>h[7SrSSKrSSKrSSKJrJr SSKJr SSK J r SSK J r \Rr /SQ/SQS .rS$S jrS%S jrS&S jrS'S jrS(SjrS)Sjr/SQ/SQS./SQ/SQS.S .rS*SjrS+SjrS,SjrS-SjrS-SjrS.SjrS/SjrS0SjrS1SjrS1SjrS2Sjr S3S jr!S4S!jr"S5S"jr#S6S#jr$g)7zj Test for ratio of Poisson intensities in two independent samples Author: Josef Perktold License: BSD-3 N)statsoptimize) HolderTuple)_zstat_generic2)_mover_confint)waldscoreexact-cmidp-cwaldccvsqrt-asqrt-vsqrt) rr r r jeffr r rr sqrt-cent sqrt-centcc)testconfintc VUnX- nUc Sn[U5eUS:waUS;a Sn[U5eSn US:Xa$[R"XW-U- 5n Xr- U - n GOUS:Xa*[R"XWSU- --U- 5n Xr- U - n GOcUS :XaC[R"XR-U- 5n Xr- U - n [RR U 5n GOUR S 5(dUR S 5(a[RRXU-5n [RR US- Xb-5nUR S 5(aPU S[RRXU-5-- n US[RRXU-5-- nUS :XaS [R"X5-n OUS:XaUn OUS:XaU n O Sn[U5e[Rn Sn OUS:Xa5Sn [R"U5[R"Xb-5- U - n OUS:Xa;Sn [R"US-5[R"Xb-S-5- U - n OsUS:Xa_Sn [RRS5n[R"XS -S -S- -5[R"Xb-5- U - n O[SU-5eU S:Xa[U SU5up[U [R"W SS5U UUUUS9nU$)aQTest for one sample poisson mean or rate Parameters ---------- count : array_like Observed count, number of events. nobs : arrat_like Currently this is total exposure time of the count variable. This will likely change. value : float, array_like This is the value of poisson rate under the null hypothesis. method : str Method to use for confidence interval. This is required, there is currently no default method. See Notes for available methods. alternative : {'two-sided', 'smaller', 'larger'} alternative hypothesis, which can be two-sided or either one of the one-sided tests. dispersion : float Dispersion scale coefficient for Poisson QMLE. Default is that the data follows a Poisson distribution. Dispersion different from 1 correspond to excess-dispersion in Poisson quasi-likelihood (GLM). Dispersion coeffficient different from one is currently only used in wald and score method. Returns ------- HolderTuple instance with test statistic, pvalue and other attributes. Notes ----- The implementatio of the hypothesis test is mainly based on the references for the confidence interval, see confint_poisson. Available methods are: - "score" : based on score test, uses variance under null value - "wald" : based on wald test, uses variance base on estimated rate. - "waldccv" : based on wald test with 0.5 count added to variance computation. This does not use continuity correction for the center of the confidence interval. - "exact-c" central confidence interval based on gamma distribution - "midp-c" : based on midp correction of central exact confidence interval. this uses numerical inversion of the test function. not vectorized. - "sqrt" : based on square root transformed counts - "sqrt-a" based on Anscombe square root transformation of counts + 3/8. See Also -------- confint_poisson 9method needs to be specified, currently no default method)rwaldccr z:excess dispersion only supported in wald and score methodsnormalrr ?r r r two-sidedlargersmallerz8alternative should be "two-sided", "larger" or "smaller"Poissonrr ?rg? unknown method %sr) statisticpvalue distributionmethod alternativeratenobs) ValueErrornprrnormsf startswithpoissoncdfpmfminimumnanisfrrclip)countr)valuer&r' dispersionnr(msgdiststdr#r$pv1pv2critress jC:\Users\julio\OneDrive\Documentos\Trabajo\Ideas Frescas\venv\Lib\site-packages\statsmodels/stats/rates.py test_poissonrB1sn A 9D ~IoQ 4 4NCS/ ! D ggj'!+,\S( 9  ggj37N3a78\S( 7 ggj(1,-\S( y)   9 % %):):8)D)Dmm5y1mmuqy!)4   X & &emm//5yAAACemm//5yAAAC + %C--F H $F I %FLCS/ !FF  6 WWU^bggai&88C? 8 WWUU]+bggai%6G.HHCO 8 zz~~e$WWUAgkR%778WWQY'(+./ ,v566 x+Iq+F wwvq!