Comparisons of Test Statistics for Noninferiority Test for the Difference between Two Independent Binominal Proportions

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1 America Joural of Biostatistics (: 3-3, ISSN Sciece Publicatios Comarisos of Test Statistics for Noiferiority Test for the Differece betwee Two Ideedet Biomial Proortios,3 Youhei Kawasaki, Faghog hag ad 3 Etsuo Miyaoka Biostatistics Grou, Develomet Divisio, Deartmet of Data Sciece, Mitsubishi Taabe Pharma Cororatio, -6 Nihobashi-Hocho -Chome, Chuo-Ku, Tokyo , Jaa Develomet ad Medical Affairs Divisio, Deartmet of Biomedical Data Scieces, GlaxoSmithKlie K.K., 6-5 Sedagaya -Chome, Shibuya-Ku, Tokyo , Jaa 3 Deartmet of Mathematics, Tokyo Uiversity of Sciece, 6 Wakamiya-Cho, Shijuku-Ku, Tokyo 6-087, Jaa Abstract: Problem statemet: Noiferiority tests are frequetly used i cliical trials to demostrate that the resose for study drugs is ot much worse tha the resose for referece drugs. Several test statistics exist. However, a detailed comariso of those test statistics is ot researched. Moreover, a little comlex calculatio might be ecessary i some of those test statistics. Aroach: I this study, we ivestigated the erformace of the existig test statistics ad roose ew test statistics. Further, we comare them with existig test methods by meas of simulatio ad devise a suitable techique of usig of these test statistics. Results: We foud that for the roosed test statistics, the actual tye I error was close to the omial level. Further, whe the samle size is moderate it is foud that, the ew test statistics have a little higher ower tha other test statistics. Coclusio: Oe of the biggest advatages of our method is that it does ot require comlicated calculatios. Key words: O-sided testig, o-iferiority trials, biomial roortios INTRODUCTION A oiferiority test, whose mai urose is to idicate whether the resose for study drugs shows cliically ot much worse tha the resose for referece drugs, is ofte coducted i cliical trials. A oiferiority test is esecially, emloyed to derive the differece betwee two biomial roortios if the resose is a ideedet biomial. The ICH-E9 guidelies ad the Euroea medicies agecy guidelies showed the framework of oiferior settig comarisos betwee treatmet grous. Research ertaiig to oiferiority tests for derivig the differeces betwee roortios has bee coducted sice a log time. However, few theses cosider the behavior of test statistics i detail. Moreover, research i this field has bee iitiated oly recetly. Duett ad Get (977 selected a examle of oiferiority test from a cliical trial. I their research, a estimator weighted by a oiferiority margi was used for the ukow arameter with test statistics. Farrigto ad Maig (990 roosed three methods for estimatig for a ukow arameter i stadard error measuremet ad they recommeded usig a restricted maximum likelihood estimator, which is a restricted value of the ull hyothesis, roosed by Miettie ad Nurmie (985. The statistical aalysis software-ower aalysis ad samle size-ca calculate ower i eight ways. Almedra-Arao (9 showed that o-iferiority test sizes are calculated for the differece betwee two ideedet roortios based o -statistic with ooled variace, for several cotiuity correctios ad the behavior of these test sizes is aalyzed. Hirotsu et al. (997 rovides cofidece itervals that correct skewess ad discusses the desig issue of the required samle size for the oiferiority test. Da ad Koch (8 roosed a method of evaluatig the oiferiority test o the basis of some cofidece itervals. They also showed the relatioshi betwee the cofidece itervals ad the oiferiority test for the differece betwee two ideedet biomial roortios. Corresodig Author: Youhei Kawasaki, Biostatistics Grou, Develomet Divisio, Deartmet of Data Sciece, Mitsubishi Taabe Pharma Cororatio, -6 Nihobashi-Hocho -Chome, Chuo-Ku, Tokyo , Jaa Tel:

