M/PhtN DdZddlmZddlmZddlZddlZddl m Z m Z ddl m Z mZddlmZddlmZdd lmZdd lmZejeZ dAd eegefd ededefdZ dBd eegefdedededeeeff dZdCdefdZdDdZ dCdZ!dEdZ"dZ#dFd Z$ dGd"Z%d#Z&dHd$Z'dId&Z(dJd(Z)dKd)Z* dLd*Z+d+Z, dMd-Z-dNd/Z.dOd0Z/dPd2Z0 dQd3Z1 dRd5Z2 dSd7Z3 dTd8Z4 dUd9Z5 dVd:Z6dWd;Z7 dXd<Z8 dYd=Z9 dZd>Z: d[d?Z;dHd@Z.step/s#4q1uu1a1}#44r ) FLOAT_INFOeps)rrrdefaultrxwcnts ` r_bound_proportion_confintr#s(!&?jnn3+?G55555 RA QA C a%%C"HH DGG DGG q a%%C"HHb11g%rlbubstepsc||}||}|dkr|n|}|dkr|n|}tj|tj|krtd||zdz }||} | dkr | |kr| }|}t|D]V} tj| tj|kr|}| }n|}||zdz }||} | dkr | |kr| }|}W||fS)a9 Private function used as a fallback by proportion_confint Used when brentq returns a non-conservative bound for the CI Parameters ---------- func : callable Callable function to use as the objective of the search lb : float Lower bound ub : float Upper bound steps : int Number of steps to use in the bisection Returns ------- est : float The estimated value. Will always produce a negative value of func func_val : float The value of the function at the estimate rzproblem with signs)npsign ValueErrorrange) rr%r&r'upperrbestbest_ptmpmid_s r_bisection_search_conservativer4<s 4 DHHE DHHEAII555DAIIbb2G wu~~''-... r'QB $r((C acDjj 5\\   73<<275>> ) )BEEB2g]d2hh !GG#**DG D=r皙?normalalphacj tj|otj|}t|tjtjf}t |ddd}t |ddd}dtjdtdtjfd }|d kr<|tj |d}|tj |d}||z } d z} |d krNtj | d | z z|z } tj dz | z} | | z } | | z}nP|d krcdtdtdtt gt fffd }tj||}tj|j} tj|j}|j}|D]\}}d}| j|}||dzkr ||z }d}d |z }|||}|dkr d| j|<nt-||d}t/j|||d\}}||dkrad}|||z d|zz z }||dkr.|dkr(|d z}|||z d|zz z }||dkr|dk(t3|||\}}|| j|<||kr d|j|<nt-||d}t/j|||d\}}||dkrad}|||z d|zz z}||dkr.|dkr(|d z}|||z d|zz z }||dkr|dk(t3|||\}}||j|<|r2|j|}d | j|z |j|<d |z | j|<|j}n|dkrtj| |||z d z} tj | |d z||z }tj| dkr(d| j| jdk<d |j| jd k<nR| dkrdn| } | d krd n|}n<|dkretj dz } || dzz}!|| dzdz z|!z }"tj |"d|"z z|!z }#| |#z} |"| z } |"| z}n|dkrytj dz } | dz}$d |$|z z}%| |$d|zz z|%z }&| tj | d| z z|z |$d|dzzz zz} | |%z} |&| z } |&| z}nR|dddkr1tjd z |d z||z d z\} }nt=d|d|d vr,tj| dd } tj|dd }|rVt|tjr tjn tj}'|'| |j!} |'||j!}|rt!| t!|fS| |fS)"u\ Confidence interval for a binomial proportion Parameters ---------- count : {int or float, array_like} number of successes, can be pandas Series or DataFrame. Arrays must contain integer values if method is "binom_test". nobs : {int or float, array_like} total number of trials. Arrays must contain integer values if method is "binom_test". alpha : float Significance level, default 0.05. Must be in (0, 1) method : {"normal", "agresti_coull", "beta", "wilson", "binom_test"} default: "normal" method to use for confidence interval. Supported 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` : Numerical inversion of binom_test Returns ------- ci_low, ci_upp : {float, ndarray, Series DataFrame} lower and upper confidence level with coverage (approximately) 1-alpha. When a pandas object is returned, then the index is taken from `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. countFN)optionalndimnobsr namerctj|jtjr|S|tjd}tj||krt|d|j|S)Nunsafe)castingz4 must have an integral dtype. Found data with dtype )r* issubdtypedtypeintegerastypeint64anyr,)r r=ys r_checkz"proportion_confint.._checks ="* - - H HHRXxH 0 0 6!q&>> #### r binom_test?r6r@cTttdrfd}nfd}|S)N binomtestcBtj|jz SN)p)rrMpvaluerr7r9r<s rrz6proportion_confint..func_factory..funcs" ?5$"===DuLLrc8tj|z SrO)rrIrRs rrz6proportion_confint..func_factory..funcs  +E42>>>FFr)hasattrr)r9r<rr7s`` r func_factoryz(proportion_confint..func_factorysyuk** GMMMMMMMM GGGGGGGKrr)Tr)r) full_outputrrbeta agresti_coullwilsong@jeffzmethod z is not available)r6rY)index) r*isscalar isinstancepdSeries DataFramer ndarraystrasarraysqrtrnormisfintrfloat