import numpy as np
from scipy.special import ndtri
from scipy.optimize import brentq
from ._discrete_distns import nchypergeom_fisher
from ._common import ConfidenceInterval
def _sample_odds_ratio(table):
"""
Given a table [[a, b], [c, d]], compute a*d/(b*c).
Return nan if the numerator and denominator are 0.
Return inf if just the denominator is 0.
"""
# table must be a 2x2 numpy array.
if table[1, 0] > 0 and table[0, 1] > 0:
oddsratio = table[0, 0] * table[1, 1] / (table[1, 0] * table[0, 1])
elif table[0, 0] == 0 or table[1, 1] == 0:
oddsratio = np.nan
else:
oddsratio = np.inf
return oddsratio
def _solve(func):
"""
Solve func(nc) = 0. func must be an increasing function.
"""
# We could just as well call the variable `x` instead of `nc`, but we
# always call this function with functions for which nc (the noncentrality
# parameter) is the variable for which we are solving.
nc = 1.0
value = func(nc)
if value == 0:
return nc
# Multiplicative factor by which to increase or decrease nc when
# searching for a bracketing interval.
factor = 2.0
# Find a bracketing interval.
if value > 0:
nc /= factor
while func(nc) > 0:
nc /= factor
lo = nc
hi = factor*nc
else:
nc *= factor
while func(nc) < 0:
nc *= factor
lo = nc/factor
hi = nc
# lo and hi bracket the solution for nc.
nc = brentq(func, lo, hi, xtol=1e-13)
return nc
def _nc_hypergeom_mean_inverse(x, M, n, N):
"""
For the given noncentral hypergeometric parameters x, M, n,and N
(table[0,0], total, row 0 sum and column 0 sum, resp., of a 2x2
contingency table), find the noncentrality parameter of Fisher's
noncentral hypergeometric distribution whose mean is x.
"""
nc = _solve(lambda nc: nchypergeom_fisher.mean(M, n, N, nc) - x)
return nc
def _hypergeom_params_from_table(table):
# The notation M, n and N is consistent with stats.hypergeom and
# stats.nchypergeom_fisher.
x = table[0, 0]
M = table.sum()
n = table[0].sum()
N = table[:, 0].sum()
return x, M, n, N
def _ci_upper(table, alpha):
"""
Compute the upper end of the confidence interval.
"""
if _sample_odds_ratio(table) == np.inf:
return np.inf
x, M, n, N = _hypergeom_params_from_table(table)
# nchypergeom_fisher.cdf is a decreasing function of nc, so we negate
# it in the lambda expression.
nc = _solve(lambda nc: -nchypergeom_fisher.cdf(x, M, n, N, nc) + alpha)
return nc
def _ci_lower(table, alpha):
"""
Compute the lower end of the confidence interval.
"""
if _sample_odds_ratio(table) == 0:
return 0
x, M, n, N = _hypergeom_params_from_table(table)
nc = _solve(lambda nc: nchypergeom_fisher.sf(x - 1, M, n, N, nc) - alpha)
return nc
def _conditional_oddsratio(table):
"""
Conditional MLE of the odds ratio for the 2x2 contingency table.
"""
x, M, n, N = _hypergeom_params_from_table(table)
# Get the bounds of the support. The support of the noncentral
# hypergeometric distribution with parameters M, n, and N is the same
# for all values of the noncentrality parameter, so we can use 1 here.
lo, hi = nchypergeom_fisher.support(M, n, N, 1)
# Check if x is at one of the extremes of the support. If so, we know
# the odds ratio is either 0 or inf.
if x == lo:
# x is at the low end of the support.
return 0
if x == hi:
# x is at the high end of the support.
return np.inf
nc = _nc_hypergeom_mean_inverse(x, M, n, N)
return nc
def _conditional_oddsratio_ci(table, confidence_level=0.95,
alternative='two-sided'):
"""
Conditional exact confidence interval for the odds ratio.
