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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().solve_power(effect_size=effect_size,
nobs1=nobs1,
alpha=alpha,
power=power,
ratio=ratio,
alternative=alternative) | 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. | solve_power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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) | 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. | power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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().solve_power(effect_size=effect_size,
nobs1=nobs1,
alpha=alpha,
power=power,
ratio=ratio,
alternative=alternative) | 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. | solve_power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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_ | 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) | power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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().solve_power(effect_size=effect_size,
df_num=df_num,
df_denom=df_denom,
alpha=alpha,
power=power,
ncc=ncc) | 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. | solve_power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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_ | 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) | power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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().solve_power(effect_size=effect_size,
df_num=df_num,
df_denom=df_denom,
alpha=alpha,
power=power,
ncc=ncc) | 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. | solve_power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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) | 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. | power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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().solve_power(effect_size=effect_size,
nobs=nobs,
alpha=alpha,
k_groups=k_groups,
power=power) | 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. | solve_power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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 | experimental, test failure in solve_power for effect_size | _solve_effect_size | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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) | 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. | power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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().solve_power(effect_size=effect_size,
nobs=nobs,
n_bins=n_bins,
alpha=alpha,
power=power) | 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. | solve_power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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) | 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. | power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
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().solve_power(effect_size=effect_size,
nobs1=nobs1,
alpha=alpha,
power=power,
ratio=ratio,
alternative=alternative) | 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. | solve_power | python | statsmodels/statsmodels | statsmodels/stats/power.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/power.py | BSD-3-Clause |
def _bound_proportion_confint(
func: Callable[[float], float], qi: float, lower: bool = True
) -> float:
"""
Try hard to find a bound different from eps/1 - eps in proportion_confint
Parameters
----------
func : callable
Callable function to use as the objective of the search
qi : float
The empirical success rate
lower : bool
Whether to fund a lower bound for the left side of the CI
Returns
-------
float
The coarse bound
"""
default = FLOAT_INFO.eps if lower else 1.0 - FLOAT_INFO.eps
def step(v):
return v / 8 if lower else v + (1.0 - v) / 8
x = step(qi)
w = func(x)
cnt = 1
while w > 0 and cnt < 10:
x = step(x)
w = func(x)
cnt += 1
return x if cnt < 10 else default | Try hard to find a bound different from eps/1 - eps in proportion_confint
Parameters
----------
func : callable
Callable function to use as the objective of the search
qi : float
The empirical success rate
lower : bool
Whether to fund a lower bound for the left side of the CI
Returns
-------
float
The coarse bound | _bound_proportion_confint | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def _bisection_search_conservative(
func: Callable[[float], float], lb: float, ub: float, steps: int = 27
) -> tuple[float, float]:
"""
Private function used as a fallback by proportion_confint
Used when brentq returns a non-conservative bound for the CI
Parameters
----------
func : callable
Callable function to use as the objective of the search
lb : float
Lower bound
ub : float
Upper bound
steps : int
Number of steps to use in the bisection
Returns
-------
est : float
The estimated value. Will always produce a negative value of func
func_val : float
The value of the function at the estimate
"""
upper = func(ub)
lower = func(lb)
best = upper if upper < 0 else lower
best_pt = ub if upper < 0 else lb
if np.sign(lower) == np.sign(upper):
raise ValueError("problem with signs")
mp = (ub + lb) / 2
mid = func(mp)
if (mid < 0) and (mid > best):
best = mid
best_pt = mp
for _ in range(steps):
if np.sign(mid) == np.sign(upper):
ub = mp
upper = mid
else:
lb = mp
mp = (ub + lb) / 2
mid = func(mp)
if (mid < 0) and (mid > best):
best = mid
best_pt = mp
return best_pt, best | Private function used as a fallback by proportion_confint
Used when brentq returns a non-conservative bound for the CI
Parameters
----------
func : callable
Callable function to use as the objective of the search
lb : float
Lower bound
ub : float
Upper bound
steps : int
Number of steps to use in the bisection
Returns
-------
est : float
The estimated value. Will always produce a negative value of func
func_val : float
The value of the function at the estimate | _bisection_search_conservative | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def proportion_confint(count, nobs, alpha:float=0.05, method="normal", alternative:str='two-sided'):
"""
Confidence interval for a binomial proportion
Parameters
----------
count : {int or float, array_like}
number of successes, can be pandas Series or DataFrame. Arrays
must contain integer values if method is "binom_test".
nobs : {int or float, array_like}
total number of trials. Arrays must contain integer values if method
is "binom_test".
alpha : float
Significance level, default 0.05. Must be in (0, 1)
method : {"normal", "agresti_coull", "beta", "wilson", "binom_test"}
default: "normal"
method to use for confidence interval. Supported methods:
- `normal` : asymptotic normal approximation
- `agresti_coull` : Agresti-Coull interval
- `beta` : Clopper-Pearson interval based on Beta distribution
- `wilson` : Wilson Score interval
- `jeffreys` : Jeffreys Bayesian Interval
- `binom_test` : Numerical inversion of binom_test
alternative : {"two-sided", "larger", "smaller"}
default: "two-sided"
specifies whether to calculate a two-sided or one-sided confidence interval.
Returns
-------
ci_low, ci_upp : {float, ndarray, Series DataFrame}
larger and smaller confidence level with coverage (approximately) 1-alpha.
When a pandas object is returned, then the index is taken from `count`.
When side is not "two-sided", lower or upper bound is set to 0 or 1 respectively.
Notes
-----
Beta, the Clopper-Pearson exact interval has coverage at least 1-alpha,
but is in general conservative. Most of the other methods have average
coverage equal to 1-alpha, but will have smaller coverage in some cases.
The "beta" and "jeffreys" interval are central, they use alpha/2 in each
tail, and alpha is not adjusted at the boundaries. In the extreme case
when `count` is zero or equal to `nobs`, then the coverage will be only
1 - alpha/2 in the case of "beta".
The confidence intervals are clipped to be in the [0, 1] interval in the
case of "normal" and "agresti_coull".
Method "binom_test" directly inverts the binomial test in scipy.stats.
which has discrete steps.
TODO: binom_test intervals raise an exception in small samples if one
interval bound is close to zero or one.
References
----------
.. [*] https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval
.. [*] Brown, Lawrence D.; Cai, T. Tony; DasGupta, Anirban (2001).
"Interval Estimation for a Binomial Proportion", Statistical
Science 16 (2): 101–133. doi:10.1214/ss/1009213286.
"""
is_scalar = np.isscalar(count) and np.isscalar(nobs)
is_pandas = isinstance(count, (pd.Series, pd.DataFrame))
count_a = array_like(count, "count", optional=False, ndim=None)
nobs_a = array_like(nobs, "nobs", optional=False, ndim=None)
def _check(x: np.ndarray, name: str) -> np.ndarray:
if np.issubdtype(x.dtype, np.integer):
return x
y = x.astype(np.int64, casting="unsafe")
if np.any(y != x):
raise ValueError(
f"{name} must have an integral dtype. Found data with "
f"dtype {x.dtype}"
)
return y
if method == "binom_test":
count_a = _check(np.asarray(count_a), "count")
nobs_a = _check(np.asarray(nobs_a), "count")
q_ = count_a / nobs_a
if alternative == 'two-sided':
if method != "binom_test":
alpha = alpha / 2.0
elif alternative not in ['larger', 'smaller']:
raise NotImplementedError(f"alternative {alternative} is not available")
if method == "normal":
std_ = np.sqrt(q_ * (1 - q_) / nobs_a)
dist = stats.norm.isf(alpha) * std_
ci_low = q_ - dist
ci_upp = q_ + dist
elif method == "binom_test" and alternative == 'two-sided':
def func_factory(count: int, nobs: int) -> Callable[[float], float]:
if hasattr(stats, "binomtest"):
def func(qi):
return stats.binomtest(count, nobs, p=qi).pvalue - alpha
else:
# Remove after min SciPy >= 1.7
def func(qi):
return stats.binom_test(count, nobs, p=qi) - alpha
return func
bcast = np.broadcast(count_a, nobs_a)
ci_low = np.zeros(bcast.shape)
ci_upp = np.zeros(bcast.shape)
index = bcast.index
for c, n in bcast:
# Enforce symmetry
reverse = False
_q = q_.flat[index]
if c > n // 2:
c = n - c
reverse = True
_q = 1 - _q
func = func_factory(c, n)
if c == 0:
ci_low.flat[index] = 0.0
else:
lower_bnd = _bound_proportion_confint(func, _q, lower=True)
val, _z = optimize.brentq(
func, lower_bnd, _q, full_output=True
)
if func(val) > 0:
power = 10
new_lb = val - (val - lower_bnd) / 2**power
while func(new_lb) > 0 and power >= 0:
power -= 1
new_lb = val - (val - lower_bnd) / 2**power
val, _ = _bisection_search_conservative(func, new_lb, _q)
ci_low.flat[index] = val
if c == n:
ci_upp.flat[index] = 1.0
else:
upper_bnd = _bound_proportion_confint(func, _q, lower=False)
val, _z = optimize.brentq(
func, _q, upper_bnd, full_output=True
)
if func(val) > 0:
power = 10
new_ub = val + (upper_bnd - val) / 2**power
while func(new_ub) > 0 and power >= 0:
power -= 1
new_ub = val - (upper_bnd - val) / 2**power
val, _ = _bisection_search_conservative(func, _q, new_ub)
ci_upp.flat[index] = val
if reverse:
temp = ci_upp.flat[index]
ci_upp.flat[index] = 1 - ci_low.flat[index]
ci_low.flat[index] = 1 - temp
index = bcast.index
elif method == "beta" or (method == "binom_test" and alternative != 'two-sided'):
ci_low = stats.beta.ppf(alpha, count_a, nobs_a - count_a + 1)
ci_upp = stats.beta.isf(alpha, count_a + 1, nobs_a - count_a)
if np.ndim(ci_low) > 0:
ci_low.flat[q_.flat == 0] = 0
ci_upp.flat[q_.flat == 1] = 1
else:
ci_low = 0 if q_ == 0 else ci_low
ci_upp = 1 if q_ == 1 else ci_upp
elif method == "agresti_coull":
crit = stats.norm.isf(alpha)
nobs_c = nobs_a + crit**2
q_c = (count_a + crit**2 / 2.0) / nobs_c
std_c = np.sqrt(q_c * (1.0 - q_c) / nobs_c)
dist = crit * std_c
ci_low = q_c - dist
ci_upp = q_c + dist
elif method == "wilson":
crit = stats.norm.isf(alpha)
crit2 = crit**2
denom = 1 + crit2 / nobs_a
center = (q_ + crit2 / (2 * nobs_a)) / denom
dist = crit * np.sqrt(
q_ * (1.0 - q_) / nobs_a + crit2 / (4.0 * nobs_a**2)
)
dist /= denom
ci_low = center - dist
ci_upp = center + dist
# method adjusted to be more forgiving of misspellings or incorrect option name
elif method[:4] == "jeff":
ci_low = stats.beta.ppf(alpha, count_a + 0.5, nobs_a - count_a + 0.5)
ci_upp = stats.beta.isf(alpha, count_a + 0.5, nobs_a - count_a + 0.5)
else:
raise NotImplementedError(f"method {method} is not available")
if method in ["normal", "agresti_coull"]:
ci_low = np.clip(ci_low, 0, 1)
ci_upp = np.clip(ci_upp, 0, 1)
if is_pandas:
container = pd.Series if isinstance(count, pd.Series) else pd.DataFrame
ci_low = container(ci_low, index=count.index)
ci_upp = container(ci_upp, index=count.index)
if alternative == 'larger':
ci_low = 0
elif alternative == 'smaller':
ci_upp = 1
if is_scalar:
return float(ci_low), float(ci_upp)
return ci_low, ci_upp | Confidence interval for a binomial proportion
Parameters
----------
count : {int or float, array_like}
number of successes, can be pandas Series or DataFrame. Arrays
must contain integer values if method is "binom_test".
nobs : {int or float, array_like}
total number of trials. Arrays must contain integer values if method
is "binom_test".
alpha : float
Significance level, default 0.05. Must be in (0, 1)
method : {"normal", "agresti_coull", "beta", "wilson", "binom_test"}
default: "normal"
method to use for confidence interval. Supported methods:
- `normal` : asymptotic normal approximation
- `agresti_coull` : Agresti-Coull interval
- `beta` : Clopper-Pearson interval based on Beta distribution
- `wilson` : Wilson Score interval
- `jeffreys` : Jeffreys Bayesian Interval
- `binom_test` : Numerical inversion of binom_test
alternative : {"two-sided", "larger", "smaller"}
default: "two-sided"
specifies whether to calculate a two-sided or one-sided confidence interval.
Returns
-------
ci_low, ci_upp : {float, ndarray, Series DataFrame}
larger and smaller confidence level with coverage (approximately) 1-alpha.
When a pandas object is returned, then the index is taken from `count`.
When side is not "two-sided", lower or upper bound is set to 0 or 1 respectively.
Notes
-----
Beta, the Clopper-Pearson exact interval has coverage at least 1-alpha,
but is in general conservative. Most of the other methods have average
coverage equal to 1-alpha, but will have smaller coverage in some cases.
The "beta" and "jeffreys" interval are central, they use alpha/2 in each
tail, and alpha is not adjusted at the boundaries. In the extreme case
when `count` is zero or equal to `nobs`, then the coverage will be only
1 - alpha/2 in the case of "beta".
The confidence intervals are clipped to be in the [0, 1] interval in the
case of "normal" and "agresti_coull".
Method "binom_test" directly inverts the binomial test in scipy.stats.
which has discrete steps.
TODO: binom_test intervals raise an exception in small samples if one
interval bound is close to zero or one.
References
----------
.. [*] https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval
.. [*] Brown, Lawrence D.; Cai, T. Tony; DasGupta, Anirban (2001).
"Interval Estimation for a Binomial Proportion", Statistical
Science 16 (2): 101–133. doi:10.1214/ss/1009213286. | proportion_confint | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def poisson_interval(interval, p):
"""
Compute P(b <= Z <= a) where Z ~ Poisson(p) and
`interval = (b, a)`.
"""
b, a = interval
prob = stats.poisson.cdf(a, p) - stats.poisson.cdf(b - 1, p)
return prob | Compute P(b <= Z <= a) where Z ~ Poisson(p) and
`interval = (b, a)`. | multinomial_proportions_confint.poisson_interval | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def truncated_poisson_factorial_moment(interval, r, p):
"""
Compute mu_r, the r-th factorial moment of a poisson random
variable of parameter `p` truncated to `interval = (b, a)`.
"""
b, a = interval
return p ** r * (1 - ((poisson_interval((a - r + 1, a), p) -
poisson_interval((b - r, b - 1), p)) /
poisson_interval((b, a), p))) | Compute mu_r, the r-th factorial moment of a poisson random
variable of parameter `p` truncated to `interval = (b, a)`. | multinomial_proportions_confint.truncated_poisson_factorial_moment | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def edgeworth(intervals):
"""
Compute the Edgeworth expansion term of Sison & Glaz's formula
(1) (approximated probability for multinomial proportions in a
given box).
"""
# Compute means and central moments of the truncated poisson
# variables.
mu_r1, mu_r2, mu_r3, mu_r4 = (
np.array([truncated_poisson_factorial_moment(interval, r, p)
for (interval, p) in zip(intervals, counts)])
for r in range(1, 5)
)
mu = mu_r1
mu2 = mu_r2 + mu - mu ** 2
mu3 = mu_r3 + mu_r2 * (3 - 3 * mu) + mu - 3 * mu ** 2 + 2 * mu ** 3
mu4 = (mu_r4 + mu_r3 * (6 - 4 * mu) +
mu_r2 * (7 - 12 * mu + 6 * mu ** 2) +
mu - 4 * mu ** 2 + 6 * mu ** 3 - 3 * mu ** 4)
# Compute expansion factors, gamma_1 and gamma_2.
g1 = mu3.sum() / mu2.sum() ** 1.5
g2 = (mu4.sum() - 3 * (mu2 ** 2).sum()) / mu2.sum() ** 2
# Compute the expansion itself.
x = (n - mu.sum()) / np.sqrt(mu2.sum())
phi = np.exp(- x ** 2 / 2) / np.sqrt(2 * np.pi)
H3 = x ** 3 - 3 * x
H4 = x ** 4 - 6 * x ** 2 + 3
H6 = x ** 6 - 15 * x ** 4 + 45 * x ** 2 - 15
f = phi * (1 + g1 * H3 / 6 + g2 * H4 / 24 + g1 ** 2 * H6 / 72)
return f / np.sqrt(mu2.sum()) | Compute the Edgeworth expansion term of Sison & Glaz's formula
(1) (approximated probability for multinomial proportions in a
given box). | multinomial_proportions_confint.edgeworth | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def approximated_multinomial_interval(intervals):
"""
Compute approximated probability for Multinomial(n, proportions)
to be in `intervals` (Sison & Glaz's formula (1)).
"""
return np.exp(
np.sum(np.log([poisson_interval(interval, p)
for (interval, p) in zip(intervals, counts)])) +
np.log(edgeworth(intervals)) -
np.log(stats.poisson._pmf(n, n))
) | Compute approximated probability for Multinomial(n, proportions)
to be in `intervals` (Sison & Glaz's formula (1)). | multinomial_proportions_confint.approximated_multinomial_interval | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def nu(c):
"""
Compute interval coverage for a given `c` (Sison & Glaz's
formula (7)).
"""
return approximated_multinomial_interval(
[(np.maximum(count - c, 0), np.minimum(count + c, n))
for count in counts]) | Compute interval coverage for a given `c` (Sison & Glaz's
formula (7)). | multinomial_proportions_confint.nu | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def multinomial_proportions_confint(counts, alpha=0.05, method='goodman'):
"""
Confidence intervals for multinomial proportions.
Parameters
----------
counts : array_like of int, 1-D
Number of observations in each category.
alpha : float in (0, 1), optional
Significance level, defaults to 0.05.
method : {'goodman', 'sison-glaz'}, optional
Method to use to compute the confidence intervals; available methods
are:
- `goodman`: based on a chi-squared approximation, valid if all
values in `counts` are greater or equal to 5 [2]_
- `sison-glaz`: less conservative than `goodman`, but only valid if
`counts` has 7 or more categories (``len(counts) >= 7``) [3]_
Returns
-------
confint : ndarray, 2-D
Array of [lower, upper] confidence levels for each category, such that
overall coverage is (approximately) `1-alpha`.
Raises
------
ValueError
If `alpha` is not in `(0, 1)` (bounds excluded), or if the values in
`counts` are not all positive or null.
NotImplementedError
If `method` is not kown.
Exception
When ``method == 'sison-glaz'``, if for some reason `c` cannot be
computed; this signals a bug and should be reported.
Notes
-----
The `goodman` method [2]_ is based on approximating a statistic based on
the multinomial as a chi-squared random variable. The usual recommendation
is that this is valid if all the values in `counts` are greater than or
equal to 5. There is no condition on the number of categories for this
method.
The `sison-glaz` method [3]_ approximates the multinomial probabilities,
and evaluates that with a maximum-likelihood estimator. The first
approximation is an Edgeworth expansion that converges when the number of
categories goes to infinity, and the maximum-likelihood estimator converges
when the number of observations (``sum(counts)``) goes to infinity. In
their paper, Sison & Glaz demo their method with at least 7 categories, so
``len(counts) >= 7`` with all values in `counts` at or above 5 can be used
as a rule of thumb for the validity of this method. This method is less
conservative than the `goodman` method (i.e. it will yield confidence
intervals closer to the desired significance level), but produces
confidence intervals of uniform width over all categories (except when the
intervals reach 0 or 1, in which case they are truncated), which makes it
most useful when proportions are of similar magnitude.
Aside from the original sources ([1]_, [2]_, and [3]_), the implementation
uses the formulas (though not the code) presented in [4]_ and [5]_.
References
----------
.. [1] Levin, Bruce, "A representation for multinomial cumulative
distribution functions," The Annals of Statistics, Vol. 9, No. 5,
1981, pp. 1123-1126.
.. [2] Goodman, L.A., "On simultaneous confidence intervals for multinomial
proportions," Technometrics, Vol. 7, No. 2, 1965, pp. 247-254.
.. [3] Sison, Cristina P., and Joseph Glaz, "Simultaneous Confidence
Intervals and Sample Size Determination for Multinomial
Proportions," Journal of the American Statistical Association,
Vol. 90, No. 429, 1995, pp. 366-369.
.. [4] May, Warren L., and William D. Johnson, "A SAS® macro for
constructing simultaneous confidence intervals for multinomial
proportions," Computer methods and programs in Biomedicine, Vol. 53,
No. 3, 1997, pp. 153-162.
.. [5] May, Warren L., and William D. Johnson, "Constructing two-sided
simultaneous confidence intervals for multinomial proportions for
small counts in a large number of cells," Journal of Statistical
Software, Vol. 5, No. 6, 2000, pp. 1-24.
"""
if alpha <= 0 or alpha >= 1:
raise ValueError('alpha must be in (0, 1), bounds excluded')
counts = np.array(counts, dtype=float)
if (counts < 0).any():
raise ValueError('counts must be >= 0')
n = counts.sum()
k = len(counts)
proportions = counts / n
if method == 'goodman':
chi2 = stats.chi2.ppf(1 - alpha / k, 1)
delta = chi2 ** 2 + (4 * n * proportions * chi2 * (1 - proportions))
region = ((2 * n * proportions + chi2 +
np.array([- np.sqrt(delta), np.sqrt(delta)])) /
(2 * (chi2 + n))).T
elif method[:5] == 'sison': # We accept any name starting with 'sison'
# Define a few functions we'll use a lot.
def poisson_interval(interval, p):
"""
Compute P(b <= Z <= a) where Z ~ Poisson(p) and
`interval = (b, a)`.
"""
b, a = interval
prob = stats.poisson.cdf(a, p) - stats.poisson.cdf(b - 1, p)
return prob
def truncated_poisson_factorial_moment(interval, r, p):
"""
Compute mu_r, the r-th factorial moment of a poisson random
variable of parameter `p` truncated to `interval = (b, a)`.
"""
b, a = interval
return p ** r * (1 - ((poisson_interval((a - r + 1, a), p) -
poisson_interval((b - r, b - 1), p)) /
poisson_interval((b, a), p)))
def edgeworth(intervals):
"""
Compute the Edgeworth expansion term of Sison & Glaz's formula
(1) (approximated probability for multinomial proportions in a
given box).
"""
# Compute means and central moments of the truncated poisson
# variables.
mu_r1, mu_r2, mu_r3, mu_r4 = (
np.array([truncated_poisson_factorial_moment(interval, r, p)
for (interval, p) in zip(intervals, counts)])
for r in range(1, 5)
)
mu = mu_r1
mu2 = mu_r2 + mu - mu ** 2
mu3 = mu_r3 + mu_r2 * (3 - 3 * mu) + mu - 3 * mu ** 2 + 2 * mu ** 3
mu4 = (mu_r4 + mu_r3 * (6 - 4 * mu) +
mu_r2 * (7 - 12 * mu + 6 * mu ** 2) +
mu - 4 * mu ** 2 + 6 * mu ** 3 - 3 * mu ** 4)
# Compute expansion factors, gamma_1 and gamma_2.
g1 = mu3.sum() / mu2.sum() ** 1.5
g2 = (mu4.sum() - 3 * (mu2 ** 2).sum()) / mu2.sum() ** 2
# Compute the expansion itself.
x = (n - mu.sum()) / np.sqrt(mu2.sum())
phi = np.exp(- x ** 2 / 2) / np.sqrt(2 * np.pi)
H3 = x ** 3 - 3 * x
H4 = x ** 4 - 6 * x ** 2 + 3
H6 = x ** 6 - 15 * x ** 4 + 45 * x ** 2 - 15
f = phi * (1 + g1 * H3 / 6 + g2 * H4 / 24 + g1 ** 2 * H6 / 72)
return f / np.sqrt(mu2.sum())
def approximated_multinomial_interval(intervals):
"""
Compute approximated probability for Multinomial(n, proportions)
to be in `intervals` (Sison & Glaz's formula (1)).
