"""
Routines for computing pvalues
Similar implementation can be found in https://github.com/scikit-hep/pyhf/blob/main/src/pyhf/infer/calculators.py
"""
import warnings
from typing import List, Tuple, Union
import numpy as np
from .distributions import (
AsymptoticTestStatisticsDistribution,
EmpricTestStatisticsDistribution,
)
__all__ = ["pvalues", "expected_pvalues"]
def __dir__():
return __all__
[docs]
def pvalues(
delta_test_statistic: float,
sig_plus_bkg_distribution: Union[
AsymptoticTestStatisticsDistribution, EmpricTestStatisticsDistribution
],
bkg_only_distribution: Union[
AsymptoticTestStatisticsDistribution, EmpricTestStatisticsDistribution
],
) -> Tuple[float, float, float]:
r"""
Calculate the p-values for the observed test statistic under the
signal + background and background-only model hypotheses.
.. math::
p_{s+b}&=& \int_{-\infty}^{-\sqrt{q_{\mu,A}} - \Delta q_\mu} \mathcal{N}(x| 0, 1) dx \\
p_{b}&=& \int_{-\infty}^{-\Delta q_\mu} \mathcal{N}(x| 0, 1) dx \\
p_{s} &=& p_{s+b}/ p_{b}
where :math:`q_\mu` stands for the test statistic and A stands for Assimov.
Args:
delta_test_statistic (``float``): :math:`\Delta q_\mu`
sig_plus_bkg_distribution (~spey.hypothesis_testing.asymptotic_calculator.AsymptoticTestStatisticsDistribution):
the distribution for the signal + background hypothesis.
bkg_only_distribution (~spey.hypothesis_testing.asymptotic_calculator.AsymptoticTestStatisticsDistribution):
The distribution for the background-only hypothesis.
Returns:
``Tuple[float, float, float]``:
The p-values for the test statistic corresponding to the :math:`p_{s+b}`,
:math:`p_{b}`, and :math:`p_{s}`.
.. seealso::
:func:`~spey.hypothesis_testing.test_statistics.compute_teststatistics`,
:func:`~spey.hypothesis_testing.asymptotic_calculator.compute_asymptotic_confidence_level`,
:func:`~spey.hypothesis_testing.utils.expected_pvalues`
"""
# delta_test_statistic is sqrt_qmu - sqrt_qmuA
# p_sb is CDF(-sqrt_qmuA - delta_test_statistic) = CDF(-sqrt_qmuA - (sqrt_qmu - sqrt_qmuA))
# CDF(-sqrt_qmu) = 1-CDF(sqrt_qmu)
# See eq 58-59 in https://arxiv.org/pdf/1007.1727.pdf
p_sb = sig_plus_bkg_distribution.pvalue(delta_test_statistic)
# p_b is CDF(delta_test_statistic) = CDF(-(sqrt_qmu - sqrt_qmuA))
p_b = bkg_only_distribution.pvalue(delta_test_statistic)
with warnings.catch_warnings(record=True):
p_s = np.true_divide(p_sb, p_b, dtype=np.float64)
return p_sb, p_b, p_s if p_b != 0.0 else 0.0
[docs]
def expected_pvalues(
sig_plus_bkg_distribution: Union[
AsymptoticTestStatisticsDistribution, EmpricTestStatisticsDistribution
],
bkg_only_distribution: Union[
AsymptoticTestStatisticsDistribution, EmpricTestStatisticsDistribution
],
) -> List[List]:
r"""
Calculate the :math:`p` values corresponding to the
median significance of variations of the signal strength from the
background only hypothesis :math:`\mu=0` at :math:`(-2,-1,0,1,2)\sigma`.
.. math::
p_{s+b}&=& \int_{-\infty}^{-\sqrt{q_{\mu,A}} - N\sigma} \mathcal{N}(x| 0, 1) dx \\
p_{b}&=& \int_{-\infty}^{-N\sigma} \mathcal{N}(x| 0, 1) dx \\
p_{s} &=& p_{s+b}/ p_{b}
where :math:`q_\mu` stands for the test statistic and A stands for Assimov.
:math:`N\sigma\in[-2,-1,0,1,2]`.
Args:
sig_plus_bkg_distribution (~spey.hypothesis_testing.asymptotic_calculator.AsymptoticTestStatisticsDistribution):
The distribution for the signal + background hypothesis.
bkg_only_distribution (~spey.hypothesis_testing.asymptotic_calculator.AsymptoticTestStatisticsDistribution):
The distribution for the background-only hypothesis.
Returns:
``List[List]``:
The p-values for the test statistic corresponding to the :math:`p_{s+b}`,
:math:`p_{b}`, and :math:`p_{s}`.
.. seealso::
:func:`~spey.hypothesis_testing.test_statistics.compute_teststatistics`,
:func:`~spey.hypothesis_testing.asymptotic_calculator.compute_asymptotic_confidence_level`,
:func:`~spey.hypothesis_testing.utils.pvalues`
"""
return list(
map(
list,
zip(
*[
pvalues(
bkg_only_distribution.expected_value(nsigma),
sig_plus_bkg_distribution,
bkg_only_distribution,
)
for nsigma in [2.0, 1.0, 0.0, -1.0, -2.0]
]
),
)
)
