.) * sum2 - sum1) / ((ng - 1.) * (ng - 2.))
else:
w = sum1 * np.ones((2, 1)) / 2.
return (q_crit / np.sqrt(2))*w
[docs]def distance_st_range(mean_all, nobs_all, var_all, df=None, triu=False):
'''pairwise distance matrix, outsourced from tukeyhsd
CHANGED: meandiffs are with sign, studentized range uses abs
q_crit added for testing
TODO: error in variance calculation when nobs_all is scalar, missing 1/n
'''
mean_all = np.asarray(mean_all)
#check if or when other ones need to be arrays
n_means = len(mean_all)
if df is None:
df = nobs_all - 1
if np.size(df) == 1: # assumes balanced samples with df = n - 1, n_i = n
df_total = n_means * df
else:
df_total = np.sum(df)
if (np.size(nobs_all) == 1) and (np.size(var_all) == 1):
#balanced sample sizes and homogenous variance
var_pairs = 1. * var_all / nobs_all * np.ones((n_means, n_means))
elif np.size(var_all) == 1:
#unequal sample sizes and homogenous variance
var_pairs = var_all * varcorrection_pairs_unbalanced(nobs_all,
srange=True)
elif np.size(var_all) > 1:
var_pairs, df_sum = varcorrection_pairs_unequal(nobs_all, var_all, df)
var_pairs /= 2.
#check division by two for studentized range
else:
raise ValueError('not supposed to be here')
#meandiffs_ = mean_all[:,None] - mean_all
meandiffs = mean_all - mean_all[:,None] #reverse sign, check with R example
std_pairs = np.sqrt(var_pairs)
idx1, idx2 = np.triu_indices(n_means, 1)
if triu:
#select all pairs from upper triangle of matrix
meandiffs = meandiffs_[idx1, idx2]
std_pairs = std_pairs_[idx1, idx2]
st_range = np.abs(meandiffs) / std_pairs #studentized range statistic
return st_range, meandiffs, std_pairs, (idx1,idx2) #return square arrays
def contrast_allpairs(nm):
'''contrast or restriction matrix for all pairs of nm variables
Parameters
----------
nm : int
Returns
-------
contr : ndarray, 2d, (nm*(nm-1)/2, nm)
contrast matrix for all pairwise comparisons
'''
contr = []
for i in range(nm):
for j in range(i+1, nm):
contr_row = np.zeros(nm)
contr_row[i] = 1
contr_row[j] = -1
contr.append(contr_row)
return np.array(contr)
def contrast_all_one(nm):
'''contrast or restriction matrix for all against first comparison
Parameters
----------
nm : int
Returns
-------
contr : ndarray, 2d, (nm-1, nm)
contrast matrix for all against first comparisons
'''
contr = np.column_stack((np.ones(nm-1), -np.eye(nm-1)))
return contr
def contrast_diff_mean(nm):
'''contrast or restriction matrix for all against mean comparison
Parameters
----------
nm : int
Returns
-------
contr : ndarray, 2d, (nm-1, nm)
contrast matrix for all against mean comparisons
'''
return np.eye(nm) - np.ones((nm,nm))/nm
def tukey_pvalues(std_range, nm, df):
#corrected but very slow with warnings about integration
from statsmodels.sandbox.distributions.multivariate import mvstdtprob
#nm = len(std_range)
contr = contrast_allpairs(nm)
corr = np.dot(contr, contr.T)/2.