2 Am. J. Biostatistics (: 3-3, hag et al. (6 roosed a ew test statistic for the oiferiority test for ordered categorical data ad they exaded their test statistic to the differece of roortios. I this study, we roose a ew test statistic, distict from the method roosed by hag et al. (6. We reset a method of derivig a estimator, focusig o the oiferiority test for the differece betwee two ideedet biomial roortios ad we detect ad verify a well-erformig estimator i this study. This -test statistic asymtotically has a stadard ormal distributio. However, several test statistics have bee roosed sice the ukow arameter ivolved i -test statistics. Pooled variace: The variace of the estimator uder the ull hyothesis i a sigificace test is: ˆ V( δ π( π MATERIALS AND METHODS Suose that X ad X are two ideedet radom variables with a biomial distributio. The first radom variable is size ad it has a biomial roortio π, deoted as X B (, π. The secod radom variable is size ad it has a biomial roortio π, deoted as X B (, π. I this study, we assume that a large biomial roortio is referred cosistetly. Here, the hyothesis of the oiferiority test for derivig the differece betwee roortios is: H : π π 0 0 H : π π > 0 ( where, the oiferiority margi is 0 >0. We assume that δ π π. The differece betwee samle roortio, δ ˆ πˆ ˆ π, is the estimator for δ, where π ˆ X / ad π ˆ X / m. Therefore, the exected value uder the ull hyothesis is: V( ˆ by: The variace is: E( δ ˆ π π π ( π π ( π 0 δ ( Therefore, the statistic of stadardized ˆδ is give CE π ˆ 0 πˆ ( πˆ πˆ ( πˆ 0 ˆ ˆ ( π π where, the ukow arameter is π π π. This variace is geerally kow as ooled variace. By relacig the ukow arameter i this variace with the estimator ˆπ, the test statistic is give by: P π ˆ 0 πˆ ( πˆ X X where, the estimator for π is π ˆ. Alterative hyothesis variace: The variace of the -test statistic is idetical to the oe used uder the alterative hyothesis. Each maximum likelihood estimator ot related to the hyothesis is used for the ukow arameter i the variace. The W test statistic is show as: W π ˆ 0 πˆ ˆ ˆ ˆ ( π π( π This is kow as the Wald test statistic. May researchers have idicated i may study that the erformace of the Wald statistic suffers whe the samle size is small. Further, Muzel ad Hsuschke (3 showed the framework of the oiferiority test for ordered categorical data. Whe the umber of categories is assumed to be two, it is regarded as a roblem with regard to the differece betwee roortios. Hece, this test statistic is derived by extedig the method roosed by Muzel ad Hsuschke (3 to the oiferiority test for derivig the differece betwee roortios. Null hyothesis variace : The variace of the oiferiority test uder the ull hyothesis is: ˆ ( π 0( π 0 π( π V( δ

3 Am. J. Biostatistics (: 3-3, Duett ad Get (977 roosed the estimator: X X π ˆ 0 (3 for the ukow arameter π. By usig this estimator, the D test statistic is show as: D π ˆ 0 ˆ ˆ ˆ 0( π 0 π ( π This is called the Duett-Get test statistic. We suggest that the roblem was that the estimator (3 exceeded the limit value. Null hyothesis variace : Miettie ad Nurmie (985 costructed a maximum likelihood estimator with a restrictio for the biomial roortio π uder the ull hyothesis. Farrigto ad Maig (990 roosed a test statistic usig this estimator. The loglikelihood fuctio uder the restricted ull hyothesis π π 0 is: l( π x l( π ( x l( π 0 0 x l( π ( x l( π The solutio π, which maximizes this fuctio is give by solvig the followig cubic equatio: Where: a aπ bπ cπ d 0 3 b { x x ( } 0 c (x x x 0 0 d x ( 0 0 Therefore, the maximum likelihood estimator is: Where: 3 w { π Cos (v / u } / 3 3 v (b / 3a bc / 6a d / a u sig(v (b / 3a c / 3a π ɶ u cos(w b / 3a Usig this restricted maximum likelihood estimator, the F test statistic ca be show as: F π ˆ 0 ( πɶ 0( π ɶ 0 πɶ ( πɶ Null hyothesis variace 3: hag et al. (6 roosed a ew test statistic for oiferiority test i ordered categorical data. They exteded it to derive the differece betwee roortios ad itroduce the C test statistic as: C (ˆ ( 0 where, the ubiased estimators are: 5 Where: ˆ πˆ 0 ( 0 ( ( π π 0 π( π 0 π( π Usig each maximum likelihood estimator for the ukow arameter i the C test statistic, the CE statistic is defied by: CE π ˆ 0 πˆ ( πˆ πˆ ( πˆ 0 ˆ ˆ ( π π Kawasaki et al. (8 alied this test statistic to the cofidece iterval for the differece betwee two ideedet biomial roortios. They showed that the ew cofidece iterval showed a greater imrovemet i erformace tha the Wald iterval. Null hyothesis variace : I the test statistic used by hag et al. (6, the estimator for the ukow arameter i variace is ot ubiased. I this study, we use these ubiased estimators for the ukow arameter to roose a ew test statistic that is defied as: CU π ˆ 0 ɶ ɶ ɶ