broadcastzerosshaper]flatr#rbrentqr4rXppfr;intervalNotImplementedErrorclip)(r9r<r7method is_scalar is_pandascount_anobs_arHq_alpha_2std_distci_lowci_upprUbcastr]cnreverse_qr lower_bndval_zpowernew_lbr3 upper_bndnew_ubtempcritnobs_cq_cstd_ccrit2denomcenter containers( ` rproportion_confintrosFv E""8r{4'8'8I529bl";<rgoodmancl|dks|dkrtdtjtdkrtdt }z }|dkrtj d||z z d}|dzdz|z|zd|z zz}dz|z|ztjtj | tj |gzd|zzz j }n|d d d krd fd fdfdfd}d} || } || dz} | d|z cxkr| ks?n| krtd| dz } | } || dz} | d|z cxkr| k:n=d|z | z | | z z } tj || z z d} tj|| d| zzz zd}tj| |gj }ntd|z|S)u$ 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 [2]_ - `sison-glaz`: less conservative than `goodman`, but only valid if `counts` has 7 or more categories (``len(counts) >= 7``) [3]_ 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 [2]_ 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 [3]_ 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 ([1]_, [2]_, and [3]_), the implementation uses the formulas (though not the code) presented in [4]_ and [5]_. References ---------- .. [1] Levin, Bruce, "A representation for multinomial cumulative distribution functions," The Annals of Statistics, Vol. 9, No. 5, 1981, pp. 1123-1126. .. [2] Goodman, L.A., "On simultaneous confidence intervals for multinomial proportions," Technometrics, Vol. 7, No. 2, 1965, pp. 247-254. .. [3] 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. .. [4] 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. .. [5] 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. rrz(alpha must be in (0, 1), bounds excludedrBzcounts must be >= 0rr)r[Nsisonc|\}}tj||tj|dz |z }|S)zj Compute P(b <= Z <= a) where Z ~ Poisson(p) and `interval = (b, a)`. r)rpoissoncdf)rqrPbaprobs rpoisson_intervalz9multinomial_proportions_confint..poisson_intervalsD DAq=$$Q**U]->->q1ua-H-HHDKrc|\}}||zd||z dz|f|||z |dz f|z ||f|z z zS)z Compute mu_r, the r-th factorial moment of a poisson random variable of parameter `p` truncated to `interval = (b, a)`. rr)rqrrPrrrs r"truncated_poisson_factorial_momentzKmultinomial_proportions_confint..truncated_poisson_factorial_moments DAq6Q#3#3QUQYNA#F#F#3#3QUAENA#F#F$G"2"2Aq61"="=#>?@ @rcfdtddD\}}}}|}||z|dzz }||dd|zz zz|zd|dzzz d|dzzz}||dd|zz zz|dd |zz d|dzzzzz|zd|dzzz d|dzzzd|dzzz }||d zz } |d|dzzz |dzz } |z tj|z } tj| dz dz tjdtjzz } | dzd| zz } | dzd| dzzz dz}| dzd | dzzz d | dzzzd z }| d| | zdz z| |zd z z| dz|zdz zz}|tj|z S)z Compute the Edgeworth expansion term of Sison & Glaz's formula (1) (approximated probability for multinomial proportions in a given box). c3xK|]3tjfdtDV4dS)c0g|]\}}||Srr).0rqrPrrs r zPmultinomial_proportions_confint..edgeworth...sBGGG+x=zEmultinomial_proportions_confint..edgeworth..s**GGGGG/29f/E/EGGGHH******rrrr)r[ g?-H)r-sumr*rfexppi)rmu_r1mu_r2mu_r3mu_r4mumu2mu3mu4g1g2r phiH3H4H6frrrs` r edgeworthz2multinomial_proportions_confint..edgeworths_******q!*** &E5% B"*rQw&C%1q2v:..3a"'kAAaKOC5ABJ//ARK!bAg+567B!G $&'"'k245aK@C SWWYY#--B''))a3!8.."2"222cggii1nDBRVVXX!3!33A&16A&&RU););;Ca!a%Ba!a1f*$q(Ba"qAv+%Q!V 3b8Bq27Q;b25a" r8IIJArwswwyy))) )rc Jtjtjtjfdt |Dtj|ztjt jz S)z Compute approximated probability for Multinomial(n, proportions) to be in `intervals` (Sison & Glaz's formula (1)). c.g|]\}}||Srr)rrqrPrs rrz^multinomial_proportions_confint..approximated_multinomial_interval..s@LLL#0Ha 0/!<<LLLr)r*rrlogrrr_pmf)rrrrrs r!approximated_multinomial_intervalzJmultinomial_proportions_confint..approximated_multinomial_intervals 6rvLLLL47 64J4JLLLMMNNyy++,,-u}))!Q//001 rc6fdDS)zm Compute interval coverage for a given `c` (Sison & Glaz's formula (7)). cpg|]2}tj|z dtj|zf3Sr)r*maximumminimum)rr9rrs rrz?multinomial_proportions_confint..nu..sN&&&*UQY**BJuqy!,D,DE&&&rr)rrrrs`rnuz+multinomial_proportions_confint..nusF 54&&&&&$&&&'' 'rrzHCouldn't find a value for `c` that solves nu(c) <= 1 - alpha < nu(c + 1)zmethod "%s" is not available)r,r*rrjrFrlenrchi2rprfT Exceptionrrrr)rr7rtk proportionsrdeltaregionrrnucnucp1gci_lowerci_upperrrrrrs` @@@@@rmultinomial_proportions_confintr5sj zzUaZZCDDD XfE * * *F 0./// A F A1*K z~~a%!)mQ// QU[0471{?KLq5;&-8rwu~~-rwu~~>??