BIOEQUIVALENCE: COMPARISON OF AVERAGE VALUES WITH GENERALIZED p -VALUES

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1 Ελληνικό Στατιτικό Ιντιτούτο Πρακτικά 7 ου Πανελληνίου Συνερίου Στατιτικής (004), ελ IOEQIAENCE: COMPAION OF AEAGE AE WIH GENEAIZED p -AE tavroula Poulopoulou ad aki Papaioaou iverity of Piraeu AAC he aim of bioequivalece tudie i the evaluatio of bioequivalece of pharmaceutical product. he product are uually two ad baic pharmacokietic parameter uch a AC, C max ad t max are ued. hee tudie aim at ivetigatig the cloee of the ditributio of the pharmacokietic parameter (repoe) for the two product, which i made maily by comparig the average value of thee repoe uder ormality or logormality. he exitig method of comparig average bioequivalece are baed o the aumptio of equal variace for the pharmacokietic parameter. I thi paper, we propoe the ue of geeralized p-value, uggeted by ui ad Weerahadi i 989, for tetig average bioequivalece i the cae of uequal variace. We preet exact mall ample ize tet of average bioequivalece for parallel tudy deig ad demotrate how to compute their p-value whe the pharmacokietic repoe follow ormal or logormal ditributio with uequal variace.. INODCION he cocept of bioavailability ad bioequivalece play a importat role i pharmaceutical reearch ad developmet, epecially i the geeric drug idutry. For thi reao, i the lat thirty year, tudie of bioavailability whoe purpoe i the evaluatio of bioequivalece of two or more product, have become very popular i the drug idutry. ice the early 70 a large ad cotatly icreaig bibliography o tatitical method for bioequivalece tudie ha appeared. he tatitical apect of thee tudie are baed o the cocept of the cloee betwee the margial ditributio of the product pharmacokietic repoe. ually the product are two: the tet product ad the referece product. aed o the fact i may cae that the ditributio of a radom variable ca be 577

2 determied by it momet, the cloee betwee the two ditributio ca be evaluated from the two firt momet of the two product margial ditributio. I thi paper, we will deal with the compario of the firt momet, i.e. the mea of the ditributio of the pharmacokietic parameter of the two product. he tudie baed o the mea are metioed i the bibliography a average bioequivalece tudie. egulatio of mot coutrie icludig Greece (ΕΟF 996), the other Europea io coutrie (EMEA 00,003), the ited tate of America (FDA) ad Japa require oly evidece for average bioequivalece i order to give approval for geeric. o we will coider two product a bioequivalet, if the average value of the pharmacokietic parameter for the two drug e.g. AC, C max, max, are cloe eough. Accordig to the FDA regulatio, if we defie by the populatio mea of oe pharmacokietic parameter, e.g. AC, of the tet product ad by the populatio mea of AC for the referece product, i order to prove the bioequivalece, we mut tet the hypothei: Η ο : ή veru Η α : < <, () where ad are give kow cotat. he Europea io ad the FDA ue =.5 ad = 0.80 =.5 ad tet thi hypothei at a 95% cofidece level. It ha bee oberved that the pharmacokietic parameter uually follow a ormal or logormal ditributio. For thi reao, the tatitical methodologie for tetig bioequivalece of two product are baed o comparig the mea value of two ormal or logormal ditributio. Aother importat aumptio which i regularly employed i that the variace of the pharmacokietic parameter for the two product are equal. der thi aumptio, erger ad Hu (996) ad Chow ad iu (99) produce exact tet for tetig bioequivalece of two product. I the cae of uequal rage betwee the reult of two product, the method that are ued do ot give exact reult [Chow ad iu (99), p. 6-85]. I thi paper we preet exact mall ample ize tet for the compario of average bioequivalece i the cae of uequal variace. he tet are baed o the cocept of geeralized p-value, which wa itroduced by ui ad Weerahadi i 989. he method of geeralized p-value i briefly preeted i ectio with a applicatio to the ehre- Fiher problem. I ectio 3 we preet two oe-ided bioequivalece tet baed o geeralized p-value uder the aumptio of uequal variace for two ditributioal cae: ormality ad logormality ad uder parallel, i.e. idepedet, deig. imilar reult have 578