$   C JcZ UnX- nUS- nUc Sn[U5eUS:XaB[RRU5[R "XT- 5-nXW- XW-4nGO(US:XaI[RRU5[R "USU- -U- 5-nXW- XW-4nGOUS:XaZ[RRU5n X S-S- -n U [R "X S-S- -5-nX- U- X-U- 4nGOyUS:Xa[ XSU-SS S 9nGObUS :XaQ[RRU5n XYS-SU-- -n U [R "XT- 5-nX- X-4nGO US :XaU[RRU5n X S-S- -n U [R "US -5-nX- U- X-U- 4nGOUS:Xa[RRU5n [R "[R"US -S- S55n [R "US -S-5n U S- n[R"X- S5S-S - U- X-S-S - U- 4nGOUS:Xan[RRU5n [R "US -5n U S- n[R"X- S5S-S - U- X-S-S - U- 4nGOUS:Xap[RRU5n [R "X S-S-S- -5n U S- n[R"X- S5S-U- X-S-U- 4nGOUS;a[RRX05U- n [RRX0S-5U- n[R"U 5R5(a6[R"U 5S:XaSn OSU [R"U 5'X4nOnURS5(aJUS-n[RRX?5U- [RRX?5U- 4nO[SU-5e[R"USS5US4nU$)uP Confidence interval for a Poisson mean or rate The function is vectorized for all methods except "midp-c", which uses an iterative method to invert the hypothesis test function. All current methods are central, that is the probability of each tail is smaller or equal to alpha / 2. The one-sided interval limits can be obtained by doubling alpha. Parameters ---------- count : array_like Observed count, number of events. exposure : arrat_like Currently this is total exposure time of the count variable. This will likely change. method : str Method to use for confidence interval This is required, there is currently no default method alpha : float in (0, 1) Significance level, nominal coverage of the confidence interval is 1 - alpha. Returns ------- tuple (low, upp) : confidence limits. Notes ----- Methods are mainly based on Barker (2002) [1]_ and Swift (2009) [3]_. Available methods are: - "exact-c" central confidence interval based on gamma distribution - "score" : based on score test, uses variance under null value - "wald" : based on wald test, uses variance base on estimated rate. - "waldccv" : based on wald test with 0.5 count added to variance computation. This does not use continuity correction for the center of the confidence interval. - "midp-c" : based on midp correction of central exact confidence interval. this uses numerical inversion of the test function. not vectorized. - "jeffreys" : based on Jeffreys' prior. computed using gamma distribution - "sqrt" : based on square root transformed counts - "sqrt-a" based on Anscombe square root transformation of counts + 3/8. - "sqrt-centcc" will likely be dropped. anscombe with continuity corrected center. (Similar to R survival cipoisson, but without the 3/8 right shift of the confidence interval). sqrt-cent is the same as sqrt-a, using a different computation, will be deleted. sqrt-v is a corrected square root method attributed to vandenbrouke, which might also be deleted. Todo: - missing dispersion, - maybe split nobs and exposure (? needed in NB). Exposure could be used to standardize rate. - modified wald, switch method if count=0. See Also -------- test_poisson References ---------- .. [1] Barker, Lawrence. 2002. “A Comparison of Nine Confidence Intervals for a Poisson Parameter When the Expected Number of Events Is ≤ 5.” The American Statistician 56 (2): 85–89. https://doi.org/10.1198/000313002317572736. .. [2] Patil, VV, and HV Kulkarni. 2012. “Comparison of Confidence Intervals for the Poisson Mean: Some New Aspects.” REVSTAT–Statistical Journal 10(2): 211–27. .. [3] Swift, Michael Bruce. 2009. “Comparison of Confidence Intervals for a Poisson Mean – Further Considerations.” Communications in Statistics - Theory and Methods 38 (5): 748–59. https://doi.org/10.1080/03610920802255856. rrrr rr r r )alphar& method_startrrr rrr rr!)gammar rgrr")r*rr,r4r+r_invert_test_confintmaximumrHppfisnananysizer.)r6exposurer&rFr9r(r:whalfcir?center center_low center_upplowuppcountcs rAconfint_poissonrXsQd A  D AIE ~Io  u%(99lDL ) 9  u%a10D(EElDL ) 7 zz~~e$q1$rwwuQw{233~"V^q$8 9 8  !%!e)H/8: 6 zz~~e$a1q5))rwwtx((nfn - ; zz~~e$q1$rwwuu}--~"V^q$8 9 = zz~~e$WWRZZ (;Q?@ WWUU]S01 q :-q1A5=B"Q&.!35 8 zz~~e$'q 6>1-1E9Q>"U*a/1 8 zz~~e$'A+!334qjj+a /! 