"""
if alternative == 'two-sided':
alpha = 0.5*(1 - confidence_level)
lower = _ci_lower(table, alpha)
upper = _ci_upper(table, alpha)
elif alternative == 'less':
lower = 0.0
upper = _ci_upper(table, 1 - confidence_level)
else:
# alternative == 'greater'
lower = _ci_lower(table, 1 - confidence_level)
upper = np.inf
return lower, upper
def _sample_odds_ratio_ci(table, confidence_level=0.95,
alternative='two-sided'):
oddsratio = _sample_odds_ratio(table)
log_or = np.log(oddsratio)
se = np.sqrt((1/table).sum())
if alternative == 'less':
z = ndtri(confidence_level)
loglow = -np.inf
loghigh = log_or + z*se
elif alternative == 'greater':
z = ndtri(confidence_level)
loglow = log_or - z*se
loghigh = np.inf
else:
# alternative is 'two-sided'
z = ndtri(0.5*confidence_level + 0.5)
loglow = log_or - z*se
loghigh = log_or + z*se
return np.exp(loglow), np.exp(loghigh)
class OddsRatioResult:
"""
Result of `scipy.stats.contingency.odds_ratio`. See the
docstring for `odds_ratio` for more details.
Attributes
----------
statistic : float
The computed odds ratio.
* If `kind` is ``'sample'``, this is sample (or unconditional)
estimate, given by
``table[0, 0]*table[1, 1]/(table[0, 1]*table[1, 0])``.
* If `kind` is ``'conditional'``, this is the conditional
maximum likelihood estimate for the odds ratio. It is
the noncentrality parameter of Fisher's noncentral
hypergeometric distribution with the same hypergeometric
parameters as `table` and whose mean is ``table[0, 0]``.
Methods
-------
confidence_interval :
Confidence interval for the odds ratio.
"""
def __init__(self, _table, _kind, statistic):
# for now, no need to make _table and _kind public, since this sort of
# information is returned in very few `scipy.stats` results
self._table = _table
self._kind = _kind
self.statistic = statistic
def __repr__(self):
return f"OddsRatioResult(statistic={self.statistic})"
def confidence_interval(self, confidence_level=0.95,
alternative='two-sided'):
"""
Confidence interval for the odds ratio.
Parameters
----------
confidence_level: float
Desired confidence level for the confidence interval.
The value must be given as a fraction between 0 and 1.
Default is 0.95 (meaning 95%).
alternative : {'two-sided', 'less', 'greater'}, optional
The alternative hypothesis of the hypothesis test to which the
confidence interval corresponds. That is, suppose the null
hypothesis is that the true odds ratio equals ``OR`` and the
confidence interval is ``(low, high)``. Then the following options
for `alternative` are available (default is 'two-sided'):
* 'two-sided': the true odds ratio is not equal to ``OR``. There
is evidence against the null hypothesis at the chosen
`confidence_level` if ``high < OR`` or ``low > OR``.
* 'less': the true odds ratio is less than ``OR``. The ``low`` end
of the confidence interval is 0, and there is evidence against
the null hypothesis at the chosen `confidence_level` if
``high < OR``.
* 'greater': the true odds ratio is greater than ``OR``. The
``high`` end of the confidence interval is ``np.inf``, and there
is evidence against the null hypothesis at the chosen
`confidence_level` if ``low > OR``.
Returns
-------
ci : ``ConfidenceInterval`` instance
The confidence interval, represented as an object with
attributes ``low`` and ``high``.
Notes
-----
When `kind` is ``'conditional'``, the limits of the confidence
interval are the conditional "exact confidence limits" as described
by Fisher [1]_. The conditional odds ratio and confidence interval are
also discussed in Section 4.1.2 of the text by Sahai and Khurshid [2]_.
When `kind` is ``'sample'``, the confidence interval is computed
under the assumption that the logarithm of the odds ratio is normally
distributed with standard error given by::
se = sqrt(1/a + 1/b + 1/c + 1/d)
where ``a``, ``b``, ``c`` and ``d`` are the elements of the
contingency table. (See, for example, [2]_, section 3.1.3.2,
or [3]_, section 2.3.3).
