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