3fnq5H15L L ' 'kkooe+h6kkooe1W-8 88C=    wws|q %(BHHSM"Z   6 " "kkooe,x7kkooe,x79,v566 **RUA 1 &B IrCcSU- nUnUS:waUS-n[XXHS9upUS:waX-n X-n US:XaJ[RRUS- U 5n [RRSUS- - U 5n OaUS:Xa&Sn [RRSU- U 5n O5US:Xa/[RRXy5n [R n [R "W S5n U W 4$)utolerance interval for a poisson observation Parameters ---------- count : array_like Observed count, number of events. exposure : arrat_like Currently this is total exposure time of the count variable. prob : float in (0, 1) Probability of poisson interval, often called "content". With known parameters, each tail would have at most probability ``1 - prob / 2`` in the two-sided interval. exposure_new : float Exposure of the new or predicted observation. method : str Method to used for confidence interval of the estimate of the poisson rate, used in `confint_poisson`. This is required, there is currently no default method. alpha : float in (0, 1) Significance level for the confidence interval of the estimate of the Poisson rate. Nominal coverage of the confidence interval is 1 - alpha. alternative : {"two-sider", "larger", "smaller") The tolerance interval can be two-sided or one-sided. Alternative "larger" provides the upper bound of the confidence interval, larger counts are outside the interval. Returns ------- tuple (low, upp) of limits of tolerance interval. The tolerance interval is a closed interval, that is both ``low`` and ``upp`` are in the interval. Notes ----- verified against R package tolerance `poistol.int` See Also -------- confint_poisson confint_quantile_poisson References ---------- .. [1] Hahn, Gerald J., and William Q. Meeker. 1991. Statistical Intervals: A Guide for Practitioners. 1st ed. Wiley Series in Probability and Statistics. Wiley. https://doi.org/10.1002/9780470316771. .. [2] Hahn, Gerald J., and Ramesh Chandra. 1981. “Tolerance Intervals for Poisson and Binomial Variables.” Journal of Quality Technology 13 (2): 100–110. https://doi.org/10.1080/00224065.1981.11980998. rrrr&rFrrrrXrr/rKr+infrJ) r6rOprob exposure_newr&rFr' prob_tailalpha_rUrVlow_predupp_preds rAtolerance_int_poissonrctsnDI Fk!uvLHCq  k!==$$Y]C8==$$QQ%6<  ==$$Q]C8  !==$$Y466zz(A&H X rCcUnUS:waUS-n[XXGS9upUS:waX-nX-n US:Xa?[RRX(5n [RRX)5n O]US:Xa"Sn [RRX)5n O5US:Xa/[RRX(5n [R n [R "W S5n U W 4$)auconfidence interval for quantile of poisson random variable Parameters ---------- count : array_like Observed count, number of events. exposure : arrat_like Currently this is total exposure time of the count variable. prob : float in (0, 1) Probability for the quantile, e.g. 0.95 to get the upper 95% quantile. With known mean mu, the quantile would be poisson.ppf(prob, mu). exposure_new : float Exposure of the new or predicted observation. method : str Method to used for confidence interval of the estimate of the poisson rate, used in `confint_poisson`. This is required, there is currently no default method. alpha : float in (0, 1) Significance level for the confidence interval of the estimate of the Poisson rate. Nominal coverage of the confidence interval is 1 - alpha. alternative : {"two-sider", "larger", "smaller") The tolerance interval can be two-sided or one-sided. Alternative "larger" provides the upper bound of the confidence interval, larger counts are outside the interval. Returns ------- tuple (low, upp) of limits of tolerance interval. The confidence interval is a closed interval, that is both ``low`` and ``upp`` are in the interval. See Also -------- confint_poisson tolerance_int_poisson References ---------- Hahn, Gerald J, and William Q Meeker. 2010. Statistical Intervals: A Guide for Practitioners. rrrZrrrrr[) r6rOr]r^r&rFr'r`rUrVrarbs rAconfint_quantile_poissonresZFk!uvLHCq  k!==$$T/==$$T/  ==$$T/  !==$$T/66zz(A&H X rCc^^^^UUUU4Sjn[TTUS9n[R"XVSSSS9n[R"XVSSSS9n[R"U5S:XdeUSUS4$)z6invert hypothesis test to get confidence interval c2>[TTUTS9ST- S-nU$)N)r7r&rr)rB)rvrFr6r&r)s rAfunc"_invert_test_confint..funcs- %Qv >q A  rC)r&r:0yE>Fxtoldispr)rXrfminr+rN) r6r)rFr&rGrjrQrUrVs ```` rArIrI sq  \ :B --e$U ;C --e$U ;C 773<1   q63q6>rCc ^^^^^^^UUUUUUU4Sjn[TTTTUTS9n [R"XSSSS9n [R"XSSSS9n [R"U 5S:XdeU SU S4$)zAinvert hypothesis test to get confidence interval for 2indep c 8>[TTTTUTTS9ST- S-nU$)N)r7r&comparerr)test_poisson_2indep) rhrirFrscount1count2 exposure1 exposure2r&s rArj)_invert_test_confint_2indep..func%sB Y VW rC)r&rsrrlFrmr)confint_poisson_2indeprrpr+rN) rurwrvrxrFr&rsrGrjrQrUrVs ``````` rA_invert_test_confint_2indepr{s{  69'3W FB --e$U ;C --e$U ;C 773<1   q63q6>rC) rr score-logwald-log exact-cond cond-midpr etest-score etest-wald)rr r rr)ratiodiff)rr r|r}sqrtccmover)rr r rc [[RXX#/5uppX- nX- X- nnSnUS:XGauUcSnUb[R"S[ 5 UnUcUcS=pEOUnUnUU- nUS;a'XU-- [R "X-U-5- nSnGOvUS;a*XU-- [R "XUS ---5- nSnGOFUS ;aZ[R"X- 5[R"U5- nU[R "S SU- -U-X-- 5-nSnGOUS ;aS[R"X- 5[R"U5- [R "SU - SU - -5- nSnGOUS ;aYS [R "U S -5[R "U S -U-5- -nU[R "SU-5-nSnGO.US;aeSSKJ n USU-- nX-n[RnURU UUUS9nUS;a'US[RRU UU5-- nSnGOURS5(a8UR!S5(aSnOSnU c0n [#XX#4UUUS.U D6unnSnGOu[%SUS35eUS:XGaTUcSnUS;a+UU- U- [R "X- UU - -5- nSnGO)US;a9UU- U- nU[R "US-U S -- US-U S -- -5-nSnOUS;aqX-nUX-- nUU- nSU[R "US -SU-U -X-- -5--nUU-nUU- U- [R "UU - UU - -5- nUU4nSnOsURS5(aCUR!S5(aSnO SnUS:XaUS-nU c0n [#XX#4UUSUS.U D6unnSnO[%SUS35e['S 5eUS:Xa[)USU5unnUU4n UU- n!UU- n"[+UWUUUUU U!U"UUUS!9 n#U#$)"uTest for comparing two sample Poisson intensity rates. Rates are defined as expected count divided by exposure. The Null and alternative hypothesis for the rates, rate1 and rate2, of two independent Poisson samples are for compare = 'diff' - H0: rate1 - rate2 - value = 0 - H1: rate1 - rate2 - value != 0 if alternative = 'two-sided' - H1: rate1 - rate2 - value > 0 if alternative = 'larger' - H1: rate1 - rate2 - value < 0 if alternative = 'smaller' for compare = 'ratio' - H0: rate1 / rate2 - value = 0 - H1: rate1 / rate2 - value != 0 if alternative = 'two-sided' - H1: rate1 / rate2 - value > 0 if alternative = 'larger' - H1: rate1 / rate2 - value < 0 if alternative = 'smaller' Parameters ---------- count1 : int Number of events in first sample, treatment group. exposure1 : float Total exposure (time * subjects) in first sample. count2 : int Number of events in second sample, control group. exposure2 : float Total exposure (time * subjects) in second sample. ratio_null: float Ratio of the two Poisson rates under the Null hypothesis. Default is 1. Deprecated, use ``value`` instead. .. deprecated:: 0.14.0 Use ``value`` instead. value : float Value of the ratio or difference of 2 independent rates under the null hypothesis. Default is equal rates, i.e. 1 for ratio and 0 for diff. .. versionadded:: 0.14.0 Replacement for ``ratio_null``. method : string Method for the test statistic and the p-value. Defaults to `'score'`. see Notes. ratio: - 'wald': method W1A, wald test, variance based on observed rates - 'score': method W2A, score test, variance based on estimate under the Null hypothesis - 'wald-log': W3A, uses log-ratio, variance based on observed rates - 'score-log' W4A, uses log-ratio, variance based on estimate under the Null hypothesis - 'sqrt': W5A, based on variance stabilizing square root transformation - 'exact-cond': exact conditional test based on binomial distribution This uses ``binom_test`` which is minlike in the two-sided case. - 'cond-midp': midpoint-pvalue of exact conditional test - 'etest' or 'etest-score: etest with score test statistic - 'etest-wald': etest with wald test statistic diff: - 'wald', - 'waldccv' - 'score' - 'etest-score' or 'etest: etest with score test statistic - 'etest-wald': etest with wald test statistic compare : {'diff', 'ratio'} Default is "ratio". If compare is `ratio`, then the hypothesis test is for the rate ratio defined by ratio = rate1 / rate2. If compare is `diff`, then the hypothesis test is for diff = rate1 - rate2. alternative : {"two-sided" (default), "larger", smaller} The alternative hypothesis, H1, has to be one of the following - 'two-sided': H1: ratio, or diff, of rates is not equal to value - 'larger' : H1: ratio, or diff, of rates is larger than value - 'smaller' : H1: ratio, or diff, of rates is smaller than value etest_kwds: dictionary Additional optional parameters to be passed to the etest_poisson_2indep function, namely y_grid. Returns ------- results : instance of HolderTuple class The two main attributes are test statistic `statistic` and p-value `pvalue`. See Also -------- tost_poisson_2indep