Source code for statsmodels.stats.power

# -*- coding: utf-8 -*-
#pylint: disable-msg=W0142
"""Statistical power, solving for nobs, ... - trial version

Created on Sat Jan 12 21:48:06 2013

Author: Josef Perktold

Example
roundtrip - root with respect to all variables

       calculated, desired
nobs   33.367204205 33.367204205
effect 0.5 0.5
alpha  0.05 0.05
power   0.8 0.8


TODO:
refactoring
 - rename beta -> power,    beta (type 2 error is beta = 1-power)  DONE
 - I think the current implementation can handle any kinds of extra keywords
   (except for maybe raising meaningful exceptions
 - streamline code, I think internally classes can be merged
   how to extend to k-sample tests?
   user interface for different tests that map to the same (internal) test class
 - sequence of arguments might be inconsistent,
   arg and/or kwds so python checks what's required and what can be None.
 - templating for docstrings ?


"""
from __future__ import print_function
from statsmodels.compat.python import iteritems
import numpy as np
from scipy import stats, optimize
from statsmodels.tools.rootfinding import brentq_expanding

def ttest_power(effect_size, nobs, alpha, df=None, alternative='two-sided'):
    '''Calculate power of a ttest
    '''
    d = effect_size
    if df is None:
        df = nobs - 1

    if alternative in ['two-sided', '2s']:
        alpha_ = alpha / 2.  #no inplace changes, doesn't work
    elif alternative in ['smaller', 'larger']:
        alpha_ = alpha
    else:
        raise ValueError("alternative has to be 'two-sided', 'larger' " +
                         "or 'smaller'")

    pow_ = 0
    if alternative in ['two-sided', '2s', 'larger']:
        crit_upp = stats.t.isf(alpha_, df)
        #print crit_upp, df, d*np.sqrt(nobs)
        # use private methods, generic methods return nan with negative d
        if np.any(np.isnan(crit_upp)):
            # avoid endless loop, https://github.com/scipy/scipy/issues/2667
            pow_ = np.nan
        else:
            pow_ = stats.nct._sf(crit_upp, df, d*np.sqrt(nobs))
    if alternative in ['two-sided', '2s', 'smaller']:
        crit_low = stats.t.ppf(alpha_, df)
        #print crit_low, df, d*np.sqrt(nobs)
        if np.any(np.isnan(crit_low)):
            pow_ = np.nan
        else:
            pow_ += stats.nct._cdf(crit_low, df, d*np.sqrt(nobs))
    return pow_

def normal_power(effect_size, nobs, alpha, alternative='two-sided', sigma=1.):
    '''Calculate power of a normal distributed test statistic

    '''
    d = effect_size

    if alternative in ['two-sided', '2s']:
        alpha_ = alpha / 2.  #no inplace changes, doesn't work
    elif alternative in ['smaller', 'larger']:
        alpha_ = alpha
    else:
        raise ValueError("alternative has to be 'two-sided', 'larger' " +
                         "or 'smaller'")

    pow_ = 0
    if alternative in ['two-sided', '2s', 'larger']:
        crit = stats.norm.isf(alpha_)
        pow_ = stats.norm.sf(crit - d*np.sqrt(nobs)/sigma)
    if alternative in ['two-sided', '2s', 'smaller']:
        crit = stats.norm.ppf(alpha_)
        pow_ += stats.norm.cdf(crit - d*np.sqrt(nobs)/sigma)
    return pow_

def ftest_anova_power(effect_size, nobs, alpha, k_groups=2, df=None):
    '''power for ftest for one way anova with k equal sized groups

    nobs total sample size, sum over all groups

    should be general nobs observations, k_groups restrictions ???
    '''
    df_num = nobs - k_groups
    df_denom = k_groups - 1
    crit = stats.f.isf(alpha, df_denom, df_num)
    pow_ = stats.ncf.sf(crit, df_denom, df_num, effect_size**2 * nobs)
    return pow_#, crit

def ftest_power(effect_size, df_num, df_denom, alpha, ncc=1):
    '''Calculate the power of a F-test.

