# 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 ?
"""
import warnings
import numpy as np
from scipy import stats, optimize, special
from statsmodels.tools.rootfinding import brentq_expanding
def nct_cdf(x, df, nc):
return special.nctdtr(df, nc, x)
def nct_sf(x, df, nc):
return 1 - special.nctdtr(df, nc, x)
def ncf_cdf(x, dfn, dfd, nc):
return special.ncfdtr(dfn, dfd, nc, x)
def ncf_sf(x, dfn, dfd, nc):
return 1 - special.ncfdtr(dfn, dfd, nc, x)
def ncf_ppf(q, dfn, dfd, nc):
return special.ncfdtri(dfn, dfd, nc, q)
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, does not 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))
# use scipy.special
pow_ = 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))
pow_ += 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
This is an generalization of `normal_power` when variance under Null and
Alternative differ.
Parameters
----------
effect size : float
difference in the estimated means or statistics under the alternative
normalized by the standard deviation (without division by sqrt(nobs).
nobs : float or int
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.
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'.
"""
d = effect_size
if alternative in ['two-sided', '2s']:
alpha_ = alpha / 2. #no inplace changes, does not 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_
[docs]def normal_power_het(diff, nobs, alpha, std_null=1., std_alternative=None,
alternative='two-sided'):
"""Calculate power of a normal distributed test statistic
This is an generalization of `normal_power` when variance under Null and
Alternative differ.
Parameters
----------
diff : float
difference in the estimated means or statistics under the alternative.
nobs : float or int
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.
std_null : float
standard deviation under the Null hypothesis without division by
sqrt(nobs)
std_alternative : float
standard deviation under the Alternative hypothesis without division
by sqrt(nobs)
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
"""
d = diff
if std_alternative is None:
std_alternative = std_null
if alternative in ['two-sided', '2s']:
alpha_ = alpha / 2. #no inplace changes, does not work
elif alternative in ['smaller', 'larger']:
alpha_ = alpha
else:
raise ValueError("alternative has to be 'two-sided', 'larger' " +
"or 'smaller'")
std_ratio = std_null / std_alternative
pow_ = 0
if alternative in ['two-sided', '2s', 'larger']:
crit = stats.norm.isf(alpha_)
pow_ = stats.norm.sf(crit * std_ratio -
d*np.sqrt(nobs) / std_alternative)
if alternative in ['two-sided', '2s', 'smaller']:
crit = stats.norm.ppf(alpha_)
pow_ += stats.norm.cdf(crit * std_ratio -
d*np.sqrt(nobs) / std_alternative)
return pow_
[docs]def normal_sample_size_one_tail(diff, power, alpha, std_null=1.,
std_alternative=None):
"""explicit sample size computation if only one tail is relevant
The sample size is based on the power in one tail assuming that the
alternative is in the tail where the test has power that increases
with sample size.
Use alpha/2 to compute the one tail approximation to the two-sided
test, i.e. consider only one tail of two-sided test.
Parameters
----------
diff : float
difference in the estimated means or statistics under the alternative.
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.
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.
Note: alpha is used for one tail. Use alpha/2 for two-sided
alternative.
std_null : float
standard deviation under the Null hypothesis without division by
sqrt(nobs)
std_alternative : float
standard deviation under the Alternative hypothesis without division
by sqrt(nobs). Defaults to None. If None, ``std_alternative`` is set
to the value of ``std_null``.
Returns
-------
nobs : float
Sample size to achieve (at least) the desired power.
If the minimum power is satisfied for all positive sample sizes, then
``nobs`` will be zero. This will be the case when power <= alpha if
std_alternative is equal to std_null.
"""
if std_alternative is None:
std_alternative = std_null
crit_power = stats.norm.isf(power)
crit = stats.norm.isf(alpha)
n1 = (np.maximum(crit * std_null - crit_power * std_alternative, 0)
/ diff)**2
return n1
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 = k_groups - 1
df_denom = nobs - k_groups
crit = stats.f.isf(alpha, df_num, df_denom)
pow_ = ncf_sf(crit, df_num, df_denom, effect_size**2 * nobs)
return pow_
def ftest_power(effect_size, df2, df1, alpha, ncc=1):
'''Calculate the power of a F-test.
Parameters
----------
effect_size : float
The effect size is here Cohen's ``f``, the square root of ``f2``.
df2 : int or float
Denominator degrees of freedom.
