def confint_noncentrality(f_stat,
df1,
df2,
alpha=0.05,
alternative="two-sided"):
"""confidence interval for noncentality parameter in F-test
This does not yet handle non-negativity constraint on nc.
Currently only two-sided alternative is supported.
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
----------
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`
"""
if alternative in ["two-sided", "2s", "ts"]:
alpha1s = alpha / 2
ci = ncfdtrinc(df1, df2, [1 - alpha1s, alpha1s], f_stat)
else:
raise NotImplementedError
return ci
def _noncentrality_f(f_stat, df1, df2, alpha=0.05):
"""noncentrality parameter for f statistic
`nc` is zero-truncated umvue
Parameters
----------
fstat : float
f-statistic, for example from a hypothesis test
df : int or float
Degrees of freedom
alpha : float in (0, 1)
Significance level for the confidence interval, covarage is 1 - alpha.
Returns
-------
HolderTuple
The main attributes are
- ``nc`` : estimate of noncentrality parameter
- ``confint`` : lower and upper bound of confidence interval for `nc``
Other attributes are estimates for nc by different methods.
References
----------
.. [1] Kubokawa, T., C.P. Robert, and A.K.Md.E. Saleh. 1993. “Estimation of
Noncentrality Parameters.” Canadian Journal of Statistics 21 (1): 45–57.
https://doi.org/10.2307/3315657.
"""
alpha_half = alpha / 2
x_s = f_stat * df1 / df2
nc_umvue = (df2 - 2) * x_s - df1
nc = np.maximum(nc_umvue, 0)
nc_krs = np.maximum(nc_umvue, x_s * 2 * (df2 - 1) / (df1 + 2))
nc_median = special.ncfdtrinc(df1, df2, 0.5, f_stat)
ci = special.ncfdtrinc(df1, df2, [1 - alpha_half, alpha_half], f_stat)
res = Holder(nc=nc,
confint=ci,
nc_umvue=nc_umvue,
nc_krs=nc_krs,
nc_median=nc_median,
name="Noncentrality for F-distributed random variable"
)
return res
def _compute_ncp_f(df1, df2, power=1 / 3):
"""
Uses the inverse function of the non-central F distribution with regard to NCP to recover the NCP corresponding
to a given level of power for a given F test.
:param df1: Numerator degrees of freedom
:param df2: Denominator degrees of freedom
:param power: Desired level of power
:return:
"""
critval = f._ppf(.95, df1, df2)
return ncfdtrinc(df1, df2, 1 - power, critval)
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
def _calc_bound(self, is_multirep, alphatest, alpha, lower_tail_prob,
upper_tail_prob, prob, noncentrality, cl_type, dfe1, dfe2,
dfh, fcrit, omega, tolerance, **kwargs):
"""Calculate power bounds """
df1_unirep = None
if 'df1_unirep' in kwargs.keys():
df1_unirep = kwargs['df1_unirep']
fmethod = Constants.FMETHOD_MISSING
if alpha > tolerance:
if cl_type == Constants.CLTYPE_DESIRED_KNOWN:
if is_multirep:
chi = chi2.ppf(lower_tail_prob, dfe1)
noncentrality = (chi / dfe1) * omega
else:
chi = chi2.ppf(lower_tail_prob, dfh)
noncentrality = (chi / dfh) * omega
elif cl_type == Constants.CLTYPE_DESIRED_ESTIMATE:
f_a = omega / dfh
bound = finv(upper_tail_prob, dfh, dfe1)
if f_a <= bound:
noncentrality = 0
else:
noncentrality = special.ncfdtrinc(dfh, dfe1,
upper_tail_prob, f_a)
if is_multirep:
prob, fmethod = probf(fcrit, dfh, dfe2, noncentrality)
else:
prob, fmethod = probf(fcrit, df1_unirep, dfe2, noncentrality)
if fmethod == Constants.FMETHOD_NORMAL_LR and prob == 1:
power_bound = alphatest
else:
power_bound = 1 - prob
power = Power()
power.power = power_bound
power.fmethod = fmethod
power.noncentrality_parameter = noncentrality
return power