Source code for statsmodels.stats.multitest

'''Multiple Testing and P-Value Correction

Author: Josef Perktold

'''

from statsmodels.compat.python import range
from statsmodels.compat.collections import OrderedDict

import numpy as np

#==============================================
#
# Part 1: Multiple Tests and P-Value Correction
#
#==============================================

def _ecdf(x):
'''no frills empirical cdf used in fdrcorrection
'''
nobs = len(x)
return np.arange(1,nobs+1)/float(nobs)

multitest_methods_names = {'b': 'Bonferroni',
's': 'Sidak',
'h': 'Holm',
'hs': 'Holm-Sidak',
'sh': 'Simes-Hochberg',
'ho': 'Hommel',
'fdr_bh': 'FDR Benjamini-Hochberg',
'fdr_by': 'FDR Benjamini-Yekutieli',
'fdr_tsbh': 'FDR 2-stage Benjamini-Hochberg',
'fdr_tsbky': 'FDR 2-stage Benjamini-Krieger-Yekutieli',
}

_alias_list = [['b', 'bonf', 'bonferroni'],
['s', 'sidak'],
['h', 'holm'],
['hs', 'holm-sidak'],
['sh', 'simes-hochberg'],
['ho', 'hommel'],
['fdr_bh', 'fdr_i', 'fdr_p', 'fdri', 'fdrp'],
['fdr_by', 'fdr_n', 'fdr_c', 'fdrn', 'fdrcorr'],
['fdr_tsbh', 'fdr_2sbh'],
['fdr_tsbky', 'fdr_2sbky', 'fdr_twostage'],
['fdr_gbs']
]

multitest_alias = OrderedDict()
for m in _alias_list:
multitest_alias[m] = m
for a in m[1:]:
multitest_alias[a] = m