This corresponds to the df_resid in Wald tests.
df1 : int or float
Numerator degrees of freedom.
This corresponds to the number of constraints in Wald tests.
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
-----
changed in 0.14: use df2, df1 instead of df_num, df_denom as arg names.
The latter had reversed meaning.
The sample size is given implicitly by ``df2`` with fixed number of
constraints given by numerator degrees of freedom ``df1``:
nobs = df2 + df1 + ncc
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 (df1) as number of constraints and d_denom (df2) as
df_resid.
'''
df_num, df_denom = df1, df2
nc = effect_size**2 * (df_denom + df_num + ncc)
crit = stats.f.isf(alpha, df_num, df_denom)
# pow_ = stats.ncf.sf(crit, df_num, df_denom, nc)
# use scipy.special for ncf
pow_ = ncf_sf(crit, df_num, df_denom, nc)
return pow_ #, crit, nc
def ftest_power_f2(effect_size, df_num, df_denom, alpha, ncc=1):
'''Calculate the power of a F-test.
Based on Cohen's `f^2` effect size.
This assumes
df_num : numerator degrees of freedom, (number of constraints)
df_denom : denominator degrees of freedom (df_resid in regression)
nobs = df_denom + df_num + ncc
nc = effect_size * nobs (noncentrality index)
Power is computed one-sided in the upper tail.
Parameters
----------
effect_size : float
Cohen's f2 effect size or noncentrality divided by nobs.
df_num : int or float
Numerator degrees of freedom.
This corresponds to the number of constraints in Wald tests.
df_denom : int or float
Denominator degrees of freedom.
This corresponds to the df_resid in Wald tests.
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
The sample size is given implicitly by ``df_denom`` with fixed number of
constraints given by numerator degrees of freedom ``df_num``:
nobs = df_denom + df_num + ncc
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 (df1) as number of constraints and d_denom (df2) as
df_resid.
'''
nc = effect_size * (df_denom + df_num + ncc)
crit = stats.f.isf(alpha, df_num, df_denom)
# pow_ = stats.ncf.sf(crit, df_num, df_denom, nc)
# use scipy.special for ncf
pow_ = ncf_sf(crit, df_num, df_denom, nc)
return pow_
#class based implementation
#--------------------------
class Power:
'''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 kwds.items() 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 : {'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, array_like}
specifies the values of the number of observations in the plot
effect_size : {scalar, array_like}
specifies the values of the effect_size in the plot
alpha : {float, 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 : str
title for the axis. Use an empty string, ``''``, to avoid a title.
plt_kwds : {None, dict}
not used yet
kwds : dict
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
-------
Figure
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
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 : str, '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.The effect size is here Cohen's f, square
root of "f2".
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 : str, '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 : str, '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 : str, '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 : str, '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 : str, '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 of a constraint
This class is not recommended, use `FTestPowerF2` with corrected interface.
This is based on Cohen's f as effect size measure.
Warning: Methods in this class have the names df_num and df_denom reversed.
See Also
--------
FTestPowerF2 :
Class with Cohen's f-squared as effect size, corrected keyword names.
Examples
--------
Sample size and power for multiple regression base on R-squared
Compute effect size from R-squared
>>> r2 = 0.1
>>> f2 = r2 / (1 - r2)
>>> f = np.sqrt(f2)
>>> r2, f2, f
(0.1, 0.11111111111111112, 0.33333333333333337)
Find sample size by solving for denominator df, wrongly named ``df_num``
>>> df1 = 1 # number of constraints in hypothesis test
>>> df2 = FTestPower().solve_power(effect_size=f, alpha=0.1, power=0.9,
df_denom=df1)
>>> ncc = 1 # default
>>> nobs = df2 + df1 + ncc
>>> df2, nobs
(76.46459758305376, 78.46459758305376)
verify power at df2
>>> FTestPower().power(effect_size=f, alpha=0.1, df_denom=df1, df_num=df2)
0.8999999972109698
"""
[docs] def power(self, effect_size, df_num, df_denom, alpha, ncc=1):
'''Calculate the power of a F-test.
The effect size is Cohen's ``f``, square root of ``f2``.
The sample size is given by ``nobs = df_denom + df_num + ncc``
Warning: The meaning of df_num and df_denom is reversed.