3 bee obtaied for cro-over deig ad will be preeted elewhere. For tatitical iue related with bioequivalece tudie ee Papaioaou (00). GENEAIZED p AE. Geeral theory. he cocept of geeralized p-value wa preeted by ui ad Weerahadi i 989 i coectio with problem of igificat tetig for oe-ided hypothee of the form Η ο : θ θ ο veru Η α : θ>θ ο or of the form Η ο : θ θ ο veru Η α : θ<θ ο, where θ i the calar parameter of our iteret, ad there appear a uiace parameter η. et Χ be a radom variable, with deity fuctio f(x ξ) where ξ=(θ, η) i a ukow vector of parameter ivolvig θ, the parameter of our iteret ad η, the vector of uiace parameter. et X be the ample vector ad x the oberved value of Χ. Our problem i to tet the ull hypothei Η ο : θ θ ο (ή Η ο : θ θ ο ). For thi purpoe we uually employ tet of ize α, amely tet with cotat igificace level α. I tetig problem which iclude uiace parameter ometime it i difficult or impoible to fid uch tet. I thee cae, the reearcher i atified with a level α tet, amely a tet with power fuctio o that up θ Hο β(θ) α. hough thi procedure eem to be more flexible tha the fixed level tetig, there appear difficultie i multiparameter problem. Oe uch a problem i, whe the p- value deped o η, the vector of uiace parameter o it might be difficult to etimate it. I order to urpa thee difficultie, we coider a geeralized tet fuctio of the form (Χ;x,ξ), which i ot oly a fuctio of Χ but alo ivolve the oberved x ad the parameter ξ. We demad from thi geeralized tet tatitic to atify the followig three requiremet:. t ob =(x;x,ξ) to be free of ξ.. for fixed x ad ξ the ditributio of (Χ;x,ξ) to be free of the uiace (.) parameter η ad to be a fuctio of a kow ditributio. 3. for fixed x ad η, (Χ;x,ξ) mut be a tochatically mootoe fuctio of θ. he quatity (X;x,ξ) play the role of a tet tatitic, which, however, deped o the oberved value x ad the parameter θ ad η. t ob, which i (x;x,ξ), may deped o θ but at θ o of Η ο it ha a kow value a i the cae with tet of igificace. A geeralized tet tatitic (Χ;x,ξ) which atifie the previou requiremet ad i particular i tochatically icreaig i θ lead to the followig geeralized extreme or rejectio regio: C x (ξ) = {X : (X;x,ξ) - (x;x,ξ) 0}= {X : (X;x,ξ) t ob } for tetig Η ο : θ θ ο. 579

4 If the geeralized tet tatitic (Χ;x,ξ) i tochatically decreaig i θ we coider the tet baed o the geeralized extreme or rejectio regio: C x (ξ) = {X : (X;x,ξ) - (x;x,ξ) 0}= {X : (X;x,ξ) t ob } o, followig the uual defiitio for the p-value, the p-value for thi tet i: p(x) = up θœ Ηο Pr(Xœ C x (ξ) θ). If the p-value i maller tha α, we will reject Η ο, a uual. equiremet 3 guaratee that the tet that we produce baed o the tatitic (Χ;x,ξ) i the uiformly mot powerful (MP) level α tet, if thi tatitic i ufficiet, ice it i kow that for the exitece of a MP tet of level α of Η ο : θ θ ο agait Η : θ>θ ο there mut be a ufficiet tatitic (Χ) for θ ad the family of pdf or pmf of to have mootoe likelihood ratio. Accordig to ehma (986), thi mea that (Χ) hould be a tochatically mootoe fuctio of θ. Moreover, equiremet 3 mea that the probability Pr((X;x,ξ) t) icreae a θ θ ο icreae if (.) i tochatically icreaig, wherea it decreae a θ θ ο icreae if (.) i tochatically decreaig i θ. o we have repectively: p(x) = up θœηο Pr((X;x,θ, η) t) = Pr((X;x,θ ο, η) t) p(x) = up θœηο Pr((X;x,θ, η) t) = Pr((X;x,θ ο, η) t). o it i poible to evaluate thee p-value ice they are idepedet from the uiace parameter.. Applicatio of geeralized p-value: he ehre-fiher problem. I order to demotrate the poible ue of the geeralized p-value procedure, we coider the ehre Fiher problem, which ca be formulated a follow. et Χ, Χ,, Χ ad Υ, Υ,, Υ be two et of idepedet obervatio from the ormal populatio Ν(, ) ad Ν(, ) repectively. We alo aume that Χ, Χ,, Χ ad Υ, Υ,, Υ are idepedet. We wih to tet the ull hypothei Η ο : 0 veru the alterative Η α : > 0 baed o the idepedet ufficiet tatitic X, Y, ad, which are or are related to the maximum likelihood etimator of the mea και ad the variace ad repectively. I thi problem we are itereted i the parameter θ = ad the uiace parameter i η = (, ). o, followig ui ad Weerahadi (989) ad Weerahadi (995) the radom quatity which ca be ued a a geeralized tet tatitic to fid the extreme regio C x,y (ξ) for the ehre Fiher problem ad which complie with the requiremet (.) i: 580