    Parameters
    ----------
    effect_size : float
        standardized effect size, mean divided by the standard deviation.
        effect size has to be positive.
    df_num : int or float
        numerator degrees of freedom.
    df_denom : int or float
        denominator degrees of freedom.
    alpha : float in interval (0,1)
        significance level, e.g. 0.05, is the probability of a type I
        error, that is wrong rejections if the Null Hypothesis is true.
    ncc : int
        degrees of freedom correction for non-centrality parameter.
        see Notes

    Returns
    -------
    power : float
        Power of the test, e.g. 0.8, is one minus the probability of a
        type II error. Power is the probability that the test correctly
        rejects the Null Hypothesis if the Alternative Hypothesis is true.

    Notes
    -----

    sample size is given implicitly by df_num

    set ncc=0 to match t-test, or f-test in LikelihoodModelResults.
    ncc=1 matches the non-centrality parameter in R::pwr::pwr.f2.test

    ftest_power with ncc=0 should also be correct for f_test in regression
    models, with df_num and d_denom as defined there. (not verified yet)
    '''
    nc = effect_size**2 * (df_denom + df_num + ncc)
    crit = stats.f.isf(alpha, df_denom, df_num)
    pow_ = stats.ncf.sf(crit, df_denom, df_num, nc)
    return pow_ #, crit, nc


#class based implementation
#--------------------------

class Power(object):
    '''Statistical Power calculations, Base Class

    so far this could all be class methods
    '''

    def __init__(self, **kwds):
        self.__dict__.update(kwds)
        # used only for instance level start values
        self.start_ttp = dict(effect_size=0.01, nobs=10., alpha=0.15,
                              power=0.6, nobs1=10., ratio=1,
                              df_num=10, df_denom=3   # for FTestPower
                              )
        # TODO: nobs1 and ratio are for ttest_ind,
        #      need start_ttp for each test/class separately,
        # possible rootfinding problem for effect_size, starting small seems to
        # work
        from collections import defaultdict
        self.start_bqexp = defaultdict(dict)
        for key in ['nobs', 'nobs1', 'df_num', 'df_denom']:
            self.start_bqexp[key] = dict(low=2., start_upp=50.)
        for key in ['df_denom']:
            self.start_bqexp[key] = dict(low=1., start_upp=50.)
        for key in ['ratio']:
            self.start_bqexp[key] = dict(low=1e-8, start_upp=2)
        for key in ['alpha']:
            self.start_bqexp[key] = dict(low=1e-12, upp=1 - 1e-12)

    def power(self, *args, **kwds):
        raise NotImplementedError

    def _power_identity(self, *args, **kwds):
        power_ = kwds.pop('power')
        return self.power(*args, **kwds) - power_

    def solve_power(self, **kwds):
        '''solve for any one of the parameters of a t-test

        for t-test the keywords are:
            effect_size, nobs, alpha, power

        exactly one needs to be ``None``, all others need numeric values

        *attaches*

        cache_fit_res : list
            Cache of the result of the root finding procedure for the latest
            call to ``solve_power``, mainly for debugging purposes.
            The first element is the success indicator, one if successful.
            The remaining elements contain the return information of the up to
            three solvers that have been tried.


        '''
        #TODO: maybe use explicit kwds,
        #    nicer but requires inspect? and not generic across tests
        #    I'm duplicating this in the subclass to get informative docstring
        key = [k for k,v in iteritems(kwds) if v is None]
        #print kwds, key;
        if len(key) != 1:
            raise ValueError('need exactly one keyword that is None')
        key = key[0]

        if key == 'power':
            del kwds['power']
            return self.power(**kwds)

        if kwds['effect_size'] == 0:
            import warnings
            from statsmodels.tools.sm_exceptions import HypothesisTestWarning
            warnings.warn('Warning: Effect size of 0 detected', HypothesisTestWarning)
            if key == 'power':
                return kwds['alpha']
            if key == 'alpha':
                return kwds['power']
            else:
                raise ValueError('Cannot detect an effect-size of 0. Try changing your effect-size.')


        self._counter = 0

        def func(x):
            kwds[key] = x
            fval = self._power_identity(**kwds)
            self._counter += 1
            #print self._counter,
            if self._counter > 500:
                raise RuntimeError('possible endless loop (500 NaNs)')
            if np.isnan(fval):
                return np.inf
            else:
                return fval

