1220 lines
45 KiB
Python
1220 lines
45 KiB
Python
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"""
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Created on Wed Mar 18 10:33:38 2020
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Author: Josef Perktold
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License: BSD-3
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"""
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import numpy as np
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from scipy import stats
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from scipy.special import ncfdtrinc
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# functions that use scipy.special instead of boost based function in stats
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from statsmodels.stats.power import ncf_cdf, ncf_ppf
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from statsmodels.stats.robust_compare import TrimmedMean, scale_transform
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from statsmodels.tools.testing import Holder
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from statsmodels.stats.base import HolderTuple
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def effectsize_oneway(means, vars_, nobs, use_var="unequal", ddof_between=0):
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"""
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Effect size corresponding to Cohen's f = nc / nobs for oneway anova
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This contains adjustment for Welch and Brown-Forsythe Anova so that
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effect size can be used with FTestAnovaPower.
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Parameters
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----------
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means : array_like
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Mean of samples to be compared
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vars_ : float or array_like
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Residual (within) variance of each sample or pooled
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If ``vars_`` is scalar, then it is interpreted as pooled variance that
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is the same for all samples, ``use_var`` will be ignored.
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Otherwise, the variances are used depending on the ``use_var`` keyword.
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nobs : int or array_like
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Number of observations for the samples.
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If nobs is scalar, then it is assumed that all samples have the same
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number ``nobs`` of observation, i.e. a balanced sample case.
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Otherwise, statistics will be weighted corresponding to nobs.
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Only relative sizes are relevant, any proportional change to nobs does
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not change the effect size.
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use_var : {"unequal", "equal", "bf"}
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If ``use_var`` is "unequal", then the variances can differ across
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samples and the effect size for Welch anova will be computed.
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ddof_between : int
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Degrees of freedom correction for the weighted between sum of squares.
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The denominator is ``nobs_total - ddof_between``
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This can be used to match differences across reference literature.
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Returns
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-------
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f2 : float
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Effect size corresponding to squared Cohen's f, which is also equal
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to the noncentrality divided by total number of observations.
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Notes
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-----
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This currently handles the following cases for oneway anova
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- balanced sample with homoscedastic variances
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- samples with different number of observations and with homoscedastic
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variances
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- samples with different number of observations and with heteroskedastic
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variances. This corresponds to Welch anova
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In the case of "unequal" and "bf" methods for unequal variances, the
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effect sizes do not directly correspond to the test statistic in Anova.
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Both have correction terms dropped or added, so the effect sizes match up
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with using FTestAnovaPower.
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If all variances are equal, then all three methods result in the same
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effect size. If variances are unequal, then the three methods produce
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small differences in effect size.
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Note, the effect size and power computation for BF Anova was not found in
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the literature. The correction terms were added so that FTestAnovaPower
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provides a good approximation to the power.
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Status: experimental
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We might add additional returns, if those are needed to support power
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and sample size applications.
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Examples
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--------
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The following shows how to compute effect size and power for each of the
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three anova methods. The null hypothesis is that the means are equal which
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corresponds to a zero effect size. Under the alternative, means differ
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with two sample means at a distance delta from the mean. We assume the
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variance is the same under the null and alternative hypothesis.
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``nobs`` for the samples defines the fraction of observations in the
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samples. ``nobs`` in the power method defines the total sample size.
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In simulations, the computed power for standard anova,
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i.e.``use_var="equal"`` overestimates the simulated power by a few percent.
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The equal variance assumption does not hold in this example.
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>>> from statsmodels.stats.oneway import effectsize_oneway
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>>> from statsmodels.stats.power import FTestAnovaPower
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>>>
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>>> nobs = np.array([10, 12, 13, 15])
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>>> delta = 0.5
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>>> means_alt = np.array([-1, 0, 0, 1]) * delta
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>>> vars_ = np.arange(1, len(means_alt) + 1)
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>>>
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>>> f2_alt = effectsize_oneway(means_alt, vars_, nobs, use_var="equal")
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>>> f2_alt
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0.04581300813008131
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>>>
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>>> kwds = {'effect_size': np.sqrt(f2_alt), 'nobs': 100, 'alpha': 0.05,
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... 'k_groups': 4}
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>>> power = FTestAnovaPower().power(**kwds)
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>>> power
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0.39165892158983273
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>>>
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>>> f2_alt = effectsize_oneway(means_alt, vars_, nobs, use_var="unequal")
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>>> f2_alt
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0.060640138408304504
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>>>
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>>> kwds['effect_size'] = np.sqrt(f2_alt)
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>>> power = FTestAnovaPower().power(**kwds)
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>>> power
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0.5047366512800622
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>>>
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>>> f2_alt = effectsize_oneway(means_alt, vars_, nobs, use_var="bf")
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>>> f2_alt
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0.04391324307956788
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>>>
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>>> kwds['effect_size'] = np.sqrt(f2_alt)
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>>> power = FTestAnovaPower().power(**kwds)
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>>> power
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0.3765792117047725
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"""
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# the code here is largely a copy of onway_generic with adjustments
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means = np.asarray(means)
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n_groups = means.shape[0]
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if np.size(nobs) == 1:
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nobs = np.ones(n_groups) * nobs
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nobs_t = nobs.sum()
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if use_var == "equal":
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if np.size(vars_) == 1:
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var_resid = vars_
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else:
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vars_ = np.asarray(vars_)
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var_resid = ((nobs - 1) * vars_).sum() / (nobs_t - n_groups)
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vars_ = var_resid # scalar, if broadcasting works
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weights = nobs / vars_
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w_total = weights.sum()
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w_rel = weights / w_total
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# meanw_t = (weights * means).sum() / w_total
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meanw_t = w_rel @ means
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f2 = np.dot(weights, (means - meanw_t)**2) / (nobs_t - ddof_between)
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if use_var.lower() == "bf":
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weights = nobs
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w_total = weights.sum()
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w_rel = weights / w_total
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meanw_t = w_rel @ means
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# TODO: reuse general case with weights
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tmp = ((1. - nobs / nobs_t) * vars_).sum()
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statistic = 1. * (nobs * (means - meanw_t)**2).sum()
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statistic /= tmp
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f2 = statistic * (1. - nobs / nobs_t).sum() / nobs_t
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# correction factor for df_num in BFM
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df_num2 = n_groups - 1
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df_num = tmp**2 / ((vars_**2).sum() +
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(nobs / nobs_t * vars_).sum()**2 -
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2 * (nobs / nobs_t * vars_**2).sum())
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f2 *= df_num / df_num2
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return f2
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def convert_effectsize_fsqu(f2=None, eta2=None):
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"""Convert squared effect sizes in f family
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f2 is signal to noise ratio, var_explained / var_residual
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eta2 is proportion of explained variance, var_explained / var_total
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uses the relationship:
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f2 = eta2 / (1 - eta2)
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Parameters
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----------
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f2 : None or float
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Squared Cohen's F effect size. If f2 is not None, then eta2 will be
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computed.
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eta2 : None or float
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Squared eta effect size. If f2 is None and eta2 is not None, then f2 is
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computed.
