# Authors: Alexandre Gramfort # Denis Engemann # Eric Larson # # License: Simplified BSD import numpy as np from functools import reduce from string import ascii_uppercase from ..externals.six import string_types # The following function is a rewriting of scipy.stats.f_oneway # Contrary to the scipy.stats.f_oneway implementation it does not # copy the data while keeping the inputs unchanged. def _f_oneway(*args): """ Performs a 1-way ANOVA. The one-way ANOVA tests the null hypothesis that 2 or more groups have the same population mean. The test is applied to samples from two or more groups, possibly with differing sizes. Parameters ---------- sample1, sample2, ... : array_like The sample measurements should be given as arguments. Returns ------- F-value : float The computed F-value of the test p-value : float The associated p-value from the F-distribution Notes ----- The ANOVA test has important assumptions that must be satisfied in order for the associated p-value to be valid. 1. The samples are independent 2. Each sample is from a normally distributed population 3. The population standard deviations of the groups are all equal. This property is known as homocedasticity. If these assumptions are not true for a given set of data, it may still be possible to use the Kruskal-Wallis H-test (`stats.kruskal`_) although with some loss of power The algorithm is from Heiman[2], pp.394-7. See scipy.stats.f_oneway that should give the same results while being less efficient References ---------- .. [1] Lowry, Richard. "Concepts and Applications of Inferential Statistics". Chapter 14. http://faculty.vassar.edu/lowry/ch14pt1.html .. [2] Heiman, G.W. Research Methods in Statistics. 2002. """ from scipy import stats sf = stats.f.sf n_classes = len(args) n_samples_per_class = np.array([len(a) for a in args]) n_samples = np.sum(n_samples_per_class) ss_alldata = reduce(lambda x, y: x + y, [np.sum(a ** 2, axis=0) for a in args]) sums_args = [np.sum(a, axis=0) for a in args] square_of_sums_alldata = reduce(lambda x, y: x + y, sums_args) ** 2 square_of_sums_args = [s ** 2 for s in sums_args] sstot = ss_alldata - square_of_sums_alldata / float(n_samples) ssbn = 0 for k, _ in enumerate(args): ssbn += square_of_sums_args[k] / n_samples_per_class[k] ssbn -= square_of_sums_alldata / float(n_samples) sswn = sstot - ssbn dfbn = n_classes - 1 dfwn = n_samples - n_classes msb = ssbn / float(dfbn) msw = sswn / float(dfwn) f = msb / msw prob = sf(dfbn, dfwn, f) return f, prob def f_oneway(*args): """Call scipy.stats.f_oneway, but return only f-value""" return _f_oneway(*args)[0] def _map_effects(n_factors, effects): """Map effects to indices""" if n_factors > len(ascii_uppercase): raise ValueError('Maximum number of factors supported is 26') factor_names = list(ascii_uppercase[:n_factors]) if isinstance(effects, string_types): if '*' in effects and ':' in effects: raise ValueError('Not "*" and ":" permitted in effects') elif '+' in effects and ':' in effects: raise ValueError('Not "+" and ":" permitted in effects') elif effects == 'all': effects = None elif len(effects) == 1 or ':' in effects: effects = [effects] elif '+' in effects: # all main effects effects = effects.split('+') elif '*' in effects: pass # handle later else: raise ValueError('"{0}" is not a valid option for "effects"' .format(effects)) if isinstance(effects, list): bad_names = [e for e in effects if e not in factor_names] if len(bad_names) > 1: raise ValueError('Effect names: {0} are not valid. They should ' 'the first `n_factors` ({1}) characters from the' 'alphabet'.format(bad_names, n_factors)) indices = list(np.arange(2 ** n_factors - 1)) names = list() for this_effect in indices: contrast_idx = _get_contrast_indices(this_effect + 1, n_factors) this_code = (n_factors - 1) - np.where(contrast_idx == 1)[0] this_name = [factor_names[e] for e in this_code] this_name.sort() names.append(':'.join(this_name)) if effects is None or isinstance(effects, string_types): effects_ = names else: effects_ = effects selection = [names.index(sel) for sel in effects_] names = [names[sel] for sel in selection] if isinstance(effects, string_types): if '*' in effects: # hierarchical order of effects # the * based effect can be used as stop index sel_ind = names.index(effects.replace('*', ':')) + 1 names = names[:sel_ind] selection = selection[:sel_ind] return selection, names def _get_contrast_indices(effect_idx, n_factors): """Henson's factor coding, see num2binvec""" binrepr = np.binary_repr(effect_idx, n_factors) return np.array([int(i) for i in binrepr], dtype=int) def _iter_contrasts(n_subjects, factor_levels, effect_picks): """ Aux Function: Setup contrasts """ from scipy.signal import detrend sc = [] n_factors = len(factor_levels) # prepare computation of Kronecker products for n_levels