References
----------
.. [1] R. A. Fisher (1935), The logic of inductive inference,
Journal of the Royal Statistical Society, Vol. 98, No. 1,
pp. 39-82.
.. [2] H. Sahai and A. Khurshid (1996), Statistics in Epidemiology:
Methods, Techniques, and Applications, CRC Press LLC, Boca
Raton, Florida.
.. [3] Alan Agresti, An Introduction to Categorical Data Analysis
(second edition), Wiley, Hoboken, NJ, USA (2007).
"""
if alternative not in ['two-sided', 'less', 'greater']:
raise ValueError("`alternative` must be 'two-sided', 'less' or "
"'greater'.")
if confidence_level < 0 or confidence_level > 1:
raise ValueError('confidence_level must be between 0 and 1')
if self._kind == 'conditional':
ci = self._conditional_odds_ratio_ci(confidence_level, alternative)
else:
ci = self._sample_odds_ratio_ci(confidence_level, alternative)
return ci
def _conditional_odds_ratio_ci(self, confidence_level=0.95,
alternative='two-sided'):
"""
Confidence interval for the conditional odds ratio.
"""
table = self._table
if 0 in table.sum(axis=0) or 0 in table.sum(axis=1):
# If both values in a row or column are zero, the p-value is 1,
# the odds ratio is NaN and the confidence interval is (0, inf).
ci = (0, np.inf)
else:
ci = _conditional_oddsratio_ci(table,
confidence_level=confidence_level,
alternative=alternative)
return ConfidenceInterval(low=ci[0], high=ci[1])
def _sample_odds_ratio_ci(self, confidence_level=0.95,
alternative='two-sided'):
"""
Confidence interval for the sample odds ratio.
"""
if confidence_level < 0 or confidence_level > 1:
raise ValueError('confidence_level must be between 0 and 1')
table = self._table
if 0 in table.sum(axis=0) or 0 in table.sum(axis=1):
# If both values in a row or column are zero, the p-value is 1,
# the odds ratio is NaN and the confidence interval is (0, inf).
ci = (0, np.inf)
else:
ci = _sample_odds_ratio_ci(table,
confidence_level=confidence_level,
alternative=alternative)
return ConfidenceInterval(low=ci[0], high=ci[1])
def odds_ratio(table, *, kind='conditional'):
r"""
Compute the odds ratio for a 2x2 contingency table.
Parameters
----------
table : array_like of ints
A 2x2 contingency table. Elements must be non-negative integers.
kind : str, optional
Which kind of odds ratio to compute, either the sample
odds ratio (``kind='sample'``) or the conditional odds ratio
(``kind='conditional'``). Default is ``'conditional'``.
Returns
-------
result : `~scipy.stats._result_classes.OddsRatioResult` instance
The returned object has two computed attributes:
statistic : float
* If `kind` is ``'sample'``, this is sample (or unconditional)
estimate, given by
``table[0, 0]*table[1, 1]/(table[0, 1]*table[1, 0])``.
* If `kind` is ``'conditional'``, this is the conditional
maximum likelihood estimate for the odds ratio. It is
the noncentrality parameter of Fisher's noncentral
hypergeometric distribution with the same hypergeometric
parameters as `table` and whose mean is ``table[0, 0]``.
The object has the method `confidence_interval` that computes
the confidence interval of the odds ratio.
See Also
--------
scipy.stats.fisher_exact
relative_risk
Notes
-----
The conditional odds ratio was discussed by Fisher (see "Example 1"
of [1]_). Texts that cover the odds ratio include [2]_ and [3]_.
.. versionadded:: 1.10.0
References
----------
.. [1] R. A. Fisher (1935), The logic of inductive inference,
Journal of the Royal Statistical Society, Vol. 98, No. 1,
pp. 39-82.
.. [2] Breslow NE, Day NE (1980). Statistical methods in cancer research.