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Returns
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-------
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res : Holder instance
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An instance of the Holder class with f2 and eta2 as attributes.
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"""
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if f2 is not None:
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eta2 = 1 / (1 + 1 / f2)
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elif eta2 is not None:
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f2 = eta2 / (1 - eta2)
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res = Holder(f2=f2, eta2=eta2)
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return res
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def _fstat2effectsize(f_stat, df):
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"""Compute anova effect size from F-statistic
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This might be combined with convert_effectsize_fsqu
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Parameters
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----------
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f_stat : array_like
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Test statistic of an F-test
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df : tuple
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degrees of freedom ``df = (df1, df2)`` where
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- df1 : numerator degrees of freedom, number of constraints
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- df2 : denominator degrees of freedom, df_resid
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Returns
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-------
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res : Holder instance
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This instance contains effect size measures f2, eta2, omega2 and eps2
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as attributes.
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Notes
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-----
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This uses the following definitions:
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- f2 = f_stat * df1 / df2
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- eta2 = f2 / (f2 + 1)
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- omega2 = (f2 - df1 / df2) / (f2 + 2)
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- eps2 = (f2 - df1 / df2) / (f2 + 1)
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This differs from effect size measures in other function which define
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``f2 = f_stat * df1 / nobs``
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or an equivalent expression for power computation. The noncentrality
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index for the hypothesis test is in those cases given by
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``nc = f_stat * df1``.
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Currently omega2 and eps2 are computed in two different ways. Those
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values agree for regular cases but can show different behavior in corner
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cases (e.g. zero division).
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"""
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df1, df2 = df
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f2 = f_stat * df1 / df2
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eta2 = f2 / (f2 + 1)
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omega2_ = (f_stat - 1) / (f_stat + (df2 + 1) / df1)
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omega2 = (f2 - df1 / df2) / (f2 + 1 + 1 / df2) # rewrite
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eps2_ = (f_stat - 1) / (f_stat + df2 / df1)
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eps2 = (f2 - df1 / df2) / (f2 + 1) # rewrite
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return Holder(f2=f2, eta2=eta2, omega2=omega2, eps2=eps2, eps2_=eps2_,
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omega2_=omega2_)
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# conversion functions for Wellek's equivalence effect size
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# these are mainly to compare with literature
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def wellek_to_f2(eps, n_groups):
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"""Convert Wellek's effect size (sqrt) to Cohen's f-squared
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This computes the following effect size :
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f2 = 1 / n_groups * eps**2
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Parameters
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----------
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eps : float or ndarray
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Wellek's effect size used in anova equivalence test
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n_groups : int
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Number of groups in oneway comparison
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Returns
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-------
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f2 : effect size Cohen's f-squared
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"""
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f2 = 1 / n_groups * eps**2
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return f2
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def f2_to_wellek(f2, n_groups):
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"""Convert Cohen's f-squared to Wellek's effect size (sqrt)
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This computes the following effect size :
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eps = sqrt(n_groups * f2)
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Parameters
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----------
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f2 : float or ndarray
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Effect size Cohen's f-squared
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n_groups : int
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Number of groups in oneway comparison
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Returns
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-------
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eps : float or ndarray
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Wellek's effect size used in anova equivalence test
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"""
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eps = np.sqrt(n_groups * f2)
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return eps
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def fstat_to_wellek(f_stat, n_groups, nobs_mean):
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"""Convert F statistic to wellek's effect size eps squared
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This computes the following effect size :
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es = f_stat * (n_groups - 1) / nobs_mean
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Parameters
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----------
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f_stat : float or ndarray
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Test statistic of an F-test.
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n_groups : int
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Number of groups in oneway comparison
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nobs_mean : float or ndarray
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Average number of observations across groups.
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Returns
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-------
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eps : float or ndarray
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Wellek's effect size used in anova equivalence test
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"""
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es = f_stat * (n_groups - 1) / nobs_mean
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return es
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def confint_noncentrality(f_stat, df, alpha=0.05,
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alternative="two-sided"):
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"""
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Confidence interval for noncentrality parameter in F-test
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This does not yet handle non-negativity constraint on nc.
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Currently only two-sided alternative is supported.
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Parameters
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----------
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f_stat : float
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df : tuple
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|||
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degrees of freedom ``df = (df1, df2)`` where
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- df1 : numerator degrees of freedom, number of constraints
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- df2 : denominator degrees of freedom, df_resid
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alpha : float, default 0.05
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alternative : {"two-sided"}
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|
|
Other alternatives have not been implements.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
float
|
|||
|
|
The end point of the confidence interval.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The algorithm inverts the cdf of the noncentral F distribution with
|
|||
|
|
respect to the noncentrality parameters.
|
|||
|
|
See Steiger 2004 and references cited in it.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Steiger, James H. 2004. “Beyond the F Test: Effect Size Confidence
|
|||
|
|
Intervals and Tests of Close Fit in the Analysis of Variance and
|
|||
|
|
Contrast Analysis.” Psychological Methods 9 (2): 164–82.
|
|||
|
|
https://doi.org/10.1037/1082-989X.9.2.164.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
confint_effectsize_oneway
|
|||
|
|
"""
|
|||
|
|
|
|||
|
|
df1, df2 = df
|
|||
|
|
if alternative in ["two-sided", "2s", "ts"]:
|
|||
|
|
alpha1s = alpha / 2
|
|||
|
|
ci = ncfdtrinc(df1, df2, [1 - alpha1s, alpha1s], f_stat)
|
|||
|
|
else:
|
|||
|
|
raise NotImplementedError
|
|||
|
|
|
|||
|
|
return ci
|
|||
|
|
|
|||
|
|
|
|||
|
|
def confint_effectsize_oneway(f_stat, df, alpha=0.05, nobs=None):
|
|||
|
|
"""
|
|||
|
|
Confidence interval for effect size in oneway anova for F distribution
|
|||
|
|
|
|||
|
|
This does not yet handle non-negativity constraint on nc.
|
|||
|
|
Currently only two-sided alternative is supported.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
f_stat : float
|
|||
|
|
df : tuple
|
|||
|
|
degrees of freedom ``df = (df1, df2)`` where
|
|||
|
|
|
|||
|
|
- df1 : numerator degrees of freedom, number of constraints
|
|||
|
|
- df2 : denominator degrees of freedom, df_resid
|
|||
|
|
|
|||
|
|
alpha : float, default 0.05
|
|||
|
|
nobs : int, default None
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
Holder
|
|||
|
|
Class with effect size and confidence attributes
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The confidence interval for the noncentrality parameter is obtained by
|
|||
|
|
inverting the cdf of the noncentral F distribution. Confidence intervals
|
|||
|
|
for other effect sizes are computed by endpoint transformation.