in factor_levels: # for each factor append # 1) column vector of length == number of levels, # 2) square matrix with diagonal == number of levels # main + interaction effects for contrasts sc.append([np.ones([n_levels, 1]), detrend(np.eye(n_levels), type='constant')]) for this_effect in effect_picks: contrast_idx = _get_contrast_indices(this_effect + 1, n_factors) c_ = sc[0][contrast_idx[n_factors - 1]] for i_contrast in range(1, n_factors): this_contrast = contrast_idx[(n_factors - 1) - i_contrast] c_ = np.kron(c_, sc[i_contrast][this_contrast]) df1 = np.linalg.matrix_rank(c_) df2 = df1 * (n_subjects - 1) yield c_, df1, df2 def f_threshold_mway_rm(n_subjects, factor_levels, effects='A*B', pvalue=0.05): """ Compute f-value thesholds for a two-way ANOVA Parameters ---------- n_subjects : int The number of subjects to be analyzed. factor_levels : list-like The number of levels per factor. effects : str A string denoting the effect to be returned. The following mapping is currently supported: * ``'A'``: main effect of A * ``'B'``: main effect of B * ``'A:B'``: interaction effect * ``'A+B'``: both main effects * ``'A*B'``: all three effects pvalue : float The p-value to be thresholded. Returns ------- f_threshold : list | float list of f-values for each effect if the number of effects requested > 2, else float. See Also -------- f_oneway f_mway_rm Notes ----- .. versionadded:: 0.10 """ from scipy.stats import f effect_picks, _ = _map_effects(len(factor_levels), effects) f_threshold = [] for _, df1, df2 in _iter_contrasts(n_subjects, factor_levels, effect_picks): f_threshold.append(f(df1, df2).isf(pvalue)) return f_threshold if len(f_threshold) > 1 else f_threshold[0] def f_mway_rm(data, factor_levels, effects='all', alpha=0.05, correction=False, return_pvals=True): """M-way repeated measures ANOVA for fully balanced designs Parameters ---------- data : ndarray 3D array where the first two dimensions are compliant with a subjects X conditions scheme where the first factor repeats slowest:: A1B1 A1B2 A2B1 A2B2 subject 1 1.34 2.53 0.97 1.74 subject ... .... .... .... .... subject k 2.45 7.90 3.09 4.76 The last dimensions is thought to carry the observations for mass univariate analysis. factor_levels : list-like The number of levels per factor. effects : str | list A string denoting the effect to be returned. The following mapping is currently supported (example with 2 factors): * ``'A'``: main effect of A * ``'B'``: main effect of B * ``'A:B'``: interaction effect * ``'A+B'``: both main effects * ``'A*B'``: all three effects * ``'all'``: all effects (equals 'A*B' in a 2 way design) If list, effect names are used: ``['A', 'B', 'A:B']``. alpha : float The significance threshold. correction : bool The correction method to be employed if one factor has more than two levels. If True, sphericity correction using the Greenhouse-Geisser method will be applied. return_pvals : bool If True, return p values corresponding to f values. Returns ------- f_vals : ndarray An array of f values with length corresponding to the number of effects estimated. The shape depends on the number of effects estimated. p_vals : ndarray If not requested via return_pvals, defaults to an empty array. See Also -------- f_oneway f_threshold_mway_rm Notes ----- .. versionadded:: 0.10 """ from scipy.stats import f if data.ndim == 2: # general purpose support, e.g. behavioural data data = data[:, :, np.newaxis] elif data.ndim > 3: # let's allow for some magic here. data = data.reshape( data.shape[0], data.shape[1], np.prod(data.shape[2:])) effect_picks, _ = _map_effects(len(factor_levels), effects) n_obs = data.shape[2] n_replications = data.shape[0] # put last axis in front to 'iterate' over mass univariate instances. data = np.rollaxis(data, 2) fvalues, pvalues = [], [] for c_, df1, df2 in _iter_contrasts(n_replications, factor_levels, effect_picks): y = np.dot(data, c_) b = np.mean(y, axis=1)[:, np.newaxis, :] ss = np.sum(np.sum(y * b, axis=2), axis=1) mse = (np.sum(np.sum(y * y, axis=2), axis=1) - ss) / (df2 / df1) fvals = ss / mse fvalues.append(fvals) if correction: # sample covariances, leave off "/ (y.shape[1] - 1)" norm because # it falls out. v = np.array([np.dot(y_.T, y_) for y_ in y]) v = (np.array([np.trace(vv) for vv in v]) ** 2 / (df1 * np.sum(np.sum(v * v, axis=2), axis=1))) eps = v df1, df2 = np.zeros(n_obs) + df1, np.zeros(n_obs) + df2 if correction: # numerical imprecision can cause eps=0.99999999999999989 # even with a single category, so never let our degrees of # freedom drop below 1. df1, df2 = [np.maximum(d[None, :] * eps, 1.) for d in (df1, df2)] if return_pvals: pvals = f(df1, df2).sf(fvals) else: pvals = np.empty(0) pvalues.append(pvals) # handle single effect returns return [np.squeeze(np.asarray(vv)) for vv in (fvalues, pvalues)]