Source code for statsmodels.stats.proportion

# -*- coding: utf-8 -*-
"""Tests and Confidence Intervals for Binomial Proportions

Created on Fri Mar 01 00:23:07 2013

Author: Josef Perktold
"""

from statsmodels.compat.python import lzip

import numpy as np
from scipy import optimize, stats

from statsmodels.stats.base import AllPairsResults, HolderTuple
from statsmodels.stats.weightstats import _zstat_generic2
from statsmodels.tools.sm_exceptions import HypothesisTestWarning
from statsmodels.tools.testing import Holder

FLOAT_INFO = np.finfo(float)

[docs]def proportion_confint(count, nobs, alpha=0.05, method='normal'): """ Confidence interval for a binomial proportion Parameters ---------- count : int or array_array_like number of successes, can be pandas Series or DataFrame nobs : int total number of trials alpha : float in (0, 1) significance level, default 0.05 method : {'normal', 'agresti_coull', 'beta', 'wilson', 'binom_test'} default: 'normal' method to use for confidence interval, currently available methods : - `normal` : asymptotic normal approximation - `agresti_coull` : Agresti-Coull interval - `beta` : Clopper-Pearson interval based on Beta distribution - `wilson` : Wilson Score interval - `jeffreys` : Jeffreys Bayesian Interval - `binom_test` : experimental, inversion of binom_test Returns ------- ci_low, ci_upp : float, ndarray, or pandas Series or DataFrame lower and upper confidence level with coverage (approximately) 1-alpha. When a pandas object is returned, then the index is taken from the `count`. Notes ----- Beta, the Clopper-Pearson exact interval has coverage at least 1-alpha, but is in general conservative. Most of the other methods have average coverage equal to 1-alpha, but will have smaller coverage in some cases. The 'beta' and 'jeffreys' interval are central, they use alpha/2 in each tail, and alpha is not adjusted at the boundaries. In the extreme case when `count` is zero or equal to `nobs`, then the coverage will be only 1 - alpha/2 in the case of 'beta'. The confidence intervals are clipped to be in the [0, 1] interval in the case of 'normal' and 'agresti_coull'. Method "binom_test" directly inverts the binomial test in scipy.stats. which has discrete steps. TODO: binom_test intervals raise an exception in small samples if one interval bound is close to zero or one. References ---------- https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval Brown, Lawrence D.; Cai, T. Tony; DasGupta, Anirban (2001). "Interval Estimation for a Binomial Proportion", Statistical Science 16 (2): 101–133. doi:10.1214/ss/1009213286. """ pd_index = getattr(count, 'index', None) if pd_index is not None and callable(pd_index): # this rules out lists, lists have an index method pd_index = None def _check(x: np.ndarray, name: str) -> np.ndarray: if np.issubdtype(x.dtype, np.integer): return x y = x.astype(np.int64, casting="unsafe") if np.any(y != x): raise ValueError( f"{name} must have an integral dtype. Found data with " f"dtype {x.dtype}" ) return y count = _check(np.asarray(count), "count") nobs = _check(np.asarray(nobs), "nobs") q_ = count / nobs alpha_2 = 0.5 * alpha if method == 'normal': std_ = np.sqrt(q_ * (1 - q_) / nobs) dist = stats.norm.isf(alpha / 2.) * std_ ci_low = q_ - dist ci_upp = q_ + dist elif method == 'binom_test': # inverting the binomial test if hasattr(stats, "binomtest"): def func(qi): return stats.binomtest(count, nobs, p=qi).pvalue - alpha else: # Remove after min SciPy >= 1.7 def func(qi): return stats.binom_test(count, nobs, p=qi) - alpha def _bound(qi, lower=True): """ Try hard to find a bound different from eps or 1 - eps Parameters ---------- qi : float The empirical success rate lower : bool Whether to fund a lower bound for the left side of the CI Returns ------- float The coarse bound """ default = FLOAT_INFO.eps if lower else 1.0 - FLOAT_INFO.eps def step(v): return v / 8 if lower else v + (1.0 - v) / 8 x = step(qi) w = func(x) cnt = 1 while w > 0 and cnt < 10: x = step(x) w = func(x) cnt += 1 return x if cnt < 10 else default if count == 0: ci_low = 0 else: lower_bnd = _bound(q_, lower=True) ci_low = optimize.brentq(func, lower_bnd, q_) if count == nobs: ci_upp = 1 else: upper_bnd = _bound(q_, lower=False) ci_upp = optimize.brentq(func, q_, upper_bnd) elif method == 'beta': ci_low = stats.beta.ppf(alpha_2, count, nobs - count + 1) ci_upp = stats.beta.isf(alpha_2, count + 1, nobs - count) if np.ndim(ci_low) > 0: ci_low[q_ == 0] = 0 ci_upp[q_ == 1] = 1 else: ci_low = ci_low if (q_ != 0) else 0 ci_upp = ci_upp if (q_ != 1) else 1 elif method == 'agresti_coull': crit = stats.norm.isf(alpha / 2.) nobs_c = nobs + crit**2 q_c = (count + crit**2 / 2.) / nobs_c std_c = np.sqrt(q_c * (1. - q_c) / nobs_c) dist = crit * std_c ci_low = q_c - dist ci_upp = q_c + dist elif method == 'wilson': crit = stats.norm.isf(alpha / 2.) crit2 = crit**2 denom = 1 + crit2 / nobs center = (q_ + crit2 / (2 * nobs)) / denom dist = crit * np.sqrt(q_ * (1. - q_) / nobs + crit2 / (4. * nobs**2)) dist /= denom ci_low = center - dist ci_upp = center + dist # method adjusted to be more forgiving of misspellings or incorrect option name elif method[:4] == 'jeff': ci_low, ci_upp = stats.beta.interval(1 - alpha, count + 0.5, nobs - count + 0.5) else: raise NotImplementedError('method "%s" is not available' % method) if method in ['normal', 'agresti_coull']: ci_low = np.clip(ci_low, 0, 1) ci_upp = np.clip(ci_upp, 0, 1) if pd_index is not None and np.ndim(ci_low) > 0: import pandas as pd if np.ndim(ci_low) == 1: ci_low = pd.Series(ci_low, index=pd_index) ci_upp = pd.Series(ci_upp, index=pd_index) if np.ndim(ci_low) == 2: ci_low = pd.DataFrame(ci_low, index=pd_index) ci_upp = pd.DataFrame(ci_upp, index=pd_index) return ci_low, ci_upp
[docs]def multinomial_proportions_confint(counts, alpha=0.05, method='goodman'): '''Confidence intervals for multinomial proportions. Parameters ---------- counts : array_like of int, 1-D Number of observations in each category. alpha : float in (0, 1), optional Significance level, defaults to 0.05. method : {'goodman', 'sison-glaz'}, optional Method to use to compute the confidence intervals; available methods are: - `goodman`: based on a chi-squared approximation, valid if all values in `counts` are greater or equal to 5 _ - `sison-glaz`: less conservative than `goodman`, but only valid if `counts` has 7 or more categories (``len(counts) >= 7``) _ Returns ------- confint : ndarray, 2-D Array of [lower, upper] confidence levels for each category, such that overall coverage is (approximately) `1-alpha`. Raises ------ ValueError If `alpha` is not in `(0, 1)` (bounds excluded), or if the values in `counts` are not all positive or null. NotImplementedError If `method` is not kown. Exception When ``method == 'sison-glaz'``, if for some reason `c` cannot be computed; this signals a bug and should be reported. Notes ----- The `goodman` method _ is based on approximating a statistic based on the multinomial as a chi-squared random variable. The usual recommendation is that this is valid if all the values in `counts` are greater than or equal to 5. There is no condition on the number of categories for this method. The `sison-glaz` method _ approximates the multinomial probabilities, and evaluates that with a maximum-likelihood estimator. The first approximation is an Edgeworth expansion that converges when the number of categories goes to infinity, and the maximum-likelihood estimator converges when the number of observations (``sum(counts)``) goes to infinity. In their paper, Sison & Glaz demo their method with at least 7 categories, so ``len(counts) >= 7`` with all values in `counts` at or above 5 can be used as a rule of thumb for the validity of this method. This method is less conservative than the `goodman` method (i.e. it will yield confidence intervals closer to the desired significance level), but produces confidence intervals of uniform width over all categories (except when the intervals reach 0 or 1, in which case they are truncated), which makes it most useful when proportions are of similar magnitude. Aside from the original sources (_, _, and _), the implementation uses the formulas (though not the code) presented in _ and _. References ---------- ..  Levin, Bruce, "A representation for multinomial cumulative distribution functions," The Annals of Statistics, Vol. 9, No. 5, 1981, pp. 1123-1126. ..  