|
|||
|
|
|
|||
|
|
|
|||
|
|
R package ``effectsize`` does not compute the confidence intervals in the
|
|||
|
|
same way. Their confidence intervals can be replicated with
|
|||
|
|
|
|||
|
|
>>> ci_nc = confint_noncentrality(f_stat, df1, df2, alpha=0.1)
|
|||
|
|
>>> ci_es = smo._fstat2effectsize(ci_nc / df1, df1, df2)
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
confint_noncentrality
|
|||
|
|
"""
|
|||
|
|
|
|||
|
|
df1, df2 = df
|
|||
|
|
if nobs is None:
|
|||
|
|
nobs = df1 + df2 + 1
|
|||
|
|
ci_nc = confint_noncentrality(f_stat, df, alpha=alpha)
|
|||
|
|
|
|||
|
|
ci_f2 = ci_nc / nobs
|
|||
|
|
ci_res = convert_effectsize_fsqu(f2=ci_f2)
|
|||
|
|
ci_res.ci_omega2 = (ci_f2 - df1 / df2) / (ci_f2 + 1 + 1 / df2)
|
|||
|
|
ci_res.ci_nc = ci_nc
|
|||
|
|
ci_res.ci_f = np.sqrt(ci_res.f2)
|
|||
|
|
ci_res.ci_eta = np.sqrt(ci_res.eta2)
|
|||
|
|
ci_res.ci_f_corrected = np.sqrt(ci_res.f2 * (df1 + 1) / df1)
|
|||
|
|
|
|||
|
|
return ci_res
|
|||
|
|
|
|||
|
|
|
|||
|
|
def anova_generic(means, variances, nobs, use_var="unequal",
|
|||
|
|
welch_correction=True, info=None):
|
|||
|
|
"""
|
|||
|
|
Oneway Anova based on summary statistics
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
means : array_like
|
|||
|
|
Mean of samples to be compared
|
|||
|
|
variances : float or array_like
|
|||
|
|
Residual (within) variance of each sample or pooled.
|
|||
|
|
If ``variances`` is scalar, then it is interpreted as pooled variance
|
|||
|
|
that is the same for all samples, ``use_var`` will be ignored.
|
|||
|
|
Otherwise, the variances are used depending on the ``use_var`` keyword.
|
|||
|
|
nobs : int or array_like
|
|||
|
|
Number of observations for the samples.
|
|||
|
|
If nobs is scalar, then it is assumed that all samples have the same
|
|||
|
|
number ``nobs`` of observation, i.e. a balanced sample case.
|
|||
|
|
Otherwise, statistics will be weighted corresponding to nobs.
|
|||
|
|
Only relative sizes are relevant, any proportional change to nobs does
|
|||
|
|
not change the effect size.
|
|||
|
|
use_var : {"unequal", "equal", "bf"}
|
|||
|
|
If ``use_var`` is "unequal", then the variances can differ across
|
|||
|
|
samples and the effect size for Welch anova will be computed.
|
|||
|
|
welch_correction : bool
|
|||
|
|
If this is false, then the Welch correction to the test statistic is
|
|||
|
|
not included. This allows the computation of an effect size measure
|
|||
|
|
that corresponds more closely to Cohen's f.
|
|||
|
|
info : not used yet
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
res : results instance
|
|||
|
|
This includes `statistic` and `pvalue`.
|
|||
|
|
|
|||
|
|
"""
|
|||
|
|
options = {"use_var": use_var,
|
|||
|
|
"welch_correction": welch_correction
|
|||
|
|
}
|
|||
|
|
if means.ndim != 1:
|
|||
|
|
raise ValueError('data (means, ...) has to be one-dimensional')
|
|||
|
|
nobs_t = nobs.sum()
|
|||
|
|
n_groups = len(means)
|
|||
|
|
# mean_t = (nobs * means).sum() / nobs_t
|
|||
|
|
if use_var == "unequal":
|
|||
|
|
weights = nobs / variances
|
|||
|
|
else:
|
|||
|
|
weights = nobs
|
|||
|
|
|
|||
|
|
w_total = weights.sum()
|
|||
|
|
w_rel = weights / w_total
|
|||
|
|
# meanw_t = (weights * means).sum() / w_total
|
|||
|
|
meanw_t = w_rel @ means
|
|||
|
|
|
|||
|
|
statistic = np.dot(weights, (means - meanw_t)**2) / (n_groups - 1.)
|
|||
|
|
df_num = n_groups - 1.
|
|||
|
|
|
|||
|
|
if use_var == "unequal":
|
|||
|
|
tmp = ((1 - w_rel)**2 / (nobs - 1)).sum() / (n_groups**2 - 1)
|
|||
|
|
if welch_correction:
|
|||
|
|
statistic /= 1 + 2 * (n_groups - 2) * tmp
|
|||
|
|
df_denom = 1. / (3. * tmp)
|
|||
|
|
|
|||
|
|
elif use_var == "equal":
|
|||
|
|
# variance of group demeaned total sample, pooled var_resid
|
|||
|
|
tmp = ((nobs - 1) * variances).sum() / (nobs_t - n_groups)
|
|||
|
|
statistic /= tmp
|
|||
|
|
df_denom = nobs_t - n_groups
|
|||
|
|
|
|||
|
|
elif use_var == "bf":
|
|||
|
|
tmp = ((1. - nobs / nobs_t) * variances).sum()
|
|||
|
|
statistic = 1. * (nobs * (means - meanw_t)**2).sum()
|
|||
|
|
statistic /= tmp
|
|||
|
|
|
|||
|
|
df_num2 = n_groups - 1
|
|||
|
|
df_denom = tmp**2 / ((1. - nobs / nobs_t) ** 2 *
|
|||
|
|
variances ** 2 / (nobs - 1)).sum()
|
|||
|
|
df_num = tmp**2 / ((variances ** 2).sum() +
|
|||
|
|
(nobs / nobs_t * variances).sum() ** 2 -
|
|||
|
|
2 * (nobs / nobs_t * variances ** 2).sum())
|
|||
|
|
pval2 = stats.f.sf(statistic, df_num2, df_denom)
|
|||
|
|
options["df2"] = (df_num2, df_denom)
|
|||
|
|
options["df_num2"] = df_num2
|
|||
|
|
options["pvalue2"] = pval2
|
|||
|
|
|
|||
|
|
else:
|
|||
|
|
raise ValueError('use_var is to be one of "unequal", "equal" or "bf"')
|
|||
|
|
|
|||
|
|
pval = stats.f.sf(statistic, df_num, df_denom)
|
|||
|
|
res = HolderTuple(statistic=statistic,
|
|||
|
|
pvalue=pval,
|
|||
|
|
df=(df_num, df_denom),
|
|||
|
|
df_num=df_num,
|
|||
|
|
df_denom=df_denom,
|
|||
|
|
nobs_t=nobs_t,
|
|||
|
|
n_groups=n_groups,
|
|||
|
|
means=means,
|
|||
|
|
nobs=nobs,
|
|||
|
|
vars_=variances,
|
|||
|
|
**options
|
|||
|
|
)
|
|||
|
|
return res
|
|||
|
|
|
|||
|
|
|
|||
|
|
def anova_oneway(data, groups=None, use_var="unequal", welch_correction=True,
|
|||
|
|
trim_frac=0):
|
|||
|
|
"""Oneway Anova
|
|||
|
|
|
|||
|
|
This implements standard anova, Welch and Brown-Forsythe, and trimmed
|
|||
|
|
(Yuen) variants of those.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
data : tuple of array_like or DataFrame or Series
|
|||
|
|
Data for k independent samples, with k >= 2.