"""
return np.exp(
np.sum(np.log([poisson_interval(interval, p)
for (interval, p) in zip(intervals, counts)])) +
np.log(edgeworth(intervals)) -
np.log(stats.poisson._pmf(n, n))
)
def nu(c):
"""
Compute interval coverage for a given `c` (Sison & Glaz's
formula (7)).
"""
return approximated_multinomial_interval(
[(np.maximum(count - c, 0), np.minimum(count + c, n))
for count in counts])
# Find the value of `c` that will give us the confidence intervals
# (solving nu(c) <= 1 - alpha < nu(c + 1).
c = 1.0
nuc = nu(c)
nucp1 = nu(c + 1)
while not (nuc <= (1 - alpha) < nucp1):
if c > n:
raise Exception("Couldn't find a value for `c` that "
"solves nu(c) <= 1 - alpha < nu(c + 1)")
c += 1
nuc = nucp1
nucp1 = nu(c + 1)
# Compute gamma and the corresponding confidence intervals.
g = (1 - alpha - nuc) / (nucp1 - nuc)
ci_lower = np.maximum(proportions - c / n, 0)
ci_upper = np.minimum(proportions + (c + 2 * g) / n, 1)
region = np.array([ci_lower, ci_upper]).T
else:
raise NotImplementedError('method "%s" is not available' % method)
return region | Confidence intervals for multinomial proportions.
Parameters
----------
counts : array_like of int, 1-D
Number of observations in each category.
alpha : float in (0, 1), optional
Significance level, defaults to 0.05.
method : {'goodman', 'sison-glaz'}, optional
Method to use to compute the confidence intervals; available methods
are:
- `goodman`: based on a chi-squared approximation, valid if all
values in `counts` are greater or equal to 5 [2]_
- `sison-glaz`: less conservative than `goodman`, but only valid if
`counts` has 7 or more categories (``len(counts) >= 7``) [3]_
Returns
-------
confint : ndarray, 2-D
Array of [lower, upper] confidence levels for each category, such that
overall coverage is (approximately) `1-alpha`.
Raises
------
ValueError
If `alpha` is not in `(0, 1)` (bounds excluded), or if the values in
`counts` are not all positive or null.
NotImplementedError
If `method` is not kown.
Exception
When ``method == 'sison-glaz'``, if for some reason `c` cannot be
computed; this signals a bug and should be reported.
Notes
-----
The `goodman` method [2]_ is based on approximating a statistic based on
the multinomial as a chi-squared random variable. The usual recommendation
is that this is valid if all the values in `counts` are greater than or
equal to 5. There is no condition on the number of categories for this
method.
The `sison-glaz` method [3]_ approximates the multinomial probabilities,
and evaluates that with a maximum-likelihood estimator. The first
approximation is an Edgeworth expansion that converges when the number of
categories goes to infinity, and the maximum-likelihood estimator converges
when the number of observations (``sum(counts)``) goes to infinity. In
their paper, Sison & Glaz demo their method with at least 7 categories, so
``len(counts) >= 7`` with all values in `counts` at or above 5 can be used
as a rule of thumb for the validity of this method. This method is less
conservative than the `goodman` method (i.e. it will yield confidence
intervals closer to the desired significance level), but produces
confidence intervals of uniform width over all categories (except when the
intervals reach 0 or 1, in which case they are truncated), which makes it
most useful when proportions are of similar magnitude.
Aside from the original sources ([1]_, [2]_, and [3]_), the implementation
uses the formulas (though not the code) presented in [4]_ and [5]_.
References
----------
.. [1] Levin, Bruce, "A representation for multinomial cumulative
distribution functions," The Annals of Statistics, Vol. 9, No. 5,
1981, pp. 1123-1126.
.. [2] Goodman, L.A., "On simultaneous confidence intervals for multinomial
proportions," Technometrics, Vol. 7, No. 2, 1965, pp. 247-254.
.. [3] Sison, Cristina P., and Joseph Glaz, "Simultaneous Confidence
Intervals and Sample Size Determination for Multinomial
Proportions," Journal of the American Statistical Association,
Vol. 90, No. 429, 1995, pp. 366-369.
.. [4] May, Warren L., and William D. Johnson, "A SAS® macro for
constructing simultaneous confidence intervals for multinomial
proportions," Computer methods and programs in Biomedicine, Vol. 53,
No. 3, 1997, pp. 153-162.
.. [5] May, Warren L., and William D. Johnson, "Constructing two-sided
simultaneous confidence intervals for multinomial proportions for
small counts in a large number of cells," Journal of Statistical
Software, Vol. 5, No. 6, 2000, pp. 1-24. | multinomial_proportions_confint | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def samplesize_confint_proportion(proportion, half_length, alpha=0.05,
method='normal'):
"""
Find sample size to get desired confidence interval length
Parameters
----------
proportion : float in (0, 1)
proportion or quantile
half_length : float in (0, 1)
desired half length of the confidence interval
alpha : float in (0, 1)
significance level, default 0.05,
coverage of the two-sided interval is (approximately) ``1 - alpha``
method : str in ['normal']
method to use for confidence interval,
currently only normal approximation
Returns
-------
n : float
sample size to get the desired half length of the confidence interval
Notes
-----
this is mainly to store the formula.
possible application: number of replications in bootstrap samples
"""
q_ = proportion
if method == 'normal':
n = q_ * (1 - q_) / (half_length / stats.norm.isf(alpha / 2.))**2
else:
raise NotImplementedError('only "normal" is available')
return n | Find sample size to get desired confidence interval length
Parameters
----------
proportion : float in (0, 1)
proportion or quantile
half_length : float in (0, 1)
desired half length of the confidence interval
alpha : float in (0, 1)
significance level, default 0.05,
coverage of the two-sided interval is (approximately) ``1 - alpha``
method : str in ['normal']
method to use for confidence interval,
currently only normal approximation
Returns
-------
n : float
sample size to get the desired half length of the confidence interval
Notes
-----
this is mainly to store the formula.
possible application: number of replications in bootstrap samples | samplesize_confint_proportion | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def proportion_effectsize(prop1, prop2, method='normal'):
"""
Effect size for a test comparing two proportions
for use in power function
Parameters
----------
prop1, prop2 : float or array_like
The proportion value(s).
Returns
-------
es : float or ndarray
effect size for (transformed) prop1 - prop2
Notes
-----
only method='normal' is implemented to match pwr.p2.test
see http://www.statmethods.net/stats/power.html
Effect size for `normal` is defined as ::
2 * (arcsin(sqrt(prop1)) - arcsin(sqrt(prop2)))
I think other conversions to normality can be used, but I need to check.
Examples
--------
>>> import statsmodels.api as sm
>>> sm.stats.proportion_effectsize(0.5, 0.4)
0.20135792079033088
>>> sm.stats.proportion_effectsize([0.3, 0.4, 0.5], 0.4)
array([-0.21015893, 0. , 0.20135792])
"""
if method != 'normal':
raise ValueError('only "normal" is implemented')
es = 2 * (np.arcsin(np.sqrt(prop1)) - np.arcsin(np.sqrt(prop2)))
return es | Effect size for a test comparing two proportions
for use in power function
Parameters
----------
prop1, prop2 : float or array_like
The proportion value(s).
Returns
-------
es : float or ndarray
effect size for (transformed) prop1 - prop2
Notes
-----
only method='normal' is implemented to match pwr.p2.test
see http://www.statmethods.net/stats/power.html
Effect size for `normal` is defined as ::
2 * (arcsin(sqrt(prop1)) - arcsin(sqrt(prop2)))
I think other conversions to normality can be used, but I need to check.
Examples
--------
>>> import statsmodels.api as sm
>>> sm.stats.proportion_effectsize(0.5, 0.4)
0.20135792079033088
>>> sm.stats.proportion_effectsize([0.3, 0.4, 0.5], 0.4)
array([-0.21015893, 0. , 0.20135792]) | proportion_effectsize | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def std_prop(prop, nobs):
"""
Standard error for the estimate of a proportion
This is just ``np.sqrt(p * (1. - p) / nobs)``
Parameters
----------
prop : array_like
proportion
nobs : int, array_like
number of observations
Returns
-------
std : array_like
standard error for a proportion of nobs independent observations
"""
return np.sqrt(prop * (1. - prop) / nobs) | Standard error for the estimate of a proportion
This is just ``np.sqrt(p * (1. - p) / nobs)``
Parameters
----------
prop : array_like
proportion
nobs : int, array_like
number of observations
Returns
-------
std : array_like
standard error for a proportion of nobs independent observations | std_prop | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def _power_ztost(mean_low, var_low, mean_upp, var_upp, mean_alt, var_alt,
alpha=0.05, discrete=True, dist='norm', nobs=None,
continuity=0, critval_continuity=0):
"""
Generic statistical power function for normal based equivalence test
This includes options to adjust the normal approximation and can use
the binomial to evaluate the probability of the rejection region
see power_ztost_prob for a description of the options
"""
# TODO: refactor structure, separate norm and binom better
if not isinstance(continuity, tuple):
continuity = (continuity, continuity)
crit = stats.norm.isf(alpha)
k_low = mean_low + np.sqrt(var_low) * crit
k_upp = mean_upp - np.sqrt(var_upp) * crit
if discrete or dist == 'binom':
k_low = np.ceil(k_low * nobs + 0.5 * critval_continuity)
k_upp = np.trunc(k_upp * nobs - 0.5 * critval_continuity)
if dist == 'norm':
#need proportion
k_low = (k_low) * 1. / nobs #-1 to match PASS
k_upp = k_upp * 1. / nobs
# else:
# if dist == 'binom':
# #need counts
# k_low *= nobs
# k_upp *= nobs
#print mean_low, np.sqrt(var_low), crit, var_low
#print mean_upp, np.sqrt(var_upp), crit, var_upp
if np.any(k_low > k_upp): #vectorize
import warnings
warnings.warn("no overlap, power is zero", HypothesisTestWarning)
std_alt = np.sqrt(var_alt)
z_low = (k_low - mean_alt - continuity[0] * 0.5 / nobs) / std_alt
z_upp = (k_upp - mean_alt + continuity[1] * 0.5 / nobs) / std_alt
if dist == 'norm':
power = stats.norm.cdf(z_upp) - stats.norm.cdf(z_low)
elif dist == 'binom':
power = (stats.binom.cdf(k_upp, nobs, mean_alt) -
stats.binom.cdf(k_low-1, nobs, mean_alt))
return power, (k_low, k_upp, z_low, z_upp) | Generic statistical power function for normal based equivalence test
This includes options to adjust the normal approximation and can use
the binomial to evaluate the probability of the rejection region
see power_ztost_prob for a description of the options | _power_ztost | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def binom_tost(count, nobs, low, upp):
"""
Exact TOST test for one proportion using binomial distribution
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials.
nobs : int
the number of trials or observations.
low, upp : floats
lower and upper limit of equivalence region
Returns
-------
pvalue : float
p-value of equivalence test
pval_low, pval_upp : floats
p-values of lower and upper one-sided tests
"""
# binom_test_stat only returns pval
tt1 = binom_test(count, nobs, alternative='larger', prop=low)
tt2 = binom_test(count, nobs, alternative='smaller', prop=upp)
return np.maximum(tt1, tt2), tt1, tt2, | Exact TOST test for one proportion using binomial distribution
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials.
nobs : int
the number of trials or observations.
low, upp : floats
lower and upper limit of equivalence region
Returns
-------
pvalue : float
p-value of equivalence test
pval_low, pval_upp : floats
p-values of lower and upper one-sided tests | binom_tost | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def binom_tost_reject_interval(low, upp, nobs, alpha=0.05):
"""
Rejection region for binomial TOST
The interval includes the end points,
`reject` if and only if `r_low <= x <= r_upp`.
The interval might be empty with `r_upp < r_low`.
Parameters
----------
low, upp : floats
lower and upper limit of equivalence region
nobs : int
the number of trials or observations.
Returns
-------
x_low, x_upp : float
lower and upper bound of rejection region
"""
x_low = stats.binom.isf(alpha, nobs, low) + 1
x_upp = stats.binom.ppf(alpha, nobs, upp) - 1
return x_low, x_upp | Rejection region for binomial TOST
The interval includes the end points,
`reject` if and only if `r_low <= x <= r_upp`.
The interval might be empty with `r_upp < r_low`.
Parameters
----------
low, upp : floats
lower and upper limit of equivalence region
nobs : int
the number of trials or observations.
Returns
-------
x_low, x_upp : float
lower and upper bound of rejection region | binom_tost_reject_interval | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def binom_test_reject_interval(value, nobs, alpha=0.05, alternative='two-sided'):
"""
Rejection region for binomial test for one sample proportion
The interval includes the end points of the rejection region.
Parameters
----------
value : float
proportion under the Null hypothesis
nobs : int
the number of trials or observations.
Returns
-------
x_low, x_upp : int
lower and upper bound of rejection region
"""
if alternative in ['2s', 'two-sided']:
alternative = '2s' # normalize alternative name
alpha = alpha / 2
if alternative in ['2s', 'smaller']:
x_low = stats.binom.ppf(alpha, nobs, value) - 1
else:
x_low = 0
if alternative in ['2s', 'larger']:
x_upp = stats.binom.isf(alpha, nobs, value) + 1
else :
x_upp = nobs
return int(x_low), int(x_upp) | Rejection region for binomial test for one sample proportion
The interval includes the end points of the rejection region.
Parameters
----------
value : float
proportion under the Null hypothesis
nobs : int
the number of trials or observations.
Returns
-------
x_low, x_upp : int
lower and upper bound of rejection region | binom_test_reject_interval | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def binom_test(count, nobs, prop=0.5, alternative='two-sided'):
"""
Perform a test that the probability of success is p.
This is an exact, two-sided test of the null hypothesis
that the probability of success in a Bernoulli experiment
is `p`.
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials.
nobs : int
the number of trials or observations.
prop : float, optional
The probability of success under the null hypothesis,
`0 <= prop <= 1`. The default value is `prop = 0.5`
alternative : str in ['two-sided', 'smaller', 'larger']
alternative hypothesis, which can be two-sided or either one of the
one-sided tests.
Returns
-------
p-value : float
The p-value of the hypothesis test
Notes
-----
This uses scipy.stats.binom_test for the two-sided alternative.
"""
if np.any(prop > 1.0) or np.any(prop < 0.0):
raise ValueError("p must be in range [0,1]")
if alternative in ['2s', 'two-sided']:
try:
pval = stats.binomtest(count, n=nobs, p=prop).pvalue
except AttributeError:
# Remove after min SciPy >= 1.7
pval = stats.binom_test(count, n=nobs, p=prop)
elif alternative in ['l', 'larger']:
pval = stats.binom.sf(count-1, nobs, prop)
elif alternative in ['s', 'smaller']:
pval = stats.binom.cdf(count, nobs, prop)
else:
raise ValueError('alternative not recognized\n'
'should be two-sided, larger or smaller')
return pval | Perform a test that the probability of success is p.
This is an exact, two-sided test of the null hypothesis
that the probability of success in a Bernoulli experiment
is `p`.
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials.
nobs : int
the number of trials or observations.
prop : float, optional
The probability of success under the null hypothesis,
`0 <= prop <= 1`. The default value is `prop = 0.5`
alternative : str in ['two-sided', 'smaller', 'larger']
alternative hypothesis, which can be two-sided or either one of the
one-sided tests.
Returns
-------
p-value : float
The p-value of the hypothesis test
Notes
-----
This uses scipy.stats.binom_test for the two-sided alternative. | binom_test | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def power_ztost_prop(low, upp, nobs, p_alt, alpha=0.05, dist='norm',
variance_prop=None, discrete=True, continuity=0,
critval_continuity=0):
"""
Power of proportions equivalence test based on normal distribution
Parameters
----------
low, upp : floats
lower and upper limit of equivalence region
nobs : int
number of observations
p_alt : float in (0,1)
proportion under the alternative
alpha : float in (0,1)
significance level of the test
dist : str in ['norm', 'binom']
This defines the distribution to evaluate the power of the test. The
critical values of the TOST test are always based on the normal
approximation, but the distribution for the power can be either the
normal (default) or the binomial (exact) distribution.
variance_prop : None or float in (0,1)
If this is None, then the variances for the two one sided tests are
based on the proportions equal to the equivalence limits.
If variance_prop is given, then it is used to calculate the variance
for the TOST statistics. If this is based on an sample, then the
estimated proportion can be used.
discrete : bool
If true, then the critical values of the rejection region are converted
to integers. If dist is "binom", this is automatically assumed.
If discrete is false, then the TOST critical values are used as
floating point numbers, and the power is calculated based on the
rejection region that is not discretized.
continuity : bool or float
adjust the rejection region for the normal power probability. This has
and effect only if ``dist='norm'``
critval_continuity : bool or float
If this is non-zero, then the critical values of the tost rejection
region are adjusted before converting to integers. This affects both
distributions, ``dist='norm'`` and ``dist='binom'``.
Returns
-------
power : float
statistical power of the equivalence test.
(k_low, k_upp, z_low, z_upp) : tuple of floats
critical limits in intermediate steps
temporary return, will be changed
Notes
-----
In small samples the power for the ``discrete`` version, has a sawtooth
pattern as a function of the number of observations. As a consequence,
small changes in the number of observations or in the normal approximation
can have a large effect on the power.
``continuity`` and ``critval_continuity`` are added to match some results
of PASS, and are mainly to investigate the sensitivity of the ztost power
to small changes in the rejection region. From my interpretation of the
equations in the SAS manual, both are zero in SAS.
works vectorized
**verification:**
The ``dist='binom'`` results match PASS,
The ``dist='norm'`` results look reasonable, but no benchmark is available.
References
----------
SAS Manual: Chapter 68: The Power Procedure, Computational Resources
PASS Chapter 110: Equivalence Tests for One Proportion.
"""
mean_low = low
var_low = std_prop(low, nobs)**2
mean_upp = upp
var_upp = std_prop(upp, nobs)**2
mean_alt = p_alt
var_alt = std_prop(p_alt, nobs)**2
if variance_prop is not None:
var_low = var_upp = std_prop(variance_prop, nobs)**2
power = _power_ztost(mean_low, var_low, mean_upp, var_upp, mean_alt, var_alt,
alpha=alpha, discrete=discrete, dist=dist, nobs=nobs,
continuity=continuity, critval_continuity=critval_continuity)
return np.maximum(power[0], 0), power[1:] | Power of proportions equivalence test based on normal distribution
Parameters
----------
low, upp : floats
lower and upper limit of equivalence region
nobs : int
number of observations
p_alt : float in (0,1)
proportion under the alternative
alpha : float in (0,1)
significance level of the test
dist : str in ['norm', 'binom']
This defines the distribution to evaluate the power of the test. The
critical values of the TOST test are always based on the normal
approximation, but the distribution for the power can be either the
normal (default) or the binomial (exact) distribution.
variance_prop : None or float in (0,1)
If this is None, then the variances for the two one sided tests are
based on the proportions equal to the equivalence limits.
If variance_prop is given, then it is used to calculate the variance
for the TOST statistics. If this is based on an sample, then the
estimated proportion can be used.
discrete : bool
If true, then the critical values of the rejection region are converted
to integers. If dist is "binom", this is automatically assumed.
If discrete is false, then the TOST critical values are used as
floating point numbers, and the power is calculated based on the
rejection region that is not discretized.
continuity : bool or float
adjust the rejection region for the normal power probability. This has
and effect only if ``dist='norm'``
critval_continuity : bool or float
If this is non-zero, then the critical values of the tost rejection
region are adjusted before converting to integers. This affects both
distributions, ``dist='norm'`` and ``dist='binom'``.
Returns
-------
power : float
statistical power of the equivalence test.
(k_low, k_upp, z_low, z_upp) : tuple of floats
critical limits in intermediate steps
temporary return, will be changed
Notes
-----
In small samples the power for the ``discrete`` version, has a sawtooth
pattern as a function of the number of observations. As a consequence,
small changes in the number of observations or in the normal approximation
can have a large effect on the power.
``continuity`` and ``critval_continuity`` are added to match some results
of PASS, and are mainly to investigate the sensitivity of the ztost power
to small changes in the rejection region. From my interpretation of the
equations in the SAS manual, both are zero in SAS.
works vectorized
**verification:**
The ``dist='binom'`` results match PASS,
The ``dist='norm'`` results look reasonable, but no benchmark is available.
References
----------
SAS Manual: Chapter 68: The Power Procedure, Computational Resources
PASS Chapter 110: Equivalence Tests for One Proportion. | power_ztost_prop | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def _table_proportion(count, nobs):
"""
Create a k by 2 contingency table for proportion
helper function for proportions_chisquare
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials.
nobs : int
the number of trials or observations.
Returns
-------
table : ndarray
(k, 2) contingency table
Notes
-----
recent scipy has more elaborate contingency table functions
"""
count = np.asarray(count)
dt = np.promote_types(count.dtype, np.float64)
count = np.asarray(count, dtype=dt)
table = np.column_stack((count, nobs - count))
expected = table.sum(0) * table.sum(1)[:, None] * 1. / table.sum()
n_rows = table.shape[0]
return table, expected, n_rows | Create a k by 2 contingency table for proportion
helper function for proportions_chisquare
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials.
nobs : int
the number of trials or observations.
Returns
-------
table : ndarray
(k, 2) contingency table
Notes
-----
recent scipy has more elaborate contingency table functions | _table_proportion | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def proportions_ztest(count, nobs, value=None, alternative='two-sided',
prop_var=False):
"""
Test for proportions based on normal (z) test
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials. If this is array_like, then
the assumption is that this represents the number of successes for
each independent sample
nobs : {int, array_like}
the number of trials or observations, with the same length as
count.
value : float, array_like or None, optional
This is the value of the null hypothesis equal to the proportion in the
case of a one sample test. In the case of a two-sample test, the
null hypothesis is that prop[0] - prop[1] = value, where prop is the
proportion in the two samples. If not provided value = 0 and the null
is prop[0] = prop[1]
alternative : str in ['two-sided', 'smaller', 'larger']
The alternative hypothesis can be either two-sided or one of the one-
sided tests, smaller means that the alternative hypothesis is
``prop < value`` and larger means ``prop > value``. In the two sample
test, smaller means that the alternative hypothesis is ``p1 < p2`` and
larger means ``p1 > p2`` where ``p1`` is the proportion of the first
sample and ``p2`` of the second one.
prop_var : False or float in (0, 1)
If prop_var is false, then the variance of the proportion estimate is
calculated based on the sample proportion. Alternatively, a proportion
can be specified to calculate this variance. Common use case is to
use the proportion under the Null hypothesis to specify the variance
of the proportion estimate.
Returns
-------
zstat : float
test statistic for the z-test
p-value : float
p-value for the z-test
Examples
--------
>>> count = 5
>>> nobs = 83
>>> value = .05
>>> stat, pval = proportions_ztest(count, nobs, value)
>>> print('{0:0.3f}'.format(pval))
0.695
>>> import numpy as np
>>> from statsmodels.stats.proportion import proportions_ztest
>>> count = np.array([5, 12])
>>> nobs = np.array([83, 99])
>>> stat, pval = proportions_ztest(count, nobs)
>>> print('{0:0.3f}'.format(pval))
0.159
Notes
-----
This uses a simple normal test for proportions. It should be the same as
running the mean z-test on the data encoded 1 for event and 0 for no event
so that the sum corresponds to the count.