@q>#$%  w      @ @ @ @ @ * * * * * * *D         ' ' ' ' ' ' ' bee1q5 1u9--------1uu!HIII FACBq1uIIE 1u9---------Y_ -:kAE1155:kQQY!O;Q??8X.//1!"@6"IJJJ Mrc|}|dkr2|d|z z|tj|dz z dzz }ntd|S)a 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 r6rrKr)zonly "normal" is available)rrgrhrr) proportion half_lengthr7rtryrs rsamplesize_confint_proportionrsX: B  !b&M[5:>>%"*+E+EEI I!">??? Hrc|dkrtddtjtj|tjtj|z z}|S)a] 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]) r6zonly "normal" is implementedr))r,r*arcsinrf)prop1prop2rtess rproportion_effectsizer"sWH7888 bi''")BGENN*C*CC DB Irc<tj|d|z z|z S)ae 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 rr*rf)propr<s rstd_proprMs"$ 7429%, - --rrcNtj|d|z z|d|z z|z zS)Nrr)p1p2ratios r_std_diff_proprbs, 72R=2R=5#88 9 99rrgc t| ts| | f} tj|} |t j|| zz} |t j|| zz }|s|dkrPt j| | zd| zz} t j|| zd| zz }|dkr| dz| z } |dz| z }t j | |krddl }| dtt j|}| |z | ddz| z z |z }||z | ddz| z z|z }|dkr@tj |tj |z }nL|dkrFtj || |tj | dz | |z }|| |||ffS) a 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 binomrJrgrrNzno overlap, power is zeror)r_tuplerrgrhr*rfceiltruncrFwarningswarnr rr)mean_lowvar_lowmean_uppvar_uppmean_altvar_altr7discreter|r< continuitycritval_continuityrk_lowk_upprstd_altz_lowz_upprs r _power_ztostr fs j% ( (. *- :>>% D rww''$. .E rww''$. .E&47?? s-?'??@@.@(@@AA 6>>bL4'EBJ%E veemJ 13HIIIggG X  1  3d : :g EE X  1  3d : :g EE v~~ u%% u(=(== h77[__U1WdH==> 5%. ..rc~t||d|}t||d|}tj||||fS)a 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 larger) alternativersmaller)rIr*r)r9r<lowupptt1tt2s r binom_tostrsI, UDhS A A AC UDic B B BC :c3  c **rctj|||dz}tj|||dz }||fS)a 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 r)rrrhrp)rrr<r7x_lowx_upps rbinom_tost_reject_intervalrsF, KOOE4 - - 1E KOOE4 - - 1E %<r two-sidedc|dvrd}|dz }|dvr%tj|||dz }nd}|dvr%tj|||dz}n|}t |t |fS)a 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 2srrr))rrrr)rr )rrrprhri)valuer<r7r rrs rbinom_test_reject_intervalrs$)))  ''' tU33a7&&& tU33a7 u::s5zz !!rrJctj|dkstj|dkrtd|dvrE tj|||j}n#t $rtj|||}YnbwxYw|dvr%tj |dz ||}n5|dvr"tj |||}ntd |S) aV 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. rrVzp must be in range [0,1]r)rrP)lr r)srzAalternative not recognized should be two-sided, larger or smaller) r*rFr,rrMrQAttributeErrorrIrsfr)r9r<rr pvals rrIrIs> vdSj5RVD3J//53444))) ;?5DD999@DD ; ; ;#ETT:::DDD ;  ' '{~~eAgtT22 ( ( ({udD11BCC C KsA""!BBc|d||zz}t||||\}}tj|||tj|dz ||z }|S)NrJ)r7r)rrrr)rrr<p_altr7rrrs rpower_binom_tostr&si }sSy!-c3EJJJLE5 [__UD% 0 0[__U1WdE::;E Lrc .|} t||dz} |} t||dz} |}t||dz}|t||dzx} } t| | | | ||||||||  }tj|dd|ddfS)a 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. r)N)r7rr|r<rrrr)rr r*r)rrr<r%r7r| variance_proprrrrrrrrrrs rpower_ztost_propr)%sTHsD!!1$GHsD!!1$GHud##Q&G $]D991<<' 7Hgxxd&;M O O OE :eAh " "E!""I --rctj|}tj|jtj}tj||}tj|||z f}|d|ddddfzdz|z }|jd}|||fS)a 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 rrrNr)r*re promote_typesrBfloat64 column_stackrrm)r9r<dttableexpectedn_rowss r_table_proportionr2}s. Ju  E %+rz 2 2B JuB ' ' 'E OUD5L1 2 2Eyy||eiill111d733b8599;;FH [^F (F ""rFcbtj|}tj|}|jdkr|tj|z}|dz|z }tj|}||dkrt dd}|dkr||z }n,|dkr|d|dz |z }nd}t |tj|dztj|z } tjd|z } |r|} | d| z z| z} tj| } ddlm } | || |S) a 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[0] - prop[1] = value, where prop is the proportion in the two samples. If not provided value = 0 and the null is prop[0] = prop[1] 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. rrNz*value must be provided for a 1-sample testrr)z-more than two samples are not implemented yetr ) r*resize ones_liker,rrrrfstatsmodels.stats.weightstatsr )r9r<rr prop_varrk_samplediffmsgp_pooled nobs_factvar_std_diffr s rproportions_ztestr?sMR Ju  E :d  D yA~~bl5))) 2: Dwt}}H } q==IJJ J1}}e| QAwa 5(=!