[docs]def multipletests(pvals, alpha=0.05, method='hs', is_sorted=False, returnsorted=False): '''test results and p-value correction for multiple tests Parameters ---------- pvals : array_like uncorrected p-values alpha : float FWER, family-wise error rate, e.g. 0.1 method : string Method used for testing and adjustment of pvalues. Can be either the full name or initial letters. Available methods are :: `bonferroni` : one-step correction `sidak` : one-step correction `holm-sidak` : step down method using Sidak adjustments `holm` : step-down method using Bonferroni adjustments `simes-hochberg` : step-up method (independent) `hommel` : closed method based on Simes tests (non-negative) `fdr_bh` : Benjamini/Hochberg (non-negative) `fdr_by` : Benjamini/Yekutieli (negative) `fdr_tsbh` : two stage fdr correction (non-negative) `fdr_tsbky` : two stage fdr correction (non-negative) is_sorted : bool If False (default), the p_values will be sorted, but the corrected pvalues are in the original order. If True, then it assumed that the pvalues are already sorted in ascending order. returnsorted : bool not tested, return sorted p-values instead of original sequence Returns ------- reject : array, boolean true for hypothesis that can be rejected for given alpha pvals_corrected : array p-values corrected for multiple tests alphacSidak: float corrected alpha for Sidak method alphacBonf: float corrected alpha for Bonferroni method Notes ----- There may be API changes for this function in the future. Except for 'fdr_twostage', the p-value correction is independent of the alpha specified as argument. In these cases the corrected p-values can also be compared with a different alpha. In the case of 'fdr_twostage', the corrected p-values are specific to the given alpha, see ``fdrcorrection_twostage``. The 'fdr_gbs' procedure is not verified against another package, p-values are derived from scratch and are not derived in the reference. In Monte Carlo experiments the method worked correctly and maintained the false discovery rate. All procedures that are included, control FWER or FDR in the independent case, and most are robust in the positively correlated case. `fdr_gbs`: high power, fdr control for independent case and only small violation in positively correlated case **Timing**: Most of the time with large arrays is spent in `argsort`. When we want to calculate the p-value for several methods, then it is more efficient to presort the pvalues, and put the results back into the original order outside of the function. Method='hommel' is very slow for large arrays, since it requires the evaluation of n partitions, where n is the number of p-values. ''' import gc pvals = np.asarray(pvals) alphaf = alpha # Notation ? if not is_sorted: sortind = np.argsort(pvals) pvals = np.take(pvals, sortind) ntests = len(pvals) alphacSidak = 1 - np.power((1. - alphaf), 1./ntests) alphacBonf = alphaf / float(ntests) if method.lower() in ['b', 'bonf', 'bonferroni']: reject = pvals <= alphacBonf pvals_corrected = pvals * float(ntests) elif method.lower() in ['s', 'sidak']: reject = pvals <= alphacSidak pvals_corrected = 1 - np.power((1. - pvals), ntests) elif method.lower() in ['hs', 'holm-sidak']: alphacSidak_all = 1 - np.power((1. - alphaf), 1./np.arange(ntests, 0, -1)) notreject = pvals > alphacSidak_all del alphacSidak_all nr_index = np.nonzero(notreject) if nr_index.size == 0: # nonreject is empty, all rejected notrejectmin = len(pvals) else: notrejectmin = np.min(nr_index) notreject[notrejectmin:] = True reject = ~notreject del notreject pvals_corrected_raw = 1 - np.power((1. - pvals), np.arange(ntests, 0, -1)) pvals_corrected = np.maximum.accumulate(pvals_corrected_raw) del pvals_corrected_raw elif method.lower() in ['h', 'holm']: notreject = pvals > alphaf / np.arange(ntests, 0, -1) nr_index = np.nonzero(notreject) if nr_index.size == 0: # nonreject is empty, all rejected notrejectmin = len(pvals) else: notrejectmin = np.min(nr_index) notreject[notrejectmin:] = True reject = ~notreject pvals_corrected_raw = pvals * np.arange(ntests, 0, -1) pvals_corrected = np.maximum.accumulate(pvals_corrected_raw) del pvals_corrected_raw gc.collect() elif method.lower() in ['sh', 'simes-hochberg']: alphash = alphaf / np.arange(ntests, 0, -1) reject = pvals <= alphash rejind = np.nonzero(reject) if rejind.size > 0: rejectmax = np.max(np.nonzero(reject)) reject[:rejectmax] = True pvals_corrected_raw = np.arange(ntests, 0, -1) * pvals pvals_corrected = np.minimum.accumulate(pvals_corrected_raw[::-1])[::-1] del pvals_corrected_raw elif method.lower() in ['ho', 'hommel']: # we need a copy because we overwrite it in a loop a = pvals.copy() for m in range(ntests, 1, -1): cim = np.min(m * pvals[-m:] / np.arange(1,m+1.)) a[-m:] = np.maximum(a[-m:], cim) a[:-m] = np.maximum(a[:-m], np.minimum(m * pvals[:-m], cim)) pvals_corrected = a reject = a <= alphaf elif method.lower() in ['fdr_bh', 'fdr_i', 'fdr_p', 'fdri', 'fdrp']: # delegate, call with sorted pvals reject, pvals_corrected = fdrcorrection(pvals, alpha=alpha, method='indep', is_sorted=True) elif method.lower() in ['fdr_by', 'fdr_n', 'fdr_c', 'fdrn', 'fdrcorr']: # delegate, call with sorted pvals reject, pvals_corrected = fdrcorrection(pvals, alpha=alpha, method='n', is_sorted=True) elif method.lower() in ['fdr_tsbky', 'fdr_2sbky', 'fdr_twostage']: # delegate, call with sorted pvals reject, pvals_corrected = fdrcorrection_twostage(pvals, alpha=alpha, method='bky', is_sorted=True)[:2] elif method.lower() in ['fdr_tsbh', 'fdr_2sbh']: # delegate, call with sorted pvals reject, pvals_corrected = fdrcorrection_twostage(pvals, alpha=alpha, method='bh', is_sorted=True)[:2] elif method.lower() in ['fdr_gbs']: #adaptive stepdown in Gavrilov, Benjamini, Sarkar, Annals of Statistics 2009 ## notreject = pvals > alphaf / np.arange(ntests, 0, -1) #alphacSidak ## notrejectmin = np.min(np.nonzero(notreject)) ## notreject[notrejectmin:] = True ## reject = ~notreject ii = np.arange(1, ntests + 1) q = (ntests + 1. - ii)/ii * pvals / (1. - pvals) pvals_corrected_raw = np.maximum.accumulate(q) #up requirementd pvals_corrected = np.minimum.accumulate(pvals_corrected_raw[::-1])[::-1] del pvals_corrected_raw reject = pvals_corrected <= alpha else: raise ValueError('method not recognized') if not pvals_corrected is None: #not necessary anymore pvals_corrected[pvals_corrected>1] = 1 if is_sorted or returnsorted: return reject, pvals_corrected, alphacSidak, alphacBonf else: pvals_corrected_ = np.empty_like(pvals_corrected) pvals_corrected_[sortind] = pvals_corrected del pvals_corrected reject_ = np.empty_like(reject) reject_[sortind] = reject return reject_, pvals_corrected_, alphacSidak, alphacBonf