In the one and two sample cases with two-sided alternative, this test
produces the same p-value as ``proportions_chisquare``, since the
chisquare is the distribution of the square of a standard normal
distribution.
"""
# TODO: verify that this really holds
# TODO: add continuity correction or other improvements for small samples
# TODO: change options similar to propotion_ztost ?
count = np.asarray(count)
nobs = np.asarray(nobs)
if nobs.size == 1:
nobs = nobs * np.ones_like(count)
prop = count * 1. / nobs
k_sample = np.size(prop)
if value is None:
if k_sample == 1:
raise ValueError('value must be provided for a 1-sample test')
value = 0
if k_sample == 1:
diff = prop - value
elif k_sample == 2:
diff = prop[0] - prop[1] - value
else:
msg = 'more than two samples are not implemented yet'
raise NotImplementedError(msg)
p_pooled = np.sum(count) * 1. / np.sum(nobs)
nobs_fact = np.sum(1. / nobs)
if prop_var:
p_pooled = prop_var
var_ = p_pooled * (1 - p_pooled) * nobs_fact
std_diff = np.sqrt(var_)
from statsmodels.stats.weightstats import _zstat_generic2
return _zstat_generic2(diff, std_diff, alternative) | Test for proportions based on normal (z) test
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials. If this is array_like, then
the assumption is that this represents the number of successes for
each independent sample
nobs : {int, array_like}
the number of trials or observations, with the same length as
count.
value : float, array_like or None, optional
This is the value of the null hypothesis equal to the proportion in the
case of a one sample test. In the case of a two-sample test, the
null hypothesis is that prop[0] - prop[1] = value, where prop is the
proportion in the two samples. If not provided value = 0 and the null
is prop[0] = prop[1]
alternative : str in ['two-sided', 'smaller', 'larger']
The alternative hypothesis can be either two-sided or one of the one-
sided tests, smaller means that the alternative hypothesis is
``prop < value`` and larger means ``prop > value``. In the two sample
test, smaller means that the alternative hypothesis is ``p1 < p2`` and
larger means ``p1 > p2`` where ``p1`` is the proportion of the first
sample and ``p2`` of the second one.
prop_var : False or float in (0, 1)
If prop_var is false, then the variance of the proportion estimate is
calculated based on the sample proportion. Alternatively, a proportion
can be specified to calculate this variance. Common use case is to
use the proportion under the Null hypothesis to specify the variance
of the proportion estimate.
Returns
-------
zstat : float
test statistic for the z-test
p-value : float
p-value for the z-test
Examples
--------
>>> count = 5
>>> nobs = 83
>>> value = .05
>>> stat, pval = proportions_ztest(count, nobs, value)
>>> print('{0:0.3f}'.format(pval))
0.695
>>> import numpy as np
>>> from statsmodels.stats.proportion import proportions_ztest
>>> count = np.array([5, 12])
>>> nobs = np.array([83, 99])
>>> stat, pval = proportions_ztest(count, nobs)
>>> print('{0:0.3f}'.format(pval))
0.159
Notes
-----
This uses a simple normal test for proportions. It should be the same as
running the mean z-test on the data encoded 1 for event and 0 for no event
so that the sum corresponds to the count.
In the one and two sample cases with two-sided alternative, this test
produces the same p-value as ``proportions_chisquare``, since the
chisquare is the distribution of the square of a standard normal
distribution. | proportions_ztest | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def proportions_ztost(count, nobs, low, upp, prop_var='sample'):
"""
Equivalence test based on normal distribution
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials. If this is array_like, then
the assumption is that this represents the number of successes for
each independent sample
nobs : int
the number of trials or observations, with the same length as
count.
low, upp : float
equivalence interval low < prop1 - prop2 < upp
prop_var : str or float in (0, 1)
prop_var determines which proportion is used for the calculation
of the standard deviation of the proportion estimate
The available options for string are 'sample' (default), 'null' and
'limits'. If prop_var is a float, then it is used directly.
Returns
-------
pvalue : float
pvalue of the non-equivalence test
t1, pv1 : tuple of floats
test statistic and pvalue for lower threshold test
t2, pv2 : tuple of floats
test statistic and pvalue for upper threshold test
Notes
-----
checked only for 1 sample case
"""
if prop_var == 'limits':
prop_var_low = low
prop_var_upp = upp
elif prop_var == 'sample':
prop_var_low = prop_var_upp = False #ztest uses sample
elif prop_var == 'null':
prop_var_low = prop_var_upp = 0.5 * (low + upp)
elif np.isreal(prop_var):
prop_var_low = prop_var_upp = prop_var
tt1 = proportions_ztest(count, nobs, alternative='larger',
prop_var=prop_var_low, value=low)
tt2 = proportions_ztest(count, nobs, alternative='smaller',
prop_var=prop_var_upp, value=upp)
return np.maximum(tt1[1], tt2[1]), tt1, tt2, | Equivalence test based on normal distribution
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials. If this is array_like, then
the assumption is that this represents the number of successes for
each independent sample
nobs : int
the number of trials or observations, with the same length as
count.
low, upp : float
equivalence interval low < prop1 - prop2 < upp
prop_var : str or float in (0, 1)
prop_var determines which proportion is used for the calculation
of the standard deviation of the proportion estimate
The available options for string are 'sample' (default), 'null' and
'limits'. If prop_var is a float, then it is used directly.
Returns
-------
pvalue : float
pvalue of the non-equivalence test
t1, pv1 : tuple of floats
test statistic and pvalue for lower threshold test
t2, pv2 : tuple of floats
test statistic and pvalue for upper threshold test
Notes
-----
checked only for 1 sample case | proportions_ztost | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def proportions_chisquare(count, nobs, value=None):
"""
Test for proportions based on chisquare test
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials. If this is array_like, then
the assumption is that this represents the number of successes for
each independent sample
nobs : int
the number of trials or observations, with the same length as
count.
value : None or float or array_like
Returns
-------
chi2stat : float
test statistic for the chisquare test
p-value : float
p-value for the chisquare test
(table, expected)
table is a (k, 2) contingency table, ``expected`` is the corresponding
table of counts that are expected under independence with given
margins
Notes
-----
Recent version of scipy.stats have a chisquare test for independence in
contingency tables.
This function provides a similar interface to chisquare tests as
``prop.test`` in R, however without the option for Yates continuity
correction.
count can be the count for the number of events for a single proportion,
or the counts for several independent proportions. If value is given, then
all proportions are jointly tested against this value. If value is not
given and count and nobs are not scalar, then the null hypothesis is
that all samples have the same proportion.
"""
nobs = np.atleast_1d(nobs)
table, expected, n_rows = _table_proportion(count, nobs)
if value is not None:
expected = np.column_stack((nobs * value, nobs * (1 - value)))
ddof = n_rows - 1
else:
ddof = n_rows
#print table, expected
chi2stat, pval = stats.chisquare(table.ravel(), expected.ravel(),
ddof=ddof)
return chi2stat, pval, (table, expected) | Test for proportions based on chisquare test
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials. If this is array_like, then
the assumption is that this represents the number of successes for
each independent sample
nobs : int
the number of trials or observations, with the same length as
count.
value : None or float or array_like
Returns
-------
chi2stat : float
test statistic for the chisquare test
p-value : float
p-value for the chisquare test
(table, expected)
table is a (k, 2) contingency table, ``expected`` is the corresponding
table of counts that are expected under independence with given
margins
Notes
-----
Recent version of scipy.stats have a chisquare test for independence in
contingency tables.
This function provides a similar interface to chisquare tests as
``prop.test`` in R, however without the option for Yates continuity
correction.
count can be the count for the number of events for a single proportion,
or the counts for several independent proportions. If value is given, then
all proportions are jointly tested against this value. If value is not
given and count and nobs are not scalar, then the null hypothesis is
that all samples have the same proportion. | proportions_chisquare | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def proportions_chisquare_allpairs(count, nobs, multitest_method='hs'):
"""
Chisquare test of proportions for all pairs of k samples
Performs a chisquare test for proportions for all pairwise comparisons.
The alternative is two-sided
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials.
nobs : int
the number of trials or observations.
multitest_method : str
This chooses the method for the multiple testing p-value correction,
that is used as default in the results.
It can be any method that is available in ``multipletesting``.
The default is Holm-Sidak 'hs'.
Returns
-------
result : AllPairsResults instance
The returned results instance has several statistics, such as p-values,
attached, and additional methods for using a non-default
``multitest_method``.
Notes
-----
Yates continuity correction is not available.
"""
#all_pairs = lmap(list, lzip(*np.triu_indices(4, 1)))
all_pairs = lzip(*np.triu_indices(len(count), 1))
pvals = [proportions_chisquare(count[list(pair)], nobs[list(pair)])[1]
for pair in all_pairs]
return AllPairsResults(pvals, all_pairs, multitest_method=multitest_method) | Chisquare test of proportions for all pairs of k samples
Performs a chisquare test for proportions for all pairwise comparisons.
The alternative is two-sided
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials.
nobs : int
the number of trials or observations.
multitest_method : str
This chooses the method for the multiple testing p-value correction,
that is used as default in the results.
It can be any method that is available in ``multipletesting``.
The default is Holm-Sidak 'hs'.
Returns
-------
result : AllPairsResults instance
The returned results instance has several statistics, such as p-values,
attached, and additional methods for using a non-default
``multitest_method``.
Notes
-----
Yates continuity correction is not available. | proportions_chisquare_allpairs | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def proportions_chisquare_pairscontrol(count, nobs, value=None,
multitest_method='hs', alternative='two-sided'):
"""
Chisquare test of proportions for pairs of k samples compared to control
Performs a chisquare test for proportions for pairwise comparisons with a
control (Dunnet's test). The control is assumed to be the first element
of ``count`` and ``nobs``. The alternative is two-sided, larger or
smaller.
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials.
nobs : int
the number of trials or observations.
multitest_method : str
This chooses the method for the multiple testing p-value correction,
that is used as default in the results.
It can be any method that is available in ``multipletesting``.
The default is Holm-Sidak 'hs'.
alternative : str in ['two-sided', 'smaller', 'larger']
alternative hypothesis, which can be two-sided or either one of the
one-sided tests.
Returns
-------
result : AllPairsResults instance
The returned results instance has several statistics, such as p-values,
attached, and additional methods for using a non-default
``multitest_method``.
Notes
-----
Yates continuity correction is not available.
``value`` and ``alternative`` options are not yet implemented.
"""
if (value is not None) or (alternative not in ['two-sided', '2s']):
raise NotImplementedError
#all_pairs = lmap(list, lzip(*np.triu_indices(4, 1)))
all_pairs = [(0, k) for k in range(1, len(count))]
pvals = [proportions_chisquare(count[list(pair)], nobs[list(pair)],
#alternative=alternative)[1]
)[1]
for pair in all_pairs]
return AllPairsResults(pvals, all_pairs, multitest_method=multitest_method) | Chisquare test of proportions for pairs of k samples compared to control
Performs a chisquare test for proportions for pairwise comparisons with a
control (Dunnet's test). The control is assumed to be the first element
of ``count`` and ``nobs``. The alternative is two-sided, larger or
smaller.
Parameters
----------
count : {int, array_like}
the number of successes in nobs trials.
nobs : int
the number of trials or observations.
multitest_method : str
This chooses the method for the multiple testing p-value correction,
that is used as default in the results.
It can be any method that is available in ``multipletesting``.
The default is Holm-Sidak 'hs'.
alternative : str in ['two-sided', 'smaller', 'larger']
alternative hypothesis, which can be two-sided or either one of the
one-sided tests.
Returns
-------
result : AllPairsResults instance
The returned results instance has several statistics, such as p-values,
attached, and additional methods for using a non-default
``multitest_method``.
Notes
-----
Yates continuity correction is not available.
``value`` and ``alternative`` options are not yet implemented. | proportions_chisquare_pairscontrol | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def confint_proportions_2indep(count1, nobs1, count2, nobs2, method=None,
compare='diff', alpha=0.05, correction=True):
"""
Confidence intervals for comparing two independent proportions.
This assumes that we have two independent binomial samples.
Parameters
----------
count1, nobs1 : float
Count and sample size for first sample.
count2, nobs2 : float
Count and sample size for the second sample.
method : str
Method for computing confidence interval. If method is None, then a
default method is used. The default might change as more methods are
added.
diff:
- 'wald',
- 'agresti-caffo'
- 'newcomb' (default)
- 'score'
ratio:
- 'log'
- 'log-adjusted' (default)
- 'score'
odds-ratio:
- 'logit'
- 'logit-adjusted' (default)
- 'score'
compare : string in ['diff', 'ratio' 'odds-ratio']
If compare is diff, then the confidence interval is for diff = p1 - p2.
If compare is ratio, then the confidence interval is for the risk ratio
defined by ratio = p1 / p2.
If compare is odds-ratio, then the confidence interval is for the
odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2).
alpha : float
Significance level for the confidence interval, default is 0.05.
The nominal coverage probability is 1 - alpha.
Returns
-------
low, upp
See Also
--------
test_proportions_2indep
tost_proportions_2indep
Notes
-----
Status: experimental, API and defaults might still change.
more ``methods`` will be added.
References
----------
.. [1] Fagerland, Morten W., Stian Lydersen, and Petter Laake. 2015.
“Recommended Confidence Intervals for Two Independent Binomial
Proportions.” Statistical Methods in Medical Research 24 (2): 224–54.
https://doi.org/10.1177/0962280211415469.
.. [2] Koopman, P. A. R. 1984. “Confidence Intervals for the Ratio of Two
Binomial Proportions.” Biometrics 40 (2): 513–17.
https://doi.org/10.2307/2531405.
.. [3] Miettinen, Olli, and Markku Nurminen. "Comparative analysis of two
rates." Statistics in medicine 4, no. 2 (1985): 213-226.
.. [4] Newcombe, Robert G. 1998. “Interval Estimation for the Difference
between Independent Proportions: Comparison of Eleven Methods.”
Statistics in Medicine 17 (8): 873–90.
https://doi.org/10.1002/(SICI)1097-0258(19980430)17:8<873::AID-
SIM779>3.0.CO;2-I.
.. [5] Newcombe, Robert G., and Markku M. Nurminen. 2011. “In Defence of
Score Intervals for Proportions and Their Differences.” Communications
in Statistics - Theory and Methods 40 (7): 1271–82.
https://doi.org/10.1080/03610920903576580.
"""
method_default = {'diff': 'newcomb',
'ratio': 'log-adjusted',
'odds-ratio': 'logit-adjusted'}
# normalize compare name
if compare.lower() == 'or':
compare = 'odds-ratio'
if method is None:
method = method_default[compare]
method = method.lower()
if method.startswith('agr'):
method = 'agresti-caffo'
p1 = count1 / nobs1
p2 = count2 / nobs2
diff = p1 - p2
addone = 1 if method == 'agresti-caffo' else 0
if compare == 'diff':
if method in ['wald', 'agresti-caffo']:
count1_, nobs1_ = count1 + addone, nobs1 + 2 * addone
count2_, nobs2_ = count2 + addone, nobs2 + 2 * addone
p1_ = count1_ / nobs1_
p2_ = count2_ / nobs2_
diff_ = p1_ - p2_
var = p1_ * (1 - p1_) / nobs1_ + p2_ * (1 - p2_) / nobs2_
z = stats.norm.isf(alpha / 2)
d_wald = z * np.sqrt(var)
low = diff_ - d_wald
upp = diff_ + d_wald
elif method.startswith('newcomb'):
low1, upp1 = proportion_confint(count1, nobs1,
method='wilson', alpha=alpha)
low2, upp2 = proportion_confint(count2, nobs2,
method='wilson', alpha=alpha)
d_low = np.sqrt((p1 - low1)**2 + (upp2 - p2)**2)
d_upp = np.sqrt((p2 - low2)**2 + (upp1 - p1)**2)
low = diff - d_low
upp = diff + d_upp
elif method == "score":
low, upp = _score_confint_inversion(count1, nobs1, count2, nobs2,
compare=compare, alpha=alpha,
correction=correction)
else:
raise ValueError('method not recognized')
elif compare == 'ratio':
# ratio = p1 / p2
if method in ['log', 'log-adjusted']:
addhalf = 0.5 if method == 'log-adjusted' else 0
count1_, nobs1_ = count1 + addhalf, nobs1 + addhalf
count2_, nobs2_ = count2 + addhalf, nobs2 + addhalf
p1_ = count1_ / nobs1_
p2_ = count2_ / nobs2_
ratio_ = p1_ / p2_
var = (1 / count1_) - 1 / nobs1_ + 1 / count2_ - 1 / nobs2_
z = stats.norm.isf(alpha / 2)
d_log = z * np.sqrt(var)
low = np.exp(np.log(ratio_) - d_log)
upp = np.exp(np.log(ratio_) + d_log)
elif method == 'score':
res = _confint_riskratio_koopman(count1, nobs1, count2, nobs2,
alpha=alpha,
correction=correction)
low, upp = res.confint
else:
raise ValueError('method not recognized')
elif compare == 'odds-ratio':
# odds_ratio = p1 / (1 - p1) / p2 * (1 - p2)
if method in ['logit', 'logit-adjusted', 'logit-smoothed']:
if method in ['logit-smoothed']:
adjusted = _shrink_prob(count1, nobs1, count2, nobs2,
shrink_factor=2, return_corr=False)[0]
count1_, nobs1_, count2_, nobs2_ = adjusted
else:
addhalf = 0.5 if method == 'logit-adjusted' else 0
count1_, nobs1_ = count1 + addhalf, nobs1 + 2 * addhalf
count2_, nobs2_ = count2 + addhalf, nobs2 + 2 * addhalf
p1_ = count1_ / nobs1_
p2_ = count2_ / nobs2_
odds_ratio_ = p1_ / (1 - p1_) / p2_ * (1 - p2_)
var = (1 / count1_ + 1 / (nobs1_ - count1_) +
1 / count2_ + 1 / (nobs2_ - count2_))
z = stats.norm.isf(alpha / 2)
d_log = z * np.sqrt(var)
low = np.exp(np.log(odds_ratio_) - d_log)
upp = np.exp(np.log(odds_ratio_) + d_log)
elif method == "score":
low, upp = _score_confint_inversion(count1, nobs1, count2, nobs2,
compare=compare, alpha=alpha,
correction=correction)
else:
raise ValueError('method not recognized')
else:
raise ValueError('compare not recognized')
return low, upp | Confidence intervals for comparing two independent proportions.
This assumes that we have two independent binomial samples.
Parameters
----------
count1, nobs1 : float
Count and sample size for first sample.
count2, nobs2 : float
Count and sample size for the second sample.
method : str
Method for computing confidence interval. If method is None, then a
default method is used. The default might change as more methods are
added.
diff:
- 'wald',
- 'agresti-caffo'
- 'newcomb' (default)
- 'score'
ratio:
- 'log'
- 'log-adjusted' (default)
- 'score'
odds-ratio:
- 'logit'
- 'logit-adjusted' (default)
- 'score'
compare : string in ['diff', 'ratio' 'odds-ratio']
If compare is diff, then the confidence interval is for diff = p1 - p2.
If compare is ratio, then the confidence interval is for the risk ratio
defined by ratio = p1 / p2.
If compare is odds-ratio, then the confidence interval is for the
odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2).
alpha : float
Significance level for the confidence interval, default is 0.05.
The nominal coverage probability is 1 - alpha.
Returns
-------
low, upp
See Also
--------
test_proportions_2indep
tost_proportions_2indep
Notes
-----
Status: experimental, API and defaults might still change.
more ``methods`` will be added.
References
----------
.. [1] Fagerland, Morten W., Stian Lydersen, and Petter Laake. 2015.
“Recommended Confidence Intervals for Two Independent Binomial
Proportions.” Statistical Methods in Medical Research 24 (2): 224–54.
https://doi.org/10.1177/0962280211415469.
.. [2] Koopman, P. A. R. 1984. “Confidence Intervals for the Ratio of Two
Binomial Proportions.” Biometrics 40 (2): 513–17.
https://doi.org/10.2307/2531405.
.. [3] Miettinen, Olli, and Markku Nurminen. "Comparative analysis of two
rates." Statistics in medicine 4, no. 2 (1985): 213-226.
.. [4] Newcombe, Robert G. 1998. “Interval Estimation for the Difference
between Independent Proportions: Comparison of Eleven Methods.”
Statistics in Medicine 17 (8): 873–90.
https://doi.org/10.1002/(SICI)1097-0258(19980430)17:8<873::AID-
SIM779>3.0.CO;2-I.
.. [5] Newcombe, Robert G., and Markku M. Nurminen. 2011. “In Defence of
Score Intervals for Proportions and Their Differences.” Communications
in Statistics - Theory and Methods 40 (7): 1271–82.
https://doi.org/10.1080/03610920903576580. | confint_proportions_2indep | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def _shrink_prob(count1, nobs1, count2, nobs2, shrink_factor=2,
return_corr=True):
"""
Shrink observed counts towards independence
Helper function for 'logit-smoothed' inference for the odds-ratio of two
independent proportions.
Parameters
----------
count1, nobs1 : float or int
count and sample size for first sample
count2, nobs2 : float or int
count and sample size for the second sample
shrink_factor : float
This corresponds to the number of observations that are added in total
proportional to the probabilities under independence.
return_corr : bool
If true, then only the correction term is returned
If false, then the corrected counts, i.e. original counts plus
correction term, are returned.
Returns
-------
count1_corr, nobs1_corr, count2_corr, nobs2_corr : float
correction or corrected counts
prob_indep :
TODO/Warning : this will change most likely
probabilities under independence, only returned if return_corr is
false.
"""
vectorized = any(np.size(i) > 1 for i in [count1, nobs1, count2, nobs2])
if vectorized:
raise ValueError("function is not vectorized")
nobs_col = np.array([count1 + count2, nobs1 - count1 + nobs2 - count2])
nobs_row = np.array([nobs1, nobs2])
nobs = nobs1 + nobs2
prob_indep = (nobs_col * nobs_row[:, None]) / nobs**2
corr = shrink_factor * prob_indep
if return_corr:
return (corr[0, 0], corr[0].sum(), corr[1, 0], corr[1].sum())
else:
return (count1 + corr[0, 0], nobs1 + corr[0].sum(),
count2 + corr[1, 0], nobs2 + corr[1].sum()), prob_indep | Shrink observed counts towards independence
Helper function for 'logit-smoothed' inference for the odds-ratio of two
independent proportions.
Parameters
----------
count1, nobs1 : float or int
count and sample size for first sample
count2, nobs2 : float or int
count and sample size for the second sample
shrink_factor : float
This corresponds to the number of observations that are added in total
proportional to the probabilities under independence.
return_corr : bool
If true, then only the correction term is returned
If false, then the corrected counts, i.e. original counts plus
correction term, are returned.
Returns
-------
count1_corr, nobs1_corr, count2_corr, nobs2_corr : float
correction or corrected counts
prob_indep :
TODO/Warning : this will change most likely
probabilities under independence, only returned if return_corr is
false. | _shrink_prob | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def score_test_proportions_2indep(count1, nobs1, count2, nobs2, value=None,
compare='diff', alternative='two-sided',
correction=True, return_results=True):
"""
Score test for two independent proportions
This uses the constrained estimate of the proportions to compute
the variance under the Null hypothesis.
Parameters
----------
count1, nobs1 :
count and sample size for first sample
count2, nobs2 :
count and sample size for the second sample
value : float
diff, ratio or odds-ratio under the null hypothesis. If value is None,
then equality of proportions under the Null is assumed,
i.e. value=0 for 'diff' or value=1 for either rate or odds-ratio.
compare : string in ['diff', 'ratio' 'odds-ratio']
If compare is diff, then the confidence interval is for diff = p1 - p2.