tstat = std_range / np.sqrt(2) * np.ones(corr.shape[0]) #need len of all pairs
return multicontrast_pvalues(tstat, corr, df=df)
def test_tukey_pvalues():
#testcase with 3 is not good because all pairs has also 3*(3-1)/2=3 elements
res = tukey_pvalues(3.649, 3, 16) #3.649*np.ones(3), 16)
assert_almost_equal(0.05, res[0], 3)
assert_almost_equal(0.05*np.ones(3), res[1], 3)
def multicontrast_pvalues(tstat, tcorr, df=None, dist='t', alternative='two-sided'):
'''pvalues for simultaneous tests
'''
from statsmodels.sandbox.distributions.multivariate import mvstdtprob
if (df is None) and (dist == 't'):
raise ValueError('df has to be specified for the t-distribution')
tstat = np.asarray(tstat)
ntests = len(tstat)
cc = np.abs(tstat)
pval_global = 1 - mvstdtprob(-cc,cc, tcorr, df)
pvals = []
for ti in cc:
limits = ti*np.ones(ntests)
pvals.append(1 - mvstdtprob(-cc,cc, tcorr, df))
return pval_global, np.asarray(pvals)
[docs]class StepDown(object):
'''a class for step down methods
This is currently for simple tree subset descend, similar to homogeneous_subsets,
but checks all leave-one-out subsets instead of assuming an ordered set.
Comment in SAS manual:
SAS only uses interval subsets of the sorted list, which is sufficient for range
tests (maybe also equal variance and balanced sample sizes are required).
For F-test based critical distances, the restriction to intervals is not sufficient.
This version uses a single critical value of the studentized range distribution
for all comparisons, and is therefore a step-down version of Tukey HSD.
The class is written so it can be subclassed, where the get_distance_matrix and
get_crit are overwritten to obtain other step-down procedures such as REGW.
iter_subsets can be overwritten, to get a recursion as in the many to one comparison
with a control such as in Dunnet's test.
A one-sided right tail test is not covered because the direction of the inequality
is hard coded in check_set. Also Peritz's check of partitions is not possible, but
I have not seen it mentioned in any more recent references.
I have only partially read the step-down procedure for closed tests by Westfall.
One change to make it more flexible, is to separate out the decision on a subset,
also because the F-based tests, FREGW in SPSS, take information from all elements of
a set and not just pairwise comparisons. I haven't looked at the details of
the F-based tests such as Sheffe yet. It looks like running an F-test on equality
of means in each subset. This would also outsource how pairwise conditions are
combined, any larger or max. This would also imply that the distance matrix cannot
be calculated in advance for tests like the F-based ones.
'''
def __init__(self, vals, nobs_all, var_all, df=None):
self.vals = vals
self.n_vals = len(vals)
self.nobs_all = nobs_all
self.var_all = var_all
self.df = df
# the following has been moved to run
#self.cache_result = {}
#self.crit = self.getcrit(0.5) #decide where to set alpha, moved to run
#self.accepted = [] #store accepted sets, not unique
[docs] def get_crit(self, alpha):
#currently tukey Q, add others
q_crit = get_tukeyQcrit(self.n_vals, self.df, alpha=alpha)
return q_crit * np.ones(self.n_vals)
[docs] def get_distance_matrix(self):
'''studentized range statistic'''
#make into property, decorate
dres = distance_st_range(self.vals, self.nobs_all, self.var_all, df=self.df)
self.distance_matrix = dres[0]
[docs] def iter_subsets(self, indices):
for ii in range(len(indices)):
idxsub = copy.copy(indices)
idxsub.pop(ii)
yield idxsub
[docs] def check_set(self, indices):
'''check whether pairwise distances of indices satisfy condition
'''
indtup = tuple(indices)
if indtup in self.cache_result:
return self.cache_result[indtup]
else:
set_distance_matrix = self.distance_matrix[np.asarray(indices)[:,None], indices]
n_elements = len(indices)
if np.any(set_distance_matrix > self.crit[n_elements-1]):
res = True
else:
res = False
self.cache_result[indtup] = res
return res
[docs] def stepdown(self, indices):
print(indices)
if self.check_set(indices): # larger than critical distance
if (len(indices) > 2): # step down into subsets if more than 2 elements
for subs in self.iter_subsets(indices):
self.stepdown(subs)
else:
self.rejected.append(tuple(indices))
else:
self.accepted.append(tuple(indices))
return indices
[docs] def run(self, alpha):
'''main function to run the test,
could be done in __call__ instead
this could have all the initialization code
'''
self.cache_result = {}
self.crit = self.get_crit(alpha) #decide where to set alpha, moved to run
self.accepted = [] #store accepted sets, not unique
self.rejected = []
self.get_distance_matrix()
self.stepdown(lrange(self.n_vals))
return list(set(self.accepted)), list(set(sd.rejected))
[docs]def homogeneous_subsets(vals, dcrit):
'''recursively check all pairs of vals for minimum distance
step down method as in Newman-Keuls and Ryan procedures. This is not a
closed procedure since not all partitions are checked.
