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# Authors: The MNE-Python contributors.
# License: BSD-3-Clause
# Copyright the MNE-Python contributors.
from functools import reduce
from string import ascii_uppercase
import numpy as np
from scipy import stats
from scipy.signal import detrend
from ..utils import _check_option
# 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 ttest_1samp_no_p(X, sigma=0, method="relative"):
"""Perform one-sample t-test.
This is a modified version of :func:`scipy.stats.ttest_1samp` that avoids
a (relatively) time-consuming p-value calculation, and can adjust
for implausibly small variance values :footcite:`RidgwayEtAl2012`.
Parameters
----------
X : array
Array to return t-values for.
sigma : float
The variance estimate will be given by ``var + sigma * max(var)`` or
``var + sigma``, depending on "method". By default this is 0 (no
adjustment). See Notes for details.
method : str
If 'relative', the minimum variance estimate will be sigma * max(var),
if 'absolute' the minimum variance estimate will be sigma.
Returns
-------
t : array
T-values, potentially adjusted using the hat method.
Notes
-----
To use the "hat" adjustment method :footcite:`RidgwayEtAl2012`, a value
of ``sigma=1e-3`` may be a reasonable choice.
References
----------
.. footbibliography::
"""
_check_option("method", method, ["absolute", "relative"])
var = np.var(X, axis=0, ddof=1)
if sigma > 0:
limit = sigma * np.max(var) if method == "relative" else sigma
var += limit
return np.mean(X, axis=0) / np.sqrt(var / X.shape[0])
def ttest_ind_no_p(a, b, equal_var=True, sigma=0.0):
"""Independent samples t-test without p calculation.
This is a modified version of :func:`scipy.stats.ttest_ind`. It operates
along the first axis. The ``sigma`` parameter provides an optional "hat"
adjustment (see :func:`ttest_1samp_no_p` and :footcite:`RidgwayEtAl2012`).
Parameters
----------
a : array-like
The first array.
b : array-like
The second array.
equal_var : bool
Assume equal variance. See :func:`scipy.stats.ttest_ind`.
sigma : float
The regularization. See :func:`ttest_1samp_no_p`.
Returns
-------
t : array
T values.
References
----------
.. footbibliography::
"""
v1 = np.var(a, axis=0, ddof=1)
v2 = np.var(b, axis=0, ddof=1)
n1 = a.shape[0]
n2 = b.shape[0]
if equal_var:
df = n1 + n2 - 2.0
var = ((n1 - 1) * v1 + (n2 - 1) * v2) / df
var = var * (1.0 / n1 + 1.0 / n2)
else:
vn1 = v1 / n1
vn2 = v2 / n2
with np.errstate(divide="ignore", invalid="ignore"):
df = (vn1 + vn2) ** 2 / (vn1**2 / (n1 - 1) + vn2**2 / (n2 - 1))
# If df is undefined, variances are zero (assumes n1 > 0 & n2 > 0).
# Hence it doesn't matter what df is as long as it's not NaN.
df = np.where(np.isnan(df), 1, df)
var = vn1 + vn2
if sigma > 0:
var += sigma * np.max(var)
denom = np.sqrt(var)
d = np.mean(a, 0) - np.mean(b, 0)
with np.errstate(divide="ignore", invalid="ignore"):
t = np.divide(d, denom)
return t
def f_oneway(*args):
"""Perform 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 :footcite:`Lowry2014`.
This is a modified version of :func:`scipy.stats.f_oneway` that avoids
computing the associated p-value.
Parameters
----------
*args : array_like
The sample measurements should be given as arguments.
Returns
-------
F-value : float
The computed F-value of the test.
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 homoscedasticity.
If these assumptions are not true for a given set of data, it may still be
possible to use the Kruskal-Wallis H-test (:func:`scipy.stats.kruskal`)
although with some loss of power.
The algorithm is from Heiman :footcite:`Heiman2002`, pp.394-7.
References
----------
.. footbibliography::
"""
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
return f
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, str):
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(f'"{effects}" is not a valid option for "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(
f"Effect names: {bad_names} are not valid. They should consist of the "
f"first `n_factors` ({n_factors}) characters from the alphabet"
)
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, str):
effects_ = names
else:
effects_ = effects
selection = [names.index(sel) for sel in effects_]
names = [names[sel] for sel in selection]
if isinstance(effects, str):
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): # noqa: D401
"""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):
"""Set up contrasts."""
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 thresholds 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
"""
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(stats.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", correction=False, return_pvals=True):
"""Compute 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']``.
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-statistics 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
"""
out_reshape = (-1,)
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
out_reshape = data.shape[2:]
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, 1.0) for d in (df1, df2))
if return_pvals:
pvals = stats.f(df1, df2).sf(fvals)
else:
pvals = np.empty(0)
pvalues.append(pvals)
# handle single effect returns
return [
np.squeeze(np.asarray([v.reshape(out_reshape) for v in vv]))
for vv in (fvalues, pvalues)
]
def _parametric_ci(arr, ci=0.95):
"""Calculate the `ci`% parametric confidence interval for `arr`."""
mean = arr.mean(0)
if len(arr) < 2: # can't compute standard error
sigma = np.full_like(mean, np.nan)
return mean, sigma
sigma = stats.sem(arr, 0)
return stats.t.interval(ci, loc=mean, scale=sigma, df=arr.shape[0])