etest_poisson_2indep Notes ----- The hypothesis tests for compare="ratio" are based on Gu et al 2018. The e-tests are also based on ... - 'wald': method W1A, wald test, variance based on separate estimates - 'score': method W2A, score test, variance based on estimate under Null - 'wald-log': W3A, wald test for log transformed ratio - 'score-log' W4A, score test for log transformed ratio - 'sqrt': W5A, based on variance stabilizing square root transformation - 'exact-cond': exact conditional test based on binomial distribution - 'cond-midp': midpoint-pvalue of exact conditional test - 'etest': etest with score test statistic - 'etest-wald': etest with wald test statistic The hypothesis test for compare="diff" are mainly based on Ng et al 2007 and ... - wald - score - etest-score - etest-wald Note the etests use the constraint maximum likelihood estimate (cmle) as parameters for the underlying Poisson probabilities. The constraint cmle parameters are the same as in the score test. The E-test in Krishnamoorty and Thomson uses a moment estimator instead of the score estimator. References ---------- .. [1] Gu, Ng, Tang, Schucany 2008: Testing the Ratio of Two Poisson Rates, Biometrical Journal 50 (2008) 2, 2008 .. [2] Ng, H. K. T., K. Gu, and M. L. Tang. 2007. “A Comparative Study of Tests for the Difference of Two Poisson Means.” Computational Statistics & Data Analysis 51 (6): 3085–99. https://doi.org/10.1016/j.csda.2006.02.004. Nrr /'ratio_null' is deprecated, use 'value' keywordrr rrr)r|)r})rr )r~rr) proportion)propr')rrbinomialetestr)r7r&r'r/method "" not recognizedrr rEz-scorer7r&rsr'#"compare" needs to be ratio or diff) r#r$r%rsr&r'ratesrrr7 rates_cmle ratio_null)mapr+asarraywarningswarn FutureWarningrlogstatsmodels.statsrr3 binom_testrbinomr1r.endswithetest_poisson_2indepr*NotImplementedErrorrr)$rurwrvrxr7rr&rsr' etest_kwdsy1n1y2n2drate1rate2rrhr_dstatr;rbpy_totalr$ method_etest count_pooled rate_pooleddtr2_cmler1_cmlerrrr@s$ rArtrt\sff%KLNBB A7BG5EJ' >F  ! MMK' )E  %-!" "EJJ !e Y cMRWWbg_%==DD x cMRWWRsAv+-=%>>DD } $FF27ObffSk1D BGGQS[3.27;< 0 if alternative = 'larger' - H1: rate1 - rate2 - value < 0 if alternative = 'smaller' for compare = 'ratio' - H0: rate1 / rate2 - value = 0 - H1: rate1 / rate2 - value != 0 if alternative = 'two-sided' - H1: rate1 / rate2 - value > 0 if alternative = 'larger' - H1: rate1 / rate2 - value < 0 if alternative = 'smaller' Parameters ---------- count1 : int Number of events in first sample exposure1 : float Total exposure (time * subjects) in first sample count2 : int Number of events in first sample exposure2 : float Total exposure (time * subjects) in first sample ratio_null: float Ratio of the two Poisson rates under the Null hypothesis. Default is 1. Deprecated, use ``value`` instead. .. deprecated:: 0.14.0 Use ``value`` instead. value : float Value of the ratio or diff of 2 independent rates under the null hypothesis. Default is equal rates, i.e. 1 for ratio and 0 for diff. .. versionadded:: 0.14.0 Replacement for ``ratio_null``. method : {"score", "wald"} Method for the test statistic that defines the rejection region. alternative : string The alternative hypothesis, H1, has to be one of the following - 'two-sided': H1: ratio of rates is not equal to ratio_null (default) - 'larger' : H1: ratio of rates is larger than ratio_null - 'smaller' : H1: ratio of rates is smaller than ratio_null y_grid : None or 1-D ndarray Grid values for counts of the Poisson distribution used for computing the pvalue. By default truncation is based on an upper tail Poisson quantiles. ygrid : None or 1-D ndarray Same as y_grid. Deprecated. If both y_grid and ygrid are provided, ygrid will be ignored. .. deprecated:: 0.14.0 Use ``y_grid`` instead. Returns ------- stat_sample : float test statistic for the sample pvalue : float References ---------- Gu, Ng, Tang, Schucany 2008: Testing the Ratio of Two Poisson Rates, Biometrical Journal 50 (2008) 