Parameters
----------
vals : array_like
values that are pairwise compared
dcrit : array_like or float
critical distance for rejecting, either float, or 2-dimensional array
with distances on the upper triangle.
Returns
-------
rejs : list of pairs
list of pair-indices with (strictly) larger than critical difference
nrejs : list of pairs
list of pair-indices with smaller than critical difference
lli : list of tuples
list of subsets with smaller than critical difference
res : tree
result of all comparisons (for checking)
this follows description in SPSS notes on Post-Hoc Tests
Because of the recursive structure, some comparisons are made several
times, but only unique pairs or sets are returned.
Examples
--------
>>> m = [0, 2, 2.5, 3, 6, 8, 9, 9.5,10 ]
>>> rej, nrej, ssli, res = homogeneous_subsets(m, 2)
>>> set_partition(ssli)
([(5, 6, 7, 8), (1, 2, 3), (4,)], [0])
>>> [np.array(m)[list(pp)] for pp in set_partition(ssli)[0]]
[array([ 8. , 9. , 9.5, 10. ]), array([ 2. , 2.5, 3. ]), array([ 6.])]
'''
nvals = len(vals)
indices_ = lrange(nvals)
rejected = []
subsetsli = []
if np.size(dcrit) == 1:
dcrit = dcrit*np.ones((nvals, nvals)) #example numbers for experimenting
def subsets(vals, indices_):
'''recursive function for constructing homogeneous subset
registers rejected and subsetli in outer scope
'''
i, j = (indices_[0], indices_[-1])
if vals[-1] - vals[0] > dcrit[i,j]:
rejected.append((indices_[0], indices_[-1]))
return [subsets(vals[:-1], indices_[:-1]),
subsets(vals[1:], indices_[1:]),
(indices_[0], indices_[-1])]
else:
subsetsli.append(tuple(indices_))
return indices_
res = subsets(vals, indices_)
all_pairs = [(i,j) for i in range(nvals) for j in range(nvals-1,i,-1)]
rejs = set(rejected)
not_rejected = list(set(all_pairs) - rejs)
return list(rejs), not_rejected, list(set(subsetsli)), res
[docs]def set_partition(ssli):
'''extract a partition from a list of tuples
this should be correctly called select largest disjoint sets.
Begun and Gabriel 1981 don't seem to be bothered by sets of accepted
hypothesis with joint elements,
e.g. maximal_accepted_sets = { {1,2,3}, {2,3,4} }
This creates a set partition from a list of sets given as tuples.
It tries to find the partition with the largest sets. That is, sets are
included after being sorted by length.
If the list doesn't include the singletons, then it will be only a
partial partition. Missing items are singletons (I think).
Examples
--------
>>> li
[(5, 6, 7, 8), (1, 2, 3), (4, 5), (0, 1)]
>>> set_partition(li)
([(5, 6, 7, 8), (1, 2, 3)], [0, 4])
'''
part = []
for s in sorted(list(set(ssli)), key=len)[::-1]:
#print(s,
s_ = set(s).copy()
if not any(set(s_).intersection(set(t)) for t in part):
#print('inside:', s
part.append(s)
#else: print(part
missing = list(set(i for ll in ssli for i in ll)
- set(i for ll in part for i in ll))
return part, missing
[docs]def set_remove_subs(ssli):
'''remove sets that are subsets of another set from a list of tuples
Parameters
----------
ssli : list of tuples
each tuple is considered as a set
Returns
-------
part : list of tuples
new list with subset tuples removed, it is sorted by set-length of tuples. The
list contains original tuples, duplicate elements are not removed.