        #TODO: I'm using the following so I get a warning when start_ttp is not defined
        try:
            start_value = self.start_ttp[key]
        except KeyError:
            start_value = 0.9
            import warnings
            from statsmodels.tools.sm_exceptions import ValueWarning
            warnings.warn('Warning: using default start_value for {0}'.format(key), ValueWarning)

        fit_kwds = self.start_bqexp[key]
        fit_res = []
        #print vars()
        try:
            val, res = brentq_expanding(func, full_output=True, **fit_kwds)
            failed = False
            fit_res.append(res)
        except ValueError:
            failed = True
            fit_res.append(None)

        success = None
        if (not failed) and res.converged:
            success = 1
        else:
            # try backup
            # TODO: check more cases to make this robust
            if not np.isnan(start_value):
                val, infodict, ier, msg = optimize.fsolve(func, start_value,
                                                          full_output=True) #scalar
                #val = optimize.newton(func, start_value) #scalar
                fval = infodict['fvec']
                fit_res.append(infodict)
            else:
                ier = -1
                fval = 1
                fit_res.append([None])

            if ier == 1 and np.abs(fval) < 1e-4 :
                success = 1
            else:
                #print infodict
                if key in ['alpha', 'power', 'effect_size']:
                    val, r = optimize.brentq(func, 1e-8, 1-1e-8,
                                             full_output=True) #scalar
                    success = 1 if r.converged else 0
                    fit_res.append(r)
                else:
                    success = 0

        if not success == 1:
            import warnings
            from statsmodels.tools.sm_exceptions import (ConvergenceWarning,
                convergence_doc)
            warnings.warn(convergence_doc, ConvergenceWarning)

        #attach fit_res, for reading only, should be needed only for debugging
        fit_res.insert(0, success)
        self.cache_fit_res = fit_res
        return val

    def plot_power(self, dep_var='nobs', nobs=None, effect_size=None,
                   alpha=0.05, ax=None, title=None, plt_kwds=None, **kwds):
        '''plot power with number of observations or effect size on x-axis

        Parameters
        ----------
        dep_var : string in ['nobs', 'effect_size', 'alpha']
            This specifies which variable is used for the horizontal axis.
            If dep_var='nobs' (default), then one curve is created for each
            value of ``effect_size``. If dep_var='effect_size' or alpha, then
            one curve is created for each value of ``nobs``.
        nobs : scalar or array_like
            specifies the values of the number of observations in the plot
        effect_size : scalar or array_like
            specifies the values of the effect_size in the plot
        alpha : float or array_like
            The significance level (type I error) used in the power
            calculation. Can only be more than a scalar, if ``dep_var='alpha'``
        ax : None or axis instance
            If ax is None, than a matplotlib figure is created. If ax is a
            matplotlib axis instance, then it is reused, and the plot elements
            are created with it.
        title : string
            title for the axis. Use an empty string, ``''``, to avoid a title.
        plt_kwds : None or dict
            not used yet
        kwds : optional keywords for power function
            These remaining keyword arguments are used as arguments to the
            power function. Many power function support ``alternative`` as a
            keyword argument, two-sample test support ``ratio``.

        Returns
        -------
        fig : matplotlib figure instance

        Notes
        -----
        This works only for classes where the ``power`` method has
        ``effect_size``, ``nobs`` and ``alpha`` as the first three arguments.
        If the second argument is ``nobs1``, then the number of observations
        in the plot are those for the first sample.
        TODO: fix this for FTestPower and GofChisquarePower

