Source code for statsmodels.sandbox.stats.multicomp

'''

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)