Goodman, L.A., "On simultaneous confidence intervals for multinomial proportions," Technometrics, Vol. 7, No. 2, 1965, pp. 247-254. ..  Sison, Cristina P., and Joseph Glaz, "Simultaneous Confidence Intervals and Sample Size Determination for Multinomial Proportions," Journal of the American Statistical Association, Vol. 90, No. 429, 1995, pp. 366-369. ..  May, Warren L., and William D. Johnson, "A SAS® macro for constructing simultaneous confidence intervals for multinomial proportions," Computer methods and programs in Biomedicine, Vol. 53, No. 3, 1997, pp. 153-162. ..  May, Warren L., and William D. Johnson, "Constructing two-sided simultaneous confidence intervals for multinomial proportions for small counts in a large number of cells," Journal of Statistical Software, Vol. 5, No. 6, 2000, pp. 1-24. ''' if alpha <= 0 or alpha >= 1: raise ValueError('alpha must be in (0, 1), bounds excluded') counts = np.array(counts, dtype=float) if (counts < 0).any(): raise ValueError('counts must be >= 0') n = counts.sum() k = len(counts) proportions = counts / n if method == 'goodman': chi2 = stats.chi2.ppf(1 - alpha / k, 1) delta = chi2 ** 2 + (4 * n * proportions * chi2 * (1 - proportions)) region = ((2 * n * proportions + chi2 + np.array([- np.sqrt(delta), np.sqrt(delta)])) / (2 * (chi2 + n))).T elif method[:5] == 'sison': # We accept any name starting with 'sison' # Define a few functions we'll use a lot. def poisson_interval(interval, p): """Compute P(b <= Z <= a) where Z ~ Poisson(p) and `interval = (b, a)`.""" b, a = interval prob = stats.poisson.cdf(a, p) - stats.poisson.cdf(b - 1, p) return prob def truncated_poisson_factorial_moment(interval, r, p): """Compute mu_r, the r-th factorial moment of a poisson random variable of parameter `p` truncated to `interval = (b, a)`.""" b, a = interval return p ** r * (1 - ((poisson_interval((a - r + 1, a), p) - poisson_interval((b - r, b - 1), p)) / poisson_interval((b, a), p))) def edgeworth(intervals): """Compute the Edgeworth expansion term of Sison & Glaz's formula (1) (approximated probability for multinomial proportions in a given box).""" # Compute means and central moments of the truncated poisson # variables. mu_r1, mu_r2, mu_r3, mu_r4 = [ np.array([truncated_poisson_factorial_moment(interval, r, p) for (interval, p) in zip(intervals, counts)]) for r in range(1, 5) ] mu = mu_r1 mu2 = mu_r2 + mu - mu ** 2 mu3 = mu_r3 + mu_r2 * (3 - 3 * mu) + mu - 3 * mu ** 2 + 2 * mu ** 3 mu4 = (mu_r4 + mu_r3 * (6 - 4 * mu) + mu_r2 * (7 - 12 * mu + 6 * mu ** 2) + mu - 4 * mu ** 2 + 6 * mu ** 3 - 3 * mu ** 4) # Compute expansion factors, gamma_1 and gamma_2. g1 = mu3.sum() / mu2.sum() ** 1.5 g2 = (mu4.sum() - 3 * (mu2 ** 2).sum()) / mu2.sum() ** 2 # Compute the expansion itself. x = (n - mu.sum()) / np.sqrt(mu2.sum()) phi = np.exp(- x ** 2 / 2) / np.sqrt(2 * np.pi) H3 = x ** 3 - 3 * x H4 = x ** 4 - 6 * x ** 2 + 3 H6 = x ** 6 - 15 * x ** 4 + 45 * x ** 2 - 15 f = phi * (1 + g1 * H3 / 6 + g2 * H4 / 24 + g1 ** 2 * H6 / 72) return f / np.sqrt(mu2.sum()) def approximated_multinomial_interval(intervals): """Compute approximated probability for Multinomial(n, proportions) to be in `intervals` (Sison & Glaz's formula (1)).""" return np.exp( np.sum(np.log([poisson_interval(interval, p) for (interval, p) in zip(intervals, counts)])) + np.log(edgeworth(intervals)) - np.log(stats.poisson._pmf(n, n)) ) def nu(c): """Compute interval coverage for a given `c` (Sison & Glaz's formula (7)).""" return approximated_multinomial_interval( [(np.maximum(count - c, 0), np.minimum(count + c, n)) for count in counts]) # Find the value of `c` that will give us the confidence intervals # (solving nu(c) <= 1 - alpha < nu(c + 1). c = 1.0 nuc = nu(c) nucp1 = nu(c + 1) while not (nuc <= (1 - alpha) < nucp1): if c > n: raise Exception("Couldn't find a value for `c` that " "solves nu(c) <= 1 - alpha < nu(c + 1)") c += 1 nuc = nucp1 nucp1 = nu(c + 1) # Compute gamma and the corresponding confidence intervals. g = (1 - alpha - nuc) / (nucp1 - nuc) ci_lower = np.maximum(proportions - c / n, 0) ci_upper = np.minimum(proportions + (c + 2 * g) / n, 1) region = np.array([ci_lower, ci_upper]).T else: raise NotImplementedError('method "%s" is not available' % method) return region