        TODO: maybe add line variable, if we want more than nobs and effectsize
        '''
        #if pwr_kwds is None:
        #    pwr_kwds = {}
        from statsmodels.graphics import utils
        from statsmodels.graphics.plottools import rainbow
        fig, ax = utils.create_mpl_ax(ax)
        import matplotlib.pyplot as plt
        colormap = plt.cm.Dark2 #pylint: disable-msg=E1101
        plt_alpha = 1 #0.75
        lw = 2
        if dep_var == 'nobs':
            colors = rainbow(len(effect_size))
            colors = [colormap(i) for i in np.linspace(0, 0.9, len(effect_size))]
            for ii, es in enumerate(effect_size):
                power = self.power(es, nobs, alpha, **kwds)
                ax.plot(nobs, power, lw=lw, alpha=plt_alpha,
                        color=colors[ii], label='es=%4.2F' % es)
                xlabel = 'Number of Observations'
        elif dep_var in ['effect size', 'effect_size', 'es']:
            colors = rainbow(len(nobs))
            colors = [colormap(i) for i in np.linspace(0, 0.9, len(nobs))]
            for ii, n in enumerate(nobs):
                power = self.power(effect_size, n, alpha, **kwds)
                ax.plot(effect_size, power, lw=lw, alpha=plt_alpha,
                        color=colors[ii], label='N=%4.2F' % n)
                xlabel = 'Effect Size'
        elif dep_var in ['alpha']:
            # experimental nobs as defining separate lines
            colors = rainbow(len(nobs))

            for ii, n in enumerate(nobs):
                power = self.power(effect_size, n, alpha, **kwds)
                ax.plot(alpha, power, lw=lw, alpha=plt_alpha,
                        color=colors[ii], label='N=%4.2F' % n)
                xlabel = 'alpha'
        else:
            raise ValueError('depvar not implemented')

        if title is None:
            title = 'Power of Test'
        ax.set_xlabel(xlabel)
        ax.set_title(title)
        ax.legend(loc='lower right')
        return fig