Volume I - The analysis of case-control studies. IARC Sci Publ.
(32):5-338. PMID: 7216345. (See section 4.2.)
.. [3] H. Sahai and A. Khurshid (1996), Statistics in Epidemiology:
Methods, Techniques, and Applications, CRC Press LLC, Boca
Raton, Florida.
.. [4] Berger, Jeffrey S. et al. "Aspirin for the Primary Prevention of
Cardiovascular Events in Women and Men: A Sex-Specific
Meta-analysis of Randomized Controlled Trials."
JAMA, 295(3):306-313, :doi:`10.1001/jama.295.3.306`, 2006.
Examples
--------
In epidemiology, individuals are classified as "exposed" or
"unexposed" to some factor or treatment. If the occurrence of some
illness is under study, those who have the illness are often
classified as "cases", and those without it are "noncases". The
counts of the occurrences of these classes gives a contingency
table::
exposed unexposed
cases a b
noncases c d
The sample odds ratio may be written ``(a/c) / (b/d)``. ``a/c`` can
be interpreted as the odds of a case occurring in the exposed group,
and ``b/d`` as the odds of a case occurring in the unexposed group.
The sample odds ratio is the ratio of these odds. If the odds ratio
is greater than 1, it suggests that there is a positive association
between being exposed and being a case.
Interchanging the rows or columns of the contingency table inverts
the odds ratio, so it is import to understand the meaning of labels
given to the rows and columns of the table when interpreting the
odds ratio.
In [4]_, the use of aspirin to prevent cardiovascular events in women
and men was investigated. The study notably concluded:
...aspirin therapy reduced the risk of a composite of
cardiovascular events due to its effect on reducing the risk of
ischemic stroke in women [...]
The article lists studies of various cardiovascular events. Let's
focus on the ischemic stoke in women.
The following table summarizes the results of the experiment in which
participants took aspirin or a placebo on a regular basis for several
years. Cases of ischemic stroke were recorded::
Aspirin Control/Placebo
Ischemic stroke 176 230
No stroke 21035 21018
The question we ask is "Is there evidence that the aspirin reduces the
risk of ischemic stroke?"
Compute the odds ratio:
>>> from scipy.stats.contingency import odds_ratio
>>> res = odds_ratio([[176, 230], [21035, 21018]])
>>> res.statistic
0.7646037659999126
For this sample, the odds of getting an ischemic stroke for those who have
been taking aspirin are 0.76 times that of those
who have received the placebo.
To make statistical inferences about the population under study,
we can compute the 95% confidence interval for the odds ratio:
>>> res.confidence_interval(confidence_level=0.95)
ConfidenceInterval(low=0.6241234078749812, high=0.9354102892100372)
The 95% confidence interval for the conditional odds ratio is
approximately (0.62, 0.94).
The fact that the entire 95% confidence interval falls below 1 supports
the authors' conclusion that the aspirin was associated with a
statistically significant reduction in ischemic stroke.
"""
if kind not in ['conditional', 'sample']:
raise ValueError("`kind` must be 'conditional' or 'sample'.")
c = np.asarray(table)
if c.shape != (2, 2):
raise ValueError(f"Invalid shape {c.shape}. The input `table` must be "
"of shape (2, 2).")
if not np.issubdtype(c.dtype, np.integer):
raise ValueError("`table` must be an array of integers, but got "
f"type {c.dtype}")
c = c.astype(np.int64)
if np.any(c < 0):
raise ValueError("All values in `table` must be nonnegative.")
if 0 in c.sum(axis=0) or 0 in c.sum(axis=1):
# If both values in a row or column are zero, the p-value is NaN and
# the odds ratio is NaN.
result = OddsRatioResult(_table=c, _kind=kind, statistic=np.nan)
return result
if kind == 'sample':
oddsratio = _sample_odds_ratio(c)
else: # kind is 'conditional'
oddsratio = _conditional_oddsratio(c)
result = OddsRatioResult(_table=c, _kind=kind, statistic=oddsratio)
return result