[docs]def samplesize_confint_proportion(proportion, half_length, alpha=0.05, method='normal'): '''find sample size to get desired confidence interval length Parameters ---------- proportion : float in (0, 1) proportion or quantile half_length : float in (0, 1) desired half length of the confidence interval alpha : float in (0, 1) significance level, default 0.05, coverage of the two-sided interval is (approximately) ``1 - alpha`` method : str in ['normal'] method to use for confidence interval, currently only normal approximation Returns ------- n : float sample size to get the desired half length of the confidence interval Notes ----- this is mainly to store the formula. possible application: number of replications in bootstrap samples ''' q_ = proportion if method == 'normal': n = q_ * (1 - q_) / (half_length / stats.norm.isf(alpha / 2.))**2 else: raise NotImplementedError('only "normal" is available') return n
[docs]def proportion_effectsize(prop1, prop2, method='normal'): ''' Effect size for a test comparing two proportions for use in power function Parameters ---------- prop1, prop2 : float or array_like The proportion value(s). Returns ------- es : float or ndarray effect size for (transformed) prop1 - prop2 Notes ----- only method='normal' is implemented to match pwr.p2.test see http://www.statmethods.net/stats/power.html Effect size for `normal` is defined as :: 2 * (arcsin(sqrt(prop1)) - arcsin(sqrt(prop2))) I think other conversions to normality can be used, but I need to check. Examples -------- >>> import statsmodels.api as sm >>> sm.stats.proportion_effectsize(0.5, 0.4) 0.20135792079033088 >>> sm.stats.proportion_effectsize([0.3, 0.4, 0.5], 0.4) array([-0.21015893, 0. , 0.20135792]) ''' if method != 'normal': raise ValueError('only "normal" is implemented') es = 2 * (np.arcsin(np.sqrt(prop1)) - np.arcsin(np.sqrt(prop2))) return es
def std_prop(prop, nobs): '''standard error for the estimate of a proportion This is just ``np.sqrt(p * (1. - p) / nobs)`` Parameters ---------- prop : array_like proportion nobs : int, array_like number of observations Returns ------- std : array_like standard error for a proportion of nobs independent observations ''' return np.sqrt(prop * (1. - prop) / nobs) def _std_diff_prop(p1, p2, ratio=1): return np.sqrt(p1 * (1 - p1) + p2 * (1 - p2) / ratio) def _power_ztost(mean_low, var_low, mean_upp, var_upp, mean_alt, var_alt, alpha=0.05, discrete=True, dist='norm', nobs=None, continuity=0, critval_continuity=0): '''Generic statistical power function for normal based equivalence test This includes options to adjust the normal approximation and can use the binomial to evaluate the probability of the rejection region see power_ztost_prob for a description of the options ''' # TODO: refactor structure, separate norm and binom better if not isinstance(continuity, tuple): continuity = (continuity, continuity) crit = stats.norm.isf(alpha) k_low = mean_low + np.sqrt(var_low) * crit k_upp = mean_upp - np.sqrt(var_upp) * crit if discrete or dist == 'binom': k_low = np.ceil(k_low * nobs + 0.5 * critval_continuity) k_upp = np.trunc(k_upp * nobs - 0.5 * critval_continuity) if dist == 'norm': #need proportion k_low = (k_low) * 1. / nobs #-1 to match PASS k_upp = k_upp * 1. / nobs # else: # if dist == 'binom': # #need counts # k_low *= nobs # k_upp *= nobs #print mean_low, np.sqrt(var_low), crit, var_low #print mean_upp, np.sqrt(var_upp), crit, var_upp if np.any(k_low > k_upp): #vectorize import warnings warnings.warn("no overlap, power is zero", HypothesisTestWarning) std_alt = np.sqrt(var_alt) z_low = (k_low - mean_alt - continuity * 0.5 / nobs) / std_alt z_upp = (k_upp - mean_alt + continuity * 0.5 / nobs) / std_alt if dist == 'norm': power = stats.norm.cdf(z_upp) - stats.norm.cdf(z_low) elif dist == 'binom': power = (stats.binom.cdf(k_upp, nobs, mean_alt) - stats.binom.cdf(k_low-1, nobs, mean_alt)) return power, (k_low, k_upp, z_low, z_upp)