Examples
--------
>>> set_remove_subs([(0, 1), (1, 2), (1, 2, 3), (0,)])
[(1, 2, 3), (0, 1)]
>>> set_remove_subs([(0, 1), (1, 2), (1,1, 1, 2, 3), (0,)])
[(1, 1, 1, 2, 3), (0, 1)]
'''
#TODO: maybe convert all tuples to sets immediately, but I don't need the extra efficiency
part = []
for s in sorted(list(set(ssli)), key=lambda x: len(set(x)))[::-1]:
#print(s,
#s_ = set(s).copy()
if not any(set(s).issubset(set(t)) for t in part):
#print('inside:', s
part.append(s)
#else: print(part
## missing = list(set(i for ll in ssli for i in ll)
## - set(i for ll in part for i in ll))
return part
if __name__ == '__main__':
examples = ['tukey', 'tukeycrit', 'fdr', 'fdrmc', 'bonf', 'randmvn',
'multicompdev', 'None']#[-1]
if 'tukey' in examples:
#Example Tukey
x = np.array([[0,0,1]]).T + np.random.randn(3, 20)
print(Tukeythreegene(*x))
#Example FDR
#------------
if ('fdr' in examples) or ('bonf' in examples):
x1 = [1,1,1,0,-1,-1,-1,0,1,1,-1,1]
print(lzip(np.arange(len(x1)), x1))
print(maxzero(x1))
#[(0, 1), (1, 1), (2, 1), (3, 0), (4, -1), (5, -1), (6, -1), (7, 0), (8, 1), (9, 1), (10, -1), (11, 1)]
#(11, array([ 3, 7, 11]))
print(maxzerodown(-np.array(x1)))
locs = np.linspace(0,1,10)
locs = np.array([0.]*6 + [0.75]*4)
rvs = locs + stats.norm.rvs(size=(20,10))
tt, tpval = stats.ttest_1samp(rvs, 0)
tpval_sortind = np.argsort(tpval)
tpval_sorted = tpval[tpval_sortind]
reject = tpval_sorted < ecdf(tpval_sorted)*0.05
reject2 = max(np.nonzero(reject))
print(reject)
res = np.array(lzip(np.round(rvs.mean(0),4),np.round(tpval,4),
reject[tpval_sortind.argsort()]),
dtype=[('mean',float),
('pval',float),
('reject', np.bool8)])
#from statsmodels.iolib import SimpleTable
print(SimpleTable(res, headers=res.dtype.names))
print(fdrcorrection_bak(tpval, alpha=0.05))
print(reject)
print('\nrandom example')
print('bonf', multipletests(tpval, alpha=0.05, method='bonf'))
print('sidak', multipletests(tpval, alpha=0.05, method='sidak'))
print('hs', multipletests(tpval, alpha=0.05, method='hs'))
print('sh', multipletests(tpval, alpha=0.05, method='sh'))
pvals = np.array('0.0020 0.0045 0.0060 0.0080 0.0085 0.0090 0.0175 0.0250 '
'0.1055 0.5350'.split(), float)
print('\nexample from lecturnotes')
for meth in ['bonf', 'sidak', 'hs', 'sh']:
print(meth)
print(multipletests(pvals, alpha=0.05, method=meth))
if 'fdrmc' in examples:
mcres = mcfdr(nobs=100, nrepl=1000, ntests=30, ntrue=30, mu=0.1, alpha=0.05, rho=0.3)
mcmeans = np.array(mcres).mean(0)
print(mcmeans)
print(mcmeans[0]/6., 1-mcmeans[1]/4.)
