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Source code for statsmodels.stats.multitest

'''Multiple Testing and P-Value Correction


Author: Josef Perktold
License: BSD-3

'''

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',
                           'fdr_gbs': 'FDR adaptive Gavrilov-Benjamini-Sarkar'
                           }

_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[0]] = m[0]
    for a in m[1:]:
        multitest_alias[a] = m[0]

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)[0]
        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)[0]
        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[0].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 negcorr implemented')
    reject = pvals_sorted <= ecdffactor*alpha
    if reject.any():
        rejectmax = max(np.nonzero(reject)[0])
        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


[docs]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' - 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
[docs]def local_fdr(zscores, null_proportion=1.0, null_pdf=None, deg=7, nbins=30): """ Calculate local FDR values for a list of Z-scores. Parameters ---------- zscores : array-like A vector of Z-scores null_proportion : float The assumed proportion of true null hypotheses null_pdf : function mapping reals to positive reals The density of null Z-scores; if None, use standard normal deg : integer The maximum exponent in the polynomial expansion of the density of non-null Z-scores nbins : integer The number of bins for estimating the marginal density of Z-scores. Returns ------- fdr : array-like A vector of FDR values References ---------- B Efron (2008). Microarrays, Empirical Bayes, and the Two-Groups Model. Statistical Science 23:1, 1-22. Examples -------- Basic use (the null Z-scores are taken to be standard normal): >>> from statsmodels.stats.multitest import local_fdr >>> import numpy as np >>> zscores = np.random.randn(30) >>> fdr = local_fdr(zscores) Use a Gaussian null distribution estimated from the data: >>> null = EmpiricalNull(zscores) >>> fdr = local_fdr(zscores, null_pdf=null.pdf) """ from statsmodels.genmod.generalized_linear_model import GLM from statsmodels.genmod.generalized_linear_model import families from statsmodels.regression.linear_model import OLS # Bins for Poisson modeling of the marginal Z-score density minz = min(zscores) maxz = max(zscores) bins = np.linspace(minz, maxz, nbins) # Bin counts zhist = np.histogram(zscores, bins)[0] # Bin centers zbins = (bins[:-1] + bins[1:]) / 2 # The design matrix at bin centers dmat = np.vander(zbins, deg + 1) # Use this to get starting values for Poisson regression md = OLS(np.log(1 + zhist), dmat).fit() # Poisson regression md = GLM(zhist, dmat, family=families.Poisson()).fit(start_params=md.params) # The design matrix for all Z-scores dmat_full = np.vander(zscores, deg + 1) # The height of the estimated marginal density of Z-scores, # evaluated at every observed Z-score. fz = md.predict(dmat_full) / (len(zscores) * (bins[1] - bins[0])) # The null density. if null_pdf is None: f0 = np.exp(-0.5 * zscores**2) / np.sqrt(2 * np.pi) else: f0 = null_pdf(zscores) # The local FDR values fdr = null_proportion * f0 / fz fdr = np.clip(fdr, 0, 1) return fdr
[docs]class NullDistribution(object): """ Estimate a Gaussian distribution for the null Z-scores. The observed Z-scores consist of both null and non-null values. The fitted distribution of null Z-scores is Gaussian, but may have non-zero mean and/or non-unit scale. Parameters ---------- zscores : array-like The observed Z-scores. null_lb : float Z-scores between `null_lb` and `null_lb` are all considered to be true null hypotheses. null_ub : float See `null_lb`. estimate_mean : bool If True, estimate the mean of the distribution. If False, the mean is fixed at zero. estimate_scale : bool If True, estimate the scale of the distribution. If False, the scale parameter is fixed at 1. estimate_null_proportion : bool If True, estimate the proportion of true null hypotheses (i.e. the proportion of z-scores with expected value zero). If False, this parameter is fixed at 1. Attributes ---------- mean : float The estimated mean of the empirical null distribution sd : float The estimated standard deviation of the empirical null distribution null_proportion : float The estimated proportion of true null hypotheses among all hypotheses References ---------- B Efron (2008). Microarrays, Empirical Bayes, and the Two-Groups Model. Statistical Science 23:1, 1-22. Notes ----- See also: http://nipy.org/nipy/labs/enn.html#nipy.algorithms.statistics.empirical_pvalue.NormalEmpiricalNull.fdr """ def __init__(self, zscores, null_lb=-1, null_ub=1, estimate_mean=True, estimate_scale=True, estimate_null_proportion=False): # Extract the null z-scores ii = np.flatnonzero((zscores >= null_lb) & (zscores <= null_ub)) if len(ii) == 0: raise RuntimeError("No Z-scores fall between null_lb and null_ub") zscores0 = zscores[ii] # Number of Z-scores, and null Z-scores n_zs, n_zs0 = len(zscores), len(zscores0) # Unpack and transform the parameters to the natural scale, hold # parameters fixed as specified. def xform(params): mean = 0. sd = 1. prob = 1. ii = 0 if estimate_mean: mean = params[ii] ii += 1 if estimate_scale: sd = np.exp(params[ii]) ii += 1 if estimate_null_proportion: prob = 1 / (1 + np.exp(-params[ii])) return mean, sd, prob from scipy.stats.distributions import norm def fun(params): """ Negative log-likelihood of z-scores. The function has three arguments, packed into a vector: mean : location parameter logscale : log of the scale parameter logitprop : logit of the proportion of true nulls The implementation follows section 4 from Efron 2008. """ d, s, p = xform(params) # Mass within the central region central_mass = (norm.cdf((null_ub - d) / s) - norm.cdf((null_lb - d) / s)) # Probability that a Z-score is null and is in the central region cp = p * central_mass # Binomial term rval = n_zs0 * np.log(cp) + (n_zs - n_zs0) * np.log(1 - cp) # Truncated Gaussian term for null Z-scores zv = (zscores0 - d) / s rval += np.sum(-zv**2 / 2) - n_zs0 * np.log(s) rval -= n_zs0 * np.log(central_mass) return -rval # Estimate the parameters from scipy.optimize import minimize # starting values are mean = 0, scale = 1, p0 ~ 1 mz = minimize(fun, np.r_[0., 0, 3], method="Nelder-Mead") mean, sd, prob = xform(mz['x']) self.mean = mean self.sd = sd self.null_proportion = prob # The fitted null density function
[docs] def pdf(self, zscores): """ Evaluates the fitted emirical null Z-score density. Parameters ---------- zscores : scalar or array-like The point or points at which the density is to be evaluated. Returns ------- The empirical null Z-score density evaluated at the given points. """ zval = (zscores - self.mean) / self.sd return np.exp(-0.5*zval**2 - np.log(self.sd) - 0.5*np.log(2*np.pi))