def fdrcorrection(pvals, alpha=0.05, method='indep', is_sorted=False): '''pvalue correction for false discovery rate This covers Benjamini/Hochberg for independent or positively correlated and Benjamini/Yekutieli for general or negatively correlated tests. Both are available in the function multipletests, as method=`fdr_bh`, resp. `fdr_by`. Parameters ---------- pvals : array_like set of p-values of the individual tests. alpha : float error rate method : {'indep', 'negcorr') Returns ------- rejected : array, bool True if a hypothesis is rejected, False if not pvalue-corrected : array pvalues adjusted for multiple hypothesis testing to limit FDR Notes ----- If there is prior information on the fraction of true hypothesis, then alpha should be set to alpha * m/m_0 where m is the number of tests, given by the p-values, and m_0 is an estimate of the true hypothesis. (see Benjamini, Krieger and Yekuteli) The two-step method of Benjamini, Krieger and Yekutiel that estimates the number of false hypotheses will be available (soon). Method names can be abbreviated to first letter, 'i' or 'p' for fdr_bh and 'n' for fdr_by. ''' pvals = np.asarray(pvals) if not is_sorted: pvals_sortind = np.argsort(pvals) pvals_sorted = np.take(pvals, pvals_sortind) else: pvals_sorted = pvals # alias if method in ['i', 'indep', 'p', 'poscorr']: ecdffactor = _ecdf(pvals_sorted) elif method in ['n', 'negcorr']: cm = np.sum(1./np.arange(1, len(pvals_sorted)+1)) #corrected this ecdffactor = _ecdf(pvals_sorted) / cm ## elif method in ['n', 'negcorr']: ## cm = np.sum(np.arange(len(pvals))) ## ecdffactor = ecdf(pvals_sorted)/cm else: raise ValueError('only indep and necorr implemented') reject = pvals_sorted <= ecdffactor*alpha if reject.any(): rejectmax = max(np.nonzero(reject)) reject[:rejectmax] = True pvals_corrected_raw = pvals_sorted / ecdffactor pvals_corrected = np.minimum.accumulate(pvals_corrected_raw[::-1])[::-1] del pvals_corrected_raw pvals_corrected[pvals_corrected>1] = 1 if not is_sorted: pvals_corrected_ = np.empty_like(pvals_corrected) pvals_corrected_[pvals_sortind] = pvals_corrected del pvals_corrected reject_ = np.empty_like(reject) reject_[pvals_sortind] = reject return reject_, pvals_corrected_ else: return reject, pvals_corrected def fdrcorrection_twostage(pvals, alpha=0.05, method='bky', iter=False, is_sorted=False): '''(iterated) two stage linear step-up procedure with estimation of number of true hypotheses Benjamini, Krieger and Yekuteli, procedure in Definition 6 Parameters ---------- pvals : array_like set of p-values of the individual tests. alpha : float error rate method : {'bky', 'bh') see Notes for details 'bky' : implements the procedure in Definition 6 of Benjamini, Krieger and Yekuteli 2006 'bh' : implements the two stage method of Benjamini and Hochberg iter ; bool Returns ------- rejected : array, bool True if a hypothesis is rejected, False if not pvalue-corrected : array pvalues adjusted for multiple hypotheses testing to limit FDR m0 : int ntest - rej, estimated number of true hypotheses alpha_stages : list of floats A list of alphas that have been used at each stage Notes ----- The returned corrected p-values are specific to the given alpha, they cannot be used for a different alpha. The returned corrected p-values are from the last stage of the fdr_bh linear step-up procedure (fdrcorrection0 with method='indep') corrected for the estimated fraction of true hypotheses. This means that the rejection decision can be obtained with ``pval_corrected <= alpha``, where ``alpha`` is the origianal significance level. (Note: This has changed from earlier versions (<0.5.0) of statsmodels.) BKY described several other multi-stage methods, which would be easy to implement. However, in their simulation the simple two-stage method (with iter=False) was the most robust to the presence of positive correlation TODO: What should be returned? ''' pvals = np.asarray(pvals) if not is_sorted: pvals_sortind = np.argsort(pvals) pvals = np.take(pvals, pvals_sortind) ntests = len(pvals) if method == 'bky': fact = (1.+alpha) alpha_prime = alpha / fact elif method == 'bh': fact = 1. alpha_prime = alpha else: raise ValueError("only 'bky' and 'bh' are available as method") alpha_stages = [alpha_prime] rej, pvalscorr = fdrcorrection(pvals, alpha=alpha_prime, method='indep', is_sorted=True) r1 = rej.sum() if (r1 == 0) or (r1 == ntests): return rej, pvalscorr * fact, ntests - r1, alpha_stages ri_old = r1 while True: ntests0 = 1.0 * ntests - ri_old alpha_star = alpha_prime * ntests / ntests0 alpha_stages.append(alpha_star) #print ntests0, alpha_star rej, pvalscorr = fdrcorrection(pvals, alpha=alpha_star, method='indep', is_sorted=True) ri = rej.sum() if (not iter) or ri == ri_old: break elif ri < ri_old: # prevent cycles and endless loops raise RuntimeError(" oops - shouldn't be here") ri_old = ri # make adjustment to pvalscorr to reflect estimated number of Non-Null cases # decision is then pvalscorr < alpha (or <=) pvalscorr *= ntests0 * 1.0 / ntests if method == 'bky': pvalscorr *= (1. + alpha) if not is_sorted: pvalscorr_ = np.empty_like(pvalscorr) pvalscorr_[pvals_sortind] = pvalscorr del pvalscorr reject = np.empty_like(rej) reject[pvals_sortind] = rej return reject, pvalscorr_, ntests - ri, alpha_stages else: return rej, pvalscorr, ntests - ri, alpha_stages