4 Am. J. Biostatistics (: 3-3, ( ( π ˆ ( πˆ ( π ˆ ( πˆ ( πˆ ɶ ( ( ( ( π ˆ ( πˆ ( πˆ πˆ ( πˆ ɶ (5 0 ( ( ( π ˆ ( πˆ ( πˆ πˆ ( πˆ ɶ (6 0 ( ( The derivatio for these ubiased estimators is illustrated i the Discussio. RESULT We show the validity ad usability of each test statistic. I this research, with regard to the validity of the test, it is assumed that the tye I error is close to the omial level. Further, usability of the test is assumed to be high ower. I Table, we evaluate whether the actual tye I error is at the omial level of.5%. I Table, we show that the actual tye I error is at the omial level of 5%. The actual tye I errors of each method are calculated by coductig a simulatio,0 times uder each coditio. The followig oits are idicated i Table ad. Table : Actual tye I error (% of test for the o iferiority hyothesis (, omial level is.5% Method (% Samle size π 0 P W D F CE CU Table : Actual tye I error (% of test for the oiferiority hyothesis (, omial level is 5.0% Method (% Samle size π 0 P W D F CE CU

5 Am. J. Biostatistics (: 3-3, Table 3: Actual ower (% of test for the o iferiority hyothesis (, omial level is.5% Method (% Samle size π π 0 P W D F CE CU The actual tye I error of W exceeded the omial level with a small samle size ad eve whe the samle size was moderate, it ofte exceeded the omial level. The actual tye I errors of CE, D ad P showed similar behaviors. Besides, the actual tye I errors of these methods are close to the omial level, excet i cases where the small samle sizes are small. We foud that the actual tye I errors of F ad CU came close to the omial level eve though the samle size was small. Further, whe the oulatio roortio was a extreme value, the actual tye I error of oly F was close to the omial level. Therefore, we recommed the use of F test statistics i cases where the oulatio roortio is assumed to be extreme. Thus, all of the above idicate that F ad CU test statistics have high validity. I Table 3, we showed the actual ower i the oe-sided test at the omial level of.5%. The actual ower of each method is calculated by a simulatio coducted,0 times uder each coditio, as we did for the tye I error. We deduced the followig oits from Table 3. It idicated that the W statistic ad the D statistic have 7 higher owers tha the others, esecially with a small samle size. However, we do ot ifer that it has usability sice the validity of the W statistic is ot assured. It shows that D, CE ad CU have higher owers whe the samle size is moderate. I articular, i cases with moderate samle size, it was foud that CU has a stable high ower. Further, we foud that F ad P had lower ower; i articular, F had lower ower eve at large samle size. We also foud that the characters of the ower of each statistic were chaged by the value of the oiferiority margi oly i a few cases. From the above result, it was iferred that D, CE ad CU test statistics have high usability. DISCUSSION The derivatio for these ubiased estimators is illustrated i this sectio. Let us cosider a oarametric two-samle situatio, where it is assumed that the variables Y, Y,, Y Y ad Y, Y,,Y Y are mutually ideedet. For the