Parameters
----------
effect_size : float
Standardized effect size. The effect size is here Cohen's ``f``,
square root of ``f2``.
df_num : int or float
Warning incorrect name
denominator degrees of freedom,
This corresponds to the number of constraints in Wald tests.
df_denom : int or float
Warning incorrect name
numerator degrees of freedom.
This corresponds to the df_resid in Wald tests.
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,
alpha=None, power=None, ncc=1, **kwargs):
'''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.
The effect size is Cohen's ``f``, square root of ``f2``.
The sample size is given by ``nobs = df_denom + df_num + ncc``.
Warning: The meaning of df_num and df_denom is reversed.
Parameters
----------
effect_size : float
Standardized effect size. The effect size is here Cohen's ``f``,
square root of ``f2``.
df_num : int or float
Warning incorrect name
denominator degrees of freedom,
This corresponds to the number of constraints in Wald tests.
Sample size is given by ``nobs = df_denom + df_num + ncc``
df_denom : int or float
Warning incorrect name
numerator degrees of freedom.
This corresponds to the df_resid in Wald tests.
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.
ncc : int
degrees of freedom correction for non-centrality parameter.
see Notes
kwargs : empty
``kwargs`` are not used and included for backwards compatibility.
If ``nobs`` is used as keyword, then a warning is issued. All
other keywords in ``kwargs`` raise a ValueError.
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 method 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.
'''
if kwargs:
if "nobs" in kwargs:
warnings.warn("nobs is not used")
else:
raise ValueError(f"incorrect keyword(s) {kwargs}")
return super(FTestPower, self).solve_power(effect_size=effect_size,
df_num=df_num,
df_denom=df_denom,
alpha=alpha,
power=power,
ncc=ncc)
class FTestPowerF2(Power):
"""Statistical Power calculations for generic F-test of a constraint
This is based on Cohen's f^2 as effect size measure.
Examples
--------
Sample size and power for multiple regression base on R-squared
Compute effect size from R-squared
>>> r2 = 0.1
>>> f2 = r2 / (1 - r2)
>>> f = np.sqrt(f2)
>>> r2, f2, f
(0.1, 0.11111111111111112, 0.33333333333333337)
Find sample size by solving for denominator degrees of freedom.
>>> df1 = 1 # number of constraints in hypothesis test
>>> df2 = FTestPowerF2().solve_power(effect_size=f2, alpha=0.1, power=0.9,
df_num=df1)
>>> ncc = 1 # default
>>> nobs = df2 + df1 + ncc
>>> df2, nobs
(76.46459758305376, 78.46459758305376)
verify power at df2
>>> FTestPowerF2().power(effect_size=f, alpha=0.1, df_num=df1, df_denom=df2)
0.8999999972109698
"""
def power(self, effect_size, df_num, df_denom, alpha, ncc=1):
'''Calculate the power of a F-test.
The effect size is Cohen's ``f^2``.
The sample size is given by ``nobs = df_denom + df_num + ncc``
Parameters
----------
effect_size : float
The effect size is here Cohen's ``f2``. This is equal to
the noncentrality of an F-test divided by nobs.
df_num : int or float
Numerator degrees of freedom,
This corresponds to the number of constraints in Wald tests.
df_denom : int or float
Denominator degrees of freedom.
This corresponds to the df_resid in Wald tests.
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
-----
The sample size is given implicitly by df_denom
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_f2(effect_size, df_num, df_denom, alpha, ncc=ncc)
return pow_
#method is only added to have explicit keywords and docstring
def solve_power(self, effect_size=None, df_num=None, df_denom=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.
The effect size is Cohen's ``f2``.
The sample size is given by ``nobs = df_denom + df_num + ncc``, and
can be found by solving for df_denom.
Parameters
----------
effect_size : float
The effect size is here Cohen's ``f2``. This is equal to
the noncentrality of an F-test divided by nobs.
df_num : int or float
Numerator degrees of freedom,
This corresponds to the number of constraints in Wald tests.
df_denom : int or float
Denominator degrees of freedom.
This corresponds to the df_resid in Wald tests.
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.
ncc : int
degrees of freedom correction for non-centrality parameter.
see Notes
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(FTestPowerF2, 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
This is based on Cohen's f as effect size measure.
See Also
--------
statsmodels.stats.oneway.effectsize_oneway
'''
[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. The effect size is here Cohen's f, square
root of "f2".
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 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.
'''
# update start values for root finding
if k_groups is not 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 : str, '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 : str, '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.
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
```