If compare is ratio, then the confidence interval is for the risk ratio
defined by ratio = p1 / p2.
If compare is odds-ratio, then the confidence interval is for the
odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2)
return_results : bool
If true, then a results instance with extra information is returned,
otherwise a tuple with statistic and pvalue is returned.
Returns
-------
results : results instance or tuple
If return_results is True, then a results instance with the
information in attributes is returned.
If return_results is False, then only ``statistic`` and ``pvalue``
are returned.
statistic : float
test statistic asymptotically normal distributed N(0, 1)
pvalue : float
p-value based on normal distribution
other attributes :
additional information about the hypothesis test
Notes
-----
Status: experimental, the type or extra information in the return might
change.
"""
value_default = 0 if compare == 'diff' else 1
if value is None:
# TODO: odds ratio does not work if value=1
value = value_default
nobs = nobs1 + nobs2
count = count1 + count2
p1 = count1 / nobs1
p2 = count2 / nobs2
if value == value_default:
# use pooled estimator if equality test
# shortcut, but required for odds ratio
prop0 = prop1 = count / nobs
# this uses index 0 from Miettinen Nurminned 1985
count0, nobs0 = count2, nobs2
p0 = p2
if compare == 'diff':
diff = value # hypothesis value
if diff != 0:
tmp3 = nobs
tmp2 = (nobs1 + 2 * nobs0) * diff - nobs - count
tmp1 = (count0 * diff - nobs - 2 * count0) * diff + count
tmp0 = count0 * diff * (1 - diff)
q = ((tmp2 / (3 * tmp3))**3 - tmp1 * tmp2 / (6 * tmp3**2) +
tmp0 / (2 * tmp3))
p = np.sign(q) * np.sqrt((tmp2 / (3 * tmp3))**2 -
tmp1 / (3 * tmp3))
a = (np.pi + np.arccos(q / p**3)) / 3
prop0 = 2 * p * np.cos(a) - tmp2 / (3 * tmp3)
prop1 = prop0 + diff
var = prop1 * (1 - prop1) / nobs1 + prop0 * (1 - prop0) / nobs0
if correction:
var *= nobs / (nobs - 1)
diff_stat = (p1 - p0 - diff)
elif compare == 'ratio':
# risk ratio
ratio = value
if ratio != 1:
a = nobs * ratio
b = -(nobs1 * ratio + count1 + nobs2 + count0 * ratio)
c = count
prop0 = (-b - np.sqrt(b**2 - 4 * a * c)) / (2 * a)
prop1 = prop0 * ratio
var = (prop1 * (1 - prop1) / nobs1 +
ratio**2 * prop0 * (1 - prop0) / nobs0)
if correction:
var *= nobs / (nobs - 1)
# NCSS looks incorrect for var, but it is what should be reported
# diff_stat = (p1 / p0 - ratio) # NCSS/PASS
diff_stat = (p1 - ratio * p0) # Miettinen Nurminen
elif compare in ['or', 'odds-ratio']:
# odds ratio
oratio = value
if oratio != 1:
# Note the constraint estimator does not handle odds-ratio = 1
a = nobs0 * (oratio - 1)
b = nobs1 * oratio + nobs0 - count * (oratio - 1)
c = -count
prop0 = (-b + np.sqrt(b**2 - 4 * a * c)) / (2 * a)
prop1 = prop0 * oratio / (1 + prop0 * (oratio - 1))
# try to avoid 0 and 1 proportions,
# those raise Zero Division Runtime Warnings
eps = 1e-10
prop0 = np.clip(prop0, eps, 1 - eps)
prop1 = np.clip(prop1, eps, 1 - eps)
var = (1 / (prop1 * (1 - prop1) * nobs1) +
1 / (prop0 * (1 - prop0) * nobs0))
if correction:
var *= nobs / (nobs - 1)
diff_stat = ((p1 - prop1) / (prop1 * (1 - prop1)) -
(p0 - prop0) / (prop0 * (1 - prop0)))
statistic, pvalue = _zstat_generic2(diff_stat, np.sqrt(var),
alternative=alternative)
if return_results:
res = HolderTuple(statistic=statistic,
pvalue=pvalue,
compare=compare,
method='score',
variance=var,
alternative=alternative,
prop1_null=prop1,
prop2_null=prop0,
)
return res
else:
return statistic, pvalue | Score test for two independent proportions
This uses the constrained estimate of the proportions to compute
the variance under the Null hypothesis.
Parameters
----------
count1, nobs1 :
count and sample size for first sample
count2, nobs2 :
count and sample size for the second sample
value : float
diff, ratio or odds-ratio under the null hypothesis. If value is None,
then equality of proportions under the Null is assumed,
i.e. value=0 for 'diff' or value=1 for either rate or odds-ratio.
compare : string in ['diff', 'ratio' 'odds-ratio']
If compare is diff, then the confidence interval is for diff = p1 - p2.
If compare is ratio, then the confidence interval is for the risk ratio
defined by ratio = p1 / p2.
If compare is odds-ratio, then the confidence interval is for the
odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2)
return_results : bool
If true, then a results instance with extra information is returned,
otherwise a tuple with statistic and pvalue is returned.
Returns
-------
results : results instance or tuple
If return_results is True, then a results instance with the
information in attributes is returned.
If return_results is False, then only ``statistic`` and ``pvalue``
are returned.
statistic : float
test statistic asymptotically normal distributed N(0, 1)
pvalue : float
p-value based on normal distribution
other attributes :
additional information about the hypothesis test
Notes
-----
Status: experimental, the type or extra information in the return might
change. | score_test_proportions_2indep | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def test_proportions_2indep(count1, nobs1, count2, nobs2, value=None,
method=None, compare='diff',
alternative='two-sided', correction=True,
return_results=True):
"""
Hypothesis test for comparing two independent proportions
This assumes that we have two independent binomial samples.
The Null and alternative hypothesis are
for compare = 'diff'
- H0: prop1 - prop2 - value = 0
- H1: prop1 - prop2 - value != 0 if alternative = 'two-sided'
- H1: prop1 - prop2 - value > 0 if alternative = 'larger'
- H1: prop1 - prop2 - value < 0 if alternative = 'smaller'
for compare = 'ratio'
- H0: prop1 / prop2 - value = 0
- H1: prop1 / prop2 - value != 0 if alternative = 'two-sided'
- H1: prop1 / prop2 - value > 0 if alternative = 'larger'
- H1: prop1 / prop2 - value < 0 if alternative = 'smaller'
for compare = 'odds-ratio'
- H0: or - value = 0
- H1: or - value != 0 if alternative = 'two-sided'
- H1: or - value > 0 if alternative = 'larger'
- H1: or - value < 0 if alternative = 'smaller'
where odds-ratio or = prop1 / (1 - prop1) / (prop2 / (1 - prop2))
Parameters
----------
count1 : int
Count for first sample.
nobs1 : int
Sample size for first sample.
count2 : int
Count for the second sample.
nobs2 : int
Sample size for the second sample.
value : float
Value of the difference, risk ratio or odds ratio of 2 independent
proportions under the null hypothesis.
Default is equal proportions, 0 for diff and 1 for risk-ratio and for
odds-ratio.
method : string
Method for computing the hypothesis test. If method is None, then a
default method is used. The default might change as more methods are
added.
diff:
- 'wald',
- 'agresti-caffo'
- 'score' if correction is True, then this uses the degrees of freedom
correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985
ratio:
- 'log': wald test using log transformation
- 'log-adjusted': wald test using log transformation,
adds 0.5 to counts
- 'score': if correction is True, then this uses the degrees of freedom
correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985
odds-ratio:
- 'logit': wald test using logit transformation
- 'logit-adjusted': wald test using logit transformation,
adds 0.5 to counts
- 'logit-smoothed': wald test using logit transformation, biases
cell counts towards independence by adding two observations in
total.
- 'score' if correction is True, then this uses the degrees of freedom
correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985
compare : {'diff', 'ratio' 'odds-ratio'}
If compare is `diff`, then the hypothesis test is for the risk
difference diff = p1 - p2.
If compare is `ratio`, then the hypothesis test is for the
risk ratio defined by ratio = p1 / p2.
If compare is `odds-ratio`, then the hypothesis test is for the
odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2)
alternative : {'two-sided', 'smaller', 'larger'}
alternative hypothesis, which can be two-sided or either one of the
one-sided tests.
correction : bool
If correction is True (default), then the Miettinen and Nurminen
small sample correction to the variance nobs / (nobs - 1) is used.
Applies only if method='score'.
return_results : bool
If true, then a results instance with extra information is returned,
otherwise a tuple with statistic and pvalue is returned.
Returns
-------
results : results instance or tuple
If return_results is True, then a results instance with the
information in attributes is returned.
If return_results is False, then only ``statistic`` and ``pvalue``
are returned.
statistic : float
test statistic asymptotically normal distributed N(0, 1)
pvalue : float
p-value based on normal distribution
other attributes :
additional information about the hypothesis test
See Also
--------
tost_proportions_2indep
confint_proportions_2indep
Notes
-----
Status: experimental, API and defaults might still change.
More ``methods`` will be added.
The current default methods are
- 'diff': 'agresti-caffo',
- 'ratio': 'log-adjusted',
- 'odds-ratio': 'logit-adjusted'
"""
method_default = {'diff': 'agresti-caffo',
'ratio': 'log-adjusted',
'odds-ratio': 'logit-adjusted'}
# normalize compare name
if compare.lower() == 'or':
compare = 'odds-ratio'
if method is None:
method = method_default[compare]
method = method.lower()
if method.startswith('agr'):
method = 'agresti-caffo'
if value is None:
# TODO: odds ratio does not work if value=1 for score test
value = 0 if compare == 'diff' else 1
count1, nobs1, count2, nobs2 = map(np.asarray,
[count1, nobs1, count2, nobs2])
p1 = count1 / nobs1
p2 = count2 / nobs2
diff = p1 - p2
ratio = p1 / p2
odds_ratio = p1 / (1 - p1) / p2 * (1 - p2)
res = None
if compare == 'diff':
if method in ['wald', 'agresti-caffo']:
addone = 1 if method == 'agresti-caffo' else 0
count1_, nobs1_ = count1 + addone, nobs1 + 2 * addone
count2_, nobs2_ = count2 + addone, nobs2 + 2 * addone
p1_ = count1_ / nobs1_
p2_ = count2_ / nobs2_
diff_stat = p1_ - p2_ - value
var = p1_ * (1 - p1_) / nobs1_ + p2_ * (1 - p2_) / nobs2_
statistic = diff_stat / np.sqrt(var)
distr = 'normal'
elif method.startswith('newcomb'):
msg = 'newcomb not available for hypothesis test'
raise NotImplementedError(msg)
elif method == 'score':
# Note score part is the same call for all compare
res = score_test_proportions_2indep(count1, nobs1, count2, nobs2,
value=value, compare=compare,
alternative=alternative,
correction=correction,
return_results=return_results)
if return_results is False:
statistic, pvalue = res[:2]
distr = 'normal'
# TODO/Note score_test_proportion_2samp returns statistic and
# not diff_stat
diff_stat = None
else:
raise ValueError('method not recognized')
elif compare == 'ratio':
if method in ['log', 'log-adjusted']:
addhalf = 0.5 if method == 'log-adjusted' else 0
count1_, nobs1_ = count1 + addhalf, nobs1 + addhalf
count2_, nobs2_ = count2 + addhalf, nobs2 + addhalf
p1_ = count1_ / nobs1_
p2_ = count2_ / nobs2_
ratio_ = p1_ / p2_
var = (1 / count1_) - 1 / nobs1_ + 1 / count2_ - 1 / nobs2_
diff_stat = np.log(ratio_) - np.log(value)
statistic = diff_stat / np.sqrt(var)
distr = 'normal'
elif method == 'score':
res = score_test_proportions_2indep(count1, nobs1, count2, nobs2,
value=value, compare=compare,
alternative=alternative,
correction=correction,
return_results=return_results)
if return_results is False:
statistic, pvalue = res[:2]
distr = 'normal'
diff_stat = None
else:
raise ValueError('method not recognized')
elif compare == "odds-ratio":
if method in ['logit', 'logit-adjusted', 'logit-smoothed']:
if method in ['logit-smoothed']:
adjusted = _shrink_prob(count1, nobs1, count2, nobs2,
shrink_factor=2, return_corr=False)[0]
count1_, nobs1_, count2_, nobs2_ = adjusted
else:
addhalf = 0.5 if method == 'logit-adjusted' else 0
count1_, nobs1_ = count1 + addhalf, nobs1 + 2 * addhalf
count2_, nobs2_ = count2 + addhalf, nobs2 + 2 * addhalf
p1_ = count1_ / nobs1_
p2_ = count2_ / nobs2_
odds_ratio_ = p1_ / (1 - p1_) / p2_ * (1 - p2_)
var = (1 / count1_ + 1 / (nobs1_ - count1_) +
1 / count2_ + 1 / (nobs2_ - count2_))
diff_stat = np.log(odds_ratio_) - np.log(value)
statistic = diff_stat / np.sqrt(var)
distr = 'normal'
elif method == 'score':
res = score_test_proportions_2indep(count1, nobs1, count2, nobs2,
value=value, compare=compare,
alternative=alternative,
correction=correction,
return_results=return_results)
if return_results is False:
statistic, pvalue = res[:2]
distr = 'normal'
diff_stat = None
else:
raise ValueError('method "%s" not recognized' % method)
else:
raise ValueError('compare "%s" not recognized' % compare)
if distr == 'normal' and diff_stat is not None:
statistic, pvalue = _zstat_generic2(diff_stat, np.sqrt(var),
alternative=alternative)
if return_results:
if res is None:
res = HolderTuple(statistic=statistic,
pvalue=pvalue,
compare=compare,
method=method,
diff=diff,
ratio=ratio,
odds_ratio=odds_ratio,
variance=var,
alternative=alternative,
value=value,
)
else:
# we already have a return result from score test
# add missing attributes
res.diff = diff
res.ratio = ratio
res.odds_ratio = odds_ratio
res.value = value
return res
else:
return statistic, pvalue | Hypothesis test for comparing two independent proportions
This assumes that we have two independent binomial samples.
The Null and alternative hypothesis are
for compare = 'diff'
- H0: prop1 - prop2 - value = 0
- H1: prop1 - prop2 - value != 0 if alternative = 'two-sided'
- H1: prop1 - prop2 - value > 0 if alternative = 'larger'
- H1: prop1 - prop2 - value < 0 if alternative = 'smaller'
for compare = 'ratio'
- H0: prop1 / prop2 - value = 0
- H1: prop1 / prop2 - value != 0 if alternative = 'two-sided'
- H1: prop1 / prop2 - value > 0 if alternative = 'larger'
- H1: prop1 / prop2 - value < 0 if alternative = 'smaller'
for compare = 'odds-ratio'
- H0: or - value = 0
- H1: or - value != 0 if alternative = 'two-sided'
- H1: or - value > 0 if alternative = 'larger'
- H1: or - value < 0 if alternative = 'smaller'
where odds-ratio or = prop1 / (1 - prop1) / (prop2 / (1 - prop2))
Parameters
----------
count1 : int
Count for first sample.
nobs1 : int
Sample size for first sample.
count2 : int
Count for the second sample.
nobs2 : int
Sample size for the second sample.
value : float
Value of the difference, risk ratio or odds ratio of 2 independent
proportions under the null hypothesis.
Default is equal proportions, 0 for diff and 1 for risk-ratio and for
odds-ratio.
method : string
Method for computing the hypothesis test. If method is None, then a
default method is used. The default might change as more methods are
added.
diff:
- 'wald',
- 'agresti-caffo'
- 'score' if correction is True, then this uses the degrees of freedom
correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985
ratio:
- 'log': wald test using log transformation
- 'log-adjusted': wald test using log transformation,
adds 0.5 to counts
- 'score': if correction is True, then this uses the degrees of freedom
correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985
odds-ratio:
- 'logit': wald test using logit transformation
- 'logit-adjusted': wald test using logit transformation,
adds 0.5 to counts
- 'logit-smoothed': wald test using logit transformation, biases
cell counts towards independence by adding two observations in
total.
- 'score' if correction is True, then this uses the degrees of freedom
correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985
compare : {'diff', 'ratio' 'odds-ratio'}
If compare is `diff`, then the hypothesis test is for the risk
difference diff = p1 - p2.
If compare is `ratio`, then the hypothesis test is for the
risk ratio defined by ratio = p1 / p2.
If compare is `odds-ratio`, then the hypothesis test is for the
odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2)
alternative : {'two-sided', 'smaller', 'larger'}
alternative hypothesis, which can be two-sided or either one of the
one-sided tests.
correction : bool
If correction is True (default), then the Miettinen and Nurminen
small sample correction to the variance nobs / (nobs - 1) is used.
Applies only if method='score'.
return_results : bool
If true, then a results instance with extra information is returned,
otherwise a tuple with statistic and pvalue is returned.
Returns
-------
results : results instance or tuple
If return_results is True, then a results instance with the
information in attributes is returned.
If return_results is False, then only ``statistic`` and ``pvalue``
are returned.
statistic : float
test statistic asymptotically normal distributed N(0, 1)
pvalue : float
p-value based on normal distribution
other attributes :
additional information about the hypothesis test
See Also
--------
tost_proportions_2indep
confint_proportions_2indep
Notes
-----
Status: experimental, API and defaults might still change.
More ``methods`` will be added.
The current default methods are
- 'diff': 'agresti-caffo',
- 'ratio': 'log-adjusted',
- 'odds-ratio': 'logit-adjusted' | test_proportions_2indep | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def tost_proportions_2indep(count1, nobs1, count2, nobs2, low, upp,
method=None, compare='diff', correction=True):
"""
Equivalence test based on two one-sided `test_proportions_2indep`
This assumes that we have two independent binomial samples.
The Null and alternative hypothesis for equivalence testing are
for compare = 'diff'
- H0: prop1 - prop2 <= low or upp <= prop1 - prop2
- H1: low < prop1 - prop2 < upp
for compare = 'ratio'
- H0: prop1 / prop2 <= low or upp <= prop1 / prop2
- H1: low < prop1 / prop2 < upp
for compare = 'odds-ratio'
- H0: or <= low or upp <= or
- H1: low < or < upp
where odds-ratio or = prop1 / (1 - prop1) / (prop2 / (1 - prop2))
Parameters
----------
count1, nobs1 :
count and sample size for first sample
count2, nobs2 :
count and sample size for the second sample
low, upp :
equivalence margin for diff, risk ratio or odds ratio
method : string
method for computing the hypothesis test. If method is None, then a
default method is used. The default might change as more methods are
added.
diff:
- 'wald',
- 'agresti-caffo'
- 'score' if correction is True, then this uses the degrees of freedom
correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985.
ratio:
- 'log': wald test using log transformation
- 'log-adjusted': wald test using log transformation,
adds 0.5 to counts
- 'score' if correction is True, then this uses the degrees of freedom
correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985.
odds-ratio:
- 'logit': wald test using logit transformation
- 'logit-adjusted': : wald test using logit transformation,
adds 0.5 to counts
- 'logit-smoothed': : wald test using logit transformation, biases
cell counts towards independence by adding two observations in
total.
- 'score' if correction is True, then this uses the degrees of freedom
correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985
compare : string in ['diff', 'ratio' 'odds-ratio']
If compare is `diff`, then the hypothesis test is for
diff = p1 - p2.
If compare is `ratio`, then the hypothesis test is for the
risk ratio defined by ratio = p1 / p2.
If compare is `odds-ratio`, then the hypothesis test is for the
odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2).
correction : bool
If correction is True (default), then the Miettinen and Nurminen
small sample correction to the variance nobs / (nobs - 1) is used.
Applies only if method='score'.
Returns
-------
pvalue : float
p-value is the max of the pvalues of the two one-sided tests
t1 : test results
results instance for one-sided hypothesis at the lower margin
t1 : test results
results instance for one-sided hypothesis at the upper margin
See Also
--------
test_proportions_2indep
confint_proportions_2indep
Notes
-----
Status: experimental, API and defaults might still change.
The TOST equivalence test delegates to `test_proportions_2indep` and has
the same method and comparison options.
"""
tt1 = test_proportions_2indep(count1, nobs1, count2, nobs2, value=low,
method=method, compare=compare,
alternative='larger',
correction=correction,
return_results=True)
tt2 = test_proportions_2indep(count1, nobs1, count2, nobs2, value=upp,
method=method, compare=compare,
alternative='smaller',
correction=correction,
return_results=True)
# idx_max = 1 if t1.pvalue < t2.pvalue else 0
idx_max = np.asarray(tt1.pvalue < tt2.pvalue, int)
statistic = np.choose(idx_max, [tt1.statistic, tt2.statistic])
pvalue = np.choose(idx_max, [tt1.pvalue, tt2.pvalue])
res = HolderTuple(statistic=statistic,
pvalue=pvalue,
compare=compare,
method=method,
results_larger=tt1,
results_smaller=tt2,
title="Equivalence test for 2 independent proportions"
)
return res | Equivalence test based on two one-sided `test_proportions_2indep`
This assumes that we have two independent binomial samples.
The Null and alternative hypothesis for equivalence testing are
for compare = 'diff'
- H0: prop1 - prop2 <= low or upp <= prop1 - prop2
- H1: low < prop1 - prop2 < upp
for compare = 'ratio'
- H0: prop1 / prop2 <= low or upp <= prop1 / prop2
- H1: low < prop1 / prop2 < upp
for compare = 'odds-ratio'
- H0: or <= low or upp <= or
- H1: low < or < upp
where odds-ratio or = prop1 / (1 - prop1) / (prop2 / (1 - prop2))
Parameters
----------
count1, nobs1 :
count and sample size for first sample
count2, nobs2 :
count and sample size for the second sample
low, upp :
equivalence margin for diff, risk ratio or odds ratio
method : string
method for computing the hypothesis test. If method is None, then a
default method is used. The default might change as more methods are
added.
diff:
- 'wald',
- 'agresti-caffo'
- 'score' if correction is True, then this uses the degrees of freedom
correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985.
ratio:
- 'log': wald test using log transformation
- 'log-adjusted': wald test using log transformation,
adds 0.5 to counts
- 'score' if correction is True, then this uses the degrees of freedom
correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985.
odds-ratio:
- 'logit': wald test using logit transformation
- 'logit-adjusted': : wald test using logit transformation,
adds 0.5 to counts
- 'logit-smoothed': : wald test using logit transformation, biases
cell counts towards independence by adding two observations in
total.
- 'score' if correction is True, then this uses the degrees of freedom
correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985
compare : string in ['diff', 'ratio' 'odds-ratio']
If compare is `diff`, then the hypothesis test is for
diff = p1 - p2.
If compare is `ratio`, then the hypothesis test is for the
risk ratio defined by ratio = p1 / p2.
If compare is `odds-ratio`, then the hypothesis test is for the
odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2).
correction : bool
If correction is True (default), then the Miettinen and Nurminen
small sample correction to the variance nobs / (nobs - 1) is used.
Applies only if method='score'.
Returns
-------
pvalue : float
p-value is the max of the pvalues of the two one-sided tests
t1 : test results
results instance for one-sided hypothesis at the lower margin
t1 : test results
results instance for one-sided hypothesis at the upper margin
See Also
--------
test_proportions_2indep
confint_proportions_2indep
Notes
-----
Status: experimental, API and defaults might still change.