print(mcmeans[:4], mcmeans[-4:])
if 'randmvn' in examples:
rvsmvn = randmvn(0.8, (5000,5))
print(np.corrcoef(rvsmvn, rowvar=0))
print(rvsmvn.var(0))
if 'tukeycrit' in examples:
print(get_tukeyQcrit(8, 8, alpha=0.05), 5.60)
print(get_tukeyQcrit(8, 8, alpha=0.01), 7.47)
if 'multicompdev' in examples:
#development of kruskal-wallis multiple-comparison
#example from matlab file exchange
X = np.array([[7.68, 1], [7.69, 1], [7.70, 1], [7.70, 1], [7.72, 1],
[7.73, 1], [7.73, 1], [7.76, 1], [7.71, 2], [7.73, 2],
[7.74, 2], [7.74, 2], [7.78, 2], [7.78, 2], [7.80, 2],
[7.81, 2], [7.74, 3], [7.75, 3], [7.77, 3], [7.78, 3],
[7.80, 3], [7.81, 3], [7.84, 3], [7.71, 4], [7.71, 4],
[7.74, 4], [7.79, 4], [7.81, 4], [7.85, 4], [7.87, 4],
[7.91, 4]])
xli = [X[X[:,1]==k,0] for k in range(1,5)]
xranks = stats.rankdata(X[:,0])
xranksli = [xranks[X[:,1]==k] for k in range(1,5)]
xnobs = np.array([len(x) for x in xli])
meanranks = [item.mean() for item in xranksli]
sumranks = [item.sum() for item in xranksli]
# equivalent function
#from scipy import special
#-np.sqrt(2.)*special.erfcinv(2-0.5) == stats.norm.isf(0.25)
stats.norm.sf(0.67448975019608171)
stats.norm.isf(0.25)
mrs = np.sort(meanranks)
v1, v2 = np.triu_indices(4,1)
print('\nsorted rank differences')
print(mrs[v2] - mrs[v1])
diffidx = np.argsort(mrs[v2] - mrs[v1])[::-1]
mrs[v2[diffidx]] - mrs[v1[diffidx]]
print('\nkruskal for all pairs')
for i,j in zip(v2[diffidx], v1[diffidx]):
print(i,j, stats.kruskal(xli[i], xli[j]))
mwu, mwupval = stats.mannwhitneyu(xli[i], xli[j], use_continuity=False)
print(mwu, mwupval*2, mwupval*2<0.05/6., mwupval*2<0.1/6.)
uni, intlab = np.unique(X[:,0], return_inverse=True)
groupnobs = np.bincount(intlab)
groupxsum = np.bincount(intlab, weights=X[:,0])
groupxmean = groupxsum * 1.0 / groupnobs
rankraw = X[:,0].argsort().argsort()
groupranksum = np.bincount(intlab, weights=rankraw)
# start at 1 for stats.rankdata :
grouprankmean = groupranksum * 1.0 / groupnobs + 1
assert_almost_equal(grouprankmean[intlab], stats.rankdata(X[:,0]), 15)
gs = GroupsStats(X, useranks=True)
print('\ngroupmeanfilter and grouprankmeans')
print(gs.groupmeanfilter)
print(grouprankmean[intlab])
#the following has changed
#assert_almost_equal(gs.groupmeanfilter, stats.rankdata(X[:,0]), 15)
xuni, xintlab = np.unique(X[:,0], return_inverse=True)
gs2 = GroupsStats(np.column_stack([X[:,0], xintlab]), useranks=True)
#assert_almost_equal(gs2.groupmeanfilter, stats.rankdata(X[:,0]), 15)
rankbincount = np.bincount(xranks.astype(int))
nties = rankbincount[rankbincount > 1]
ntot = float(len(xranks));
tiecorrection = 1 - (nties**3 - nties).sum()/(ntot**3 - ntot)
assert_almost_equal(tiecorrection, stats.tiecorrect(xranks),15)
print('\ntiecorrection for data and ranks')
print(tiecorrection)
print(tiecorrect(xranks))
tot = X.shape[0]
t=500 #168
f=(tot*(tot+1.)/12.)-(t/(6.*(tot-1.)))