5 (X,Y; x, y,η) = ( X - Y - θ) (.) hi ca be expreed a follow: = Ζ or = - ) ( Z, (.3) where Ζ ~ Ν(0,), ~ X ad ~ X, ad Ζ, ad are idepedet. Note that the radom quatity ha the ame ditributio with Ζ ( ) ad (x,y; x, y,η) = x - y - θ. We ee that the family of ditributio of (Χ,Υ;x,y,η) for fixed x ad y i tochatically icreaig i θ ad a a reult we have the geeralized extreme regio: C x,y (θ, η) = {(X,Y) : (X,Y;x,y,η) - (x, y;x,y,η) 0} Accordig to thi extreme regio we ca compute a p-value, which will ot deped o the uiace parameter. For the ehre Fiher problem the geeralized p-value i: p(x) = Pr( t ob θ = 0) = Pr( x - y - θ θ = 0) =Pr y x ) ( =E (y-x) G, where = [ ] ) )( ( Z which follow tudet t ditributio with ( -) d.f. ad i idepedet from Β = ) ( which follow eta with parameter ( ) ad ( ). G i the p.d.f. of tudet t ditributio with ( -) d.f. ad where Ε Β deote expectatio with repect to Β. he geeralized p-value i computed uig the firt expreio of i (.3) by geeratig radom value from the r.v. Ζ, ad (Mote Carlo). For detail ee Poulopoulou, (004). 3. IOEQIAENCE E IN CAE OF NEQA AIENCE,. ig the previou method of geeralized p-value, we ca produce exact tet for tetig the bioequivalece of two product whe the pharmacokietic repoe of the two product have uequal variace. I thi ectio X ad Y will deote ay of the pharmacokietic parameter metioed i ectio. 58

6 3. NOMAIY AMPION. et Χ, Χ,, Χ be a radom ample of a ormal populatio with mea ad variace, ad let Υ, Υ,, Υ be a idepedet radom ample from a ormal populatio with mea ad variace. he bioequivalece aumptio () ca be traformed to Η ο * : 0 veru Η α * : > 0 ad Η ο * : 0 veru Η α * : < 0, where = ad = o we ca hadle thi problem a oe ehre Fiher peudo problem. aed o the fact that X Y ~ Ν ( ) ( ),, which deped o the parameter of our iteret ad the uiace parameter ( ) ad X Y ~ Ν ( ) ( ),, which deped o parameter of our iteret ad the uiace parameter ( ) ad i view of (.) we ca coider the followig radom quatitie, a geeralized radom quatitie for tetig the bioequivalece aumptio: (X,Y; x, y,,,, ) = Υ) (Χ (3.) (X,Y; x, y,,,, ) = Υ) (Χ hee tatitic ca be expreed a follow: ~ Ζ ( ) ad ~ Ζ ( ), (3.) where Ζ ~ Ν(0, ), = ~ X ad = ~ X ad the radom variable Ζ, ad are idepedet. ice ad are tochatically icreaig i ad repectively, the geeralized p-value for tetig the left-had ided ull hypothei of form Η ο * : 0 veru Η α * : > 0 i: p(x) = Pr( t ob = 0) = Pr y - x ) (, ad the geeralized p-value for tetig the right-had ided ull hypothei of the form Η ο * : 0 veru Η α * : < 0 i: 58

7 p(x) = Pr( t ob = 0) = Pr ( where Β = ) (, = = [ ] y ) x -. ) Z ( )( whe = 0 ad = 0. I thi cae ad follow the tudet t ditributio with d.f. ad are idepedet of, which follow eta with parameter ( ) ad ( ). he p-value ca be computed by Mote Carlo uig (3.). For detail ee Poulopoulou (004). I aalogy with the O tet the previou tet of the bioequivalece aumptio ca be called wo oe ided geeralized p value tet (OGP). 3. ogormality aumptio. et our data follow logormal ditributio with uequal parameter. More pecifically let Χ be the logormal quatity aociated with obervatio from the et drug ad uppoe Χ i a logormal radom variable with parameter η ad. et alo Υ = l(x ). imilarly let Χ be the logormal quatity for the eferece drug with parameter η ad ad Υ = l(χ ). he mea for the tet ad the referece product are: = exp(η ) ad = exp(η ) repectively. o the bioequivalece aumptio become: = exp[η -(η )] which i equivalet to: θ η -(η ) θ, where θ = l( ) ad θ = l( ). o the aumptio that mut be teted, ettig θ = η ad θ = η, i the cae of the logormal model with uequal variace ca be eparated to two oe-ide bioequivalece hypothei: Η ο : θ θ θ veru Η α : θ θ >θ ad Η ο : θ θ θ veru Η α : θ θ < θ. aed o the fact that ad the uiace parameter Y ~ Ν ( η, ) ad that, which deped o the parameter η of our iteret Y ~ Ν ( η, ), which deped o the parameter η of our iteret ad the uiace parameter, the we ca coider the geeralized radom quatitie: (Υ ; y,, )= y Y η Z = y ( ) Y (Y ; y,, η )= y = y Z (, ) 583