The TOST equivalence test delegates to `test_proportions_2indep` and has
the same method and comparison options. | tost_proportions_2indep | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def _std_2prop_power(diff, p2, ratio=1, alpha=0.05, value=0):
"""
Compute standard error under null and alternative for 2 proportions
helper function for power and sample size computation
"""
if value != 0:
msg = 'non-zero diff under null, value, is not yet implemented'
raise NotImplementedError(msg)
nobs_ratio = ratio
p1 = p2 + diff
# The following contains currently redundant variables that will
# be useful for different options for the null variance
p_pooled = (p1 + p2 * ratio) / (1 + ratio)
# probabilities for the variance for the null statistic
p1_vnull, p2_vnull = p_pooled, p_pooled
p2_alt = p2
p1_alt = p2_alt + diff
std_null = _std_diff_prop(p1_vnull, p2_vnull, ratio=nobs_ratio)
std_alt = _std_diff_prop(p1_alt, p2_alt, ratio=nobs_ratio)
return p_pooled, std_null, std_alt | Compute standard error under null and alternative for 2 proportions
helper function for power and sample size computation | _std_2prop_power | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def power_proportions_2indep(diff, prop2, nobs1, ratio=1, alpha=0.05,
value=0, alternative='two-sided',
return_results=True):
"""
Power for ztest that two independent proportions are equal
This assumes that the variance is based on the pooled proportion
under the null and the non-pooled variance under the alternative
Parameters
----------
diff : float
difference between proportion 1 and 2 under the alternative
prop2 : float
proportion for the reference case, prop2, proportions for the
first case will be computed using p2 and diff
p1 = p2 + diff
nobs1 : float or int
number of observations in sample 1
ratio : float
sample size ratio, nobs2 = ratio * nobs1
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.
value : float
currently only `value=0`, i.e. equality testing, is supported
alternative : string, 'two-sided' (default), 'larger', 'smaller'
Alternative hypothesis whether the power is calculated for a
two-sided (default) or one sided test. The one-sided test can be
either 'larger', 'smaller'.
return_results : bool
If true, then a results instance with extra information is returned,
otherwise only the computed power is returned.
Returns
-------
results : results instance or float
If return_results is True, then a results instance with the
information in attributes is returned.
If return_results is False, then only the power is returned.
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.
Other attributes in results instance include :
p_pooled
pooled proportion, used for std_null
std_null
standard error of difference under the null hypothesis (without
sqrt(nobs1))
std_alt
standard error of difference under the alternative hypothesis
(without sqrt(nobs1))
"""
# TODO: avoid possible circular import, check if needed
from statsmodels.stats.power import normal_power_het
p_pooled, std_null, std_alt = _std_2prop_power(diff, prop2, ratio=ratio,
alpha=alpha, value=value)
pow_ = normal_power_het(diff, nobs1, alpha, std_null=std_null,
std_alternative=std_alt,
alternative=alternative)
if return_results:
res = Holder(power=pow_,
p_pooled=p_pooled,
std_null=std_null,
std_alt=std_alt,
nobs1=nobs1,
nobs2=ratio * nobs1,
nobs_ratio=ratio,
alpha=alpha,
)
return res
else:
return pow_ | Power for ztest that two independent proportions are equal
This assumes that the variance is based on the pooled proportion
under the null and the non-pooled variance under the alternative
Parameters
----------
diff : float
difference between proportion 1 and 2 under the alternative
prop2 : float
proportion for the reference case, prop2, proportions for the
first case will be computed using p2 and diff
p1 = p2 + diff
nobs1 : float or int
number of observations in sample 1
ratio : float
sample size ratio, nobs2 = ratio * nobs1
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.
value : float
currently only `value=0`, i.e. equality testing, is supported
alternative : string, 'two-sided' (default), 'larger', 'smaller'
Alternative hypothesis whether the power is calculated for a
two-sided (default) or one sided test. The one-sided test can be
either 'larger', 'smaller'.
return_results : bool
If true, then a results instance with extra information is returned,
otherwise only the computed power is returned.
Returns
-------
results : results instance or float
If return_results is True, then a results instance with the
information in attributes is returned.
If return_results is False, then only the power is returned.
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.
Other attributes in results instance include :
p_pooled
pooled proportion, used for std_null
std_null
standard error of difference under the null hypothesis (without
sqrt(nobs1))
std_alt
standard error of difference under the alternative hypothesis
(without sqrt(nobs1)) | power_proportions_2indep | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def samplesize_proportions_2indep_onetail(diff, prop2, power, ratio=1,
alpha=0.05, value=0,
alternative='two-sided'):
"""
Required sample size assuming normal distribution based on one tail
This uses an explicit computation for the sample size that is required
to achieve a given power corresponding to the appropriate tails of the
normal distribution. This ignores the far tail in a two-sided test
which is negligible in the common case when alternative and null are
far apart.
Parameters
----------
diff : float
Difference between proportion 1 and 2 under the alternative
prop2 : float
proportion for the reference case, prop2, proportions for the
first case will be computing using p2 and diff
p1 = p2 + diff
power : float
Power for which sample size is computed.
ratio : float
Sample size ratio, nobs2 = ratio * nobs1
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.
value : float
Currently only `value=0`, i.e. equality testing, is supported
alternative : string, 'two-sided' (default), 'larger', 'smaller'
Alternative hypothesis whether the power is calculated for a
two-sided (default) or one sided test. In the case of a one-sided
alternative, it is assumed that the test is in the appropriate tail.
Returns
-------
nobs1 : float
Number of observations in sample 1.
"""
# TODO: avoid possible circular import, check if needed
from statsmodels.stats.power import normal_sample_size_one_tail
if alternative in ['two-sided', '2s']:
alpha = alpha / 2
_, std_null, std_alt = _std_2prop_power(diff, prop2, ratio=ratio,
alpha=alpha, value=value)
nobs = normal_sample_size_one_tail(diff, power, alpha, std_null=std_null,
std_alternative=std_alt)
return nobs | Required sample size assuming normal distribution based on one tail
This uses an explicit computation for the sample size that is required
to achieve a given power corresponding to the appropriate tails of the
normal distribution. This ignores the far tail in a two-sided test
which is negligible in the common case when alternative and null are
far apart.
Parameters
----------
diff : float
Difference between proportion 1 and 2 under the alternative
prop2 : float
proportion for the reference case, prop2, proportions for the
first case will be computing using p2 and diff
p1 = p2 + diff
power : float
Power for which sample size is computed.
ratio : float
Sample size ratio, nobs2 = ratio * nobs1
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.
value : float
Currently only `value=0`, i.e. equality testing, is supported
alternative : string, 'two-sided' (default), 'larger', 'smaller'
Alternative hypothesis whether the power is calculated for a
two-sided (default) or one sided test. In the case of a one-sided
alternative, it is assumed that the test is in the appropriate tail.
Returns
-------
nobs1 : float
Number of observations in sample 1. | samplesize_proportions_2indep_onetail | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def _score_confint_inversion(count1, nobs1, count2, nobs2, compare='diff',
alpha=0.05, correction=True):
"""
Compute score confidence interval by inverting score test
Parameters
----------
count1, nobs1 :
Count and sample size for first sample.
count2, nobs2 :
Count and sample size for the second sample.
compare : string in ['diff', 'ratio' 'odds-ratio']
If compare is `diff`, then the confidence interval is for
diff = p1 - p2.
If compare is `ratio`, then the confidence interval is for the
risk ratio defined by ratio = p1 / p2.
If compare is `odds-ratio`, then the confidence interval is for the
odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2).
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.
correction : bool
If correction is True (default), then the Miettinen and Nurminen
small sample correction to the variance nobs / (nobs - 1) is used.
Applies only if method='score'.
Returns
-------
low : float
Lower confidence bound.
upp : float
Upper confidence bound.
"""
def func(v):
r = test_proportions_2indep(count1, nobs1, count2, nobs2,
value=v, compare=compare, method='score',
correction=correction,
alternative="two-sided")
return r.pvalue - alpha
rt0 = test_proportions_2indep(count1, nobs1, count2, nobs2,
value=0, compare=compare, method='score',
correction=correction,
alternative="two-sided")
# use default method to get starting values
# this will not work if score confint becomes default
# maybe use "wald" as alias that works for all compare statistics
use_method = {"diff": "wald", "ratio": "log", "odds-ratio": "logit"}
rci0 = confint_proportions_2indep(count1, nobs1, count2, nobs2,
method=use_method[compare],
compare=compare, alpha=alpha)
# Note diff might be negative
ub = rci0[1] + np.abs(rci0[1]) * 0.5
lb = rci0[0] - np.abs(rci0[0]) * 0.25
if compare == 'diff':
param = rt0.diff
# 1 might not be the correct upper bound because
# rootfinding is for the `diff` and not for a probability.
ub = min(ub, 0.99999)
elif compare == 'ratio':
param = rt0.ratio
ub *= 2 # add more buffer
if compare == 'odds-ratio':
param = rt0.odds_ratio
# root finding for confint bounds
upp = optimize.brentq(func, param, ub)
low = optimize.brentq(func, lb, param)
return low, upp | Compute score confidence interval by inverting score test
Parameters
----------
count1, nobs1 :
Count and sample size for first sample.
count2, nobs2 :
Count and sample size for the second sample.
compare : string in ['diff', 'ratio' 'odds-ratio']
If compare is `diff`, then the confidence interval is for
diff = p1 - p2.
If compare is `ratio`, then the confidence interval is for the
risk ratio defined by ratio = p1 / p2.
If compare is `odds-ratio`, then the confidence interval is for the
odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2).
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.
correction : bool
If correction is True (default), then the Miettinen and Nurminen
small sample correction to the variance nobs / (nobs - 1) is used.
Applies only if method='score'.
Returns
-------
low : float
Lower confidence bound.
upp : float
Upper confidence bound. | _score_confint_inversion | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def _confint_riskratio_koopman(count1, nobs1, count2, nobs2, alpha=0.05,
correction=True):
"""
Score confidence interval for ratio or proportions, Koopman/Nam
signature not consistent with other functions
When correction is True, then the small sample correction nobs / (nobs - 1)
by Miettinen/Nurminen is used.
"""
# The names below follow Nam
x0, x1, n0, n1 = count2, count1, nobs2, nobs1
x = x0 + x1
n = n0 + n1
z = stats.norm.isf(alpha / 2)**2
if correction:
# Mietinnen/Nurminen small sample correction
z *= n / (n - 1)
# z = stats.chi2.isf(alpha, 1)
# equ 6 in Nam 1995
a1 = n0 * (n0 * n * x1 + n1 * (n0 + x1) * z)
a2 = - n0 * (n0 * n1 * x + 2 * n * x0 * x1 + n1 * (n0 + x0 + 2 * x1) * z)
a3 = 2 * n0 * n1 * x0 * x + n * x0 * x0 * x1 + n0 * n1 * x * z
a4 = - n1 * x0 * x0 * x
p_roots_ = np.sort(np.roots([a1, a2, a3, a4]))
p_roots = p_roots_[:2][::-1]
# equ 5
ci = (1 - (n1 - x1) * (1 - p_roots) / (x0 + n1 - n * p_roots)) / p_roots
res = Holder()
res.confint = ci
res._p_roots = p_roots_ # for unit tests, can be dropped
return res | Score confidence interval for ratio or proportions, Koopman/Nam
signature not consistent with other functions
When correction is True, then the small sample correction nobs / (nobs - 1)
by Miettinen/Nurminen is used. | _confint_riskratio_koopman | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def _confint_riskratio_paired_nam(table, alpha=0.05):
"""
Confidence interval for marginal risk ratio for matched pairs
need full table
success fail marginal
success x11 x10 x1.
fail x01 x00 x0.
marginal x.1 x.0 n
The confidence interval is for the ratio p1 / p0 where
p1 = x1. / n and
p0 - x.1 / n
Todo: rename p1 to pa and p2 to pb, so we have a, b for treatment and
0, 1 for success/failure
current namings follow Nam 2009
status
testing:
compared to example in Nam 2009
internal polynomial coefficients in calculation correspond at around
4 decimals
confidence interval agrees only at 2 decimals
"""
x11, x10, x01, x00 = np.ravel(table)
n = np.sum(table) # nobs
p10, p01 = x10 / n, x01 / n
p1 = (x11 + x10) / n
p0 = (x11 + x01) / n
q00 = 1 - x00 / n
z2 = stats.norm.isf(alpha / 2)**2
# z = stats.chi2.isf(alpha, 1)
# before equ 3 in Nam 2009
g1 = (n * p0 + z2 / 2) * p0
g2 = - (2 * n * p1 * p0 + z2 * q00)
g3 = (n * p1 + z2 / 2) * p1
a0 = g1**2 - (z2 * p0 / 2)**2
a1 = 2 * g1 * g2
a2 = g2**2 + 2 * g1 * g3 + z2**2 * (p1 * p0 - 2 * p10 * p01) / 2
a3 = 2 * g2 * g3
a4 = g3**2 - (z2 * p1 / 2)**2
p_roots = np.sort(np.roots([a0, a1, a2, a3, a4]))
# p_roots = np.sort(np.roots([1, a1 / a0, a2 / a0, a3 / a0, a4 / a0]))
ci = [p_roots.min(), p_roots.max()]
res = Holder()
res.confint = ci
res.p = p1, p0
res._p_roots = p_roots # for unit tests, can be dropped
return res | Confidence interval for marginal risk ratio for matched pairs
need full table
success fail marginal
success x11 x10 x1.
fail x01 x00 x0.
marginal x.1 x.0 n
The confidence interval is for the ratio p1 / p0 where
p1 = x1. / n and
p0 - x.1 / n
Todo: rename p1 to pa and p2 to pb, so we have a, b for treatment and
0, 1 for success/failure
current namings follow Nam 2009
status
testing:
compared to example in Nam 2009
internal polynomial coefficients in calculation correspond at around
4 decimals
confidence interval agrees only at 2 decimals | _confint_riskratio_paired_nam | python | statsmodels/statsmodels | statsmodels/stats/proportion.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/proportion.py | BSD-3-Clause |
def _make_asymptotic_function(params):
"""
Generates an asymptotic distribution callable from a param matrix
Polynomial is a[0] * x**(-1/2) + a[1] * x**(-1) + a[2] * x**(-3/2)
Parameters
----------
params : ndarray
Array with shape (nalpha, 3) where nalpha is the number of
significance levels
"""
def f(n):
poly = np.array([1, np.log(n), np.log(n) ** 2])
return np.exp(poly.dot(params.T))
return f | Generates an asymptotic distribution callable from a param matrix
Polynomial is a[0] * x**(-1/2) + a[1] * x**(-1) + a[2] * x**(-3/2)
Parameters
----------
params : ndarray
Array with shape (nalpha, 3) where nalpha is the number of
significance levels | _make_asymptotic_function | python | statsmodels/statsmodels | statsmodels/stats/_lilliefors.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/_lilliefors.py | BSD-3-Clause |
def ksstat(x, cdf, alternative='two_sided', args=()):
"""
Calculate statistic for the Kolmogorov-Smirnov test for goodness of fit
This calculates the test statistic for a test of the distribution G(x) of
an observed variable against a given distribution F(x). Under the null
hypothesis the two distributions are identical, G(x)=F(x). The
alternative hypothesis can be either 'two_sided' (default), 'less'
or 'greater'. The KS test is only valid for continuous distributions.
Parameters
----------
x : array_like, 1d
array of observations
cdf : str or callable
string: name of a distribution in scipy.stats
callable: function to evaluate cdf
alternative : 'two_sided' (default), 'less' or 'greater'
defines the alternative hypothesis (see explanation)
args : tuple, sequence
distribution parameters for call to cdf
Returns
-------
D : float
KS test statistic, either D, D+ or D-
See Also
--------
scipy.stats.kstest
Notes
-----
In the one-sided test, the alternative is that the empirical
cumulative distribution function of the random variable is "less"
or "greater" than the cumulative distribution function F(x) of the
hypothesis, G(x)<=F(x), resp. G(x)>=F(x).
In contrast to scipy.stats.kstest, this function only calculates the
statistic which can be used either as distance measure or to implement
case specific p-values.
"""
nobs = float(len(x))
if isinstance(cdf, str):
cdf = getattr(stats.distributions, cdf).cdf
elif hasattr(cdf, 'cdf'):
cdf = getattr(cdf, 'cdf')
x = np.sort(x)
cdfvals = cdf(x, *args)
d_plus = (np.arange(1.0, nobs + 1) / nobs - cdfvals).max()
d_min = (cdfvals - np.arange(0.0, nobs) / nobs).max()
if alternative == 'greater':
return d_plus
elif alternative == 'less':
return d_min
return np.max([d_plus, d_min]) | Calculate statistic for the Kolmogorov-Smirnov test for goodness of fit
This calculates the test statistic for a test of the distribution G(x) of
an observed variable against a given distribution F(x). Under the null
hypothesis the two distributions are identical, G(x)=F(x). The
alternative hypothesis can be either 'two_sided' (default), 'less'
or 'greater'. The KS test is only valid for continuous distributions.
Parameters
----------
x : array_like, 1d
array of observations
cdf : str or callable
string: name of a distribution in scipy.stats
callable: function to evaluate cdf
alternative : 'two_sided' (default), 'less' or 'greater'
defines the alternative hypothesis (see explanation)
args : tuple, sequence
distribution parameters for call to cdf
Returns
-------
D : float
KS test statistic, either D, D+ or D-
See Also
--------
scipy.stats.kstest
Notes
-----
In the one-sided test, the alternative is that the empirical
cumulative distribution function of the random variable is "less"
or "greater" than the cumulative distribution function F(x) of the
hypothesis, G(x)<=F(x), resp. G(x)>=F(x).
In contrast to scipy.stats.kstest, this function only calculates the
statistic which can be used either as distance measure or to implement
case specific p-values. | ksstat | python | statsmodels/statsmodels | statsmodels/stats/_lilliefors.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/_lilliefors.py | BSD-3-Clause |
def get_lilliefors_table(dist='norm'):
"""
Generates tables for significance levels of Lilliefors test statistics
Tables for available normal and exponential distribution testing,
as specified in Lilliefors references above
Parameters
----------
dist : str
distribution being tested in set {'norm', 'exp'}.
Returns
-------
lf : TableDist object.
table of critical values
"""
# function just to keep things together
# for this test alpha is sf probability, i.e. right tail probability
alpha = 1 - np.array(PERCENTILES) / 100.0
alpha = alpha[::-1]
dist = 'normal' if dist == 'norm' else dist
if dist not in critical_values:
raise ValueError("Invalid dist parameter. Must be 'norm' or 'exp'")
cv_data = critical_values[dist]
acv_data = asymp_critical_values[dist]
size = np.array(sorted(cv_data), dtype=float)
crit_lf = np.array([cv_data[key] for key in sorted(cv_data)])
crit_lf = crit_lf[:, ::-1]
asym_params = np.array([acv_data[key] for key in sorted(acv_data)])
asymp_fn = _make_asymptotic_function(asym_params[::-1])
lf = TableDist(alpha, size, crit_lf, asymptotic=asymp_fn)
return lf | Generates tables for significance levels of Lilliefors test statistics
Tables for available normal and exponential distribution testing,
as specified in Lilliefors references above
Parameters
----------
dist : str
distribution being tested in set {'norm', 'exp'}.
Returns
-------
lf : TableDist object.
table of critical values | get_lilliefors_table | python | statsmodels/statsmodels | statsmodels/stats/_lilliefors.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/_lilliefors.py | BSD-3-Clause |
def pval_lf(d_max, n):
"""
Approximate pvalues for Lilliefors test
This is only valid for pvalues smaller than 0.1 which is not checked in
this function.
Parameters
----------
d_max : array_like
two-sided Kolmogorov-Smirnov test statistic
n : int or float
sample size
Returns
-------
p-value : float or ndarray
pvalue according to approximation formula of Dallal and Wilkinson.
Notes
-----
This is mainly a helper function where the calling code should dispatch
on bound violations. Therefore it does not check whether the pvalue is in
the valid range.
Precision for the pvalues is around 2 to 3 decimals. This approximation is
also used by other statistical packages (e.g. R:fBasics) but might not be
the most precise available.
References
----------
DallalWilkinson1986
"""
# todo: check boundaries, valid range for n and Dmax
if n > 100:
d_max *= (n / 100.) ** 0.49
n = 100
pval = np.exp(-7.01256 * d_max ** 2 * (n + 2.78019)
+ 2.99587 * d_max * np.sqrt(n + 2.78019) - 0.122119
+ 0.974598 / np.sqrt(n) + 1.67997 / n)
return pval | Approximate pvalues for Lilliefors test
This is only valid for pvalues smaller than 0.1 which is not checked in
this function.
Parameters
----------
d_max : array_like
two-sided Kolmogorov-Smirnov test statistic
n : int or float
sample size
Returns
-------
p-value : float or ndarray
pvalue according to approximation formula of Dallal and Wilkinson.
Notes
-----
This is mainly a helper function where the calling code should dispatch
on bound violations. Therefore it does not check whether the pvalue is in
the valid range.
Precision for the pvalues is around 2 to 3 decimals. This approximation is
also used by other statistical packages (e.g. R:fBasics) but might not be
the most precise available.
References
----------
DallalWilkinson1986 | pval_lf | python | statsmodels/statsmodels | statsmodels/stats/_lilliefors.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/_lilliefors.py | BSD-3-Clause |
def kstest_fit(x, dist='norm', pvalmethod="table"):
"""
Test assumed normal or exponential distribution using Lilliefors' test.
Lilliefors' test is a Kolmogorov-Smirnov test with estimated parameters.
Parameters
----------
x : array_like, 1d
Data to test.
dist : {'norm', 'exp'}, optional
The assumed distribution.
pvalmethod : {'approx', 'table'}, optional
The method used to compute the p-value of the test statistic. In
general, 'table' is preferred and makes use of a very large simulation.
'approx' is only valid for normality. if `dist = 'exp'` `table` is
always used. 'approx' uses the approximation formula of Dalal and
Wilkinson, valid for pvalues < 0.1. If the pvalue is larger than 0.1,
then the result of `table` is returned.
Returns
-------
ksstat : float
Kolmogorov-Smirnov test statistic with estimated mean and variance.
pvalue : float
If the pvalue is lower than some threshold, e.g. 0.05, then we can
reject the Null hypothesis that the sample comes from a normal
distribution.
Notes
-----
'table' uses an improved table based on 10,000,000 simulations. The
critical values are approximated using
log(cv_alpha) = b_alpha + c[0] log(n) + c[1] log(n)**2
where cv_alpha is the critical value for a test with size alpha,
b_alpha is an alpha-specific intercept term and c[1] and c[2] are
coefficients that are shared all alphas.
Values in the table are linearly interpolated. Values outside the
range are be returned as bounds, 0.990 for large and 0.001 for small
pvalues.
For implementation details, see lilliefors_critical_value_simulation.py in
the test directory.
"""
pvalmethod = string_like(pvalmethod,
"pvalmethod",
options=("approx", "table"))
x = np.asarray(x)
if x.ndim == 2 and x.shape[1] == 1:
x = x[:, 0]
elif x.ndim != 1:
raise ValueError("Invalid parameter `x`: must be a one-dimensional"
" array-like or a single-column DataFrame")
nobs = len(x)
if dist == 'norm':
z = (x - x.mean()) / x.std(ddof=1)
test_d = stats.norm.cdf
lilliefors_table = lilliefors_table_norm
elif dist == 'exp':
z = x / x.mean()
test_d = stats.expon.cdf
lilliefors_table = lilliefors_table_expon
pvalmethod = 'table'
else:
raise ValueError("Invalid dist parameter, must be 'norm' or 'exp'")
min_nobs = 4 if dist == 'norm' else 3
if nobs < min_nobs:
raise ValueError('Test for distribution {} requires at least {} '
'observations'.format(dist, min_nobs))
d_ks = ksstat(z, test_d, alternative='two_sided')
if pvalmethod == 'approx':
pval = pval_lf(d_ks, nobs)
# check pval is in desired range
if pval > 0.1:
pval = lilliefors_table.prob(d_ks, nobs)
else: # pvalmethod == 'table'
pval = lilliefors_table.prob(d_ks, nobs)
return d_ks, pval | Test assumed normal or exponential distribution using Lilliefors' test.