[docs]def binom_tost(count, nobs, low, upp): '''exact TOST test for one proportion using binomial distribution Parameters ---------- count : {int, array_like} the number of successes in nobs trials. nobs : int the number of trials or observations. low, upp : floats lower and upper limit of equivalence region Returns ------- pvalue : float p-value of equivalence test pval_low, pval_upp : floats p-values of lower and upper one-sided tests ''' # binom_test_stat only returns pval tt1 = binom_test(count, nobs, alternative='larger', prop=low) tt2 = binom_test(count, nobs, alternative='smaller', prop=upp) return np.maximum(tt1, tt2), tt1, tt2,
[docs]def binom_tost_reject_interval(low, upp, nobs, alpha=0.05): '''rejection region for binomial TOST The interval includes the end points, `reject` if and only if `r_low <= x <= r_upp`. The interval might be empty with `r_upp < r_low`. Parameters ---------- low, upp : floats lower and upper limit of equivalence region nobs : int the number of trials or observations. Returns ------- x_low, x_upp : float lower and upper bound of rejection region ''' x_low = stats.binom.isf(alpha, nobs, low) + 1 x_upp = stats.binom.ppf(alpha, nobs, upp) - 1 return x_low, x_upp
[docs]def binom_test_reject_interval(value, nobs, alpha=0.05, alternative='two-sided'): """ Rejection region for binomial test for one sample proportion The interval includes the end points of the rejection region. Parameters ---------- value : float proportion under the Null hypothesis nobs : int the number of trials or observations. Returns ------- x_low, x_upp : int lower and upper bound of rejection region """ if alternative in ['2s', 'two-sided']: alternative = '2s' # normalize alternative name alpha = alpha / 2 if alternative in ['2s', 'smaller']: x_low = stats.binom.ppf(alpha, nobs, value) - 1 else: x_low = 0 if alternative in ['2s', 'larger']: x_upp = stats.binom.isf(alpha, nobs, value) + 1 else : x_upp = nobs return int(x_low), int(x_upp)
[docs]def binom_test(count, nobs, prop=0.5, alternative='two-sided'): """ Perform a test that the probability of success is p. This is an exact, two-sided test of the null hypothesis that the probability of success in a Bernoulli experiment is `p`. Parameters ---------- count : {int, array_like} the number of successes in nobs trials. nobs : int the number of trials or observations. prop : float, optional The probability of success under the null hypothesis, `0 <= prop <= 1`. The default value is `prop = 0.5` alternative : str in ['two-sided', 'smaller', 'larger'] alternative hypothesis, which can be two-sided or either one of the one-sided tests. Returns ------- p-value : float The p-value of the hypothesis test Notes ----- This uses scipy.stats.binom_test for the two-sided alternative. """ if np.any(prop > 1.0) or np.any(prop < 0.0): raise ValueError("p must be in range [0,1]") if alternative in ['2s', 'two-sided']: try: pval = stats.binomtest(count, n=nobs, p=prop).pvalue except AttributeError: # Remove after min SciPy >= 1.7 pval = stats.binom_test(count, n=nobs, p=prop) elif alternative in ['l', 'larger']: pval = stats.binom.sf(count-1, nobs, prop) elif alternative in ['s', 'smaller']: pval = stats.binom.cdf(count, nobs, prop) else: raise ValueError('alternative not recognized\n' 'should be two-sided, larger or smaller') return pval
[docs]def power_binom_tost(low, upp, nobs, p_alt=None, alpha=0.05): if p_alt is None: p_alt = 0.5 * (low + upp) x_low, x_upp = binom_tost_reject_interval(low, upp, nobs, alpha=alpha) power = (stats.binom.cdf(x_upp, nobs, p_alt) - stats.binom.cdf(x_low-1, nobs, p_alt)) return power
[docs]def power_ztost_prop(low, upp, nobs, p_alt, alpha=0.05, dist='norm', variance_prop=None, discrete=True, continuity=0, critval_continuity=0): '''Power of proportions equivalence test based on normal distribution Parameters ---------- low, upp : floats lower and upper limit of equivalence region nobs : int number of observations p_alt : float in (0,1) proportion under the alternative alpha : float in (0,1) significance level of the test dist : str in ['norm', 'binom'] This defines the distribution to evaluate the power of the test. The critical values of the TOST test are always based on the normal approximation, but the distribution for the power can be either the normal (default) or the binomial (exact) distribution. variance_prop : None or float in (0,1) If this is None, then the variances for the two one sided tests are based on