#&&&ve}}r!BF4LL0HrDy!!I q8| $y 0Dwt}}H====== ?4; 7 77rsamplec|dkr|}|}n4|dkrdx}}n)|dkr d||zzx}}ntj|r|x}}t||d||}t||d||}tj|d |d ||fS) aY 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 limitsr@FnullrJr )r r7rrr)r*isrealr?r) r9r<rrr7 prop_var_low prop_var_upprrs rproportions_ztostrGsF8  X  &++ || V  &)S3Y&77 || 8  /&.. | E4X%1 > > >C E4Y%1 > > >C :c!fc!f % %sC 00rc.tj|}t||\}}}|%tj||z|d|z zf}|dz }n|}t j|||\}}||||ffS)a 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. Nr)ddof)r* atleast_1dr2r-r chisquareravel) r9r<rr/r0r1rIchi2statr#s rproportions_chisquarerN9sT =  D/t<<E8V ?D5L$!e)2D#EFFz_U[[]]HNN4D4D*.000NHd TE8, ,,rhscttjtd}fd|D}t |||S)a 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. 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. rc g|]>}tt|t|d?SrrNlistrpairr9r<s rrz2proportions_chisquare_allpairs..sN & & &#5d#4d4::6F G G J & & &rmultitest_method)rr* triu_indicesrr)r9r<rX all_pairspvalss`` rproportions_chisquare_allpairsr\qse>boc%jj!445I & & & & &$ & & &E 5)>N O O OOrc||dvrtdtdtD}fd|D}t|||S)a 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. 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. Nrrcg|]}d|fSrr)rrs rrz6proportions_chisquare_pairscontrol..s666A!Q666rrc g|]>}tt|t|d?SrRrSrUs rrz6proportions_chisquare_pairscontrol..sW & & &#5d#4d4::6F%%%&( & & &rrW)rrr-rr)r9r<rrXr rZr[s`` r"proportions_chisquare_pairscontrolrasP {2EEE!!66q#e**!5!5666I & & & & &% & & &E 5)>N O O OOrr9c bdddd}|dkrd}|||}|}|drd }||z } ||z } | | z } |d krd nd } |d krL|d vr|| z|d| zz}} || z|d| zz}}| |z }||z }||z }|d |z z|z |d |z z|z z}tj|dz }|t j|z}||z }||z}n.|dr~t||d|\}}t||d|\}}t j| |z dz|| z dzz}t j| |z dz|| z dzz}| |z }| |z}n|dkrt|||||||\}}nztd|dkr |dvr|dkrdnd }||z||z}} ||z||z}}| |z }||z }||z } d | z d |z z d |z zd |z z }tj|dz }|t j|z}!t j t j | |!z }t j t j | |!z}n|dkr!t||||||}"|"j \}}nhtd|dkrC|dvr|dvr#t||||ddd }#|#\} }}}n$|dkrdnd }||z|d|zz}} ||z|d|zz}}| |z }||z }|d |z z |z d |z z}$d | z d || z z zd |z zd ||z z z}tj|dz }|t j|z}!t j t j |$|!z }t j t j |$|!z}n>|dkrt|||||||\}}ntdtd||fS)u Confidence intervals for comparing two independent proportions. This assumes that we have two independent binomial samples. Parameters ---------- count1, nobs1 : float Count and sample size for first sample. count2, nobs2 : float Count and sample size for the second sample. method : str Method for computing confidence interval. If method is None, then a default method is used. The default might change as more methods are added. diff: - 'wald', - 'agresti-caffo' - 'newcomb' (default) - 'score' ratio: - 'log' - 'log-adjusted' (default) - 'score' odds-ratio: - 'logit' - 'logit-adjusted' (default) - 'score' compare : string in ['diff', 'ratio' 'odds-ratio'] If compare is diff, then the confidence interval is for diff = p1 - p2. If compare is ratio, then the confidence interval is for the risk ratio defined by ratio = p1 / p2. If compare is odds-ratio, then the confidence interval is for the odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2). alpha : float Significance level for the confidence interval, default is 0.05. The nominal coverage probability is 1 - alpha. Returns ------- low, upp See Also -------- test_proportions_2indep tost_proportions_2indep Notes ----- Status: experimental, API and defaults might still change. more ``methods`` will be added. References ---------- .. [1] Fagerland, Morten W., Stian Lydersen, and Petter Laake. 