Lilliefors' test is a Kolmogorov-Smirnov test with estimated parameters.
Parameters
----------
x : array_like, 1d
Data to test.
dist : {'norm', 'exp'}, optional
The assumed distribution.
pvalmethod : {'approx', 'table'}, optional
The method used to compute the p-value of the test statistic. In
general, 'table' is preferred and makes use of a very large simulation.
'approx' is only valid for normality. if `dist = 'exp'` `table` is
always used. 'approx' uses the approximation formula of Dalal and
Wilkinson, valid for pvalues < 0.1. If the pvalue is larger than 0.1,
then the result of `table` is returned.
Returns
-------
ksstat : float
Kolmogorov-Smirnov test statistic with estimated mean and variance.
pvalue : float
If the pvalue is lower than some threshold, e.g. 0.05, then we can
reject the Null hypothesis that the sample comes from a normal
distribution.
Notes
-----
'table' uses an improved table based on 10,000,000 simulations. The
critical values are approximated using
log(cv_alpha) = b_alpha + c[0] log(n) + c[1] log(n)**2
where cv_alpha is the critical value for a test with size alpha,
b_alpha is an alpha-specific intercept term and c[1] and c[2] are
coefficients that are shared all alphas.
Values in the table are linearly interpolated. Values outside the
range are be returned as bounds, 0.990 for large and 0.001 for small
pvalues.
For implementation details, see lilliefors_critical_value_simulation.py in
the test directory. | kstest_fit | python | statsmodels/statsmodels | statsmodels/stats/_lilliefors.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/_lilliefors.py | BSD-3-Clause |
def threshold(self, tfdr):
"""
Returns the threshold statistic for a given target FDR.
"""
if np.min(self._ufdr) <= tfdr:
return self._unq[self._ufdr <= tfdr][0]
else:
return np.inf | Returns the threshold statistic for a given target FDR. | threshold | python | statsmodels/statsmodels | statsmodels/stats/_knockoff.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/_knockoff.py | BSD-3-Clause |
def _design_knockoff_sdp(exog):
"""
Use semidefinite programming to construct a knockoff design
matrix.
Requires cvxopt to be installed.
"""
try:
from cvxopt import solvers, matrix
except ImportError:
raise ValueError("SDP knockoff designs require installation of cvxopt")
nobs, nvar = exog.shape
# Standardize exog
xnm = np.sum(exog**2, 0)
xnm = np.sqrt(xnm)
exog = exog / xnm
Sigma = np.dot(exog.T, exog)
c = matrix(-np.ones(nvar))
h0 = np.concatenate((np.zeros(nvar), np.ones(nvar)))
h0 = matrix(h0)
G0 = np.concatenate((-np.eye(nvar), np.eye(nvar)), axis=0)
G0 = matrix(G0)
h1 = 2 * Sigma
h1 = matrix(h1)
i, j = np.diag_indices(nvar)
G1 = np.zeros((nvar*nvar, nvar))
G1[i*nvar + j, i] = 1
G1 = matrix(G1)
solvers.options['show_progress'] = False
sol = solvers.sdp(c, G0, h0, [G1], [h1])
sl = np.asarray(sol['x']).ravel()
xcov = np.dot(exog.T, exog)
exogn = _get_knmat(exog, xcov, sl)
return exog, exogn, sl | Use semidefinite programming to construct a knockoff design
matrix.
Requires cvxopt to be installed. | _design_knockoff_sdp | python | statsmodels/statsmodels | statsmodels/stats/_knockoff.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/_knockoff.py | BSD-3-Clause |
def _design_knockoff_equi(exog):
"""
Construct an equivariant design matrix for knockoff analysis.
Follows the 'equi-correlated knockoff approach of equation 2.4 in
Barber and Candes.
Constructs a pair of design matrices exogs, exogn such that exogs
is a scaled/centered version of the input matrix exog, exogn is
another matrix of the same shape with cov(exogn) = cov(exogs), and
the covariances between corresponding columns of exogn and exogs
are as small as possible.
"""
nobs, nvar = exog.shape
if nobs < 2*nvar:
msg = "The equivariant knockoff can ony be used when n >= 2*p"
raise ValueError(msg)
# Standardize exog
xnm = np.sum(exog**2, 0)
xnm = np.sqrt(xnm)
exog = exog / xnm
xcov = np.dot(exog.T, exog)
ev, _ = np.linalg.eig(xcov)
evmin = np.min(ev)
sl = min(2*evmin, 1)
sl = sl * np.ones(nvar)
exogn = _get_knmat(exog, xcov, sl)
return exog, exogn, sl | Construct an equivariant design matrix for knockoff analysis.
Follows the 'equi-correlated knockoff approach of equation 2.4 in
Barber and Candes.
Constructs a pair of design matrices exogs, exogn such that exogs
is a scaled/centered version of the input matrix exog, exogn is
another matrix of the same shape with cov(exogn) = cov(exogs), and
the covariances between corresponding columns of exogn and exogs
are as small as possible. | _design_knockoff_equi | python | statsmodels/statsmodels | statsmodels/stats/_knockoff.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/_knockoff.py | BSD-3-Clause |
def conf_int(self, alpha=0.05):
"""
Returns the confidence interval of the value, `effect` of the constraint.
This is currently only available for t and z tests.
Parameters
----------
alpha : float, optional
The significance level for the confidence interval.
ie., The default `alpha` = .05 returns a 95% confidence interval.
Returns
-------
ci : ndarray, (k_constraints, 2)
The array has the lower and the upper limit of the confidence
interval in the columns.
"""
if self.effect is not None:
# confidence intervals
q = self.dist.ppf(1 - alpha / 2., *self.dist_args)
lower = self.effect - q * self.sd
upper = self.effect + q * self.sd
return np.column_stack((lower, upper))
else:
raise NotImplementedError('Confidence Interval not available') | Returns the confidence interval of the value, `effect` of the constraint.
This is currently only available for t and z tests.
Parameters
----------
alpha : float, optional
The significance level for the confidence interval.
ie., The default `alpha` = .05 returns a 95% confidence interval.
Returns
-------
ci : ndarray, (k_constraints, 2)
The array has the lower and the upper limit of the confidence
interval in the columns. | conf_int | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def summary(self, xname=None, alpha=0.05, title=None):
"""Summarize the Results of the hypothesis test
Parameters
----------
xname : list[str], optional
Default is `c_##` for ## in the number of regressors
alpha : float
significance level for the confidence intervals. Default is
alpha = 0.05 which implies a confidence level of 95%.
title : str, optional
Title for the params table. If not None, then this replaces the
default title
Returns
-------
smry : str or Summary instance
This contains a parameter results table in the case of t or z test
in the same form as the parameter results table in the model
results summary.
For F or Wald test, the return is a string.
"""
if self.effect is not None:
# TODO: should also add some extra information, e.g. robust cov ?
# TODO: can we infer names for constraints, xname in __init__ ?
if title is None:
title = 'Test for Constraints'
elif title == '':
# do not add any title,
# I think SimpleTable skips on None - check
title = None
# we have everything for a params table
use_t = (self.distribution == 't')
yname='constraints' # Not used in params_frame
if xname is None:
xname = self.c_names
from statsmodels.iolib.summary import summary_params
pvalues = np.atleast_1d(self.pvalue)
summ = summary_params((self, self.effect, self.sd, self.statistic,
pvalues, self.conf_int(alpha)),
yname=yname, xname=xname, use_t=use_t,
title=title, alpha=alpha)
return summ
elif hasattr(self, 'fvalue'):
# TODO: create something nicer for these casee
return ('<F test: F=%s, p=%s, df_denom=%.3g, df_num=%.3g>' %
(repr(self.fvalue), self.pvalue, self.df_denom,
self.df_num))
elif self.distribution == 'chi2':
return ('<Wald test (%s): statistic=%s, p-value=%s, df_denom=%.3g>' %
(self.distribution, self.statistic, self.pvalue,
self.df_denom))
else:
# generic
return ('<Wald test: statistic=%s, p-value=%s>' %
(self.statistic, self.pvalue)) | Summarize the Results of the hypothesis test
Parameters
----------
xname : list[str], optional
Default is `c_##` for ## in the number of regressors
alpha : float
significance level for the confidence intervals. Default is
alpha = 0.05 which implies a confidence level of 95%.
title : str, optional
Title for the params table. If not None, then this replaces the
default title
Returns
-------
smry : str or Summary instance
This contains a parameter results table in the case of t or z test
in the same form as the parameter results table in the model
results summary.
For F or Wald test, the return is a string. | summary | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def summary_frame(self, xname=None, alpha=0.05):
"""Return the parameter table as a pandas DataFrame
This is only available for t and normal tests
"""
if self.effect is not None:
# we have everything for a params table
use_t = (self.distribution == 't')
yname='constraints' # Not used in params_frame
if xname is None:
xname = self.c_names
from statsmodels.iolib.summary import summary_params_frame
summ = summary_params_frame((self, self.effect, self.sd,
self.statistic,self.pvalue,
self.conf_int(alpha)), yname=yname,
xname=xname, use_t=use_t,
alpha=alpha)
return summ
else:
# TODO: create something nicer
raise NotImplementedError('only available for t and z') | Return the parameter table as a pandas DataFrame
This is only available for t and normal tests | summary_frame | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def _get_matrix(self):
"""
Gets the contrast_matrix property
"""
if not hasattr(self, "_contrast_matrix"):
self.compute_matrix()
return self._contrast_matrix | Gets the contrast_matrix property | _get_matrix | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def compute_matrix(self):
"""
Construct a contrast matrix C so that
colspan(dot(D, C)) = colspan(dot(D, dot(pinv(D), T)))
where pinv(D) is the generalized inverse of D=design.
"""
T = self.term
if T.ndim == 1:
T = T[:,None]
self.T = clean0(T)
self.D = self.design
self._contrast_matrix = contrastfromcols(self.T, self.D)
try:
self.rank = self.matrix.shape[1]
except (AttributeError, IndexError):
self.rank = 1 | Construct a contrast matrix C so that
colspan(dot(D, C)) = colspan(dot(D, dot(pinv(D), T)))
where pinv(D) is the generalized inverse of D=design. | compute_matrix | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def contrastfromcols(L, D, pseudo=None):
"""
From an n x p design matrix D and a matrix L, tries
to determine a p x q contrast matrix C which
determines a contrast of full rank, i.e. the
n x q matrix
dot(transpose(C), pinv(D))
is full rank.
L must satisfy either L.shape[0] == n or L.shape[1] == p.
If L.shape[0] == n, then L is thought of as representing
columns in the column space of D.
If L.shape[1] == p, then L is thought of as what is known
as a contrast matrix. In this case, this function returns an estimable
contrast corresponding to the dot(D, L.T)
Note that this always produces a meaningful contrast, not always
with the intended properties because q is always non-zero unless
L is identically 0. That is, it produces a contrast that spans
the column space of L (after projection onto the column space of D).
Parameters
----------
L : array_like
D : array_like
"""
L = np.asarray(L)
D = np.asarray(D)
n, p = D.shape
if L.shape[0] != n and L.shape[1] != p:
raise ValueError("shape of L and D mismatched")
if pseudo is None:
pseudo = np.linalg.pinv(D) # D^+ \approx= ((dot(D.T,D))^(-1),D.T)
if L.shape[0] == n:
C = np.dot(pseudo, L).T
else:
C = L
C = np.dot(pseudo, np.dot(D, C.T)).T
Lp = np.dot(D, C.T)
if len(Lp.shape) == 1:
Lp.shape = (n, 1)
if np.linalg.matrix_rank(Lp) != Lp.shape[1]:
Lp = fullrank(Lp)
C = np.dot(pseudo, Lp).T
return np.squeeze(C) | From an n x p design matrix D and a matrix L, tries
to determine a p x q contrast matrix C which
determines a contrast of full rank, i.e. the
n x q matrix
dot(transpose(C), pinv(D))
is full rank.
L must satisfy either L.shape[0] == n or L.shape[1] == p.
If L.shape[0] == n, then L is thought of as representing
columns in the column space of D.
If L.shape[1] == p, then L is thought of as what is known
as a contrast matrix. In this case, this function returns an estimable
contrast corresponding to the dot(D, L.T)
Note that this always produces a meaningful contrast, not always
with the intended properties because q is always non-zero unless
L is identically 0. That is, it produces a contrast that spans
the column space of L (after projection onto the column space of D).
Parameters
----------
L : array_like
D : array_like | contrastfromcols | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def col_names(self):
"""column names for summary table
"""
pr_test = "P>%s" % self.distribution
col_names = [self.distribution, pr_test, 'df constraint']
if self.distribution == 'F':
col_names.append('df denom')
return col_names | column names for summary table | col_names | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def _get_pairs_labels(k_level, level_names):
"""helper function for labels for pairwise comparisons
"""
idx_pairs_all = np.triu_indices(k_level, 1)
labels = [f'{level_names[name[1]]}-{level_names[name[0]]}'
for name in zip(*idx_pairs_all)]
return labels | helper function for labels for pairwise comparisons | _get_pairs_labels | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def _contrast_pairs(k_params, k_level, idx_start):
"""create pairwise contrast for reference coding
currently not used,
using encoding contrast matrix is more general, but requires requires
factor information from a formula's model_spec.
Parameters
----------
k_params : int
number of parameters
k_level : int
number of levels or categories (including reference case)
idx_start : int
Index of the first parameter of this factor. The restrictions on the
factor are inserted as a block in the full restriction matrix starting
at column with index `idx_start`.
Returns
-------
contrasts : ndarray
restriction matrix with k_params columns and number of rows equal to
the number of restrictions.
"""
k_level_m1 = k_level - 1
idx_pairs = np.triu_indices(k_level_m1, 1)
k = len(idx_pairs[0])
c_pairs = np.zeros((k, k_level_m1))
c_pairs[np.arange(k), idx_pairs[0]] = -1
c_pairs[np.arange(k), idx_pairs[1]] = 1
c_reference = np.eye(k_level_m1)
c = np.concatenate((c_reference, c_pairs), axis=0)
k_all = c.shape[0]
contrasts = np.zeros((k_all, k_params))
contrasts[:, idx_start : idx_start + k_level_m1] = c
return contrasts | create pairwise contrast for reference coding
currently not used,
using encoding contrast matrix is more general, but requires requires
factor information from a formula's model_spec.
Parameters
----------
k_params : int
number of parameters
k_level : int
number of levels or categories (including reference case)
idx_start : int
Index of the first parameter of this factor. The restrictions on the
factor are inserted as a block in the full restriction matrix starting
at column with index `idx_start`.
Returns
-------
contrasts : ndarray
restriction matrix with k_params columns and number of rows equal to
the number of restrictions. | _contrast_pairs | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def t_test_multi(result, contrasts, method='hs', alpha=0.05, ci_method=None,
contrast_names=None):
"""perform t_test and add multiplicity correction to results dataframe
Parameters
----------
result results instance
results of an estimated model
contrasts : ndarray
restriction matrix for t_test
method : str or list of strings
method for multiple testing p-value correction, default is'hs'.
alpha : float
significance level for multiple testing reject decision.
ci_method : None
not used yet, will be for multiplicity corrected confidence intervals
contrast_names : {list[str], None}
If contrast_names are provided, then they are used in the index of the
returned dataframe, otherwise some generic default names are created.
Returns
-------
res_df : pandas DataFrame
The dataframe contains the results of the t_test and additional columns
for multiplicity corrected p-values and boolean indicator for whether
the Null hypothesis is rejected.
"""
tt = result.t_test(contrasts)
res_df = tt.summary_frame(xname=contrast_names)
if type(method) is not list:
method = [method]
for meth in method:
mt = multipletests(tt.pvalue, method=meth, alpha=alpha)
res_df['pvalue-%s' % meth] = mt[1]
res_df['reject-%s' % meth] = mt[0]
return res_df | perform t_test and add multiplicity correction to results dataframe
Parameters
----------
result results instance
results of an estimated model
contrasts : ndarray
restriction matrix for t_test
method : str or list of strings
method for multiple testing p-value correction, default is'hs'.
alpha : float
significance level for multiple testing reject decision.
ci_method : None
not used yet, will be for multiplicity corrected confidence intervals
contrast_names : {list[str], None}
If contrast_names are provided, then they are used in the index of the
returned dataframe, otherwise some generic default names are created.
Returns
-------
res_df : pandas DataFrame
The dataframe contains the results of the t_test and additional columns
for multiplicity corrected p-values and boolean indicator for whether
the Null hypothesis is rejected. | t_test_multi | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def _embed_constraints(contrasts, k_params, idx_start, index=None):
"""helper function to expand constraints to a full restriction matrix
Parameters
----------
contrasts : ndarray
restriction matrix for t_test
k_params : int
number of parameters
idx_start : int
Index of the first parameter of this factor. The restrictions on the
factor are inserted as a block in the full restriction matrix starting
at column with index `idx_start`.
index : slice or ndarray
Column index if constraints do not form a block in the full restriction
matrix, i.e. if parameters that are subject to restrictions are not
consecutive in the list of parameters.
If index is not None, then idx_start is ignored.
Returns
-------
contrasts : ndarray
restriction matrix with k_params columns and number of rows equal to
the number of restrictions.
"""
k_c, k_p = contrasts.shape
c = np.zeros((k_c, k_params))
if index is None:
c[:, idx_start : idx_start + k_p] = contrasts
else:
c[:, index] = contrasts
return c | helper function to expand constraints to a full restriction matrix
Parameters
----------
contrasts : ndarray
restriction matrix for t_test
k_params : int
number of parameters
idx_start : int
Index of the first parameter of this factor. The restrictions on the
factor are inserted as a block in the full restriction matrix starting
at column with index `idx_start`.
index : slice or ndarray
Column index if constraints do not form a block in the full restriction
matrix, i.e. if parameters that are subject to restrictions are not
consecutive in the list of parameters.
If index is not None, then idx_start is ignored.
Returns
-------
contrasts : ndarray
restriction matrix with k_params columns and number of rows equal to
the number of restrictions. | _embed_constraints | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def _constraints_factor(encoding_matrix, comparison='pairwise', k_params=None,
idx_start=None):
"""helper function to create constraints based on encoding matrix
Parameters
----------
encoding_matrix : ndarray
contrast matrix for the encoding of a factor as defined by patsy.
The number of rows should be equal to the number of levels or categories
of the factor, the number of columns should be equal to the number
of parameters for this factor.
comparison : str
Currently only 'pairwise' is implemented. The restriction matrix
can be used for testing the hypothesis that all pairwise differences
are zero.
k_params : int
number of parameters
idx_start : int
Index of the first parameter of this factor. The restrictions on the
factor are inserted as a block in the full restriction matrix starting
at column with index `idx_start`.
Returns
-------
contrast : ndarray
Contrast or restriction matrix that can be used in hypothesis test
of model results. The number of columns is k_params.
"""
cm = encoding_matrix
k_level, k_p = cm.shape
import statsmodels.sandbox.stats.multicomp as mc
if comparison in ['pairwise', 'pw', 'pairs']:
c_all = -mc.contrast_allpairs(k_level)
else:
raise NotImplementedError('currentlyonly pairwise comparison')
contrasts = c_all.dot(cm)
if k_params is not None:
if idx_start is None:
raise ValueError("if k_params is not None, then idx_start is "
"required")
contrasts = _embed_constraints(contrasts, k_params, idx_start)
return contrasts | helper function to create constraints based on encoding matrix
Parameters
----------
encoding_matrix : ndarray
contrast matrix for the encoding of a factor as defined by patsy.
The number of rows should be equal to the number of levels or categories
of the factor, the number of columns should be equal to the number
of parameters for this factor.
comparison : str
Currently only 'pairwise' is implemented. The restriction matrix
can be used for testing the hypothesis that all pairwise differences
are zero.
k_params : int
number of parameters
idx_start : int
Index of the first parameter of this factor. The restrictions on the
factor are inserted as a block in the full restriction matrix starting
at column with index `idx_start`.
Returns
-------
contrast : ndarray
Contrast or restriction matrix that can be used in hypothesis test
of model results. The number of columns is k_params. | _constraints_factor | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def t_test_pairwise(result, term_name, method='hs', alpha=0.05,
factor_labels=None, ignore=False):
"""
Perform pairwise t_test with multiple testing corrected p-values.
This uses the formula's model_spec encoding contrast matrix and should
work for all encodings of a main effect.
Parameters
----------
result : result instance
The results of an estimated model with a categorical main effect.
term_name : str
name of the term for which pairwise comparisons are computed.
Term names for categorical effects are created by patsy and
correspond to the main part of the exog names.
method : {str, list[str]}
multiple testing p-value correction, default is 'hs',
see stats.multipletesting
alpha : float
significance level for multiple testing reject decision.
factor_labels : {list[str], None}
Labels for the factor levels used for pairwise labels. If not
provided, then the labels from the formula's model_spec are used.
ignore : bool
Turn off some of the exceptions raised by input checks.
Returns
-------
MultiCompResult
The results are stored as attributes, the main attributes are the
following two. Other attributes are added for debugging purposes
or as background information.
- result_frame : pandas DataFrame with t_test results and multiple
testing corrected p-values.
- contrasts : matrix of constraints of the null hypothesis in the
t_test.
Notes
-----
Status: experimental. Currently only checked for treatment coding with
and without specified reference level.
Currently there are no multiple testing corrected confidence intervals
available.
"""
mgr = FormulaManager()
model_spec = result.model.data.model_spec
term_idx = mgr.get_term_names(model_spec).index(term_name)
term = model_spec.terms[term_idx]
idx_start = model_spec.term_slices[term].start
if not ignore and len(term.factors) > 1:
raise ValueError('interaction effects not yet supported')
factor = term.factors[0]
cat = mgr.get_factor_categories(factor, model_spec)
# cat = model_spec.encoder_state[factor][1]["categories"]
# model_spec.factor_infos[factor].categories
if factor_labels is not None:
if len(factor_labels) == len(cat):
cat = factor_labels
else:
raise ValueError("factor_labels has the wrong length, should be %d" % len(cat))
k_level = len(cat)
cm = mgr.get_contrast_matrix(term, factor, model_spec)
k_params = len(result.params)
labels = _get_pairs_labels(k_level, cat)
import statsmodels.sandbox.stats.multicomp as mc
c_all_pairs = -mc.contrast_allpairs(k_level)
contrasts_sub = c_all_pairs.dot(cm)
contrasts = _embed_constraints(contrasts_sub, k_params, idx_start)
res_df = t_test_multi(result, contrasts, method=method, ci_method=None,
alpha=alpha, contrast_names=labels)
res = MultiCompResult(result_frame=res_df,
contrasts=contrasts,
term=term,
contrast_labels=labels,
term_encoding_matrix=cm)
return res | Perform pairwise t_test with multiple testing corrected p-values.
This uses the formula's model_spec encoding contrast matrix and should
work for all encodings of a main effect.
Parameters
----------
result : result instance
The results of an estimated model with a categorical main effect.
term_name : str
name of the term for which pairwise comparisons are computed.
Term names for categorical effects are created by patsy and
correspond to the main part of the exog names.
method : {str, list[str]}
multiple testing p-value correction, default is 'hs',
see stats.multipletesting
alpha : float
significance level for multiple testing reject decision.
factor_labels : {list[str], None}
Labels for the factor levels used for pairwise labels. If not
provided, then the labels from the formula's model_spec are used.
ignore : bool
Turn off some of the exceptions raised by input checks.