|
|||
|
|
The data can be provided as a tuple or list of arrays or in long
|
|||
|
|
format with outcome observations in ``data`` and group membership in
|
|||
|
|
``groups``.
|
|||
|
|
groups : ndarray or Series
|
|||
|
|
If data is in long format, then groups is needed as indicator to which
|
|||
|
|
group or sample and observations belongs.
|
|||
|
|
use_var : {"unequal", "equal" or "bf"}
|
|||
|
|
`use_var` specified how to treat heteroscedasticity, unequal variance,
|
|||
|
|
across samples. Three approaches are available
|
|||
|
|
|
|||
|
|
"unequal" : Variances are not assumed to be equal across samples.
|
|||
|
|
Heteroscedasticity is taken into account with Welch Anova and
|
|||
|
|
Satterthwaite-Welch degrees of freedom.
|
|||
|
|
This is the default.
|
|||
|
|
"equal" : Variances are assumed to be equal across samples.
|
|||
|
|
This is the standard Anova.
|
|||
|
|
"bf" : Variances are not assumed to be equal across samples.
|
|||
|
|
The method is Browne-Forsythe (1971) for testing equality of means
|
|||
|
|
with the corrected degrees of freedom by Merothra. The original BF
|
|||
|
|
degrees of freedom are available as additional attributes in the
|
|||
|
|
results instance, ``df_denom2`` and ``p_value2``.
|
|||
|
|
|
|||
|
|
welch_correction : bool
|
|||
|
|
If this is false, then the Welch correction to the test statistic is
|
|||
|
|
not included. This allows the computation of an effect size measure
|
|||
|
|
that corresponds more closely to Cohen's f.
|
|||
|
|
trim_frac : float in [0, 0.5)
|
|||
|
|
Optional trimming for Anova with trimmed mean and winsorized variances.
|
|||
|
|
With the default trim_frac equal to zero, the oneway Anova statistics
|
|||
|
|
are computed without trimming. If `trim_frac` is larger than zero,
|
|||
|
|
then the largest and smallest observations in each sample are trimmed.
|
|||
|
|
The number of trimmed observations is the fraction of number of
|
|||
|
|
observations in the sample truncated to the next lower integer.
|
|||
|
|
`trim_frac` has to be smaller than 0.5, however, if the fraction is
|
|||
|
|
so large that there are not enough observations left over, then `nan`
|
|||
|
|
will be returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
res : results instance
|
|||
|
|
The returned HolderTuple instance has the following main attributes
|
|||
|
|
and some additional information in other attributes.
|
|||
|
|
|
|||
|
|
statistic : float
|
|||
|
|
Test statistic for k-sample mean comparison which is approximately
|
|||
|
|
F-distributed.
|
|||
|
|
pvalue : float
|
|||
|
|
If ``use_var="bf"``, then the p-value is based on corrected
|
|||
|
|
degrees of freedom following Mehrotra 1997.
|
|||
|
|
pvalue2 : float
|
|||
|
|
This is the p-value based on degrees of freedom as in
|
|||
|
|
Brown-Forsythe 1974 and is only available if ``use_var="bf"``.
|
|||
|
|
df = (df_denom, df_num) : tuple of floats
|
|||
|
|
Degreeds of freedom for the F-distribution depend on ``use_var``.
|
|||
|
|
If ``use_var="bf"``, then `df_denom` is for Mehrotra p-values
|
|||
|
|
`df_denom2` is available for Brown-Forsythe 1974 p-values.
|
|||
|
|
`df_num` is the same numerator degrees of freedom for both
|
|||
|
|
p-values.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
Welch's anova is correctly sized (not liberal or conservative) in smaller
|
|||
|
|
samples if the distribution of the samples is not very far away from the
|
|||
|
|
normal distribution. The test can become liberal if the data is strongly
|
|||
|
|
skewed. Welch's Anova can also be correctly sized for discrete
|
|||
|
|
distributions with finite support, like Lickert scale data.
|
|||
|
|
The trimmed version is robust to many non-normal distributions, it stays
|
|||
|
|
correctly sized in many cases, and is more powerful in some cases with
|
|||
|
|
skewness or heavy tails.
|
|||
|
|
|
|||
|
|
Trimming is currently based on the integer part of ``nobs * trim_frac``.
|
|||
|
|
The default might change to including fractional observations as in the
|
|||
|
|
original articles by Yuen.
|
|||
|
|
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
anova_generic
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
Brown, Morton B., and Alan B. Forsythe. 1974. “The Small Sample Behavior
|
|||
|
|
of Some Statistics Which Test the Equality of Several Means.”
|
|||
|
|
Technometrics 16 (1) (February 1): 129–132. doi:10.2307/1267501.
|
|||
|
|
|
|||
|
|
Mehrotra, Devan V. 1997. “Improving the Brown-Forsythe Solution to the
|
|||
|
|
Generalized Behrens-Fisher Problem.” Communications in Statistics -
|
|||
|
|
Simulation and Computation 26 (3): 1139–1145.
|
|||
|
|
doi:10.1080/03610919708813431.
|
|||
|
|
"""
|
|||
|
|
if groups is not None:
|
|||
|
|
uniques = np.unique(groups)
|
|||
|
|
data = [data[groups == uni] for uni in uniques]
|
|||
|
|
else:
|
|||
|
|
# uniques = None # not used yet, add to info?
|
|||
|
|
pass
|
|||
|
|
args = list(map(np.asarray, data))
|
|||
|
|
if any([x.ndim != 1 for x in args]):
|
|||
|
|
raise ValueError('data arrays have to be one-dimensional')
|
|||
|
|
|
|||
|
|
nobs = np.array([len(x) for x in args], float)
|
|||
|
|
# n_groups = len(args) # not used
|
|||
|
|
# means = np.array([np.mean(x, axis=0) for x in args], float)
|
|||
|
|
# vars_ = np.array([np.var(x, ddof=1, axis=0) for x in args], float)
|
|||
|
|
|
|||
|
|
if trim_frac == 0:
|
|||
|
|
means = np.array([x.mean() for x in args])
|
|||
|
|
vars_ = np.array([x.var(ddof=1) for x in args])
|
|||
|
|
else:
|
|||
|
|
tms = [TrimmedMean(x, trim_frac) for x in args]
|
|||
|
|
means = np.array([tm.mean_trimmed for tm in tms])
|
|||
|
|
# R doesn't use uncorrected var_winsorized
|
|||
|
|
# vars_ = np.array([tm.var_winsorized for tm in tms])
|
|||
|
|
vars_ = np.array([tm.var_winsorized * (tm.nobs - 1) /
|
|||
|
|
(tm.nobs_reduced - 1) for tm in tms])
|
|||
|
|
# nobs_original = nobs # store just in case
|
|||
|
|
nobs = np.array([tm.nobs_reduced for tm in tms])
|
|||
|
|
|
|||
|
|
res = anova_generic(means, vars_, nobs, use_var=use_var,
|
|||
|
|
welch_correction=welch_correction)
|
|||
|
|
|
|||
|
|
return res
|
|||
|
|
|
|||
|
|
|
|||
|
|
def equivalence_oneway_generic(f_stat, n_groups, nobs, equiv_margin, df,
|
|||
|
|
alpha=0.05, margin_type="f2"):
|
|||
|
|
"""Equivalence test for oneway anova (Wellek and extensions)
|
|||
|
|
|
|||
|
|
This is an helper function when summary statistics are available.