2015. “Recommended Confidence Intervals for Two Independent Binomial Proportions.” Statistical Methods in Medical Research 24 (2): 224–54. https://doi.org/10.1177/0962280211415469. .. [2] Koopman, P. A. R. 1984. “Confidence Intervals for the Ratio of Two Binomial Proportions.” Biometrics 40 (2): 513–17. https://doi.org/10.2307/2531405. .. [3] Miettinen, Olli, and Markku Nurminen. "Comparative analysis of two rates." Statistics in medicine 4, no. 2 (1985): 213-226. .. [4] Newcombe, Robert G. 1998. “Interval Estimation for the Difference between Independent Proportions: Comparison of Eleven Methods.” Statistics in Medicine 17 (8): 873–90. https://doi.org/10.1002/(SICI)1097-0258(19980430)17:8<873::AID- SIM779>3.0.CO;2-I. .. [5] Newcombe, Robert G., and Markku M. Nurminen. 2011. “In Defence of Score Intervals for Proportions and Their Differences.” Communications in Statistics - Theory and Methods 40 (7): 1271–82. https://doi.org/10.1080/03610920903576580. newcomb log-adjustedlogit-adjustedr9r odds-ratioorrgNagr agresti-cafforrr9waldrjr)rZ)rtr7score)comparer7 correctionmethod not recognizedrrrdrJ)r7rologitrelogit-smoothedrtF shrink_factor return_corrzcompare not recognized)r startswithrrgrhr*rfr_score_confint_inversionr,rr_confint_riskratio_koopmanconfint _shrink_prob)%count1nobs1count2nobs2rtrnr7romethod_defaultrrr9addonecount1_nobs1_count2_nobs2_p1_p2_diff_varzd_waldrrlow1upp1low2upp2d_lowd_uppaddhalfratio_d_logresadjusted odds_ratio_s% rconfint_proportions_2indeprsr^(-$466N}}$ ~( \\^^F !  %B %B 7DO++QQF& . . .$vouq6z/AVG$vouq6z/AVGF"CF"C#IES/F*SAG_v-EEC uqy))A%F&.C&.CC   y ) ) 6+FE3;5JJJJD$+FE3;5JJJJD$GR$YNdRi!^;<.s.LLRWQZZ!^LLLLLLrzfunction is not vectorizedNr))rrr)rrr)rFr,r*rr) r~rrrrwrx vectorizednobs_colnobs_rowr< prob_indepcorrs rr}r}s7@LLfeVU-KLLLLLJ75666x&%&.5*@6*IJKKHx''H 5=DXaaag..$'9J : %DHT DGKKMM4:tAw{{}}EEd#UT!W[[]]%:d#UT!W[[]]%:<=GH Hrc |dkrdnd} || }||z} ||z} ||z } ||z } || kr| | z x}}||}}| }|dkr|}|dkr| }|d|zz|z| z | z }||z| z d|zz |z| z}||zd|z z}|d|zz dz||zd|dzzz z |d|zz z}tj|tj|d|zz dz|d|zz z z}tjtj||dzz zdz }d|ztj|z|d|zz z }||z}|d|z z|z |d|z z|z z}|r || | dz z z}| |z |z }ng|dkr|}|dkrH| |z}||z|z|z||zz }| }| tj|dzd |z|zz z d|zz }||z}|d|z z|z |dz|zd|z z|z z}|r || | dz z z}| ||zz }n|d vr|} | dkrW|| dz z}|| z|z| | dz zz }| }| tj|dzd |z|zz zd|zz }|| zd|| dz zzz }d }!tj||!d|!z }tj||!d|!z }d|d|z z|zz d|d|z z|zz z}|r || | dz z z}| |z |d|z zz ||z |d|z zz z }t|tj|| \}"}#|rt|"|#|d ||||}$|$S|"|#fS)a Score test for two independent proportions This uses the constrained estimate of the proportions to compute the variance under the Null hypothesis. Parameters ---------- count1, nobs1 : count and sample size for first sample count2, nobs2 : count and sample size for the second sample value : float diff, ratio or odds-ratio under the null hypothesis. If value is None, then equality of proportions under the Null is assumed, i.e. value=0 for 'diff' or value=1 for either rate or odds-ratio. compare : string in ['diff', 'ratio' 'odds-ratio'] If compare is diff, then the confidence interval is for diff = p1 - p2. If compare is ratio, then the confidence interval is for the risk ratio defined by ratio = p1 / p2. If compare is odds-ratio, then the confidence interval is for the odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2) return_results : bool If true, then a results instance with extra information is returned, otherwise a tuple with statistic and pvalue is returned. Returns ------- results : results instance or tuple If return_results is True, then a results instance with the information in attributes is returned. If return_results is False, then only ``statistic`` and ``pvalue`` are