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Comparison of treatments in a combination therapy trialH. I. Patel aa Berlex Laboratories, Inc., Wayne, New Jersey

To cite this Article Patel, H. I.(1991) 'Comparison of treatments in a combination therapy trial', Journal ofBiopharmaceutical Statistics, 1: 2, 171 183

To link to this Article: DOI: 10.1080/10543409108835016URL: http://dx.doi.org/10.1080/10543409108835016

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K e y words. Restricted null hypothesis; Noncentral bivariate t ; Con-fidencc intcrva!; Op:i;r,a! allocatioc

Abstract/>:+&,,,,;, ;h r 7;;;n i i . L --i i3'-,ir,d hi. 1 - - A ""-:"--- 1 \ E,..., ..,, ...... .:hi p.,,,,,, .,I ~ z j ~ dYICiji iCi : LQ: t e ~ t in casuperiority of a combination to each of its components is tile mi-. . . . . ,i'ormiy most powerfui nionoione iesi, i i is conservative in the neign-borhood o t origin. Considering a restricted nuii hypothesis, we ana--!yze a general h e a r rncde!. ! t: p viiiiie assockited ~ l i ? hhe prnposedtest and a confidence intervai for the shorTest distance between thecombination therapy and component therapies are computed. Anal-y s i s of a design with iwii or more iombinations is proposed. Anumerical exampie is given and a problem of optimai allocation ofthe sample sires anrd orher relevant design issues are discussed.

Recen:!~, Laska an:! Pileisner ( 1) and Snapinn ( 2 ) ,among ~t h c r s , ave studieda problem of inference in a combination trial where a combination therapyis compared with its components, Suppose we restrict our attention to twocomponent therapies A and B and their combination C and to an efficacy

Copyright 8 991 by Marcel Dekker, Inc.

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m.C L ' ; T ~ 6 -- - t. , . i Oti< Zt W ~ K < ~ t O i 73; { f , >

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Comparison of Treatnients in Combination Therapy Trial 173

A Single Combination DesignIn this section we derive an asymptotic test for HiF given in Eq. (3 ) for the

. * , \. ! ! - . . - / . > - A ?.-*.-I . . ": . - 3 - .g c ~ l G ~ ~ - tl i lca[ ~~l~~~~~d:yl - n r J aklu ; s { y j - r - i ~ I T nY LZ - ' < - , -15 V . LV l l l pULG.-.------*.- +I-nIIGpow" of th e test and a confi:?znce intervai for the shortest distance betweentine combination therapy and the components.

Assuming normality for the distribution of y , we ,.btain the least squares(1s) estimator of ( 6 ; : 19,) and its dispersion matrix as ( 0 , . N1) and r = (y,,)d ~ ,espectively, where V = ; i f ) is a known matrix. The estimate of CT

-1 .-

is ci; = ( y ' y - ~ ' A ( A ' A ) ~ 'r j ) / v , based on v, degrees of freedom (d.i.).Let W, = i,/1/%nd p = ~ , , / \ ' ( v , , v ~ ~ ) ,here v,, = e2v,,. Under H z , the

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where J'yz,,2 j is tiie deiisiiy of Givariaie distri"uiiiion of the Stan-dardized variates Z, and Z2 with correlation p , ILJsinp quadrature, a SAS pro-gram is wriiter! to compute the bivarlate normal probability integral over acfinite or infinite rectangie. The program is given in Appeiadix 11. Note thatGupta ( 5 )has given extensive prvbabiiiiy tables for thi s distribution, bu t they:-sii.6-c 3,-,t7 '4 + p'7.cj az

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Numerical ExampleDiastolic blood pressure (mmHg) data were generated for a paraiiei-group- .dzsjgn ;wo incnorheraFizs anii their combicalioE.re!X rfpresenl h;iSE-line and :J the treatment response. Then for the three treatment groups, ran-dom samples of varying sizes were generated from the truncated bivariatenormal populations of ( X , Y) with means (+ = 102 , p i ) , = 1 , 2, 3 and thedispersion matrix (:;! ;;;j, where ,a,s the j t h element of the mean vector p i= (82, 87, 86). The last V i i i elemznts of p' represent the momtherapies 1and 2 and the first element represents the cornbinatioii. The x values weregenerated first from the marginal distribution of X over the range (90, 115)and then the y values were generated from the conditional distributions of Y

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Table I . Simulated Diastolic Blood Pressure ( ~ n m f i g ) at a arid Surnrrtary Statistics