|
|||
|
|
Use `equivalence_oneway` instead.
|
|||
|
|
|
|||
|
|
The null hypothesis is that the means differ by more than `equiv_margin`
|
|||
|
|
in the anova distance measure.
|
|||
|
|
If the Null is rejected, then the data supports that means are equivalent,
|
|||
|
|
i.e. within a given distance.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
f_stat : float
|
|||
|
|
F-statistic
|
|||
|
|
n_groups : int
|
|||
|
|
Number of groups in oneway comparison.
|
|||
|
|
nobs : ndarray
|
|||
|
|
Array of number of observations in groups.
|
|||
|
|
equiv_margin : float
|
|||
|
|
Equivalence margin in terms of effect size. Effect size can be chosen
|
|||
|
|
with `margin_type`. default is squared Cohen's f.
|
|||
|
|
df : tuple
|
|||
|
|
degrees of freedom ``df = (df1, df2)`` where
|
|||
|
|
|
|||
|
|
- df1 : numerator degrees of freedom, number of constraints
|
|||
|
|
- df2 : denominator degrees of freedom, df_resid
|
|||
|
|
|
|||
|
|
alpha : float in (0, 1)
|
|||
|
|
Significance level for the hypothesis test.
|
|||
|
|
margin_type : "f2" or "wellek"
|
|||
|
|
Type of effect size used for equivalence margin.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
results : instance of HolderTuple class
|
|||
|
|
The two main attributes are test statistic `statistic` and p-value
|
|||
|
|
`pvalue`.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
Equivalence in this function is defined in terms of a squared distance
|
|||
|
|
measure similar to Mahalanobis distance.
|
|||
|
|
Alternative definitions for the oneway case are based on maximum difference
|
|||
|
|
between pairs of means or similar pairwise distances.
|
|||
|
|
|
|||
|
|
The equivalence margin is used for the noncentrality parameter in the
|
|||
|
|
noncentral F distribution for the test statistic. In samples with unequal
|
|||
|
|
variances estimated using Welch or Brown-Forsythe Anova, the f-statistic
|
|||
|
|
depends on the unequal variances and corrections to the test statistic.
|
|||
|
|
This means that the equivalence margins are not fully comparable across
|
|||
|
|
methods for treating unequal variances.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
Wellek, Stefan. 2010. Testing Statistical Hypotheses of Equivalence and
|
|||
|
|
Noninferiority. 2nd ed. Boca Raton: CRC Press.
|
|||
|
|
|
|||
|
|
Cribbie, Robert A., Chantal A. Arpin-Cribbie, and Jamie A. Gruman. 2009.
|
|||
|
|
“Tests of Equivalence for One-Way Independent Groups Designs.” The Journal
|
|||
|
|
of Experimental Education 78 (1): 1–13.
|
|||
|
|
https://doi.org/10.1080/00220970903224552.
|
|||
|
|
|
|||
|
|
Jan, Show-Li, and Gwowen Shieh. 2019. “On the Extended Welch Test for
|
|||
|
|
Assessing Equivalence of Standardized Means.” Statistics in
|
|||
|
|
Biopharmaceutical Research 0 (0): 1–8.
|
|||
|
|
https://doi.org/10.1080/19466315.2019.1654915.
|
|||
|
|
|
|||
|
|
"""
|
|||
|
|
nobs_t = nobs.sum()
|
|||
|
|
nobs_mean = nobs_t / n_groups
|
|||
|
|
|
|||
|
|
if margin_type == "wellek":
|
|||
|
|
nc_null = nobs_mean * equiv_margin**2
|
|||
|
|
es = f_stat * (n_groups - 1) / nobs_mean
|
|||
|
|
type_effectsize = "Wellek's psi_squared"
|
|||
|
|
elif margin_type in ["f2", "fsqu", "fsquared"]:
|
|||
|
|
nc_null = nobs_t * equiv_margin
|
|||
|
|
es = f_stat / nobs_t
|
|||
|
|
type_effectsize = "Cohen's f_squared"
|
|||
|
|
else:
|
|||
|
|
raise ValueError('`margin_type` should be "f2" or "wellek"')
|
|||
|
|
crit_f = ncf_ppf(alpha, df[0], df[1], nc_null)
|
|||
|
|
|
|||
|
|
if margin_type == "wellek":
|
|||
|
|
# TODO: do we need a sqrt
|
|||
|
|
crit_es = crit_f * (n_groups - 1) / nobs_mean
|
|||
|
|
elif margin_type in ["f2", "fsqu", "fsquared"]:
|
|||
|
|
crit_es = crit_f / nobs_t
|
|||
|
|
|
|||
|
|
reject = (es < crit_es)
|
|||
|
|
|
|||
|
|
pv = ncf_cdf(f_stat, df[0], df[1], nc_null)
|
|||
|
|
pwr = ncf_cdf(crit_f, df[0], df[1], 1e-13) # scipy, cannot be 0
|
|||
|
|
res = HolderTuple(statistic=f_stat,
|
|||
|
|
pvalue=pv,
|
|||
|
|
effectsize=es, # match es type to margin_type
|
|||
|
|
crit_f=crit_f,
|
|||
|
|
crit_es=crit_es,
|
|||
|
|
reject=reject,
|
|||
|
|
power_zero=pwr,
|
|||
|
|
df=df,
|
|||
|
|
f_stat=f_stat,
|
|||
|
|
type_effectsize=type_effectsize
|
|||
|
|
)
|
|||
|
|
return res
|
|||
|
|
|
|||
|
|
|
|||
|
|
def equivalence_oneway(data, equiv_margin, groups=None, use_var="unequal",
|
|||
|
|
welch_correction=True, trim_frac=0, margin_type="f2"):
|
|||
|
|
"""equivalence test for oneway anova (Wellek's Anova)
|
|||
|
|
|
|||
|
|
The null hypothesis is that the means differ by more than `equiv_margin`
|
|||
|
|
in the anova distance measure.
|
|||
|
|
If the Null is rejected, then the data supports that means are equivalent,
|
|||
|
|
i.e. within a given distance.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
data : tuple of array_like or DataFrame or Series
|
|||
|
|
Data for k independent samples, with k >= 2.
|
|||
|
|
The data can be provided as a tuple or list of arrays or in long
|
|||
|
|
format with outcome observations in ``data`` and group membership in
|
|||
|
|
``groups``.
|
|||
|
|
equiv_margin : float
|
|||
|
|
Equivalence margin in terms of effect size. Effect size can be chosen
|
|||
|
|
with `margin_type`. default is squared Cohen's f.
|
|||
|
|
groups : ndarray or Series
|
|||
|
|
If data is in long format, then groups is needed as indicator to which
|
|||
|
|
group or sample and observations belongs.