[docs]class TTestPower(Power): '''Statistical Power calculations for one sample or paired sample t-test '''
[docs] def power(self, effect_size, nobs, alpha, df=None, alternative='two-sided'): '''Calculate the power of a t-test for one sample or paired samples. Parameters ---------- effect_size : float standardized effect size, mean divided by the standard deviation. effect size has to be positive. nobs : int or float sample size, number of observations. alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. df : int or float degrees of freedom. By default this is None, and the df from the one sample or paired ttest is used, ``df = nobs1 - 1`` alternative : string, 'two-sided' (default), 'larger', 'smaller' extra argument to choose whether the power is calculated for a two-sided (default) or one sided test. The one-sided test can be either 'larger', 'smaller'. . Returns ------- power : float Power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. ''' # for debugging #print 'calling ttest power with', (effect_size, nobs, alpha, df, alternative) return ttest_power(effect_size, nobs, alpha, df=df, alternative=alternative)
#method is only added to have explicit keywords and docstring
[docs] def solve_power(self, effect_size=None, nobs=None, alpha=None, power=None, alternative='two-sided'): '''solve for any one parameter of the power of a one sample t-test for the one sample t-test the keywords are: effect_size, nobs, alpha, power Exactly one needs to be ``None``, all others need numeric values. This test can also be used for a paired t-test, where effect size is defined in terms of the mean difference, and nobs is the number of pairs. Parameters ---------- effect_size : float standardized effect size, mean divided by the standard deviation. effect size has to be positive. nobs : int or float sample size, number of observations. alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. power : float in interval (0,1) power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. alternative : string, 'two-sided' (default) or 'one-sided' extra argument to choose whether the power is calculated for a two-sided (default) or one sided test. 'one-sided' assumes we are in the relevant tail. Returns ------- value : float The value of the parameter that was set to None in the call. The value solves the power equation given the remaining parameters. *attaches* cache_fit_res : list Cache of the result of the root finding procedure for the latest call to ``solve_power``, mainly for debugging purposes. The first element is the success indicator, one if successful. The remaining elements contain the return information of the up to three solvers that have been tried. Notes ----- The function uses scipy.optimize for finding the value that satisfies the power equation. It first uses ``brentq`` with a prior search for bounds. If this fails to find a root, ``fsolve`` is used. If ``fsolve`` also fails, then, for ``alpha``, ``power`` and ``effect_size``, ``brentq`` with fixed bounds is used. However, there can still be cases where this fails. ''' # for debugging #print 'calling ttest solve with', (effect_size, nobs, alpha, power, alternative) return super(TTestPower, self).solve_power(effect_size=effect_size, nobs=nobs, alpha=alpha, power=power, alternative=alternative)
[docs]class TTestIndPower(Power): '''Statistical Power calculations for t-test for two independent sample currently only uses pooled variance '''
[docs] def power(self, effect_size, nobs1, alpha, ratio=1, df=None, alternative='two-sided'): '''Calculate the power of a t-test for two independent sample Parameters ---------- effect_size : float standardized effect size, difference between the two means divided by the standard deviation. `effect_size` has to be positive. nobs1 : int or float number of observations of sample 1. The number of observations of sample two is ratio times the size of sample 1, i.e. ``nobs2 = nobs1 * ratio`` alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. ratio : float ratio of the number of observations in sample 2 relative to sample 1. see description of nobs1 The default for ratio is 1; to solve for ratio given the other arguments, it has to be explicitly set to None. df : int or float degrees of freedom. By default this is None, and the df from the ttest with pooled variance is used, ``df = (nobs1 - 1 + nobs2 - 1)`` alternative : string, 'two-sided' (default), 'larger', 'smaller' extra argument to choose whether the power is calculated for a two-sided (default) or one sided test. The one-sided test can be either 'larger', 'smaller'. Returns ------- power : float Power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. ''' nobs2 = nobs1*ratio #pooled variance if df is None: df = (nobs1 - 1 + nobs2 - 1) nobs = 1./ (1. / nobs1 + 1. / nobs2) #print 'calling ttest power with', (effect_size, nobs, alpha, df, alternative) return ttest_power(effect_size, nobs, alpha, df=df, alternative=alternative)
#method is only added to have explicit keywords and docstring