Mean 100.1 85.5 101.1S . D . 5.44 7.22 4.07

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Extension te a Desigr, with Two cr o re CambifiatiofisSuppose the early phase studies, expioratory in nature, s:aggest that a fewspecific dose combinations he frurther evaliuated i n comparison with theircomponents In a conrirnlatory study. A ruii factoriai design with K~K com-binations formed with R, doses of therapy i and R2 doses of therapy 2 hasa place in carly phases of the dmg deve!opmen: to understand the efficacyand safety profiles of the combination doses. Tnis, along with other existingknowledge for the class of drugs under investigation, should help choose asubset of combination doses for a more definitive study. Recently, Hung (7),

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!78 Paw!

we wrife H,, as H,,, : C , @ (:, ' 8 = C2@ G , where @ stands for th e-.Kroneckei- product, 'l'tien for iarge vt., under H o ! , U = (C2fj)'

/c 7L2A6 fir1\2 1 \L?L' / has 2 chi-squared distribr?tin Vvith2(( - !) d . , , whereC 2 8 is the Is estimator of C ' 2H , If i J is no t significant at a chosen level ofsignificance, one can proceed lo test the hypothesis H z . it is recommendedthat H,, be tested at the 0.2 or s o i n other level that is reasoriabfy higher tharithe conventional level uf 0.05. Such a high !eve! of significance for testingH,, can be justified because one cannot generally expect the same magnitudeso f 6 's in lo w and high dose combinations. When H,, is rejected, following

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Assuming that the response variable Y is normally distributed with knownVariai-,Ce ", suppose ii..e %ant to find an optima! a!!ocatim ~f the samplesizes n, , n2,an d ni with n, = n,) for designing a com bination therapy trial.Here x i and iz , 2re the srmr?!e sizes for th e cnmponent therapy groups an d I? : - m ax y, - d ) > 1 - tr (91I i

..A---* i l b i L ,I /-.n\.-"I-Ai i i i i ii, i i i i,- ,..-.,,-,.;c,,!.. .ii. . . -,,A - -r vL l --.lii--Uiiii 7, , i i , zed y , -.re t" s.;.-pls z p z n .fo r the ccmbination therapy and rhe component therapies I an d 2; respec-tively. With such an allocation, a lower ( 1 - a ) 100% confidence bound forthe shortest distance frox the csmhination theranv~ to the individual corn-ponents wiii be of width greater than or equal io d .The transformation V = -Y for th e response variable changes [Eq. (9)j

to Eq. (2 .3) of Bechhofer and Tamhane (8) with P = 2 , y,, = ,x,,!n, A =d l v & / ~md t) = 2 . the^ fn!!owing the tables given by Rechhofer and T a m --hane (8), we can compute n! = n, and n,.. l o i i ius t ra te th is m ethod, iet ussuppose w e want to design a trial for the treatment of hypertension. Let cr =7 mmI-Cg, $ = 3 mii;Hg, and a = 0.05. Using 2 one-sided interval, va$mof -7" and A , corresponding to P = 2 and H = 2 ar e 9" = 6.399 and h =4 .651 . S ince ( ~ r r / d ) '= 11 7.8 , ii = 118. From this fi,. = 47 and f i , = ii, =36 .

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Acknowiedgments

The author sincere!y thanks Professor A j ~ t . Tamhane, Nurthwesten Unl-versity, for his constructive suggestions and help in the preparation of thispaper. The author also wishes to thank an associate editor for his useful com-ments.

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Gamparisrr~ f Trrarmerrss in Combination Therap? Trial

Conditioning on Yo = y , using the independence of Y, nd Y2 nd then un-cnnditinning on y , we get

where @(. j is the cdf and &,) is the pGE of a N ( 0 , 1) random variable. Usingquadrature, one can evaluate the integral (for given x) first and then the ex-ternal integral.

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- . . * . . " . .' This t'njpi.am .iiiPiitcS blvi;ria;z r x i *r,> h.,p:!, i -- ?7tl : l i - : i j.liil:liil pi 5. ;e l bc q u a i iv 5 . 3.k .. 2 3;vfier, !oxjjer iirr.it e.: is .3: i ~ ~ S ~ A S i ~ S c 2 ~:L iaL : \1 .1~\3 3 :-,i 3iii: * 2: ::-* -+.;+ ::: ;- * .-'-:,g :**:r y -i' ".,.< :3 * y: :? >j

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