|
|||
|
|
use_var : {"unequal", "equal" or "bf"}
|
|||
|
|
`use_var` specified how to treat heteroscedasticity, unequal variance,
|
|||
|
|
across samples. Three approaches are available
|
|||
|
|
|
|||
|
|
"unequal" : Variances are not assumed to be equal across samples.
|
|||
|
|
Heteroscedasticity is taken into account with Welch Anova and
|
|||
|
|
Satterthwaite-Welch degrees of freedom.
|
|||
|
|
This is the default.
|
|||
|
|
"equal" : Variances are assumed to be equal across samples.
|
|||
|
|
This is the standard Anova.
|
|||
|
|
"bf: Variances are not assumed to be equal across samples.
|
|||
|
|
The method is Browne-Forsythe (1971) for testing equality of means
|
|||
|
|
with the corrected degrees of freedom by Merothra. The original BF
|
|||
|
|
degrees of freedom are available as additional attributes in the
|
|||
|
|
results instance, ``df_denom2`` and ``p_value2``.
|
|||
|
|
|
|||
|
|
welch_correction : bool
|
|||
|
|
If this is false, then the Welch correction to the test statistic is
|
|||
|
|
not included. This allows the computation of an effect size measure
|
|||
|
|
that corresponds more closely to Cohen's f.
|
|||
|
|
trim_frac : float in [0, 0.5)
|
|||
|
|
Optional trimming for Anova with trimmed mean and winsorized variances.
|
|||
|
|
With the default trim_frac equal to zero, the oneway Anova statistics
|
|||
|
|
are computed without trimming. If `trim_frac` is larger than zero,
|
|||
|
|
then the largest and smallest observations in each sample are trimmed.
|
|||
|
|
The number of trimmed observations is the fraction of number of
|
|||
|
|
observations in the sample truncated to the next lower integer.
|
|||
|
|
`trim_frac` has to be smaller than 0.5, however, if the fraction is
|
|||
|
|
so large that there are not enough observations left over, then `nan`
|
|||
|
|
will be returned.
|
|||
|
|
margin_type : "f2" or "wellek"
|
|||
|
|
Type of effect size used for equivalence margin, either squared
|
|||
|
|
Cohen's f or Wellek's psi. Default is "f2".
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
results : instance of HolderTuple class
|
|||
|
|
The two main attributes are test statistic `statistic` and p-value
|
|||
|
|
`pvalue`.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
anova_oneway
|
|||
|
|
equivalence_scale_oneway
|
|||
|
|
"""
|
|||
|
|
|
|||
|
|
# use anova to compute summary statistics and f-statistic
|
|||
|
|
res0 = anova_oneway(data, groups=groups, use_var=use_var,
|
|||
|
|
welch_correction=welch_correction,
|
|||
|
|
trim_frac=trim_frac)
|
|||
|
|
f_stat = res0.statistic
|
|||
|
|
res = equivalence_oneway_generic(f_stat, res0.n_groups, res0.nobs_t,
|
|||
|
|
equiv_margin, res0.df, alpha=0.05,
|
|||
|
|
margin_type=margin_type)
|
|||
|
|
|
|||
|
|
return res
|
|||
|
|
|
|||
|
|
|
|||
|
|
def _power_equivalence_oneway_emp(f_stat, n_groups, nobs, eps, df, alpha=0.05):
|
|||
|
|
"""Empirical power of oneway equivalence test
|
|||
|
|
|
|||
|
|
This only returns post-hoc, empirical power.
|
|||
|
|
|
|||
|
|
Warning: eps is currently effect size margin as defined as in Wellek, and
|
|||
|
|
not the signal to noise ratio (Cohen's f family).
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
f_stat : float
|
|||
|
|
F-statistic from oneway anova, used to compute empirical effect size
|
|||
|
|
n_groups : int
|
|||
|
|
Number of groups in oneway comparison.
|
|||
|
|
nobs : ndarray
|
|||
|
|
Array of number of observations in groups.
|
|||
|
|
eps : float
|
|||
|
|
Equivalence margin in terms of effect size given by Wellek's psi.
|
|||
|
|
df : tuple
|
|||
|
|
Degrees of freedom for F distribution.
|
|||
|
|
alpha : float in (0, 1)
|
|||
|
|
Significance level for the hypothesis test.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
pow : float
|
|||
|
|
Ex-post, post-hoc or empirical power at f-statistic of the equivalence
|
|||
|
|
test.
|
|||
|
|
"""
|
|||
|
|
|
|||
|
|
res = equivalence_oneway_generic(f_stat, n_groups, nobs, eps, df,
|
|||
|
|
alpha=alpha, margin_type="wellek")
|
|||
|
|
|
|||
|
|
nobs_mean = nobs.sum() / n_groups
|
|||
|
|
fn = f_stat # post-hoc power, empirical power at estimate
|
|||
|
|
esn = fn * (n_groups - 1) / nobs_mean # Wellek psi
|
|||
|
|
pow_ = ncf_cdf(res.crit_f, df[0], df[1], nobs_mean * esn)
|
|||
|
|
|
|||
|
|
return pow_
|
|||
|
|
|
|||
|
|
|
|||
|
|
def power_equivalence_oneway(f2_alt, equiv_margin, nobs_t, n_groups=None,
|
|||
|
|
df=None, alpha=0.05, margin_type="f2"):
|
|||
|
|
"""
|
|||
|
|
Power of oneway equivalence test
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
f2_alt : float
|
|||
|
|
Effect size, squared Cohen's f, under the alternative.
|
|||
|
|
equiv_margin : float
|
|||
|
|
Equivalence margin in terms of effect size. Effect size can be chosen
|
|||
|
|
with `margin_type`. default is squared Cohen's f.
|
|||
|
|
nobs_t : ndarray
|
|||
|
|
Total number of observations summed over all groups.
|
|||
|
|
n_groups : int
|
|||
|
|
Number of groups in oneway comparison. If margin_type is "wellek",
|
|||
|
|
then either ``n_groups`` or ``df`` has to be given.
|
|||
|
|
df : tuple
|
|||
|
|
Degrees of freedom for F distribution,
|
|||
|
|
``df = (n_groups - 1, nobs_t - n_groups)``
|
|||
|
|
alpha : float in (0, 1)
|
|||
|
|
Significance level for the hypothesis test.
|
|||
|
|
margin_type : "f2" or "wellek"
|
|||
|
|
Type of effect size used for equivalence margin, either squared
|
|||
|
|
Cohen's f or Wellek's psi. Default is "f2".