2, 2008 Ng, H. K. T., K. Gu, and M. L. Tang. 2007. “A Comparative Study of Tests for the Difference of Two Poisson Means.” Computational Statistics & Data Analysis 51 (6): 3085–99. https://doi.org/10.1016/j.csda.2006.02.004. rrNrrrcP>XT-- [R"X-T-T-5- $Nrx1x2rrs rA stat_func'etest_poisson_2indep..stat_funcs(#X "'S31F)GGGrCrcV>XT-- [R"XTS---T-5- $)Nrrrs rArrs-#X 36k1AC1G)HHHrCzmethod not recognizedrrTr7rc>[UTUTTS9$)N)r7)r)rrrrr7s rArrs"2r2r??rCcr>UT- UT- p2X#- T- nU[R"UT- UT- -T-5-nU$rr) rrrrrrrrr7s rArrsH!BwRu - URZ 7# =>> rCzygrid is deprecated, use y_gridgvIh%<=dz.y_grid needs to be None or 1-dimensional arraygV瞯<)rz2-sided2s)rl)rszinvalid alternative)rr+rrrrr*rrr/r4maxarangendimr1abssum)#rurwrvrxrr7r&rsr'ygridy_gridrrrrh rate2_cmle rate1_cmlertmp_rrmean1mean2 stat_sample thresholdpdf1pdf2 stat_spacemaskr$rrrrs# ` @@@@rArrspf%KLNBB RA C'  %-E  # MMK' )E eg^q3w/ ^ Y  Hx  I45 5 F  =E"b"b4H$'!:z Y  @ @x   45 5 E E JE JEB#K  7G)VuF~MM%%eS->?  3' 9q=)F# ;;! MN N ==  VU +D ==  VU +D6!T'?F47O>%(DAqjVaZ025:F''!vz*Q&1*-==>C266D&3,//"&&:L1LMB { "26ai"% HCB x ::>>%(Dslv|4r9B>F''!v|,qFSL/AABC266D&3,//"&&:L1LMB x ::>>%(DWWfslv|<=F v 5 ;dTk IJJCc\D47N2E+u4H+u4HA+x{+B w #H!"5IC!"5ICc3IBxx/?@A AjjA"BqE *L II F  X ::>>%(D]F"''%*uz"9::D$ -B< I9{ "::>>%(D]F''6C<2q50FSLBE3IIJC#:D$ -B. I+w 26ai& HCB Iw #H!"5IC!"5ICc3HB I xx/?@A A!"GHHrCc SSKJn [[RXU/5upnX2-n Xt- SU- SX1-- --n U S:XaU nO4U S:XaXt- SXS- -S--nXU- XU----nO[ SU S35e[R "U5n[R "U 5n[R"X- 5[R"U5- nU "UX&UUUS 9nS nU (a[UUUUUU UUS S 9 nU$U$) u Power of test of ratio of 2 independent poisson rates. This is based on Zhu and Zhu and Lakkis. It does not directly correspond to `test_poisson_2indep`. Parameters ---------- rate1 : float Poisson rate for the first sample, treatment group, under the alternative hypothesis. rate2 : float Poisson rate for the second sample, reference group, under the alternative hypothesis. nobs1 : float or int Number of observations in sample 1. nobs_ratio : float Sample size ratio, nobs2 = nobs_ratio * nobs1. exposure : float Exposure for each observation. Total exposure is nobs1 * exposure and nobs2 * exposure. 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 Rate ratio, rate1 / rate2, under the null hypothesis. dispersion : float Dispersion coefficient for quasi-Poisson. Dispersion different from one can capture over or under dispersion relative to Poisson distribution. method_var : {"score", "alt"} The variance of the test statistic for the null hypothesis given the rates under the alternative can be either equal to the rates under the alternative ``method_var="alt"``, or estimated under the constrained of the null hypothesis, ``method_var="score"``. 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 False, then only the power is returned. If return_results is True, then a results instance with the information in attributes 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 : 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)) References ---------- .. [1] Zhu, Haiyuan. 2017. “Sample Size Calculation for Comparing Two Poisson or Negative Binomial Rates in Noninferiority or Equivalence Trials.” Statistics in Biopharmaceutical Research, March. https://doi.org/10.1080/19466315.2016.1225594 .. [2] Zhu, Haiyuan, and Hassan Lakkis. 2014. “Sample Size Calculation for Comparing Two Negative Binomial Rates.” Statistics in Medicine 33 (3): 376–87. https://doi.org/10.1002/sim.5947. .. [3] PASS documentation rnormal_power_hetraltr r method_var  not recognizedstd_nullstd_alternativer'Npower) rp_pooledrstd_altnobs1nobs2 nobs_ratiorFtuple_) statsmodels.stats.powerrrr+rrrrr)rrrrrOr7rFr8r' method_varreturn_resultsrrv1v0rrespow_rr@s rApower_poisson_ratio_2indeprs-l9bjj5*?@E%  E  !e)a:3E.F"F GBU  w   "a%*<&*>)BB "eE/A&BCC&!c.>*>)BB "eE/A&BCC!K |?"KLL VVEM "RVVC[ 0F VVEM "RVVC[ 0F776?L776?LggbkO !&&%/;/;2A CD q'ab'%%#!  