'''Multiple Testing and P-Value Correction Author: Josef Perktold License: BSD-3 ''' import numpy as np from statsmodels.stats._knockoff import RegressionFDR __all__ = ['fdrcorrection', 'fdrcorrection_twostage', 'local_fdr', 'multipletests', 'NullDistribution', 'RegressionFDR'] # ============================================== # # 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 = {} 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', maxiter=1, is_sorted=False, returnsorted=False): """ Test results and p-value correction for multiple tests Parameters ---------- pvals : array_like, 1-d uncorrected p-values. Must be 1-dimensional. alpha : float FWER, family-wise error rate, e.g. 0.1 method : str 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) maxiter : int or bool Maximum number of iterations for two-stage fdr, `fdr_tsbh` and `fdr_tsbky`. It is ignored by all other methods. maxiter=1 (default) corresponds to the two stage method. maxiter=-1 corresponds to full iterations which is maxiter=len(pvals). maxiter=0 uses only a single stage fdr correction using a 'bh' or 'bky' prior fraction of assumed true hypotheses. 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 : ndarray, boolean true for hypothesis that can be rejected for given alpha pvals_corrected : ndarray 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 = -np.expm1(ntests * np.log1p(-pvals)) 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 # It's eqivalent to 1 - np.power((1. - pvals), # np.arange(ntests, 0, -1)) # but prevents the issue of the floating point precision pvals_corrected_raw = -np.expm1(np.arange(ntests, 0, -1) * np.log1p(-pvals)) 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', maxiter=maxiter, 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', maxiter=maxiter, 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 pvals_corrected is not 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. Parameters ---------- pvals : array_like, 1d Set of p-values of the individual tests. alpha : float, optional Family-wise error rate. Defaults to ``0.05``. method : {'i', 'indep', 'p', 'poscorr', 'n', 'negcorr'}, optional Which method to use for FDR correction. ``{'i', 'indep', 'p', 'poscorr'}`` all refer to ``fdr_bh`` (Benjamini/Hochberg for independent or positively correlated tests). ``{'n', 'negcorr'}`` both refer to ``fdr_by`` (Benjamini/Yekutieli for general or negatively correlated tests). Defaults to ``'indep'``. is_sorted : bool, optional 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. Returns ------- rejected : ndarray, bool True if a hypothesis is rejected, False if not pvalue-corrected : ndarray 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). Both methods exposed via this function (Benjamini/Hochberg, Benjamini/Yekutieli) are also available in the function ``multipletests``, as ``method="fdr_bh"`` and ``method="fdr_by"``, respectively. See also -------- multipletests ''' pvals = np.asarray(pvals) assert pvals.ndim == 1, "pvals must be 1-dimensional, that is of shape (n,)" 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 def fdrcorrection_twostage(pvals, alpha=0.05, method='bky', maxiter=1, iter=None, 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 maxiter : int or bool Maximum number of iterations. maxiter=1 (default) corresponds to the two stage method. maxiter=-1 corresponds to full iterations which is maxiter=len(pvals). maxiter=0 uses only a single stage fdr correction using a 'bh' or 'bky' prior fraction of assumed true hypotheses. Boolean maxiter is allowed for backwards compatibility with the deprecated ``iter`` keyword. maxiter=False is two-stage fdr (maxiter=1) maxiter=True is full iteration (maxiter=-1 or maxiter=len(pvals)) .. versionadded:: 0.14 Replacement for ``iter`` with additional features. iter : bool ``iter`` is deprecated use ``maxiter`` instead. If iter is True, then only one iteration step is used, this is the two-step method. If iter is False, then iterations are stopped at convergence which occurs in a finite number of steps (at most len(pvals) steps). .. deprecated:: 0.14 Use ``maxiter`` instead of ``iter``. Returns ------- rejected : ndarray, bool True if a hypothesis is rejected, False if not pvalue-corrected : ndarray pvalues adjusted for multiple hypotheses testing to limit FDR m0 : int ntest - rej, estimated number of true (not rejected) 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 original 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 iter is not None: import warnings msg = "iter keyword is deprecated, use maxiter keyword instead." warnings.warn(msg, FutureWarning) if iter is False: maxiter = 1 elif iter is True or maxiter in [-1, None] : maxiter = len(pvals) # otherwise we use maxiter 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 reject = rej pvalscorr *= fact ri = r1 else: ri_old = ri = r1 ntests0 = ntests # needed if maxiter=0 # while True: for it in range(maxiter): 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 (it >= maxiter - 1) or ri == ri_old: break elif ri < ri_old: # prevent cycles and endless loops raise RuntimeError(" oops - should not 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) pvalscorr[pvalscorr>1] = 1 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 def local_fdr(zscores, null_proportion=1.0, null_pdf=None, deg=7, nbins=30, alpha=0): """ 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 : int The maximum exponent in the polynomial expansion of the density of non-null Z-scores nbins : int The number of bins for estimating the marginal density of Z-scores. alpha : float Use Poisson ridge regression with parameter alpha to estimate the density of non-null 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) # Rescale the design matrix sd = dmat.std(0) ii = sd >1e-8 dmat[:, ii] /= sd[ii] start = OLS(np.log(1 + zhist), dmat).fit().params # Poisson regression if alpha > 0: md = GLM(zhist, dmat, family=families.Poisson()).fit_regularized(L1_wt=0, alpha=alpha, start_params=start) else: md = GLM(zhist, dmat, family=families.Poisson()).fit(start_params=start) # The design matrix for all Z-scores dmat_full = np.vander(zscores, deg + 1) dmat_full[:, ii] /= sd[ii] # 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 class NullDistribution: """ 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_ub` 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 def pdf(self, zscores): """ Evaluates the fitted empirical 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))