6 Am. J. Biostatistics (: 3-3, urose of formulatig a oarametric test, a ivotal robability is advocated by some authors. The oarametric test for oiferiority may be formulated as: P(Y < Y P(Y Y The oarametric test for oiferiority may be formulated as: The test statistic: T N (ˆ 0 is asymtotically ormally distributed with exectatio 0 ad variace: 0 0 N N H : / δ H : < / δ 0 0 (7 where, δ 0 is the oiferiority margi ad δ 0 <0. Let ϕ be a fuctio of two real variables: ad let: x < y ϕ (x, y / x y 0 x > y U i j where, U ij ϕ(y i, Y j. The ubiased estimator of P becomes: Let U ij ˆ U be the variace of U ij d let 0 ad deote the covariace: V(U, ij Cov(U,U, j l 0 ij il Cov(U,U,i k. 0 ij kj I additio 0 ad 0 are rereseted as: E(U U E(U E(U, j l, 0 ij il ij il E(U U E(U E(U,i k. 0 ij kj ij kj 3 The variace of ˆ is give by: V( ˆ [ ( (m ] where, N let T / ; the: N ˆ ˆ 0 0 N N 0 0 is a asymtotically ormal distributio. However, we caot use this test statistic. We should relace the ukow arameter i the -test statistics by estimators. Muzel ad Hsuschke (3 roosed that the test statistics for hyothesis (7 is: where, the estimator is: ˆ U U M ( ˆ 0 ˆ ˆ i (8 i where, deote U U. U i. /. Similarly, let us i. U j /, j 0.. j j U.j Uij ad: i ˆ U U (9 Moreover the emirical estimators of P ad P 3 are: ˆ U.,ˆ U i 3. j i j hag et al. (6 oited out that oe roblem with this is that it used the variace uder a alterative hyothesis. They roosed the test statistic:

7 Am. J. Biostatistics (: 3-3, C ( ˆ ( 0 m i which ( ad used a variace uder a ull hyothesis. This test statistic C follows the asymtotic stadard ormal distributio. However, we caot use it as it is. They used exressios (8 ad (9 ad roosed the CE test statistic: CE (ˆ 0 ˆ 0 ˆ 0 0( 0 m ˆ where, ˆ ˆ ˆ (. hag et al. (6 call the CE test statistic a emirical test statistic. We derive ubiased estimators for the ukow arameter with C test statistics. The ubiased estimators of P ad P 3 are give by: ɶ U U ij il ( i j j l ɶ U U. 3 ij kj ( i j i k We show that the ubiased estimator of is: (ˆ ˆ ( (ˆ ɶ ( (ˆ ɶ ɶ (0 3 ( ( Moreover, the ubiased estimators of 0 ad 0 ca be: (ˆ ˆ ( (ˆ ɶ ( (ˆ ɶ ɶ ( 3 0 ( ( (ˆ ˆ ( (ˆ ɶ ( (ˆ ɶ ɶ ( 3 0 ( ( Because ˆ, ɶ ad ɶ 3 are ubiased ad cosistet, ɶ 0 ad ɶ 0 are ubiased ad cosistet. Therefore, the CU test statistic is roosed as: CU ( ˆ 0 ɶ 0 ɶ 0 0( 0 m ɶ We let Y ad Y be two ideedet Beroulli radom variables with π ad π resectively. Through simle calculatio, we obtai: ( π π Therefore the estimator of P is give by: ˆ ˆ π 9