AwrCcUnUnUn [RU5n [R[R"U5U-X-- U - 5[R[R"U5U-X-- U - 5-S- n U $z power for equivalence test r)r,r4r0r+r) rrr)rFrrrs0_lows0_upps1r?rs rA_power_equivalence_het_v0r$sF F B 88E?D "''$-&(4=8B>? "''$-&(4=8B>? @BC D  KrCcUnUnUn [RU5n [R[R"U5U-X-- U - 5n [R [R"U5U-X--U - 5n SX-- n XU 4$r )r,r4r-r+rr0)rrr)rFrrrr!r"r#r?p1p2rs rArrsF F B 88E?D "''$-&(4=8B> ?B 2774=6)DM9R? @B =D R<rCc XU--SU-- nUS:Xa XU- -=pO [USX-X&SS9upn XU- -n XU- -n U[R"U 5[R"U 54$)NrrTr)rr+r)rrrrFrOr8r7r  rates_pooledr r rrrs rA_std_2poisson_powerr*sJ..1z>BLU*,,,R* 1e(*GZ' ' + + bggbk 11rCc SSKJn [[RXU/5upnX- n [ UUUUUUS9upn U "X- X$U U US9nU(a[ UX-U4U U UX2-UUSS9 nU$U$)uT Power of ztest for the difference between two independent poisson rates. Parameters ---------- rate1 : float Poisson rate for the first sample, treatment group, under the alternative hypothesis. rate2 : float Poisson rate for the second sample, reference group, under the alternative hypothesis. nobs1 : float or int Number of observations in sample 1. nobs_ratio : float Sample size ratio, nobs2 = nobs_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 Difference between rates 1 and 2 under the null hypothesis. method_var : {"score", "alt"} The variance of the test statistic for the null hypothesis given the rates uder the alternative, can be either equal to the rates under the alternative ``method_var="alt"``, or estimated under the constrained of the null hypothesis, ``method_var="score"``. 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 False, then only the power is returned. If return_results is True, then a results instance with the information in attributes 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 : 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)) References ---------- .. [1] Stucke, Kathrin, and Meinhard Kieser. 2013. “Sample Size Calculations for Noninferiority Trials with Poisson Distributed Count Data.” Biometrical Journal 55 (2): 203–16. https://doi.org/10.1002/bimj.201200142. .. [2] PASS manual chapter 436 rr)rrFr7r rr) r rates_altrrrrrrFr)r rrr+rr*r)rrrrrFr7r r'r rrrrrrr@s rApower_poisson_diff_2indepr-sH9bjj5*?@E% =D.    A DL%,3(3 5D|U+$!   rCc2UnSU- nU*U-U-SU--nXS-Xd-X ---SX$--- nXbU--n US:XaU *U- n O,U*[R"US-SU-U -- 5- SU-- n X-n SU- U - SSU- U- --SU-U- U--n U n X-X4$)z variance based on constrained cmle, for score test version for ratio comparison of two negative binomial samples value = rate1 / rate2 under the null rrrrEr)rrrrOr7r8rate0abrr0r1rir2s rA_var_cmle_negbinr5@s EZJ  x%'1z>:A  %-*2D"D E j  "A U""AQR!Vb2771a4!a%!)+,,Q 7 B X a!e)j"88 9 j.J & 3 4A B >2 !!rCc FSSKJn [[RXU/5upnX2-n SU- SX1-- -U- SU-U- U--n U S:XaU nOTU S:Xa)SXS--S-XC-U-XU---- nUSU-U- U-- nO%U S:Xa[ XUXEUS9SnO[ S U S 35e[R"U5n[R"U 5n[R"X- 5[R"U5- nU "UX&UUUS 9nU (a[UUUUU UUS S 9nU$U$)u Power of test of ratio of 2 independent negative binomial rates. Parameters ---------- rate1 : float Poisson rate for the first sample, treatment group, under the alternative hypothesis. rate2 : float Poisson rate for the second sample, reference group, under the alternative hypothesis. nobs1 : float or int Number of observations in sample 1. low : float Lower equivalence margin for the rate ratio, rate1 / rate2. upp : float Upper equivalence margin for the rate ratio, rate1 / rate2. nobs_ratio : float Sample size ratio, nobs2 = nobs_ratio * nobs1. exposure : float Exposure for each observation. Total exposure is nobs1 * exposure and nobs2 * exposure. value : float Rate ratio, rate1 / rate2, under the null hypothesis. 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. dispersion : float >= 0. Dispersion parameter for Negative Binomial distribution. The Poisson limiting case corresponds to ``dispersion=0``. method_var : {"score", "alt"} The variance of the test statistic for the null hypothesis given the rates under the alternative, can be either equal to the rates under the alternative ``method_var="alt"``, or estimated under the constrained of the null hypothesis, ``method_var="score"``, or based on a moment constrained estimate, ``method_var="ftotal"``. see references. 