returned. statistic : float test statistic asymptotically normal distributed N(0, 1) pvalue : float p-value based on normal distribution other attributes : additional information about the hypothesis test Notes ----- Status: experimental, the type or extra information in the return might change. r9rrNr)rrrr[)rhrgg|=r rm) statisticrQrnrtvariancer  prop1_null prop2_null) r*r+rfrarccoscosrsr r)%r~rrrrrnr roreturn_results value_defaultr<r9rrprop0rcount0nobs0p0r9tmp3tmp2tmp1tmp0qrPrr diff_statrrroratiorrrQrs% rscore_test_proportions_2indeprsf!F**AAM } 5=D VOE %B %B  $EEF B& 199DAI%-4uABAAR"'!Q$Q"2333A>EEMEE "U*ax%1u9-56  % 44!8$ $C%"*_ ( ( ( Q;;!$A&&1*)==AAR"'!Q$Q"2333A>EFNa%6A:*>&>?EsAG,,sAG,,EQY'%/0EQY'%/01  % 44!8$ $C5jUa%i%895jUa%i%89: ( 273<<4?AAAIv !I!'")!(#&&1%*%* &  rc dddd} |dkrd}|| |}|}|drd}| |d krd nd }ttj||||g\}}}}||z } ||z } | | z } | | z }| d | z z | z d | z z}d}|d kr|d vrh|dkrd nd }||z|d |zz}}||z|d |zz}}||z }||z }||z |z }|d |z z|z |d |z z|z z}|tj|z }d}ni|drd}t ||dkr/t|||||||||  }| dur |dd \}}d}d}ntd|dkr|dvr|dkrdnd }||z||z}}||z||z}}||z }||z }||z }d |z d |z z d |z zd |z z }tj |tj |z }|tj|z }d}nm|dkr/t|||||||||  }| dur |dd \}}d}d}n8td|dkr|dvr|dvr#t||||d dd } | \}}}}n$|dkrdnd }||z|d |zz}}||z|d |zz}}||z }||z }|d |z z |z d |z z}!d |z d ||z z zd |z zd ||z z z}tj |!tj |z }|tj|z }d}nX|dkr.t|||||||||  }| dur |dd \}}d}d}n$td|ztd|z|dkr)|'t|tj||\}}| r:|t||||| ||||| }n| |_ ||_||_||_|S||fS)a Hypothesis test for comparing two independent proportions This assumes that we have two independent binomial samples. The Null and alternative hypothesis are for compare = 'diff' - H0: prop1 - prop2 - value = 0 - H1: prop1 - prop2 - value != 0 if alternative = 'two-sided' - H1: prop1 - prop2 - value > 0 if alternative = 'larger' - H1: prop1 - prop2 - value < 0 if alternative = 'smaller' for compare = 'ratio' - H0: prop1 / prop2 - value = 0 - H1: prop1 / prop2 - value != 0 if alternative = 'two-sided' - H1: prop1 / prop2 - value > 0 if alternative = 'larger' - H1: prop1 / prop2 - value < 0 if alternative = 'smaller' for compare = 'odds-ratio' - H0: or - value = 0 - H1: or - value != 0 if alternative = 'two-sided' - H1: or - value > 0 if alternative = 'larger' - H1: or - value < 0 if alternative = 'smaller' where odds-ratio or = prop1 / (1 - prop1) / (prop2 / (1 - prop2)) Parameters ---------- count1 : int Count for first sample. nobs1 : int Sample size for first sample. count2 : int Count for the second sample. nobs2 : int Sample size for the second sample. value : float Value of the difference, risk ratio or odds ratio of 2 independent proportions under the null hypothesis. Default is equal proportions, 0 for diff and 1 for risk-ratio and for odds-ratio. method : string Method for computing the hypothesis test. If method is None, then a default method is used. The default might change as more methods are added. diff: - 'wald', - 'agresti-caffo' - 'score' if correction is True, then this uses the degrees of freedom correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985 ratio: - 'log': wald test using log transformation - 'log-adjusted': wald test using log transformation, adds 0.5 to counts - 'score': if correction is True, then this uses the degrees of freedom correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985 odds-ratio: - 'logit': wald test using logit transformation - 'logit-adjusted': wald test using logit transformation, adds 0.5 to counts - 'logit-smoothed': wald test using logit transformation, biases cell counts towards independence by adding two observations in total. - 'score' if correction is True, then this uses the degrees of freedom correction ``nobs / (nobs - 1)`` as in Miettinen Nurminen 1985 compare : {'diff', 'ratio' 'odds-ratio'} If