[docs] def solve_power(self, effect_size=None, nobs1=None, alpha=None, power=None, ratio=1., alternative='two-sided'): '''solve for any one parameter of the power of a two sample t-test for t-test the keywords are: effect_size, nobs1, alpha, power, ratio exactly one needs to be ``None``, all others need numeric values Parameters ---------- effect_size : float standardized effect size, difference between the two means divided by the standard deviation. `effect_size` has to be positive. nobs1 : int or float number of observations of sample 1. The number of observations of sample two is ratio times the size of sample 1, i.e. ``nobs2 = nobs1 * ratio`` alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. power : float in interval (0,1) power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. ratio : float ratio of the number of observations in sample 2 relative to sample 1. see description of nobs1 The default for ratio is 1; to solve for ratio given the other arguments it has to be explicitly set to None. alternative : string, 'two-sided' (default), 'larger', 'smaller' extra argument to choose whether the power is calculated for a two-sided (default) or one sided test. The one-sided test can be either 'larger', 'smaller'. Returns ------- value : float The value of the parameter that was set to None in the call. The value solves the power equation given the remaining parameters. Notes ----- The function uses scipy.optimize for finding the value that satisfies the power equation. It first uses ``brentq`` with a prior search for bounds. If this fails to find a root, ``fsolve`` is used. If ``fsolve`` also fails, then, for ``alpha``, ``power`` and ``effect_size``, ``brentq`` with fixed bounds is used. However, there can still be cases where this fails. ''' return super(TTestIndPower, self).solve_power(effect_size=effect_size, nobs1=nobs1, alpha=alpha, power=power, ratio=ratio, alternative=alternative)
[docs]class NormalIndPower(Power): '''Statistical Power calculations for z-test for two independent samples. currently only uses pooled variance ''' def __init__(self, ddof=0, **kwds): self.ddof = ddof super(NormalIndPower, self).__init__(**kwds)
[docs] def power(self, effect_size, nobs1, alpha, ratio=1, alternative='two-sided'): '''Calculate the power of a z-test for two independent sample Parameters ---------- effect_size : float standardized effect size, difference between the two means divided by the standard deviation. effect size has to be positive. nobs1 : int or float number of observations of sample 1. The number of observations of sample two is ratio times the size of sample 1, i.e. ``nobs2 = nobs1 * ratio`` ``ratio`` can be set to zero in order to get the power for a one sample test. alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. ratio : float ratio of the number of observations in sample 2 relative to sample 1. see description of nobs1 alternative : string, 'two-sided' (default), 'larger', 'smaller' extra argument to choose whether the power is calculated for a two-sided (default) or one sided test. The one-sided test can be either 'larger', 'smaller'. Returns ------- power : float Power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. ''' ddof = self.ddof # for correlation, ddof=3 # get effective nobs, factor for std of test statistic if ratio > 0: nobs2 = nobs1*ratio #equivalent to nobs = n1*n2/(n1+n2)=n1*ratio/(1+ratio) nobs = 1./ (1. / (nobs1 - ddof) + 1. / (nobs2 - ddof)) else: nobs = nobs1 - ddof return normal_power(effect_size, nobs, alpha, alternative=alternative)
#method is only added to have explicit keywords and docstring
[docs] def solve_power(self, effect_size=None, nobs1=None, alpha=None, power=None, ratio=1., alternative='two-sided'): '''solve for any one parameter of the power of a two sample z-test for z-test the keywords are: effect_size, nobs1, alpha, power, ratio exactly one needs to be ``None``, all others need numeric values Parameters ---------- effect_size : float standardized effect size, difference between the two means divided by the standard deviation. If ratio=0, then this is the standardized mean in the one sample test. nobs1 : int or float number of observations of sample 1. The number of observations of sample two is ratio times the size of sample 1, i.e. ``nobs2 = nobs1 * ratio`` ``ratio`` can be set to zero in order to get the power for a one sample test. alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. power : float in interval (0,1) power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. ratio : float ratio of the number of observations in sample 2 relative to sample 1. see description of nobs1 The default for ratio is 1; to solve for ration given the other arguments it has to be explicitly set to None. alternative : string, 'two-sided' (default), 'larger', 'smaller' extra argument to choose whether the power is calculated for a two-sided (default) or one sided test. The one-sided test can be either 'larger', 'smaller'. Returns ------- value : float The value of the parameter that was set to None in the call. The value solves the power equation given the remaining parameters. Notes ----- The function uses scipy.optimize for finding the value that satisfies the power equation. It first uses ``brentq`` with a prior search for bounds. If this fails to find a root, ``fsolve`` is used. If ``fsolve`` also fails, then, for ``alpha``, ``power`` and ``effect_size``, ``brentq`` with fixed bounds is used. However, there can still be cases where this fails. ''' return super(NormalIndPower, self).solve_power(effect_size=effect_size, nobs1=nobs1, alpha=alpha, power=power, ratio=ratio, alternative=alternative)
[docs]class FTestPower(Power): '''Statistical Power calculations for generic F-test '''