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 False, then only the power is returned. If return_results is True, then a results instance with the information in attributes 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 : 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)) References ---------- .. [1] Zhu, Haiyuan. 2017. “Sample Size Calculation for Comparing Two Poisson or Negative Binomial Rates in Noninferiority or Equivalence Trials.” Statistics in Biopharmaceutical Research, March. https://doi.org/10.1080/19466315.2016.1225594 .. [2] Zhu, Haiyuan, and Hassan Lakkis. 2014. “Sample Size Calculation for Comparing Two Negative Binomial Rates.” Statistics in Medicine 33 (3): 376–87. https://doi.org/10.1002/sim.5947. .. [3] PASS documentation rrrrftotalrr rOr7r8rrrr)rrrrrrrFr) r rrr+rr5rrrr)rrrrrOr7rFr8r'r r rrr r rrrrr@s rApower_negbin_ratio_2indepr9_srn9bjj5*?@E%  E u9qJ./ /8 ; z>Z '* 4 5BU  x %$$q (  "U *e56H.H IK q:~+j88 w  eJ'/)35568"K |?"KLLwwr{HggbkG   .B Bx,3(3 5D!  KrCc Z[[RXU/5upnXR-n SU- SXP-- -U- SU-U- U--n U S:XaU =pOU S:XaQSX5--S-Xe-U-XU---- n U SU-U- U-- n SXE--S-Xe-U-XU---- nUSU-U- U-- nO4U S:Xa[XUXcUS9Sn [XUXdUS9SnO[ SU S 35e[R "X- 5[R "U5- n[R "X- 5[R "U5- n[R "U 5n[R "U5n[R "U 5n[UUX'UUUS 9nU (a[US[SS UUUUU UUS S 9 nU$US$)uf Power of equivalence test of ratio of 2 indep. negative binomial rates. Parameters ---------- rate1 : float Poisson rate for the first sample, treatment group, under the alternative hypothesis. rate2 : float Poisson rate for the second sample, reference group, under the alternative hypothesis. nobs1 : float or int Number of observations in sample 1. low : float Lower equivalence margin for the rate ratio, rate1 / rate2. upp : float Upper equivalence margin for the rate ratio, rate1 / rate2. nobs_ratio : float Sample size ratio, nobs2 = nobs_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. dispersion : float >= 0. Dispersion parameter for Negative Binomial distribution. The Poisson limiting case corresponds to ``dispersion=0``. method_var : {"score", "alt"} The variance of the test statistic for the null hypothesis given the rates under the alternative, can be either equal to the rates under the alternative ``method_var="alt"``, or estimated under the constrained of the null hypothesis, ``method_var="score"``, or based on a moment constrained estimate, ``method_var="ftotal"``. see references. 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 False, then only the power is returned. If return_results is True, then a results instance with the information in attributes 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 : 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)) References ---------- .. [1] Zhu, Haiyuan. 2017. “Sample Size Calculation for Comparing Two Poisson or Negative Binomial Rates in Noninferiority or Equivalence Trials.” Statistics in Biopharmaceutical Research, March. https://doi.org/10.1080/19466315.2016.1225594 .. [2] Zhu, Haiyuan, and Hassan Lakkis. 2014. “Sample Size Calculation for Comparing Two Negative Binomial Rates.” Statistics in Medicine 33 (3): 376–87. https://doi.org/10.1002/sim.5947. .. [3] PASS documentation rrr7rr r8rrrrNrr) rr+rr5rrrrrrrs rApower_equivalence_neginb_2indepr;sZbjj5*?@E%  E u9qJ./ /8 ; z>Z '* 4 5BU x c&&*  "S (E4F,F GI1z>Z/*<<c&&*  "S (E4F,F GI1z>Z/*<< w !% +3-799:<"% +3-799:<"K |?"KLL VVEM "RVVC[ 0F VVEM "RVVC[ 0F776?L776?LggbkO !&&%/;/;2A CD q'ab'%%#!  AwrC)Nrr)N皙?)gffffff??Nr<r)r=Nr<r)r<r r )r<r rr)NNNrrN)rF)NNr rrNN)r r)r rr<r )rrrr<rrrT)rrr<rrF)r<NNN)rr<rrrr )rr<rr rT)rrr)rrrr<g{Gz?rrT)rrr<rrF)%__doc__numpyr+rscipyrrstatsmodels.stats.baserstatsmodels.stats.weightstatsr"statsmodels.stats._inference_toolsrr,method_names_poisson_1samprBrXrcrerIr{method_names_poisson_2indeprtrrrrrzrrr$rr*r-r5r9r;rCrArGs!.9= zz   6?JIXtnDF-1&1NbBD04)4BJ:B&/& 6  * +%PEI#'-4<@Yx(KO=D8< $}B18aJELBLCG(/j^}B;