the proportions equal to the equivalence limits. If variance_prop is given, then it is used to calculate the variance for the TOST statistics. If this is based on an sample, then the estimated proportion can be used. discrete : bool If true, then the critical values of the rejection region are converted to integers. If dist is "binom", this is automatically assumed. If discrete is false, then the TOST critical values are used as floating point numbers, and the power is calculated based on the rejection region that is not discretized. continuity : bool or float adjust the rejection region for the normal power probability. This has and effect only if ``dist='norm'`` critval_continuity : bool or float If this is non-zero, then the critical values of the tost rejection region are adjusted before converting to integers. This affects both distributions, ``dist='norm'`` and ``dist='binom'``. Returns ------- power : float statistical power of the equivalence test. (k_low, k_upp, z_low, z_upp) : tuple of floats critical limits in intermediate steps temporary return, will be changed Notes ----- In small samples the power for the ``discrete`` version, has a sawtooth pattern as a function of the number of observations. As a consequence, small changes in the number of observations or in the normal approximation can have a large effect on the power. ``continuity`` and ``critval_continuity`` are added to match some results of PASS, and are mainly to investigate the sensitivity of the ztost power to small changes in the rejection region. From my interpretation of the equations in the SAS manual, both are zero in SAS. works vectorized **verification:** The ``dist='binom'`` results match PASS, The ``dist='norm'`` results look reasonable, but no benchmark is available. References ---------- SAS Manual: Chapter 68: The Power Procedure, Computational Resources PASS Chapter 110: Equivalence Tests for One Proportion. ''' mean_low = low var_low = std_prop(low, nobs)**2 mean_upp = upp var_upp = std_prop(upp, nobs)**2 mean_alt = p_alt var_alt = std_prop(p_alt, nobs)**2 if variance_prop is not None: var_low = var_upp = std_prop(variance_prop, nobs)**2 power = _power_ztost(mean_low, var_low, mean_upp, var_upp, mean_alt, var_alt, alpha=alpha, discrete=discrete, dist=dist, nobs=nobs, continuity=continuity, critval_continuity=critval_continuity) return np.maximum(power, 0), power[1:]
def _table_proportion(count, nobs): '''create a k by 2 contingency table for proportion helper function for proportions_chisquare Parameters ---------- count : {int, array_like} the number of successes in nobs trials. nobs : int the number of trials or observations. Returns ------- table : ndarray (k, 2) contingency table Notes ----- recent scipy has more elaborate contingency table functions ''' count = np.asarray(count) dt = np.promote_types(count.dtype, np.float64) count = np.asarray(count, dtype=dt) table = np.column_stack((count, nobs - count)) expected = table.sum(0) * table.sum(1)[:, None] * 1. / table.sum() n_rows = table.shape return table, expected, n_rows
[docs]def proportions_ztest(count, nobs, value=None, alternative='two-sided', prop_var=False): """ Test for proportions based on normal (z) test Parameters ---------- count : {int, array_like} the number of successes in nobs trials. If this is array_like, then the assumption is that this represents the number of successes for each independent sample nobs : {int, array_like} the number of trials or observations, with the same length as count. value : float, array_like or None, optional This is the value of the null hypothesis equal to the proportion in the case of a one sample test. In the case of a two-sample test, the null hypothesis is that prop - prop = value, where prop is the proportion in the two samples. If not provided value = 0 and the null is prop = prop alternative : str in ['two-sided', 'smaller', 'larger'] The alternative hypothesis can be either two-sided or one of the one- sided tests, smaller means that the alternative hypothesis is ``prop < value`` and larger means ``prop > value``. In the two sample test, smaller means that the alternative hypothesis is ``p1 < p2`` and larger means ``p1 > p2`` where ``p1`` is the proportion of the first sample and ``p2`` of the second one. prop_var : False or float in (0, 1) If prop_var is false, then the variance of the proportion estimate is calculated based on the sample proportion. Alternatively, a proportion can be specified to calculate this variance. Common use case is to use the proportion under the Null hypothesis to specify the variance of the proportion estimate. Returns ------- zstat : float test statistic for the z-test p-value : float p-value for the z-test Examples -------- >>> count = 5 >>> nobs = 83 >>> value = .05 >>> stat, pval = proportions_ztest(count, nobs, value) >>> print('{0:0.3f}'.format(pval)) 0.695 >>> import numpy as np >>> from statsmodels.stats.proportion import proportions_ztest >>> count = np.array([5, 12]) >>> nobs = np.array([83, 99]) >>> stat, pval = proportions_ztest(count, nobs) >>> print('{0:0.3f}'.format(pval)) 0.159 Notes ----- This uses a simple normal test for proportions. It should be the same as running the mean z-test on the data encoded 1 for event and 0 for no event so that the sum corresponds to the count. In the one and two sample cases with two-sided alternative, this test produces the same p-value as ``proportions_chisquare``, since the chisquare is the distribution of the square of a standard normal distribution. """ # TODO: verify that this really holds # TODO: add continuity correction or other improvements for small samples # TODO: change options similar to propotion_ztost ? count = np.asarray(count) nobs = np.asarray(nobs) if nobs.size == 1: nobs = nobs * np.ones_like(count) prop = count * 1. / nobs k_sample = np.size(prop) if value is None: if k_sample == 1: raise ValueError('value must be provided for a 1-sample test') value = 0 if k_sample == 1: diff = prop - value elif k_sample == 2: diff = prop - prop - value else: msg = 'more than two samples are not implemented yet' raise NotImplementedError(msg) p_pooled = np.sum(count) * 1. / np.sum(nobs) nobs_fact = np.sum(1. / nobs) if prop_var: p_pooled = prop_var var_ = p_pooled * (1 - p_pooled) * nobs_fact std_diff = np.sqrt(var_) from statsmodels.stats.weightstats import _zstat_generic2 return _zstat_generic2(diff, std_diff, alternative)
[docs]def proportions_ztost(count, nobs, low, upp, prop_var='sample'): '''Equivalence test based on normal distribution Parameters ---------- count : {int, array_like} the number of successes in nobs trials. If this is array_like, then the assumption is that this represents the number of successes for each independent sample nobs : int the number of trials or observations, with the same length as count. low, upp : float equivalence interval low < prop1 - prop2 < upp prop_var : str or float in (0, 1) prop_var determines which proportion is used for the calculation of the standard deviation of the proportion estimate The available options for string are 'sample' (default), 'null' and 'limits'. If prop_var is a float, then it is used directly. Returns ------- pvalue : float pvalue of the non-equivalence test t1, pv1 : tuple of floats test statistic and pvalue for lower threshold test t2, pv2 : tuple of floats test statistic and pvalue for upper threshold test Notes ----- checked only for 1 sample case ''' if prop_var == 'limits': prop_var_low = low prop_var_upp = upp elif prop_var == 'sample': prop_var_low = prop_var_upp = False #ztest uses sample elif prop_var == 'null': prop_var_low = prop_var_upp = 0.5 * (low + upp) elif np.isreal(prop_var): prop_var_low = prop_var_upp = prop_var tt1 = proportions_ztest(count, nobs, alternative='larger', prop_var=prop_var_low, value=low) tt2 = proportions_ztest(count, nobs, alternative='smaller', prop_var=prop_var_upp, value=upp) return np.maximum(tt1, tt2), tt1, tt2,