compare is `diff`, then the hypothesis test is for the risk difference diff = p1 - p2. If compare is `ratio`, then the hypothesis test is for the risk ratio defined by ratio = p1 / p2. If compare is `odds-ratio`, then the hypothesis test is for the odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2) alternative : {'two-sided', 'smaller', 'larger'} alternative hypothesis, which can be two-sided or either one of the one-sided tests. correction : bool If correction is True (default), then the Miettinen and Nurminen small sample correction to the variance nobs / (nobs - 1) is used. Applies only if method='score'. return_results : bool If true, then a results instance with extra information is returned, otherwise a tuple with statistic and pvalue is returned. Returns ------- results : results instance or tuple If return_results is True, then a results instance with the information in attributes is returned. If return_results is False, then only ``statistic`` and ``pvalue`` are returned. statistic : float test statistic asymptotically normal distributed N(0, 1) pvalue : float p-value based on normal distribution other attributes : additional information about the hypothesis test See Also -------- tost_proportions_2indep confint_proportions_2indep Notes ----- Status: experimental, API and defaults might still change. More ``methods`` will be added. The current default methods are - 'diff': 'agresti-caffo', - 'ratio': 'log-adjusted', - 'odds-ratio': 'logit-adjusted' rjrdrerfrhrgNrir9rrrkr)r6rcz)newcomb not available for hypothesis testrm)rrnr rorFrprrqrJrrrurvzmethod "%s" not recognizedzcompare "%s" not recognizedr) rrQrnrtr9r odds_ratiorr r)rrymapr*rerfrrrr,rr}r rr9rrr)"r~rrrrrtrnr rorrrrr9rrrrrrrrrrrrrdistrr:rQrrrrs" rtest_proportions_2indeprOsD.-$466N}}$ ~( \\^^F !  }&&A#&rz(.vu'E$G$G FE65 %B %B 7D GEq2v#q2v.J C& . . . O33QQF$vouq6z/AVG$vouq6z/AVGF"CF"Cc E)IS/F*SAG_v-EEC!BGCLL0IEE   y ) ) 6=C%c** * w  /vu6;W>>G Xw &&rc ddlm}t|||||\} } } ||||| | |} |rt| | | | |||z||} | S| S)aq Power for ztest that two independent proportions are equal This assumes that the variance is based on the pooled proportion under the null and the non-pooled variance under the alternative Parameters ---------- diff : float difference between proportion 1 and 2 under the alternative prop2 : float proportion for the reference case, prop2, proportions for the first case will be computed using p2 and diff p1 = p2 + diff nobs1 : float or int number of observations in sample 1 ratio : float sample size ratio, nobs2 = ratio * nobs1 alpha : float in interval (0,1) Significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. value : float currently only `value=0`, i.e. equality testing, is supported alternative : string, 'two-sided' (default), 'larger', 'smaller' Alternative hypothesis whether the power is calculated for a two-sided (default) or one sided test. The one-sided test can be either 'larger', 'smaller'. return_results : bool If true, then a results instance with extra information is returned, otherwise only the computed power is returned. Returns ------- results : results instance or float If return_results is True, then a results instance with the information in attributes is returned. If return_results is False, then only the power is returned. power : float Power of the test, e.g. 0.8, is one minus the probability of a type II error. Power is the probability that the test correctly rejects the Null Hypothesis if the Alternative Hypothesis is true. Other attributes in results instance include : p_pooled pooled proportion, used for std_null std_null standard error of difference under the null hypothesis (without sqrt(nobs1)) std_alt standard error of difference under