from math import sqrt import numpy as np from scipy._lib._util import _validate_int from scipy.optimize import brentq from scipy.special import ndtri from ._discrete_distns import binom from ._common import ConfidenceInterval class BinomTestResult: """ Result of `scipy.stats.binomtest`. Attributes ---------- k : int The number of successes (copied from `binomtest` input). n : int The number of trials (copied from `binomtest` input). alternative : str Indicates the alternative hypothesis specified in the input to `binomtest`. It will be one of ``'two-sided'``, ``'greater'``, or ``'less'``. statistic: float The estimate of the proportion of successes. pvalue : float The p-value of the hypothesis test. """ def __init__(self, k, n, alternative, statistic, pvalue): self.k = k self.n = n self.alternative = alternative self.statistic = statistic self.pvalue = pvalue # add alias for backward compatibility self.proportion_estimate = statistic def __repr__(self): s = ("BinomTestResult(" f"k={self.k}, " f"n={self.n}, " f"alternative={self.alternative!r}, " f"statistic={self.statistic}, " f"pvalue={self.pvalue})") return s def proportion_ci(self, confidence_level=0.95, method='exact'): """ Compute the confidence interval for ``statistic``. Parameters ---------- confidence_level : float, optional Confidence level for the computed confidence interval of the estimated proportion. Default is 0.95. method : {'exact', 'wilson', 'wilsoncc'}, optional Selects the method used to compute the confidence interval for the estimate of the proportion: 'exact' : Use the Clopper-Pearson exact method [1]_. 'wilson' : Wilson's method, without continuity correction ([2]_, [3]_). 'wilsoncc' : Wilson's method, with continuity correction ([2]_, [3]_). Default is ``'exact'``. Returns ------- ci : ``ConfidenceInterval`` object The object has attributes ``low`` and ``high`` that hold the lower and upper bounds of the confidence interval. References ---------- .. [1] C. J. Clopper and E. S. Pearson, The use of confidence or fiducial limits illustrated in the case of the binomial, Biometrika, Vol. 26, No. 4, pp 404-413 (Dec. 1934). .. [2] E. B. Wilson, Probable inference, the law of succession, and statistical inference, J. Amer. Stat. Assoc., 22, pp 209-212 (1927). .. [3] Robert G. Newcombe, Two-sided confidence intervals for the single proportion: comparison of seven methods, Statistics in Medicine, 17, pp 857-872 (1998). Examples -------- >>> from scipy.stats import binomtest >>> result = binomtest(k=7, n=50, p=0.1) >>> result.statistic 0.14 >>> result.proportion_ci() ConfidenceInterval(low=0.05819170033997342, high=0.26739600249700846) """ if method not in ('exact', 'wilson', 'wilsoncc'): raise ValueError(f"method ('{method}') must be one of 'exact', " "'wilson' or 'wilsoncc'.") if not (0 <= confidence_level <= 1): raise ValueError(f'confidence_level ({confidence_level}) must be in ' 'the interval [0, 1].') if method == 'exact': low, high = _binom_exact_conf_int(self.k, self.n, confidence_level, self.alternative) else: # method is 'wilson' or 'wilsoncc' low, high = _binom_wilson_conf_int(self.k, self.n, confidence_level, self.alternative, correction=method == 'wilsoncc') return ConfidenceInterval(low=low, high=high) def _findp(func): try: p = brentq(func, 0, 1) except RuntimeError: raise RuntimeError('numerical solver failed to converge when ' 'computing the confidence limits') from None except ValueError as exc: raise ValueError('brentq raised a ValueError; report this to the ' 'SciPy developers') from exc return p def _binom_exact_conf_int(k, n, confidence_level, alternative): """ Compute the estimate and confidence interval for the binomial test. Returns proportion, prop_low, prop_high """ if alternative == 'two-sided': alpha = (1 - confidence_level) / 2 if k == 0: plow = 0.0 else: plow = _findp(lambda p: binom.sf(k-1, n, p) - alpha) if k == n: phigh = 1.0 else: phigh = _findp(lambda p: binom.cdf(k, n, p) - alpha) elif alternative == 'less': alpha = 1 - confidence_level plow = 0.0 if k == n: phigh = 1.0 else: phigh = _findp(lambda p: binom.cdf(k, n, p) - alpha) elif alternative == 'greater': alpha = 1 - confidence_level if k == 0: plow = 0.0 else: plow = _findp(lambda p: binom.sf(k-1, n, p) - alpha) phigh = 1.0 return plow, phigh def _binom_wilson_conf_int(k, n, confidence_level, alternative, correction): # This function assumes that the arguments have already been validated. # In particular, `alternative` must be one of 'two-sided', 'less' or # 'greater'. p = k / n if alternative == 'two-sided': z = ndtri(0.5 + 0.5*confidence_level) else: z = ndtri(confidence_level) # For reference, the formulas implemented here are from # Newcombe (1998) (ref. [3] in the proportion_ci docstring). denom = 2*(n + z**2) center = (2*n*p + z**2)/denom q = 1 - p if correction: if alternative == 'less' or k == 0: lo = 0.0 else: dlo = (1 + z*sqrt(z**2 - 2 - 1/n + 4*p*(n*q + 1))) / denom lo = center - dlo if alternative == 'greater' or k == n: hi = 1.0 else: dhi = (1 + z*sqrt(z**2 + 2 - 1/n + 4*p*(n*q - 1))) / denom hi = center + dhi else: delta = z/denom * sqrt(4*n*p*q + z**2) if alternative == 'less' or k == 0: lo = 0.0 else: lo = center - delta if alternative == 'greater' or k == n: hi = 1.0 else: hi = center + delta return lo, hi def binomtest(k, n, p=0.5, alternative='two-sided'): """ Perform a test that the probability of success is p. The binomial test [1]_ is a test of the null hypothesis that the probability of success in a Bernoulli experiment is `p`. Details of the test can be found in many texts on statistics, such as section 24.5 of [2]_. Parameters ---------- k : int The number of successes. n : int The number of trials. p : float, optional The hypothesized probability of success, i.e. the expected proportion of successes. The value must be in the interval ``0 <= p <= 1``. The default value is ``p = 0.5``. alternative : {'two-sided', 'greater', 'less'}, optional Indicates the alternative hypothesis. The default value is 'two-sided'. Returns ------- result : `~scipy.stats._result_classes.BinomTestResult` instance The return value is an object with the following attributes: k : int The number of successes (copied from `binomtest` input). n : int The number of trials (copied from `binomtest` input). alternative : str Indicates the alternative hypothesis specified in the input to `binomtest`. It will be one of ``'two-sided'``, ``'greater'``, or ``'less'``. statistic : float The estimate of the proportion of successes. pvalue : float The p-value of the hypothesis test. The object has the following methods: proportion_ci(confidence_level=0.95, method='exact') : Compute the confidence interval for ``statistic``. Notes ----- .. versionadded:: 1.7.0 References ---------- .. [1] Binomial test, https://en.wikipedia.org/wiki/Binomial_test .. [2] Jerrold H. Zar, Biostatistical Analysis (fifth edition), Prentice Hall, Upper Saddle River, New Jersey USA (2010) Examples -------- >>> from scipy.stats import binomtest A car manufacturer claims that no more than 10% of their cars are unsafe. 15 cars are inspected for safety, 3 were found to be unsafe. Test the manufacturer's claim: >>> result = binomtest(3, n=15, p=0.1, alternative='greater') >>> result.pvalue 0.18406106910639114 The null hypothesis cannot be rejected at the 5% level of significance because the returned p-value is greater than the critical value of 5%. The test statistic is equal to the estimated proportion, which is simply ``3/15``: >>> result.statistic 0.2 We can use the `proportion_ci()` method of the result to compute the confidence interval of the estimate: >>> result.proportion_ci(confidence_level=0.95) ConfidenceInterval(low=0.05684686759024681, high=1.0) """ k = _validate_int(k, 'k', minimum=0) n = _validate_int(n, 'n', minimum=1) if k > n: raise ValueError(f'k ({k}) must not be greater than n ({n}).') if not (0 <= p <= 1): raise ValueError(f"p ({p}) must be in range [0,1]") if alternative not in ('two-sided', 'less', 'greater'): raise ValueError(f"alternative ('{alternative}') not recognized; \n" "must be 'two-sided', 'less' or 'greater'") if alternative == 'less': pval = binom.cdf(k, n, p) elif alternative == 'greater': pval = binom.sf(k-1, n, p) else: # alternative is 'two-sided' d = binom.pmf(k, n, p) rerr = 1 + 1e-7 if k == p * n: # special case as shortcut, would also be handled by `else` below pval = 1. elif k < p * n: ix = _binary_search_for_binom_tst(lambda x1: -binom.pmf(x1, n, p), -d*rerr, np.ceil(p * n), n) # y is the number of terms between mode and n that are <= d*rerr. # ix gave us the first term where a(ix) <= d*rerr < a(ix-1) # if the first equality doesn't hold, y=n-ix. Otherwise, we # need to include ix as well as the equality holds. Note that # the equality will hold in very very rare situations due to rerr. y = n - ix + int(d*rerr == binom.pmf(ix, n, p)) pval = binom.cdf(k, n, p) + binom.sf(n - y, n, p) else: ix = _binary_search_for_binom_tst(lambda x1: binom.pmf(x1, n, p), d*rerr, 0, np.floor(p * n)) # y is the number of terms between 0 and mode that are <= d*rerr. # we need to add a 1 to account for the 0 index. # For comparing this with old behavior, see # tst_binary_srch_for_binom_tst method in test_morestats. y = ix + 1 pval = binom.cdf(y-1, n, p) + binom.sf(k-1, n, p) pval = min(1.0, pval) result = BinomTestResult(k=k, n=n, alternative=alternative, statistic=k/n, pvalue=pval) return result def _binary_search_for_binom_tst(a, d, lo, hi): """ Conducts an implicit binary search on a function specified by `a`. Meant to be used on the binomial PMF for the case of two-sided tests to obtain the value on the other side of the mode where the tail probability should be computed. The values on either side of the mode are always in order, meaning binary search is applicable. Parameters ---------- a : callable The function over which to perform binary search. Its values for inputs lo and hi should be in ascending order. d : float The value to search. lo : int The lower end of range to search. hi : int The higher end of the range to search. Returns ------- int The index, i between lo and hi such that a(i)<=d d: hi = mid-1 else: return mid if a(lo) <= d: return lo else: return lo-1