8 Am. J. Biostatistics (: 3-3, The hyothesis for oiferiority, exressio (7, ca be rereseted as: H : π π 0 0 H : π π >, 0 where, 0 δ 0. The imortat comoets of the asymtotical variace ˆ are: 0 π( π, 0 π( π, [ ( π π ] Therefore, the CE test statistic is: CE Where: ˆ 0 π( π ˆ 0 π( π ˆ [ ( π π ] π ˆ 0 ˆ 0 ˆ 0 0( 0 m ˆ We ca obtai other exressios of the relatioshi betwee the emirical estimator ad ubiased estimator for ad P 3 as: ɶ ˆ ( ˆ ˆ (3 0 ɶ ˆ ( ˆ ˆ ( Substitutig (3 ad ( ito (0- ad oticig that ˆ ˆ ˆ for the Beroulli variable, the 0 0 ubiased estimator of, 0 ad 0 ca be derived as exressios (-6 resectively. CONCLUSION I this study, we ivestigated the validity ad usability of test statistics i the oiferiority test for the differece betwee two ideedet biomial roortios. It was deduced that the ower of the P test statistic is geerally low. We suose that this is a result of the use of the variace with a assumed ull hyothesis for a sigificace test. We foud that the W test statistic showed higher ower tha the P test statistic. However, it also showed that the actual level frequetly exceeded the omial level. Therefore, the W test statistic does ot fulfill the validity of testig. Hece, usig this method oly because its ower is high might lead to a wrog coclusio. The ower of the D test statistic erformed better. However, it is best if this test statistic is used judiciously sice the estimator of a uisace arameter used i this test statistic may exceed the limit value. We have deduced that the F test statistic is the method that asses the validity i the oiferiority test. Esecially, we also foud that this is also the oly method i which the tye I error comes close to the omial level whe the oulatio roortio is a extreme value. However, we also foud that the ower of this method is comaratively low. Moreover, the method of calculatig this test statistic is a little comlicated sice this method uses a restricted maximum likelihood estimator. I coclusio, we rove that the roosed CE ad CU test statistics are methods that show that their tye I errors are comaratively closer to the omial level ad also that they have reasoably higher owers; This is articularly true i the case of the CU test statistic, which uses a ubiased estimator that shows a stable ositive behavior i the hyothesis test. I additio, oe of the biggest advatages of our method is that it does ot require comlicated calculatios. REFERENCES Almedra-Arao, F., 9. Behavior of the asymtotic ooled -statistic. JP J. Biostat., 3: htt://hmj.com/abstract/36.htm Da, R.S. ad G.G. Koch, 8. Methods for oesided testig of the differece betwee roortios ad samle size cosideratios related to oiferiority cliical trials. Pharm. Stat., 7: 30-. DOI: 0./st.87 Duett, C.W. ad M. Get, 977. Sigificace testig to establish equivalece betwee treatmets with secial referece to data i the form of tables. Biometrics, 33: htt:// Farrigto, C.P. ad G. Maig, 990. Test statistics ad samle size formulae for comarative biomial trials with ull hyothesis of ozero risk differece or oetity relative risk. Stat. Med., 9: 7-5. DOI: 0./sim

9 Am. J. Biostatistics (: 3-3, Hirotsu, C., W. Hashimoto, K. Nishihar ad E. Adachi, 997. Calculatio of the cofidece iterval with skewess correctio for the differece of two biomial robabilities. JP. J. Alied Stat., 6: DOI: 0.503/jastat.6.83 Kawasaki, Y., S. Midorikawa ad E. Miyaoka, 8. Comarisos of cofidece itervals for the differece betwee two ideedet biomial roortios. Proceedigs of the Iteratioal Associatio for Statistical Comutig, : Miettie, O. ad M. Nurmie, 985. Comarative aalysis of two rates. Stat. Med., : 3-6. DOI: 0./sim.780 Muzel, U. ad D. Hsuschke, 3. A oarametric test for rovig o iferiority i cliical trials with ordered categorical data. Pharma. Stat., : DOI: 0./st.7 hag, F., H. Fuig ad E. Miyaoka, 6. No iferior oarametric test at order categorical data. Proceedigs of the Orgaized Sessios Jaaese Joit Statistical Meetig, : 68. htts:// ate.h?id06 3