Returns
-------
MultiCompResult
The results are stored as attributes, the main attributes are the
following two. Other attributes are added for debugging purposes
or as background information.
- result_frame : pandas DataFrame with t_test results and multiple
testing corrected p-values.
- contrasts : matrix of constraints of the null hypothesis in the
t_test.
Notes
-----
Status: experimental. Currently only checked for treatment coding with
and without specified reference level.
Currently there are no multiple testing corrected confidence intervals
available. | t_test_pairwise | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def _offset_constraint(r_matrix, params_est, params_alt):
"""offset to the value of a linear constraint for new params
usage:
(cm, v) is original constraint
vo = offset_constraint(cm, res2.params, params_alt)
fs = res2.wald_test((cm, v + vo))
nc = fs.statistic * fs.df_num
"""
diff_est = r_matrix @ params_est
diff_alt = r_matrix @ params_alt
return diff_est - diff_alt | offset to the value of a linear constraint for new params
usage:
(cm, v) is original constraint
vo = offset_constraint(cm, res2.params, params_alt)
fs = res2.wald_test((cm, v + vo))
nc = fs.statistic * fs.df_num | _offset_constraint | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def wald_test_noncent(params, r_matrix, value, results, diff=None, joint=True):
"""Moncentrality parameter for a wald test in model results
The null hypothesis is ``diff = r_matrix @ params - value = 0``
Parameters
----------
params : ndarray
parameters of the model at which to evaluate noncentrality. This can
be estimated parameters or parameters under an alternative.
r_matrix : ndarray
Restriction matrix or contrasts for the Null hypothesis
value : None or ndarray
Value of the linear combination of parameters under the null
hypothesis. If value is None, then it will be replaced by zero.
results : Results instance of a model
The results instance is used to compute the covariance matrix of the
linear constraints using `cov_params.
diff : None or ndarray
If diff is not None, then it will be used instead of
``diff = r_matrix @ params - value``
joint : bool
If joint is True, then the noncentrality parameter for the joint
hypothesis will be returned.
If joint is True, then an array of noncentrality parameters will be
returned, where elements correspond to rows of the restriction matrix.
This correspond to the `t_test` in models and is not a quadratic form.
Returns
-------
nc : float or ndarray
Noncentrality parameter for Wald tests, correspondig to `wald_test`
or `t_test` depending on whether `joint` is true or not.
It needs to be divided by nobs to obtain effect size.
Notes
-----
Status : experimental, API will likely change
"""
if diff is None:
diff = r_matrix @ params - value # at parameter under alternative
cov_c = results.cov_params(r_matrix=r_matrix)
if joint:
nc = diff @ np.linalg.solve(cov_c, diff)
else:
nc = diff / np.sqrt(np.diag(cov_c))
return nc | Moncentrality parameter for a wald test in model results
The null hypothesis is ``diff = r_matrix @ params - value = 0``
Parameters
----------
params : ndarray
parameters of the model at which to evaluate noncentrality. This can
be estimated parameters or parameters under an alternative.
r_matrix : ndarray
Restriction matrix or contrasts for the Null hypothesis
value : None or ndarray
Value of the linear combination of parameters under the null
hypothesis. If value is None, then it will be replaced by zero.
results : Results instance of a model
The results instance is used to compute the covariance matrix of the
linear constraints using `cov_params.
diff : None or ndarray
If diff is not None, then it will be used instead of
``diff = r_matrix @ params - value``
joint : bool
If joint is True, then the noncentrality parameter for the joint
hypothesis will be returned.
If joint is True, then an array of noncentrality parameters will be
returned, where elements correspond to rows of the restriction matrix.
This correspond to the `t_test` in models and is not a quadratic form.
Returns
-------
nc : float or ndarray
Noncentrality parameter for Wald tests, correspondig to `wald_test`
or `t_test` depending on whether `joint` is true or not.
It needs to be divided by nobs to obtain effect size.
Notes
-----
Status : experimental, API will likely change | wald_test_noncent | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def wald_test_noncent_generic(params, r_matrix, value, cov_params, diff=None,
joint=True):
"""noncentrality parameter for a wald test
The null hypothesis is ``diff = r_matrix @ params - value = 0``
Parameters
----------
params : ndarray
parameters of the model at which to evaluate noncentrality. This can
be estimated parameters or parameters under an alternative.
r_matrix : ndarray
Restriction matrix or contrasts for the Null hypothesis
value : None or ndarray
Value of the linear combination of parameters under the null
hypothesis. If value is None, then it will be replace by zero.
cov_params : ndarray
covariance matrix of the parameter estimates
diff : None or ndarray
If diff is not None, then it will be used instead of
``diff = r_matrix @ params - value``
joint : bool
If joint is True, then the noncentrality parameter for the joint
hypothesis will be returned.
If joint is True, then an array of noncentrality parameters will be
returned, where elements correspond to rows of the restriction matrix.
This correspond to the `t_test` in models and is not a quadratic form.
Returns
-------
nc : float or ndarray
Noncentrality parameter for Wald tests, correspondig to `wald_test`
or `t_test` depending on whether `joint` is true or not.
It needs to be divided by nobs to obtain effect size.
Notes
-----
Status : experimental, API will likely change
"""
if value is None:
value = 0
if diff is None:
# at parameter under alternative
diff = r_matrix @ params - value
c = r_matrix
cov_c = c.dot(cov_params).dot(c.T)
if joint:
nc = diff @ np.linalg.solve(cov_c, diff)
else:
nc = diff / np.sqrt(np.diag(cov_c))
return nc | noncentrality parameter for a wald test
The null hypothesis is ``diff = r_matrix @ params - value = 0``
Parameters
----------
params : ndarray
parameters of the model at which to evaluate noncentrality. This can
be estimated parameters or parameters under an alternative.
r_matrix : ndarray
Restriction matrix or contrasts for the Null hypothesis
value : None or ndarray
Value of the linear combination of parameters under the null
hypothesis. If value is None, then it will be replace by zero.
cov_params : ndarray
covariance matrix of the parameter estimates
diff : None or ndarray
If diff is not None, then it will be used instead of
``diff = r_matrix @ params - value``
joint : bool
If joint is True, then the noncentrality parameter for the joint
hypothesis will be returned.
If joint is True, then an array of noncentrality parameters will be
returned, where elements correspond to rows of the restriction matrix.
This correspond to the `t_test` in models and is not a quadratic form.
Returns
-------
nc : float or ndarray
Noncentrality parameter for Wald tests, correspondig to `wald_test`
or `t_test` depending on whether `joint` is true or not.
It needs to be divided by nobs to obtain effect size.
Notes
-----
Status : experimental, API will likely change | wald_test_noncent_generic | python | statsmodels/statsmodels | statsmodels/stats/contrast.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/contrast.py | BSD-3-Clause |
def _HCCM(results, scale):
'''
sandwich with pinv(x) * diag(scale) * pinv(x).T
where pinv(x) = (X'X)^(-1) X
and scale is (nobs,)
'''
H = np.dot(results.model.pinv_wexog,
scale[:,None]*results.model.pinv_wexog.T)
return H | sandwich with pinv(x) * diag(scale) * pinv(x).T
where pinv(x) = (X'X)^(-1) X
and scale is (nobs,) | _HCCM | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def cov_hc0(results):
"""
See statsmodels.RegressionResults
"""
het_scale = results.resid**2 # or whitened residuals? only OLS?
cov_hc0 = _HCCM(results, het_scale)
return cov_hc0 | See statsmodels.RegressionResults | cov_hc0 | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def _get_sandwich_arrays(results, cov_type=''):
"""Helper function to get scores from results
Parameters
"""
if isinstance(results, tuple):
# assume we have jac and hessian_inv
jac, hessian_inv = results
xu = jac = np.asarray(jac)
hessian_inv = np.asarray(hessian_inv)
elif hasattr(results, 'model'):
if hasattr(results, '_results'):
# remove wrapper
results = results._results
# assume we have a results instance
if hasattr(results.model, 'jac'):
xu = results.model.jac(results.params)
hessian_inv = np.linalg.inv(results.model.hessian(results.params))
elif hasattr(results.model, 'score_obs'):
xu = results.model.score_obs(results.params)
hessian_inv = np.linalg.inv(results.model.hessian(results.params))
else:
xu = results.model.wexog * results.wresid[:, None]
hessian_inv = np.asarray(results.normalized_cov_params)
# experimental support for freq_weights
if hasattr(results.model, 'freq_weights') and not cov_type == 'clu':
# we do not want to square the weights in the covariance calculations
# assumes that freq_weights are incorporated in score_obs or equivalent
# assumes xu/score_obs is 2D
# temporary asarray
xu /= np.sqrt(np.asarray(results.model.freq_weights)[:, None])
else:
raise ValueError('need either tuple of (jac, hessian_inv) or results' +
'instance')
return xu, hessian_inv | Helper function to get scores from results
Parameters | _get_sandwich_arrays | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def _HCCM1(results, scale):
'''
sandwich with pinv(x) * scale * pinv(x).T
where pinv(x) = (X'X)^(-1) X
and scale is (nobs, nobs), or (nobs,) with diagonal matrix diag(scale)
Parameters
----------
results : result instance
need to contain regression results, uses results.model.pinv_wexog
scale : ndarray (nobs,) or (nobs, nobs)
scale matrix, treated as diagonal matrix if scale is one-dimensional
Returns
-------
H : ndarray (k_vars, k_vars)
robust covariance matrix for the parameter estimates
'''
if scale.ndim == 1:
H = np.dot(results.model.pinv_wexog,
scale[:,None]*results.model.pinv_wexog.T)
else:
H = np.dot(results.model.pinv_wexog,
np.dot(scale, results.model.pinv_wexog.T))
return H | sandwich with pinv(x) * scale * pinv(x).T
where pinv(x) = (X'X)^(-1) X
and scale is (nobs, nobs), or (nobs,) with diagonal matrix diag(scale)
Parameters
----------
results : result instance
need to contain regression results, uses results.model.pinv_wexog
scale : ndarray (nobs,) or (nobs, nobs)
scale matrix, treated as diagonal matrix if scale is one-dimensional
Returns
-------
H : ndarray (k_vars, k_vars)
robust covariance matrix for the parameter estimates | _HCCM1 | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def _HCCM2(hessian_inv, scale):
'''
sandwich with (X'X)^(-1) * scale * (X'X)^(-1)
scale is (kvars, kvars)
this uses results.normalized_cov_params for (X'X)^(-1)
Parameters
----------
results : result instance
need to contain regression results, uses results.normalized_cov_params
scale : ndarray (k_vars, k_vars)
scale matrix
Returns
-------
H : ndarray (k_vars, k_vars)
robust covariance matrix for the parameter estimates
'''
if scale.ndim == 1:
scale = scale[:,None]
xxi = hessian_inv
H = np.dot(np.dot(xxi, scale), xxi.T)
return H | sandwich with (X'X)^(-1) * scale * (X'X)^(-1)
scale is (kvars, kvars)
this uses results.normalized_cov_params for (X'X)^(-1)
Parameters
----------
results : result instance
need to contain regression results, uses results.normalized_cov_params
scale : ndarray (k_vars, k_vars)
scale matrix
Returns
-------
H : ndarray (k_vars, k_vars)
robust covariance matrix for the parameter estimates | _HCCM2 | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def weights_bartlett(nlags):
'''Bartlett weights for HAC
this will be moved to another module
Parameters
----------
nlags : int
highest lag in the kernel window, this does not include the zero lag
Returns
-------
kernel : ndarray, (nlags+1,)
weights for Bartlett kernel
'''
#with lag zero
return 1 - np.arange(nlags+1)/(nlags+1.) | Bartlett weights for HAC
this will be moved to another module
Parameters
----------
nlags : int
highest lag in the kernel window, this does not include the zero lag
Returns
-------
kernel : ndarray, (nlags+1,)
weights for Bartlett kernel | weights_bartlett | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def weights_uniform(nlags):
'''uniform weights for HAC
this will be moved to another module
Parameters
----------
nlags : int
highest lag in the kernel window, this does not include the zero lag
Returns
-------
kernel : ndarray, (nlags+1,)
weights for uniform kernel
'''
#with lag zero
return np.ones(nlags+1) | uniform weights for HAC
this will be moved to another module
Parameters
----------
nlags : int
highest lag in the kernel window, this does not include the zero lag
Returns
-------
kernel : ndarray, (nlags+1,)
weights for uniform kernel | weights_uniform | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def S_hac_simple(x, nlags=None, weights_func=weights_bartlett):
'''inner covariance matrix for HAC (Newey, West) sandwich
assumes we have a single time series with zero axis consecutive, equal
spaced time periods
Parameters
----------
x : ndarray (nobs,) or (nobs, k_var)
data, for HAC this is array of x_i * u_i
nlags : int or None
highest lag to include in kernel window. If None, then
nlags = floor(4(T/100)^(2/9)) is used.
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
Returns
-------
S : ndarray, (k_vars, k_vars)
inner covariance matrix for sandwich
Notes
-----
used by cov_hac_simple
options might change when other kernels besides Bartlett are available.
'''
if x.ndim == 1:
x = x[:,None]
n_periods = x.shape[0]
if nlags is None:
nlags = int(np.floor(4 * (n_periods / 100.)**(2./9.)))
weights = weights_func(nlags)
S = weights[0] * np.dot(x.T, x) #weights[0] just for completeness, is 1
for lag in range(1, nlags+1):
s = np.dot(x[lag:].T, x[:-lag])
S += weights[lag] * (s + s.T)
return S | inner covariance matrix for HAC (Newey, West) sandwich
assumes we have a single time series with zero axis consecutive, equal
spaced time periods
Parameters
----------
x : ndarray (nobs,) or (nobs, k_var)
data, for HAC this is array of x_i * u_i
nlags : int or None
highest lag to include in kernel window. If None, then
nlags = floor(4(T/100)^(2/9)) is used.
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
Returns
-------
S : ndarray, (k_vars, k_vars)
inner covariance matrix for sandwich
Notes
-----
used by cov_hac_simple
options might change when other kernels besides Bartlett are available. | S_hac_simple | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def S_white_simple(x):
'''inner covariance matrix for White heteroscedastistity sandwich
Parameters
----------
x : ndarray (nobs,) or (nobs, k_var)
data, for HAC this is array of x_i * u_i
Returns
-------
S : ndarray, (k_vars, k_vars)
inner covariance matrix for sandwich
Notes
-----
this is just dot(X.T, X)
'''
if x.ndim == 1:
x = x[:,None]
return np.dot(x.T, x) | inner covariance matrix for White heteroscedastistity sandwich
Parameters
----------
x : ndarray (nobs,) or (nobs, k_var)
data, for HAC this is array of x_i * u_i
Returns
-------
S : ndarray, (k_vars, k_vars)
inner covariance matrix for sandwich
Notes
-----
this is just dot(X.T, X) | S_white_simple | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def S_hac_groupsum(x, time, nlags=None, weights_func=weights_bartlett):
'''inner covariance matrix for HAC over group sums sandwich
This assumes we have complete equal spaced time periods.
The number of time periods per group need not be the same, but we need
at least one observation for each time period
For a single categorical group only, or a everything else but time
dimension. This first aggregates x over groups for each time period, then
applies HAC on the sum per period.
Parameters
----------
x : ndarray (nobs,) or (nobs, k_var)
data, for HAC this is array of x_i * u_i
time : ndarray, (nobs,)
timeindes, assumed to be integers range(n_periods)
nlags : int or None
highest lag to include in kernel window. If None, then
nlags = floor[4(T/100)^(2/9)] is used.
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
Returns
-------
S : ndarray, (k_vars, k_vars)
inner covariance matrix for sandwich
References
----------
Daniel Hoechle, xtscc paper
Driscoll and Kraay
'''
#needs groupsums
x_group_sums = group_sums(x, time).T #TODO: transpose return in grou_sum
return S_hac_simple(x_group_sums, nlags=nlags, weights_func=weights_func) | inner covariance matrix for HAC over group sums sandwich
This assumes we have complete equal spaced time periods.
The number of time periods per group need not be the same, but we need
at least one observation for each time period
For a single categorical group only, or a everything else but time
dimension. This first aggregates x over groups for each time period, then
applies HAC on the sum per period.
Parameters
----------
x : ndarray (nobs,) or (nobs, k_var)
data, for HAC this is array of x_i * u_i
time : ndarray, (nobs,)
timeindes, assumed to be integers range(n_periods)
nlags : int or None
highest lag to include in kernel window. If None, then
nlags = floor[4(T/100)^(2/9)] is used.
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
Returns
-------
S : ndarray, (k_vars, k_vars)
inner covariance matrix for sandwich
References
----------
Daniel Hoechle, xtscc paper
Driscoll and Kraay | S_hac_groupsum | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def S_crosssection(x, group):
'''inner covariance matrix for White on group sums sandwich
I guess for a single categorical group only,
categorical group, can also be the product/intersection of groups
This is used by cov_cluster and indirectly verified
'''
x_group_sums = group_sums(x, group).T #TODO: why transposed
return S_white_simple(x_group_sums) | inner covariance matrix for White on group sums sandwich
I guess for a single categorical group only,
categorical group, can also be the product/intersection of groups
This is used by cov_cluster and indirectly verified | S_crosssection | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def cov_crosssection_0(results, group):
'''this one is still wrong, use cov_cluster instead'''
#TODO: currently used version of groupsums requires 2d resid
scale = S_crosssection(results.resid[:,None], group)
scale = np.squeeze(scale)
cov = _HCCM1(results, scale)
return cov | this one is still wrong, use cov_cluster instead | cov_crosssection_0 | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def cov_cluster(results, group, use_correction=True):
'''cluster robust covariance matrix
Calculates sandwich covariance matrix for a single cluster, i.e. grouped
variables.
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
use_correction : bool
If true (default), then the small sample correction factor is used.
Returns
-------
cov : ndarray, (k_vars, k_vars)
cluster robust covariance matrix for parameter estimates
Notes
-----
same result as Stata in UCLA example and same as Peterson
'''
#TODO: currently used version of groupsums requires 2d resid
xu, hessian_inv = _get_sandwich_arrays(results, cov_type='clu')
if not hasattr(group, 'dtype') or group.dtype != np.dtype('int'):
clusters, group = np.unique(group, return_inverse=True)
else:
clusters = np.unique(group)
scale = S_crosssection(xu, group)
nobs, k_params = xu.shape
n_groups = len(clusters) #replace with stored group attributes if available
cov_c = _HCCM2(hessian_inv, scale)
if use_correction:
cov_c *= (n_groups / (n_groups - 1.) *
((nobs-1.) / float(nobs - k_params)))
return cov_c | cluster robust covariance matrix
Calculates sandwich covariance matrix for a single cluster, i.e. grouped
variables.
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
use_correction : bool
If true (default), then the small sample correction factor is used.
Returns
-------
cov : ndarray, (k_vars, k_vars)
cluster robust covariance matrix for parameter estimates
Notes
-----
same result as Stata in UCLA example and same as Peterson | cov_cluster | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def cov_cluster_2groups(results, group, group2=None, use_correction=True):
'''cluster robust covariance matrix for two groups/clusters
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
use_correction : bool
If true (default), then the small sample correction factor is used.
Returns
-------
cov_both : ndarray, (k_vars, k_vars)
cluster robust covariance matrix for parameter estimates, for both
clusters
cov_0 : ndarray, (k_vars, k_vars)
cluster robust covariance matrix for parameter estimates for first
cluster
cov_1 : ndarray, (k_vars, k_vars)
cluster robust covariance matrix for parameter estimates for second
cluster
Notes
-----
verified against Peterson's table, (4 decimal print precision)
'''
if group2 is None:
if group.ndim !=2 or group.shape[1] != 2:
raise ValueError('if group2 is not given, then groups needs to be ' +
'an array with two columns')
group0 = group[:, 0]
group1 = group[:, 1]
else:
group0 = group
group1 = group2
group = (group0, group1)
cov0 = cov_cluster(results, group0, use_correction=use_correction)
#[0] because we get still also returns bse
cov1 = cov_cluster(results, group1, use_correction=use_correction)
# cov of cluster formed by intersection of two groups
cov01 = cov_cluster(results,
combine_indices(group)[0],
use_correction=use_correction)
#robust cov matrix for union of groups
cov_both = cov0 + cov1 - cov01
#return all three (for now?)
return cov_both, cov0, cov1 | cluster robust covariance matrix for two groups/clusters
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
use_correction : bool
If true (default), then the small sample correction factor is used.
Returns
-------
cov_both : ndarray, (k_vars, k_vars)
cluster robust covariance matrix for parameter estimates, for both
clusters
cov_0 : ndarray, (k_vars, k_vars)
cluster robust covariance matrix for parameter estimates for first
cluster
cov_1 : ndarray, (k_vars, k_vars)
cluster robust covariance matrix for parameter estimates for second
cluster
Notes
-----
verified against Peterson's table, (4 decimal print precision) | cov_cluster_2groups | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def cov_white_simple(results, use_correction=True):
'''
heteroscedasticity robust covariance matrix (White)
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
Returns
-------
cov : ndarray, (k_vars, k_vars)
heteroscedasticity robust covariance matrix for parameter estimates
Notes
-----
This produces the same result as cov_hc0, and does not include any small
sample correction.
verified (against LinearRegressionResults and Peterson)
See Also
--------
cov_hc1, cov_hc2, cov_hc3 : heteroscedasticity robust covariance matrices
with small sample corrections
'''
xu, hessian_inv = _get_sandwich_arrays(results)
sigma = S_white_simple(xu)
cov_w = _HCCM2(hessian_inv, sigma) #add bread to sandwich
if use_correction:
nobs, k_params = xu.shape
cov_w *= nobs / float(nobs - k_params)
return cov_w | heteroscedasticity robust covariance matrix (White)
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
Returns
-------
cov : ndarray, (k_vars, k_vars)
heteroscedasticity robust covariance matrix for parameter estimates
Notes
-----
This produces the same result as cov_hc0, and does not include any small
sample correction.
verified (against LinearRegressionResults and Peterson)
See Also
--------
cov_hc1, cov_hc2, cov_hc3 : heteroscedasticity robust covariance matrices
with small sample corrections | cov_white_simple | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def cov_hac_simple(results, nlags=None, weights_func=weights_bartlett,
use_correction=True):
'''
heteroscedasticity and autocorrelation robust covariance matrix (Newey-West)
Assumes we have a single time series with zero axis consecutive, equal
spaced time periods
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
nlags : int or None
highest lag to include in kernel window. If None, then
nlags = floor[4(T/100)^(2/9)] is used.
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
Returns
-------
cov : ndarray, (k_vars, k_vars)
HAC robust covariance matrix for parameter estimates
Notes
-----
verified only for nlags=0, which is just White
just guessing on correction factor, need reference
options might change when other kernels besides Bartlett are available.
'''
xu, hessian_inv = _get_sandwich_arrays(results)
sigma = S_hac_simple(xu, nlags=nlags, weights_func=weights_func)
cov_hac = _HCCM2(hessian_inv, sigma)
if use_correction:
nobs, k_params = xu.shape
cov_hac *= nobs / float(nobs - k_params)
return cov_hac | heteroscedasticity and autocorrelation robust covariance matrix (Newey-West)
Assumes we have a single time series with zero axis consecutive, equal
spaced time periods
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
nlags : int or None
highest lag to include in kernel window. If None, then
nlags = floor[4(T/100)^(2/9)] is used.