[docs] def power(self, effect_size, df_num, df_denom, alpha, ncc=1): '''Calculate the power of a F-test. Parameters ---------- effect_size : float standardized effect size, mean divided by the standard deviation. effect size has to be positive. df_num : int or float numerator degrees of freedom. df_denom : int or float denominator degrees of freedom. alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. ncc : int degrees of freedom correction for non-centrality parameter. see Notes Returns ------- power : float Power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. Notes ----- sample size is given implicitly by df_num set ncc=0 to match t-test, or f-test in LikelihoodModelResults. ncc=1 matches the non-centrality parameter in R::pwr::pwr.f2.test ftest_power with ncc=0 should also be correct for f_test in regression models, with df_num and d_denom as defined there. (not verified yet) ''' pow_ = ftest_power(effect_size, df_num, df_denom, alpha, ncc=ncc) #print effect_size, df_num, df_denom, alpha, pow_ return pow_
#method is only added to have explicit keywords and docstring
[docs] def solve_power(self, effect_size=None, df_num=None, df_denom=None, nobs=None, alpha=None, power=None, ncc=1): '''solve for any one parameter of the power of a F-test for the one sample F-test the keywords are: effect_size, df_num, df_denom, alpha, power Exactly one needs to be ``None``, all others need numeric values. Parameters ---------- effect_size : float standardized effect size, mean divided by the standard deviation. effect size has to be positive. nobs : int or float sample size, number of observations. alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. power : float in interval (0,1) power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. alternative : string, 'two-sided' (default) or 'one-sided' extra argument to choose whether the power is calculated for a two-sided (default) or one sided test. 'one-sided' assumes we are in the relevant tail. Returns ------- value : float The value of the parameter that was set to None in the call. The value solves the power equation given the remainding parameters. Notes ----- The function uses scipy.optimize for finding the value that satisfies the power equation. It first uses ``brentq`` with a prior search for bounds. If this fails to find a root, ``fsolve`` is used. If ``fsolve`` also fails, then, for ``alpha``, ``power`` and ``effect_size``, ``brentq`` with fixed bounds is used. However, there can still be cases where this fails. ''' return super(FTestPower, self).solve_power(effect_size=effect_size, df_num=df_num, df_denom=df_denom, alpha=alpha, power=power, ncc=ncc)
[docs]class FTestAnovaPower(Power): '''Statistical Power calculations F-test for one factor balanced ANOVA '''
[docs] def power(self, effect_size, nobs, alpha, k_groups=2): '''Calculate the power of a F-test for one factor ANOVA. Parameters ---------- effect_size : float standardized effect size, mean divided by the standard deviation. effect size has to be positive. nobs : int or float sample size, number of observations. alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. k_groups : int or float number of groups in the ANOVA or k-sample comparison. Default is 2. Returns ------- power : float Power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. ''' return ftest_anova_power(effect_size, nobs, alpha, k_groups=k_groups)
#method is only added to have explicit keywords and docstring
[docs] def solve_power(self, effect_size=None, nobs=None, alpha=None, power=None, k_groups=2): '''solve for any one parameter of the power of a F-test for the one sample F-test the keywords are: effect_size, nobs, alpha, power Exactly one needs to be ``None``, all others need numeric values. Parameters ---------- effect_size : float standardized effect size, mean divided by the standard deviation. effect size has to be positive. nobs : int or float sample size, number of observations. alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. power : float in interval (0,1) power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. Returns ------- value : float The value of the parameter that was set to None in the call. The value solves the power equation given the remainding parameters. Notes ----- The function uses scipy.optimize for finding the value that satisfies the power equation. It first uses ``brentq`` with a prior search for bounds. If this fails to find a root, ``fsolve`` is used. If ``fsolve`` also fails, then, for ``alpha``, ``power`` and ``effect_size``, ``brentq`` with fixed bounds is used. However, there can still be cases where this fails. ''' # update start values for root finding if not k_groups is None: self.start_ttp['nobs'] = k_groups * 10 self.start_bqexp['nobs'] = dict(low=k_groups * 2, start_upp=k_groups * 10) # first attempt at special casing if effect_size is None: return self._solve_effect_size(effect_size=effect_size, nobs=nobs, alpha=alpha, k_groups=k_groups, power=power) return super(FTestAnovaPower, self).solve_power(effect_size=effect_size, nobs=nobs, alpha=alpha, k_groups=k_groups, power=power)
def _solve_effect_size(self, effect_size=None, nobs=None, alpha=None, power=None, k_groups=2): '''experimental, test failure in solve_power for effect_size ''' def func(x): effect_size = x return self._power_identity(effect_size=effect_size, nobs=nobs, alpha=alpha, k_groups=k_groups, power=power) val, r = optimize.brentq(func, 1e-8, 1-1e-8, full_output=True) if not r.converged: print(r) return val
[docs]class GofChisquarePower(Power): '''Statistical Power calculations for one sample chisquare test '''