[docs]def proportions_chisquare(count, nobs, value=None): '''test for proportions based on chisquare test Parameters ---------- count : {int, array_like} the number of successes in nobs trials. If this is array_like, then the assumption is that this represents the number of successes for each independent sample nobs : int the number of trials or observations, with the same length as count. value : None or float or array_like Returns ------- chi2stat : float test statistic for the chisquare test p-value : float p-value for the chisquare test (table, expected) table is a (k, 2) contingency table, ``expected`` is the corresponding table of counts that are expected under independence with given margins Notes ----- Recent version of scipy.stats have a chisquare test for independence in contingency tables. This function provides a similar interface to chisquare tests as ``prop.test`` in R, however without the option for Yates continuity correction. count can be the count for the number of events for a single proportion, or the counts for several independent proportions. If value is given, then all proportions are jointly tested against this value. If value is not given and count and nobs are not scalar, then the null hypothesis is that all samples have the same proportion. ''' nobs = np.atleast_1d(nobs) table, expected, n_rows = _table_proportion(count, nobs) if value is not None: expected = np.column_stack((nobs * value, nobs * (1 - value))) ddof = n_rows - 1 else: ddof = n_rows #print table, expected chi2stat, pval = stats.chisquare(table.ravel(), expected.ravel(), ddof=ddof) return chi2stat, pval, (table, expected)
[docs]def proportions_chisquare_allpairs(count, nobs, multitest_method='hs'): '''chisquare test of proportions for all pairs of k samples Performs a chisquare test for proportions for all pairwise comparisons. The alternative is two-sided Parameters ---------- count : {int, array_like} the number of successes in nobs trials. nobs : int the number of trials or observations. prop : float, optional The probability of success under the null hypothesis, `0 <= prop <= 1`. The default value is `prop = 0.5` multitest_method : str This chooses the method for the multiple testing p-value correction, that is used as default in the results. It can be any method that is available in ``multipletesting``. The default is Holm-Sidak 'hs'. Returns ------- result : AllPairsResults instance The returned results instance has several statistics, such as p-values, attached, and additional methods for using a non-default ``multitest_method``. Notes ----- Yates continuity correction is not available. ''' #all_pairs = lmap(list, lzip(*np.triu_indices(4, 1))) all_pairs = lzip(*np.triu_indices(len(count), 1)) pvals = [proportions_chisquare(count[list(pair)], nobs[list(pair)]) for pair in all_pairs] return AllPairsResults(pvals, all_pairs, multitest_method=multitest_method)
[docs]def proportions_chisquare_pairscontrol(count, nobs, value=None, multitest_method='hs', alternative='two-sided'): '''chisquare test of proportions for pairs of k samples compared to control Performs a chisquare test for proportions for pairwise comparisons with a control (Dunnet's test). The control is assumed to be the first element of ``count`` and ``nobs``. The alternative is two-sided, larger or smaller. Parameters ---------- count : {int, array_like} the number of successes in nobs trials. nobs : int the number of trials or observations. prop : float, optional The probability of success under the null hypothesis, `0 <= prop <= 1`. The default value is `prop = 0.5` multitest_method : str This chooses the method for the multiple testing p-value correction, that is used as default in the results. It can be any method that is available in ``multipletesting``. The default is Holm-Sidak 'hs'. alternative : str in ['two-sided', 'smaller', 'larger'] alternative hypothesis, which can be two-sided or either one of the one-sided tests. Returns ------- result : AllPairsResults instance The returned results instance has several statistics, such as p-values, attached, and additional methods for using a non-default ``multitest_method``. Notes ----- Yates continuity correction is not available. ``value`` and ``alternative`` options are not yet implemented. ''' if (value is not None) or (alternative not in ['two-sided', '2s']): raise NotImplementedError #all_pairs = lmap(list, lzip(*np.triu_indices(4, 1))) all_pairs = [(0, k) for k in range(1, len(count))] pvals = [proportions_chisquare(count[list(pair)], nobs[list(pair)], #alternative=alternative) ) for pair in all_pairs] return AllPairsResults(pvals, all_pairs, multitest_method=multitest_method)