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
Returns
-------
cov : ndarray, (k_vars, k_vars)
HAC robust covariance matrix for parameter estimates
Notes
-----
verified only for nlags=0, which is just White
just guessing on correction factor, need reference
options might change when other kernels besides Bartlett are available. | cov_hac_simple | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def lagged_groups(x, lag, groupidx):
'''
assumes sorted by time, groupidx is tuple of start and end values
not optimized, just to get a working version, loop over groups
'''
out0 = []
out_lagged = []
for lo, up in groupidx:
if lo+lag < up: #group is longer than lag
out0.append(x[lo+lag:up])
out_lagged.append(x[lo:up-lag])
if out0 == []:
raise ValueError('all groups are empty taking lags')
return np.vstack(out0), np.vstack(out_lagged) | assumes sorted by time, groupidx is tuple of start and end values
not optimized, just to get a working version, loop over groups | lagged_groups | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def S_nw_panel(xw, weights, groupidx):
'''inner covariance matrix for HAC for panel data
no denominator nobs used
no reference for this, just accounting for time indices
'''
nlags = len(weights)-1
S = weights[0] * np.dot(xw.T, xw) #weights just for completeness
for lag in range(1, nlags+1):
xw0, xwlag = lagged_groups(xw, lag, groupidx)
s = np.dot(xw0.T, xwlag)
S += weights[lag] * (s + s.T)
return S | inner covariance matrix for HAC for panel data
no denominator nobs used
no reference for this, just accounting for time indices | S_nw_panel | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def cov_nw_panel(results, nlags, groupidx, weights_func=weights_bartlett,
use_correction='hac'):
'''Panel HAC robust covariance matrix
Assumes we have a panel of time series with consecutive, equal spaced time
periods. Data is assumed to be in long format with time series of each
individual stacked into one array. Panel can be unbalanced.
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
nlags : int or None
Highest lag to include in kernel window. Currently, no default
because the optimal length will depend on the number of observations
per cross-sectional unit.
groupidx : list of tuple
each tuple should contain the start and end index for an individual.
(groupidx might change in future).
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
use_correction : 'cluster' or 'hac' or False
If False, then no small sample correction is used.
If 'cluster' (default), then the same correction as in cov_cluster is
used.
If 'hac', then the same correction as in single time series, cov_hac
is used.
Returns
-------
cov : ndarray, (k_vars, k_vars)
HAC robust covariance matrix for parameter estimates
Notes
-----
For nlags=0, this is just White covariance, cov_white.
If kernel is uniform, `weights_uniform`, with nlags equal to the number
of observations per unit in a balance panel, then cov_cluster and
cov_hac_panel are identical.
Tested against STATA `newey` command with same defaults.
Options might change when other kernels besides Bartlett and uniform are
available.
'''
if nlags == 0: #so we can reproduce HC0 White
weights = [1, 0] #to avoid the scalar check in hac_nw
else:
weights = weights_func(nlags)
xu, hessian_inv = _get_sandwich_arrays(results)
S_hac = S_nw_panel(xu, weights, groupidx)
cov_hac = _HCCM2(hessian_inv, S_hac)
if use_correction:
nobs, k_params = xu.shape
if use_correction == 'hac':
cov_hac *= nobs / float(nobs - k_params)
elif use_correction in ['c', 'clu', 'cluster']:
n_groups = len(groupidx)
cov_hac *= n_groups / (n_groups - 1.)
cov_hac *= ((nobs-1.) / float(nobs - k_params))
return cov_hac | Panel HAC robust covariance matrix
Assumes we have a panel of time series with consecutive, equal spaced time
periods. Data is assumed to be in long format with time series of each
individual stacked into one array. Panel can be unbalanced.
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
nlags : int or None
Highest lag to include in kernel window. Currently, no default
because the optimal length will depend on the number of observations
per cross-sectional unit.
groupidx : list of tuple
each tuple should contain the start and end index for an individual.
(groupidx might change in future).
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
use_correction : 'cluster' or 'hac' or False
If False, then no small sample correction is used.
If 'cluster' (default), then the same correction as in cov_cluster is
used.
If 'hac', then the same correction as in single time series, cov_hac
is used.
Returns
-------
cov : ndarray, (k_vars, k_vars)
HAC robust covariance matrix for parameter estimates
Notes
-----
For nlags=0, this is just White covariance, cov_white.
If kernel is uniform, `weights_uniform`, with nlags equal to the number
of observations per unit in a balance panel, then cov_cluster and
cov_hac_panel are identical.
Tested against STATA `newey` command with same defaults.
Options might change when other kernels besides Bartlett and uniform are
available. | cov_nw_panel | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def cov_nw_groupsum(results, nlags, time, weights_func=weights_bartlett,
use_correction=0):
'''Driscoll and Kraay Panel robust covariance matrix
Robust covariance matrix for panel data of Driscoll and Kraay.
Assumes we have a panel of time series where the time index is available.
The time index is assumed to represent equal spaced periods. At least one
observation per period is required.
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
nlags : int or None
Highest lag to include in kernel window. Currently, no default
because the optimal length will depend on the number of observations
per cross-sectional unit.
time : ndarray of int
this should contain the coding for the time period of each observation.
time periods should be integers in range(maxT) where maxT is obs of i
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
use_correction : 'cluster' or 'hac' or False
If False, then no small sample correction is used.
If 'hac' (default), then the same correction as in single time series, cov_hac
is used.
If 'cluster', then the same correction as in cov_cluster is
used.
Returns
-------
cov : ndarray, (k_vars, k_vars)
HAC robust covariance matrix for parameter estimates
Notes
-----
Tested against STATA xtscc package, which uses no small sample correction
This first averages relevant variables for each time period over all
individuals/groups, and then applies the same kernel weighted averaging
over time as in HAC.
Warning:
In the example with a short panel (few time periods and many individuals)
with mainly across individual variation this estimator did not produce
reasonable results.
Options might change when other kernels besides Bartlett and uniform are
available.
References
----------
Daniel Hoechle, xtscc paper
Driscoll and Kraay
'''
xu, hessian_inv = _get_sandwich_arrays(results)
#S_hac = S_nw_panel(xw, weights, groupidx)
S_hac = S_hac_groupsum(xu, time, nlags=nlags, weights_func=weights_func)
cov_hac = _HCCM2(hessian_inv, S_hac)
if use_correction:
nobs, k_params = xu.shape
if use_correction == 'hac':
cov_hac *= nobs / float(nobs - k_params)
elif use_correction in ['c', 'cluster']:
n_groups = len(np.unique(time))
cov_hac *= n_groups / (n_groups - 1.)
cov_hac *= ((nobs-1.) / float(nobs - k_params))
return cov_hac | Driscoll and Kraay Panel robust covariance matrix
Robust covariance matrix for panel data of Driscoll and Kraay.
Assumes we have a panel of time series where the time index is available.
The time index is assumed to represent equal spaced periods. At least one
observation per period is required.
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
nlags : int or None
Highest lag to include in kernel window. Currently, no default
because the optimal length will depend on the number of observations
per cross-sectional unit.
time : ndarray of int
this should contain the coding for the time period of each observation.
time periods should be integers in range(maxT) where maxT is obs of i
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
use_correction : 'cluster' or 'hac' or False
If False, then no small sample correction is used.
If 'hac' (default), then the same correction as in single time series, cov_hac
is used.
If 'cluster', then the same correction as in cov_cluster is
used.
Returns
-------
cov : ndarray, (k_vars, k_vars)
HAC robust covariance matrix for parameter estimates
Notes
-----
Tested against STATA xtscc package, which uses no small sample correction
This first averages relevant variables for each time period over all
individuals/groups, and then applies the same kernel weighted averaging
over time as in HAC.
Warning:
In the example with a short panel (few time periods and many individuals)
with mainly across individual variation this estimator did not produce
reasonable results.
Options might change when other kernels besides Bartlett and uniform are
available.
References
----------
Daniel Hoechle, xtscc paper
Driscoll and Kraay | cov_nw_groupsum | python | statsmodels/statsmodels | statsmodels/stats/sandwich_covariance.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/sandwich_covariance.py | BSD-3-Clause |
def effectsize_oneway(means, vars_, nobs, use_var="unequal", ddof_between=0):
"""
Effect size corresponding to Cohen's f = nc / nobs for oneway anova
This contains adjustment for Welch and Brown-Forsythe Anova so that
effect size can be used with FTestAnovaPower.
Parameters
----------
means : array_like
Mean of samples to be compared
vars_ : float or array_like
Residual (within) variance of each sample or pooled
If ``vars_`` 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.
ddof_between : int
Degrees of freedom correction for the weighted between sum of squares.
The denominator is ``nobs_total - ddof_between``
This can be used to match differences across reference literature.
Returns
-------
f2 : float
Effect size corresponding to squared Cohen's f, which is also equal
to the noncentrality divided by total number of observations.
Notes
-----
This currently handles the following cases for oneway anova
- balanced sample with homoscedastic variances
- samples with different number of observations and with homoscedastic
variances
- samples with different number of observations and with heteroskedastic
variances. This corresponds to Welch anova
In the case of "unequal" and "bf" methods for unequal variances, the
effect sizes do not directly correspond to the test statistic in Anova.
Both have correction terms dropped or added, so the effect sizes match up
with using FTestAnovaPower.
If all variances are equal, then all three methods result in the same
effect size. If variances are unequal, then the three methods produce
small differences in effect size.
Note, the effect size and power computation for BF Anova was not found in
the literature. The correction terms were added so that FTestAnovaPower
provides a good approximation to the power.
Status: experimental
We might add additional returns, if those are needed to support power
and sample size applications.
Examples
--------
The following shows how to compute effect size and power for each of the
three anova methods. The null hypothesis is that the means are equal which
corresponds to a zero effect size. Under the alternative, means differ
with two sample means at a distance delta from the mean. We assume the
variance is the same under the null and alternative hypothesis.
``nobs`` for the samples defines the fraction of observations in the
samples. ``nobs`` in the power method defines the total sample size.
In simulations, the computed power for standard anova,
i.e.``use_var="equal"`` overestimates the simulated power by a few percent.
The equal variance assumption does not hold in this example.
>>> from statsmodels.stats.oneway import effectsize_oneway
>>> from statsmodels.stats.power import FTestAnovaPower
>>>
>>> nobs = np.array([10, 12, 13, 15])
>>> delta = 0.5
>>> means_alt = np.array([-1, 0, 0, 1]) * delta
>>> vars_ = np.arange(1, len(means_alt) + 1)
>>>
>>> f2_alt = effectsize_oneway(means_alt, vars_, nobs, use_var="equal")
>>> f2_alt
0.04581300813008131
>>>
>>> kwds = {'effect_size': np.sqrt(f2_alt), 'nobs': 100, 'alpha': 0.05,
... 'k_groups': 4}
>>> power = FTestAnovaPower().power(**kwds)
>>> power
0.39165892158983273
>>>
>>> f2_alt = effectsize_oneway(means_alt, vars_, nobs, use_var="unequal")
>>> f2_alt
0.060640138408304504
>>>
>>> kwds['effect_size'] = np.sqrt(f2_alt)
>>> power = FTestAnovaPower().power(**kwds)
>>> power
0.5047366512800622
>>>
>>> f2_alt = effectsize_oneway(means_alt, vars_, nobs, use_var="bf")
>>> f2_alt
0.04391324307956788
>>>
>>> kwds['effect_size'] = np.sqrt(f2_alt)
>>> power = FTestAnovaPower().power(**kwds)
>>> power
0.3765792117047725
"""
# the code here is largely a copy of onway_generic with adjustments
means = np.asarray(means)
n_groups = means.shape[0]
if np.size(nobs) == 1:
nobs = np.ones(n_groups) * nobs
nobs_t = nobs.sum()
if use_var == "equal":
if np.size(vars_) == 1:
var_resid = vars_
else:
vars_ = np.asarray(vars_)
var_resid = ((nobs - 1) * vars_).sum() / (nobs_t - n_groups)
vars_ = var_resid # scalar, if broadcasting works
weights = nobs / vars_
w_total = weights.sum()
w_rel = weights / w_total
# meanw_t = (weights * means).sum() / w_total
meanw_t = w_rel @ means
f2 = np.dot(weights, (means - meanw_t)**2) / (nobs_t - ddof_between)
if use_var.lower() == "bf":
weights = nobs
w_total = weights.sum()
w_rel = weights / w_total
meanw_t = w_rel @ means
# TODO: reuse general case with weights
tmp = ((1. - nobs / nobs_t) * vars_).sum()
statistic = 1. * (nobs * (means - meanw_t)**2).sum()
statistic /= tmp
f2 = statistic * (1. - nobs / nobs_t).sum() / nobs_t
# correction factor for df_num in BFM
df_num2 = n_groups - 1
df_num = tmp**2 / ((vars_**2).sum() +
(nobs / nobs_t * vars_).sum()**2 -
2 * (nobs / nobs_t * vars_**2).sum())
f2 *= df_num / df_num2
return f2 | Effect size corresponding to Cohen's f = nc / nobs for oneway anova
This contains adjustment for Welch and Brown-Forsythe Anova so that
effect size can be used with FTestAnovaPower.
Parameters
----------
means : array_like
Mean of samples to be compared
vars_ : float or array_like
Residual (within) variance of each sample or pooled
If ``vars_`` 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.
ddof_between : int
Degrees of freedom correction for the weighted between sum of squares.
The denominator is ``nobs_total - ddof_between``
This can be used to match differences across reference literature.
Returns
-------
f2 : float
Effect size corresponding to squared Cohen's f, which is also equal
to the noncentrality divided by total number of observations.
Notes
-----
This currently handles the following cases for oneway anova
- balanced sample with homoscedastic variances
- samples with different number of observations and with homoscedastic
variances
- samples with different number of observations and with heteroskedastic
variances. This corresponds to Welch anova
In the case of "unequal" and "bf" methods for unequal variances, the
effect sizes do not directly correspond to the test statistic in Anova.
Both have correction terms dropped or added, so the effect sizes match up
with using FTestAnovaPower.
If all variances are equal, then all three methods result in the same
effect size. If variances are unequal, then the three methods produce
small differences in effect size.
Note, the effect size and power computation for BF Anova was not found in
the literature. The correction terms were added so that FTestAnovaPower
provides a good approximation to the power.
Status: experimental
We might add additional returns, if those are needed to support power
and sample size applications.
Examples
--------
The following shows how to compute effect size and power for each of the
three anova methods. The null hypothesis is that the means are equal which
corresponds to a zero effect size. Under the alternative, means differ
with two sample means at a distance delta from the mean. We assume the
variance is the same under the null and alternative hypothesis.
``nobs`` for the samples defines the fraction of observations in the
samples. ``nobs`` in the power method defines the total sample size.
In simulations, the computed power for standard anova,
i.e.``use_var="equal"`` overestimates the simulated power by a few percent.
The equal variance assumption does not hold in this example.
>>> from statsmodels.stats.oneway import effectsize_oneway
>>> from statsmodels.stats.power import FTestAnovaPower
>>>
>>> nobs = np.array([10, 12, 13, 15])
>>> delta = 0.5
>>> means_alt = np.array([-1, 0, 0, 1]) * delta
>>> vars_ = np.arange(1, len(means_alt) + 1)
>>>
>>> f2_alt = effectsize_oneway(means_alt, vars_, nobs, use_var="equal")
>>> f2_alt
0.04581300813008131
>>>
>>> kwds = {'effect_size': np.sqrt(f2_alt), 'nobs': 100, 'alpha': 0.05,
... 'k_groups': 4}
>>> power = FTestAnovaPower().power(**kwds)
>>> power
0.39165892158983273
>>>
>>> f2_alt = effectsize_oneway(means_alt, vars_, nobs, use_var="unequal")
>>> f2_alt
0.060640138408304504
>>>
>>> kwds['effect_size'] = np.sqrt(f2_alt)
>>> power = FTestAnovaPower().power(**kwds)
>>> power
0.5047366512800622
>>>
>>> f2_alt = effectsize_oneway(means_alt, vars_, nobs, use_var="bf")
>>> f2_alt
0.04391324307956788
>>>
>>> kwds['effect_size'] = np.sqrt(f2_alt)
>>> power = FTestAnovaPower().power(**kwds)
>>> power
0.3765792117047725 | effectsize_oneway | python | statsmodels/statsmodels | statsmodels/stats/oneway.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/oneway.py | BSD-3-Clause |
def convert_effectsize_fsqu(f2=None, eta2=None):
"""Convert squared effect sizes in f family
f2 is signal to noise ratio, var_explained / var_residual
eta2 is proportion of explained variance, var_explained / var_total
uses the relationship:
f2 = eta2 / (1 - eta2)
Parameters
----------
f2 : None or float
Squared Cohen's F effect size. If f2 is not None, then eta2 will be
computed.
eta2 : None or float
Squared eta effect size. If f2 is None and eta2 is not None, then f2 is
computed.
Returns
-------
res : Holder instance
An instance of the Holder class with f2 and eta2 as attributes.
"""
if f2 is not None:
eta2 = 1 / (1 + 1 / f2)
elif eta2 is not None:
f2 = eta2 / (1 - eta2)
res = Holder(f2=f2, eta2=eta2)
return res | Convert squared effect sizes in f family
f2 is signal to noise ratio, var_explained / var_residual
eta2 is proportion of explained variance, var_explained / var_total
uses the relationship:
f2 = eta2 / (1 - eta2)
Parameters
----------
f2 : None or float
Squared Cohen's F effect size. If f2 is not None, then eta2 will be
computed.
eta2 : None or float
Squared eta effect size. If f2 is None and eta2 is not None, then f2 is
computed.
Returns
-------
res : Holder instance
An instance of the Holder class with f2 and eta2 as attributes. | convert_effectsize_fsqu | python | statsmodels/statsmodels | statsmodels/stats/oneway.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/oneway.py | BSD-3-Clause |
def _fstat2effectsize(f_stat, df):
"""Compute anova effect size from F-statistic
This might be combined with convert_effectsize_fsqu
Parameters
----------
f_stat : array_like
Test statistic of an F-test
df : tuple
degrees of freedom ``df = (df1, df2)`` where
- df1 : numerator degrees of freedom, number of constraints
- df2 : denominator degrees of freedom, df_resid
Returns
-------
res : Holder instance
This instance contains effect size measures f2, eta2, omega2 and eps2
as attributes.
Notes
-----
This uses the following definitions:
- f2 = f_stat * df1 / df2
- eta2 = f2 / (f2 + 1)
- omega2 = (f2 - df1 / df2) / (f2 + 2)
- eps2 = (f2 - df1 / df2) / (f2 + 1)
This differs from effect size measures in other function which define
``f2 = f_stat * df1 / nobs``
or an equivalent expression for power computation. The noncentrality
index for the hypothesis test is in those cases given by
``nc = f_stat * df1``.
Currently omega2 and eps2 are computed in two different ways. Those
values agree for regular cases but can show different behavior in corner
cases (e.g. zero division).
"""
df1, df2 = df
f2 = f_stat * df1 / df2
eta2 = f2 / (f2 + 1)
omega2_ = (f_stat - 1) / (f_stat + (df2 + 1) / df1)
omega2 = (f2 - df1 / df2) / (f2 + 1 + 1 / df2) # rewrite
eps2_ = (f_stat - 1) / (f_stat + df2 / df1)
eps2 = (f2 - df1 / df2) / (f2 + 1) # rewrite
return Holder(f2=f2, eta2=eta2, omega2=omega2, eps2=eps2, eps2_=eps2_,
omega2_=omega2_) | Compute anova effect size from F-statistic
This might be combined with convert_effectsize_fsqu
Parameters
----------
f_stat : array_like
Test statistic of an F-test
df : tuple
degrees of freedom ``df = (df1, df2)`` where
- df1 : numerator degrees of freedom, number of constraints
- df2 : denominator degrees of freedom, df_resid
Returns
-------
res : Holder instance
This instance contains effect size measures f2, eta2, omega2 and eps2
as attributes.
Notes
-----
This uses the following definitions:
- f2 = f_stat * df1 / df2
- eta2 = f2 / (f2 + 1)
- omega2 = (f2 - df1 / df2) / (f2 + 2)
- eps2 = (f2 - df1 / df2) / (f2 + 1)
This differs from effect size measures in other function which define
``f2 = f_stat * df1 / nobs``
or an equivalent expression for power computation. The noncentrality
index for the hypothesis test is in those cases given by
``nc = f_stat * df1``.
Currently omega2 and eps2 are computed in two different ways. Those
values agree for regular cases but can show different behavior in corner
cases (e.g. zero division). | _fstat2effectsize | python | statsmodels/statsmodels | statsmodels/stats/oneway.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/oneway.py | BSD-3-Clause |
def wellek_to_f2(eps, n_groups):
"""Convert Wellek's effect size (sqrt) to Cohen's f-squared
This computes the following effect size :
f2 = 1 / n_groups * eps**2
Parameters
----------
eps : float or ndarray
Wellek's effect size used in anova equivalence test
n_groups : int
Number of groups in oneway comparison
Returns
-------
f2 : effect size Cohen's f-squared
"""
f2 = 1 / n_groups * eps**2
return f2 | Convert Wellek's effect size (sqrt) to Cohen's f-squared
This computes the following effect size :
f2 = 1 / n_groups * eps**2
Parameters
----------
eps : float or ndarray
Wellek's effect size used in anova equivalence test
n_groups : int
Number of groups in oneway comparison
Returns
-------
f2 : effect size Cohen's f-squared | wellek_to_f2 | python | statsmodels/statsmodels | statsmodels/stats/oneway.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/oneway.py | BSD-3-Clause |
def f2_to_wellek(f2, n_groups):
"""Convert Cohen's f-squared to Wellek's effect size (sqrt)
This computes the following effect size :
eps = sqrt(n_groups * f2)
Parameters
----------
f2 : float or ndarray
Effect size Cohen's f-squared
n_groups : int
Number of groups in oneway comparison
Returns
-------
eps : float or ndarray
Wellek's effect size used in anova equivalence test
"""
eps = np.sqrt(n_groups * f2)
return eps | Convert Cohen's f-squared to Wellek's effect size (sqrt)
This computes the following effect size :
eps = sqrt(n_groups * f2)
Parameters
----------
f2 : float or ndarray
Effect size Cohen's f-squared
n_groups : int
Number of groups in oneway comparison
Returns
-------
eps : float or ndarray
Wellek's effect size used in anova equivalence test | f2_to_wellek | python | statsmodels/statsmodels | statsmodels/stats/oneway.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/oneway.py | BSD-3-Clause |
def fstat_to_wellek(f_stat, n_groups, nobs_mean):
"""Convert F statistic to wellek's effect size eps squared
This computes the following effect size :
es = f_stat * (n_groups - 1) / nobs_mean
Parameters
----------
f_stat : float or ndarray
Test statistic of an F-test.
n_groups : int
Number of groups in oneway comparison
nobs_mean : float or ndarray
Average number of observations across groups.
Returns
-------
eps : float or ndarray
Wellek's effect size used in anova equivalence test
"""
es = f_stat * (n_groups - 1) / nobs_mean
return es | Convert F statistic to wellek's effect size eps squared
This computes the following effect size :
es = f_stat * (n_groups - 1) / nobs_mean
Parameters
----------
f_stat : float or ndarray
Test statistic of an F-test.
n_groups : int
Number of groups in oneway comparison
nobs_mean : float or ndarray
Average number of observations across groups.
Returns
-------
eps : float or ndarray
Wellek's effect size used in anova equivalence test | fstat_to_wellek | python | statsmodels/statsmodels | statsmodels/stats/oneway.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/oneway.py | BSD-3-Clause |
def confint_noncentrality(f_stat, df, alpha=0.05,
alternative="two-sided"):
"""
Confidence interval for noncentrality parameter in F-test
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
alternative : {"two-sided"}
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 | Confidence interval for noncentrality parameter in F-test
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
alternative : {"two-sided"}
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 | confint_noncentrality | python | statsmodels/statsmodels | statsmodels/stats/oneway.py | https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/oneway.py | BSD-3-Clause |
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