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
pow_alt : float
|
|||
|
|
Power of the equivalence test at given equivalence effect size under
|
|||
|
|
the alternative.
|
|||
|
|
"""
|
|||
|
|
|
|||
|
|
# one of n_groups or df has to be specified
|
|||
|
|
if df is None:
|
|||
|
|
if n_groups is None:
|
|||
|
|
raise ValueError("either df or n_groups has to be provided")
|
|||
|
|
df = (n_groups - 1, nobs_t - n_groups)
|
|||
|
|
|
|||
|
|
# esn = fn * (n_groups - 1) / nobs_mean # Wellek psi
|
|||
|
|
|
|||
|
|
# fix for scipy, ncf does not allow nc == 0, fixed in scipy master
|
|||
|
|
if f2_alt == 0:
|
|||
|
|
f2_alt = 1e-13
|
|||
|
|
# effect size, critical value at margin
|
|||
|
|
# f2_null = equiv_margin
|
|||
|
|
if margin_type in ["f2", "fsqu", "fsquared"]:
|
|||
|
|
f2_null = equiv_margin
|
|||
|
|
elif margin_type == "wellek":
|
|||
|
|
if n_groups is None:
|
|||
|
|
raise ValueError("If margin_type is wellek, then n_groups has "
|
|||
|
|
"to be provided")
|
|||
|
|
# f2_null = (n_groups - 1) * n_groups / nobs_t * equiv_margin**2
|
|||
|
|
nobs_mean = nobs_t / n_groups
|
|||
|
|
f2_null = nobs_mean * equiv_margin**2 / nobs_t
|
|||
|
|
f2_alt = nobs_mean * f2_alt**2 / nobs_t
|
|||
|
|
else:
|
|||
|
|
raise ValueError('`margin_type` should be "f2" or "wellek"')
|
|||
|
|
|
|||
|
|
crit_f_margin = ncf_ppf(alpha, df[0], df[1], nobs_t * f2_null)
|
|||
|
|
pwr_alt = ncf_cdf(crit_f_margin, df[0], df[1], nobs_t * f2_alt)
|
|||
|
|
return pwr_alt
|
|||
|
|
|
|||
|
|
|
|||
|
|
def simulate_power_equivalence_oneway(means, nobs, equiv_margin, vars_=None,
|
|||
|
|
k_mc=1000, trim_frac=0,
|
|||
|
|
options_var=None, margin_type="f2"
|
|||
|
|
): # , anova_options=None): #TODO
|
|||
|
|
"""Simulate Power for oneway equivalence test (Wellek's Anova)
|
|||
|
|
|
|||
|
|
This function is experimental and written to evaluate asymptotic power
|
|||
|
|
function. This function will change without backwards compatibility
|
|||
|
|
constraints. The only part that is stable is `pvalue` attribute in results.
|
|||
|
|
|
|||
|
|
Effect size for equivalence margin
|
|||
|
|
|
|||
|
|
"""
|
|||
|
|
if options_var is None:
|
|||
|
|
options_var = ["unequal", "equal", "bf"]
|
|||
|
|
if vars_ is not None:
|
|||
|
|
stds = np.sqrt(vars_)
|
|||
|
|
else:
|
|||
|
|
stds = np.ones(len(means))
|
|||
|
|
|
|||
|
|
nobs_mean = nobs.mean()
|
|||
|
|
n_groups = len(nobs)
|
|||
|
|
res_mc = []
|
|||
|
|
f_mc = []
|
|||
|
|
reject_mc = []
|
|||
|
|
other_mc = []
|
|||
|
|
for _ in range(k_mc):
|
|||
|
|
y0, y1, y2, y3 = (m + std * np.random.randn(n)
|
|||
|
|
for (n, m, std) in zip(nobs, means, stds))
|
|||
|
|
|
|||
|
|
res_i = []
|
|||
|
|
f_i = []
|
|||
|
|
reject_i = []
|
|||
|
|
other_i = []
|
|||
|
|
for uv in options_var:
|
|||
|
|
# for welch in options_welch:
|
|||
|
|
# res1 = sma.anova_generic(means, vars_, nobs, use_var=uv,
|
|||
|
|
# welch_correction=welch)
|
|||
|
|
res0 = anova_oneway([y0, y1, y2, y3], use_var=uv,
|
|||
|
|
trim_frac=trim_frac)
|
|||
|
|
f_stat = res0.statistic
|
|||
|
|
res1 = equivalence_oneway_generic(f_stat, n_groups, nobs.sum(),
|
|||
|
|
equiv_margin, res0.df,
|
|||
|
|
alpha=0.05,
|
|||
|
|
margin_type=margin_type)
|
|||
|
|
res_i.append(res1.pvalue)
|
|||
|
|
es_wellek = f_stat * (n_groups - 1) / nobs_mean
|
|||
|
|
f_i.append(es_wellek)
|
|||
|
|
reject_i.append(res1.reject)
|
|||
|
|
other_i.extend([res1.crit_f, res1.crit_es, res1.power_zero])
|
|||
|
|
res_mc.append(res_i)
|
|||
|
|
f_mc.append(f_i)
|
|||
|
|
reject_mc.append(reject_i)
|
|||
|
|
other_mc.append(other_i)
|
|||
|
|
|
|||
|
|
f_mc = np.asarray(f_mc)
|
|||
|
|
other_mc = np.asarray(other_mc)
|
|||
|
|
res_mc = np.asarray(res_mc)
|
|||
|
|
reject_mc = np.asarray(reject_mc)
|
|||
|
|
res = Holder(f_stat=f_mc,
|
|||
|
|
other=other_mc,
|
|||
|
|
pvalue=res_mc,
|
|||
|
|
reject=reject_mc
|
|||
|
|
)
|
|||
|
|
return res
|
|||
|
|
|
|||
|
|
|
|||
|
|
def test_scale_oneway(data, method="bf", center="median", transform="abs",
|
|||
|
|
trim_frac_mean=0.1, trim_frac_anova=0.0):
|
|||
|
|
"""Oneway Anova test for equal scale, variance or dispersion
|
|||
|
|
|
|||
|
|
This hypothesis test performs a oneway anova test on transformed data and
|
|||
|
|
includes Levene and Brown-Forsythe tests for equal variances as special
|
|||
|
|
cases.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
data : tuple of array_like or DataFrame or Series
|
|||
|
|
Data for k independent samples, with k >= 2. The data can be provided
|
|||
|
|
as a tuple or list of arrays or in long format with outcome
|
|||
|
|
observations in ``data`` and group membership in ``groups``.
|
|||
|
|
method : {"unequal", "equal" or "bf"}
|
|||
|
|
How to treat heteroscedasticity across samples. This is used as
|
|||
|
|
`use_var` option in `anova_oneway` and refers to the variance of the
|
|||
|
|
transformed data, i.e. assumption is on 4th moment if squares are used
|
|||
|
|
as transform.
|
|||
|
|
Three approaches are available:
|
|||
|
|
|
|||
|
|
"unequal" : Variances are not assumed to be equal across samples.
|
|||
|
|
Heteroscedasticity is taken into account with Welch Anova and
|
|||
|
|
Satterthwaite-Welch degrees of freedom.
|
|||
|
|
This is the default.
|
|||
|
|
"equal" : Variances are assumed to be equal across samples.
|
|||
|
|
This is the standard Anova.
|
|||
|
|
"bf" : Variances are not assumed to be equal across samples.
|
|||
|
|
The method is Browne-Forsythe (1971) for testing equality of means
|
|||
|
|
with the corrected degrees of freedom by Merothra. The original BF
|
|||
|
|
degrees of freedom are available as additional attributes in the
|
|||
|
|
results instance, ``df_denom2`` and ``p_value2``.