[docs] def power(self, effect_size, nobs, alpha, n_bins, ddof=0):#alternative='two-sided'): '''Calculate the power of a chisquare test for one sample Only two-sided alternative is implemented Parameters ---------- effect_size : float standardized effect size, according to Cohen's definition. see :func:`statsmodels.stats.gof.chisquare_effectsize` nobs : int or float sample size, number of observations. alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. n_bins : int number of bins or cells in the distribution. Returns ------- power : float Power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. ''' from statsmodels.stats.gof import chisquare_power return chisquare_power(effect_size, nobs, n_bins, alpha, ddof=0)
#method is only added to have explicit keywords and docstring
[docs] def solve_power(self, effect_size=None, nobs=None, alpha=None, power=None, n_bins=2): '''solve for any one parameter of the power of a one sample chisquare-test for the one sample chisquare-test the keywords are: effect_size, nobs, alpha, power Exactly one needs to be ``None``, all others need numeric values. n_bins needs to be defined, a default=2 is used. Parameters ---------- effect_size : float standardized effect size, according to Cohen's definition. see :func:`statsmodels.stats.gof.chisquare_effectsize` nobs : int or float sample size, number of observations. alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. power : float in interval (0,1) power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. n_bins : int number of bins or cells in the distribution Returns ------- value : float The value of the parameter that was set to None in the call. The value solves the power equation given the remaining parameters. Notes ----- The function uses scipy.optimize for finding the value that satisfies the power equation. It first uses ``brentq`` with a prior search for bounds. If this fails to find a root, ``fsolve`` is used. If ``fsolve`` also fails, then, for ``alpha``, ``power`` and ``effect_size``, ``brentq`` with fixed bounds is used. However, there can still be cases where this fails. ''' return super(GofChisquarePower, self).solve_power(effect_size=effect_size, nobs=nobs, n_bins=n_bins, alpha=alpha, power=power)
class _GofChisquareIndPower(Power): '''Statistical Power calculations for chisquare goodness-of-fit test TODO: this is not working yet for 2sample case need two nobs in function no one-sided chisquare test, is there one? use normal distribution? -> drop one-sided options? ''' def power(self, effect_size, nobs1, alpha, ratio=1, alternative='two-sided'): '''Calculate the power of a chisquare for two independent sample Parameters ---------- effect_size : float standardize effect size, difference between the two means divided by the standard deviation. effect size has to be positive. nobs1 : int or float number of observations of sample 1. The number of observations of sample two is ratio times the size of sample 1, i.e. ``nobs2 = nobs1 * ratio`` alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. ratio : float ratio of the number of observations in sample 2 relative to sample 1. see description of nobs1 The default for ratio is 1; to solve for ration given the other arguments it has to be explicitely set to None. alternative : string, 'two-sided' (default) or 'one-sided' extra argument to choose whether the power is calculated for a two-sided (default) or one sided test. 'one-sided' assumes we are in the relevant tail. Returns ------- power : float Power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. ''' from statsmodels.stats.gof import chisquare_power nobs2 = nobs1*ratio #equivalent to nobs = n1*n2/(n1+n2)=n1*ratio/(1+ratio) nobs = 1./ (1. / nobs1 + 1. / nobs2) return chisquare_power(effect_size, nobs, alpha) #method is only added to have explicit keywords and docstring def solve_power(self, effect_size=None, nobs1=None, alpha=None, power=None, ratio=1., alternative='two-sided'): '''solve for any one parameter of the power of a two sample z-test for z-test the keywords are: effect_size, nobs1, alpha, power, ratio exactly one needs to be ``None``, all others need numeric values Parameters ---------- effect_size : float standardize effect size, difference between the two means divided by the standard deviation. nobs1 : int or float number of observations of sample 1. The number of observations of sample two is ratio times the size of sample 1, i.e. ``nobs2 = nobs1 * ratio`` alpha : float in interval (0,1) significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. power : float in interval (0,1) power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. ratio : float ratio of the number of observations in sample 2 relative to sample 1. see description of nobs1 The default for ratio is 1; to solve for ration given the other arguments it has to be explicitely set to None. alternative : string, 'two-sided' (default) or 'one-sided' extra argument to choose whether the power is calculated for a two-sided (default) or one sided test. 'one-sided' assumes we are in the relevant tail. Returns ------- value : float The value of the parameter that was set to None in the call. The value solves the power equation given the remainding parameters. Notes ----- The function uses scipy.optimize for finding the value that satisfies the power equation. It first uses ``brentq`` with a prior search for bounds. If this fails to find a root, ``fsolve`` is used. If ``fsolve`` also fails, then, for ``alpha``, ``power`` and ``effect_size``, ``brentq`` with fixed bounds is used. However, there can still be cases where this fails. ''' return super(_GofChisquareIndPower, self).solve_power(effect_size=effect_size, nobs1=nobs1, alpha=alpha, power=power, ratio=ratio, alternative=alternative) #shortcut functions tt_solve_power = TTestPower().solve_power tt_ind_solve_power = TTestIndPower().solve_power zt_ind_solve_power = NormalIndPower().solve_power