the alternative hypothesis (without sqrt(nobs1)) r)normal_power_hetrr7r)rstd_alternativer )rr;rrrrrr7)statsmodels.stats.powerrrr )r9rrrr7rr rrr;rrpow_rs rpower_proportions_2indeprst988888"24e9>e#M#M#MHh  D%,3(3 5 5 5D 4&&$  5= %   rctddlm}|dvr|dz }t|||||\}} } ||||| | } | S)a Required sample size assuming normal distribution based on one tail This uses an explicit computation for the sample size that is required to achieve a given power corresponding to the appropriate tails of the normal distribution. This ignores the far tail in a two-sided test which is negligible in the common case when alternative and null are far apart. Parameters ---------- diff : float Difference between proportion 1 and 2 under the alternative prop2 : float proportion for the reference case, prop2, proportions for the first case will be computing using p2 and diff p1 = p2 + diff power : float Power for which sample size is computed. ratio : float Sample size ratio, nobs2 = ratio * nobs1 alpha : float in interval (0,1) Significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. value : float Currently only `value=0`, i.e. equality testing, is supported alternative : string, 'two-sided' (default), 'larger', 'smaller' Alternative hypothesis whether the power is calculated for a two-sided (default) or one sided test. In the case of a one-sided alternative, it is assumed that the test is in the appropriate tail. Returns ------- nobs1 : float Number of observations in sample 1. r)normal_sample_size_one_tailr^r)r)rr)rrr) r9rrrr7rr rr3rrr<s r%samplesize_proportions_2indep_onetailrTsPDCCCCC))) +D%u27uFFFAx ' &tUEH7> @ @ @D Krc  fd}tddd }dddd } t|  } | d tj| d d zz} | dtj| dd zz } dkr|j} t | d} ndkr |j} | dz} dkr|j} tj || | }tj || | }||fS)a Compute score confidence interval by inverting score test Parameters ---------- count1, nobs1 : Count and sample size for first sample. count2, nobs2 : Count and sample size for the second sample. compare : string in ['diff', 'ratio' 'odds-ratio'] If compare is `diff`, then the confidence interval is for diff = p1 - p2. If compare is `ratio`, then the confidence interval is for the risk ratio defined by ratio = p1 / p2. If compare is `odds-ratio`, then the confidence interval is for the odds-ratio defined by or = p1 / (1 - p1) / (p2 / (1 - p2). alpha : float in interval (0,1) Significance level, e.g. 0.05, is the probability of a type I error, that is wrong rejections if the Null Hypothesis is true. correction : bool If correction is True (default), then the Miettinen and Nurminen small sample correction to the variance nobs / (nobs - 1) is used. Applies only if method='score'. Returns ------- low : float Lower confidence bound. upp : float Upper confidence bound. c Ht|dd }|jz S)Nrmrrrnrtror )rrQ) rrr7rnror~rrrs rrz&_score_confint_inversion..funcs; #FE65*+WW/90; = = =x%rrrmrrrlrrsrf)rtrnr7rrJg?r9gwJ?rr)rg) rrr*absr9minrrrro)r~rrrrnr7rorrt0 use_methodrci0r&r%paramrrs``````` rrzrzslD            "&%()77-7.9 ; ; ;C!5HHJ %feVU-7-@.5U D D DD a26$q'??S( (B a26$q'??T) )B&W   G    a, /$r * *C /$E * *C 8Orcf||||f\}}}} ||z} || z} tj|dz dz} |r | | | dz z z} ||| z|z| ||zz| zzz} | || z| zd| z|z|zz| ||zd|zzz| zzz}d|z| z|z| z| |z|z|zz|| z| z| zz}| |z|z| z}tjtj| |||g}|ddddd}d| |z d|z z|| z| |zz z z |z }t }||_||_|S)z Score confidence interval for ratio or proportions, Koopman/Nam signature not consistent with other functions When correction is True, then the small sample correction nobs / (nobs - 1) by Miettinen/Nurminen is used. r)rN) rrgrhr*sortrootsr r|_p_roots)r~rrrr7rox0x1n0n1r rra1a2a3a4p_roots_p_rootscirs rr{r{sVUE1NBB RA RA uqy!!1$A Q!a%[ rAv{R27^a// 0B b1q1urzB.rBwR7G1H11LL MB R"r A B b 0 027Q;? BB R! 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