|
|||
|
|
|
|||
|
|
center : "median", "mean", "trimmed" or float
|
|||
|
|
Statistic used for centering observations. If a float, then this
|
|||
|
|
value is used to center. Default is median.
|
|||
|
|
transform : "abs", "square" or callable
|
|||
|
|
Transformation for the centered observations. If a callable, then this
|
|||
|
|
function is called on the centered data.
|
|||
|
|
Default is absolute value.
|
|||
|
|
trim_frac_mean=0.1 : float in [0, 0.5)
|
|||
|
|
Trim fraction for the trimmed mean when `center` is "trimmed"
|
|||
|
|
trim_frac_anova : float in [0, 0.5)
|
|||
|
|
Optional trimming for Anova with trimmed mean and Winsorized variances.
|
|||
|
|
With the default trim_frac equal to zero, the oneway Anova statistics
|
|||
|
|
are computed without trimming. If `trim_frac` is larger than zero,
|
|||
|
|
then the largest and smallest observations in each sample are trimmed.
|
|||
|
|
see ``trim_frac`` option in `anova_oneway`
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
res : results instance
|
|||
|
|
The returned HolderTuple instance has the following main attributes
|
|||
|
|
and some additional information in other attributes.
|
|||
|
|
|
|||
|
|
statistic : float
|
|||
|
|
Test statistic for k-sample mean comparison which is approximately
|
|||
|
|
F-distributed.
|
|||
|
|
pvalue : float
|
|||
|
|
If ``method="bf"``, then the p-value is based on corrected
|
|||
|
|
degrees of freedom following Mehrotra 1997.
|
|||
|
|
pvalue2 : float
|
|||
|
|
This is the p-value based on degrees of freedom as in
|
|||
|
|
Brown-Forsythe 1974 and is only available if ``method="bf"``.
|
|||
|
|
df : (df_denom, df_num)
|
|||
|
|
Tuple containing degrees of freedom for the F-distribution depend
|
|||
|
|
on ``method``. If ``method="bf"``, then `df_denom` is for Mehrotra
|
|||
|
|
p-values `df_denom2` is available for Brown-Forsythe 1974 p-values.
|
|||
|
|
`df_num` is the same numerator degrees of freedom for both
|
|||
|
|
p-values.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
anova_oneway
|
|||
|
|
scale_transform
|
|||
|
|
"""
|
|||
|
|
|
|||
|
|
data = map(np.asarray, data)
|
|||
|
|
xxd = [scale_transform(x, center=center, transform=transform,
|
|||
|
|
trim_frac=trim_frac_mean) for x in data]
|
|||
|
|
|
|||
|
|
res = anova_oneway(xxd, groups=None, use_var=method,
|
|||
|
|
welch_correction=True, trim_frac=trim_frac_anova)
|
|||
|
|
res.data_transformed = xxd
|
|||
|
|
return res
|
|||
|
|
|
|||
|
|
|
|||
|
|
def equivalence_scale_oneway(data, equiv_margin, method='bf', center='median',
|
|||
|
|
transform='abs', trim_frac_mean=0.,
|
|||
|
|
trim_frac_anova=0.):
|
|||
|
|
"""Oneway Anova test for equivalence of scale, variance or dispersion
|
|||
|
|
|
|||
|
|
This hypothesis test performs a oneway equivalence anova test on
|
|||
|
|
transformed data.
|
|||
|
|
|
|||
|
|
Note, the interpretation of the equivalence margin `equiv_margin` will
|
|||
|
|
depend on the transformation of the data. Transformations like
|
|||
|
|
absolute deviation are not scaled to correspond to the variance under
|
|||
|
|
normal distribution.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
data : tuple of array_like or DataFrame or Series
|
|||
|
|
Data for k independent samples, with k >= 2. The data can be provided
|
|||
|
|
as a tuple or list of arrays or in long format with outcome
|
|||
|
|
observations in ``data`` and group membership in ``groups``.
|
|||
|
|
equiv_margin : float
|
|||
|
|
Equivalence margin in terms of effect size. Effect size can be chosen
|
|||
|
|
with `margin_type`. default is squared Cohen's f.
|
|||
|
|
method : {"unequal", "equal" or "bf"}
|
|||
|
|
How to treat heteroscedasticity across samples. This is used as
|
|||
|
|
`use_var` option in `anova_oneway` and refers to the variance of the
|
|||
|
|
transformed data, i.e. assumption is on 4th moment if squares are used
|
|||
|
|
as transform.
|
|||
|
|
Three approaches are available:
|
|||
|
|
|
|||
|
|
"unequal" : Variances are not assumed to be equal across samples.
|
|||
|
|
Heteroscedasticity is taken into account with Welch Anova and
|
|||
|
|
Satterthwaite-Welch degrees of freedom.
|
|||
|
|
This is the default.
|
|||
|
|
"equal" : Variances are assumed to be equal across samples.
|
|||
|
|
This is the standard Anova.
|
|||
|
|
"bf" : Variances are not assumed to be equal across samples.
|
|||
|
|
The method is Browne-Forsythe (1971) for testing equality of means
|
|||
|
|
with the corrected degrees of freedom by Merothra. The original BF
|
|||
|
|
degrees of freedom are available as additional attributes in the
|
|||
|
|
results instance, ``df_denom2`` and ``p_value2``.
|
|||
|
|
center : "median", "mean", "trimmed" or float
|
|||
|
|
Statistic used for centering observations. If a float, then this
|
|||
|
|
value is used to center. Default is median.
|
|||
|
|
transform : "abs", "square" or callable
|
|||
|
|
Transformation for the centered observations. If a callable, then this
|
|||
|
|
function is called on the centered data.
|
|||
|
|
Default is absolute value.
|
|||
|
|
trim_frac_mean : float in [0, 0.5)
|
|||
|
|
Trim fraction for the trimmed mean when `center` is "trimmed"
|
|||
|
|
trim_frac_anova : float in [0, 0.5)
|
|||
|
|
Optional trimming for Anova with trimmed mean and Winsorized variances.
|
|||
|
|
With the default trim_frac equal to zero, the oneway Anova statistics
|
|||
|
|
are computed without trimming. If `trim_frac` is larger than zero,
|
|||
|
|
then the largest and smallest observations in each sample are trimmed.
|
|||
|
|
see ``trim_frac`` option in `anova_oneway`
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
results : instance of HolderTuple class
|
|||
|
|
The two main attributes are test statistic `statistic` and p-value
|
|||
|
|
`pvalue`.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
anova_oneway
|
|||
|
|
scale_transform
|
|||
|
|
equivalence_oneway
|
|||
|
|
"""
|
|||
|
|
data = map(np.asarray, data)
|
|||
|
|
xxd = [scale_transform(x, center=center, transform=transform,
|
|||
|
|
trim_frac=trim_frac_mean) for x in data]
|
|||
|
|
|
|||
|
|
res = equivalence_oneway(xxd, equiv_margin, use_var=method,
|
|||
|
|
welch_correction=True, trim_frac=trim_frac_anova)
|
|||
|
|
res.x_transformed = xxd
|
|||
|
|
return res
|