f=(tot*(tot+1.)/12.)/stats.tiecorrect(xranks)
print('\npairs of mean rank differences')
for i,j in zip(v2[diffidx], v1[diffidx]):
#pdiff = np.abs(mrs[i] - mrs[j])
pdiff = np.abs(meanranks[i] - meanranks[j])
se = np.sqrt(f * np.sum(1./xnobs[[i,j]] )) #np.array([8,8]))) #Fixme groupnobs[[i,j]] ))
print(i,j, pdiff, se, pdiff/se, pdiff/se>2.6310)
multicomp = MultiComparison(*X.T)
multicomp.kruskal()
gsr = GroupsStats(X, useranks=True)
print('\nexamples for kruskal multicomparison')
for i in range(10):
x1, x2 = (np.random.randn(30,2) + np.array([0, 0.5])).T
skw = stats.kruskal(x1, x2)
mc2=MultiComparison(np.r_[x1, x2], np.r_[np.zeros(len(x1)), np.ones(len(x2))])
newskw = mc2.kruskal()
print(skw, np.sqrt(skw[0]), skw[1]-newskw, (newskw/skw[1]-1)*100)
tablett, restt, arrtt = multicomp.allpairtest(stats.ttest_ind)
tablemw, resmw, arrmw = multicomp.allpairtest(stats.mannwhitneyu)
print('')
print(tablett)
print('')
print(tablemw)
tablemwhs, resmw, arrmw = multicomp.allpairtest(stats.mannwhitneyu, method='hs')
print('')
print(tablemwhs)
if 'last' in examples:
xli = (np.random.randn(60,4) + np.array([0, 0, 0.5, 0.5])).T
#Xrvs = np.array(catstack(xli))
xrvs, xrvsgr = catstack(xli)
multicompr = MultiComparison(xrvs, xrvsgr)
tablett, restt, arrtt = multicompr.allpairtest(stats.ttest_ind)
print(tablett)
xli=[[8,10,9,10,9],[7,8,5,8,5],[4,8,7,5,7]]
x,l = catstack(xli)
gs4 = GroupsStats(np.column_stack([x,l]))
print(gs4.groupvarwithin())
#test_tukeyhsd() #moved to test_multi.py
gmeans = np.array([ 7.71375, 7.76125, 7.78428571, 7.79875])
gnobs = np.array([8, 8, 7, 8])
sd = StepDown(gmeans, gnobs, 0.001, [27])
#example from BKY
pvals = [0.0001, 0.0004, 0.0019, 0.0095, 0.0201, 0.0278, 0.0298, 0.0344, 0.0459,
0.3240, 0.4262, 0.5719, 0.6528, 0.7590, 1.000 ]
#same number of rejection as in BKY paper:
#single step-up:4, two-stage:8, iterated two-step:9
#also alpha_star is the same as theirs for TST
print(fdrcorrection0(pvals, alpha=0.05, method='indep'))
print(fdrcorrection_twostage(pvals, alpha=0.05, iter=False))
res_tst = fdrcorrection_twostage(pvals, alpha=0.05, iter=False)
assert_almost_equal([0.047619, 0.0649], res_tst[-1][:2],3) #alpha_star for stage 2
assert_equal(8, res_tst[0].sum())
print(fdrcorrection_twostage(pvals, alpha=0.05, iter=True))
print('fdr_gbs', multipletests(pvals, alpha=0.05, method='fdr_gbs'))
#multicontrast_pvalues(tstat, tcorr, df)
test_tukey_pvalues()
tukey_pvalues(3.649, 3, 16)