8 where Z = ( Y η ) ~ N(0, ), Z = ( Y η ) ~ N(0, ), = ( ) ~ X ad =( ) ~ X which are alo idepedet. o for tetig the ull hypothee Η ο ad Η ο we defie the geeralized radom quatitie: = (θ θ ) θ ad = (θ θ ) θ der bioequivalece the ditributio of ad are give by: Z Z ~ y y (θ ( ) θ ) θ, ( ) (3.3) Z Z ~ y y (θ ( ) θ ) θ. ( ) he quatitie ad have oberved value free of uiace parameter. ob = ob ob (θ θ ) θ = θ θ (θ θ ) θ = θ ad ob = ob ob (θ θ ) θ = θ θ (θ θ ) θ = θ. We alo oberve that ad tochatically decreaig with θ θ. herefore the geeralized p-value for tetig Η ο i give by: p(x) = Pr( θ θ θ = θ ) wherea for tetig Η ο we have the geeralized p-value : p(x) = Pr( θ θ θ = θ ) he p-value ca be computed by Mote Carlo uig (3.3). For detail ee Poulopoulou (004). ee alo Krihamoorthy ad Mathew (003). ΠΕΡΙΛΗΨΗ Ο κοπός των ελετών βιοϊουναίας είναι η αποτίηη της βιοϊουναίας υο προϊόντων, όπου για την αποτίηη αυτή χρηιοποιούνται βαικοί φαρακευτικοί παράετροι όπως AC, C max και t max. Στόχος των ελετών αυτών είναι η ιερεύνηη της ιουναίας των περιθώριων κατανοών των φαρακοκινητικών αποκρίεων για τα υο προϊόντα, η οποία πραγατοποιείται κυρίως ε την ύγκριη των έων τιών των φαρακοκινητικών αυτών αποκρίεων κάτω από την υπόθεη της κανονικότητας ή της λογαριθοκανονικότητας. Αρκετοί έθοοι έχουν προταθεί για την αποτίηη της έης βιοϊουναίας, οι οποίες κυρίως βαίζονται την ύγκριη των έων τιών των φαρακοκινητικών παραέτρων κάτω από την υπόθεη ίων ιακυάνεων των φαρακοκινητικών παραέτρων των υο προϊόντων. Σε αυτήν την εργαία προτείνουε την εφαρογή της ιέας των γενικευένων p-value, που προτάθηκε από τους ui και Weerahadi το 989, ε προβλήατα ελέγχου έης βιοϊουναίας για την παραγωγή ελέγχων ύγκριης της έης βιοιαθειότητας την περίπτωη άνιων ιακυάνεων. Ειικότερα παρουιάζουε ακριβείς ελέγχους ικρών ειγάτων της έης βιοϊουναίας για παράλληλους χειαούς ελετών βιοϊουναίας, τις περιπτώεις που οι φαρακοκινητικές αποκρίεις ακολουθούν κανονική ή λογαριθοκανονική κατανοών ε άνιες ιακυάνεις. 584

9 EFEENCE erger,.. ad Hu, J. C. (996). ioequivalece trial, iterectio uio tet ad equivalece cofidece et. tatitical ciece Chow. G. ad iu J. P. (99) Deig ad Aalyi of ioavailability ad ioequivalece tudie. Dekker, New York. Helleic Drug Orgaizatio (EOF)(996). Guidelie for ioavailability ad ioequivalece tudie. ef. o (i Greek). Krihamoorthy, K. ad Mathew,. (003). Iferece o the mea of logormal ditributio uig geeralized p value ad geeralized cofidece iterval. J. tatit. Pla. ad Ifer., 5, 03. ehma, E.. (986). etig tatitical Hypothei d ed. Wiley, New York. Papaioaou,. (00) tatitical iue i drug developmet. Proceedig, of the 5 th Pahelleic tatitic Coferece i Ioaia, Greece. Creek tatitic Ititute. Poulopoulou,. (004). ioequivalece tudie ad Geeralized p-value. M. c hei, Dept. of tatitic ad Iurace ciece, iverity of Piraeu (i Greek). he Europea Agecy for the Evaluatio of Medicial Product (ΕΜEΑ) (998) tatitical Priciple for Cliical rial.8 March 998, CPMPICH he Europea Agecy for the Evaluatio of Medicial Product (ΕΜEΑ) (000) Note for Guidace o the ivetigatio of bioavailability ad bioequivalece.4 Dec. 000, CPMPEWPQWP4098. ui K.W. ad Weerahadi. (989) Geeralized p value i igificace tetig of hypothee i the preece of uiace parameter. J. Amer. tatit. Aoc. 84, Food ad Drug Admiitratio (FDA) (99). Guidace o tatitical Procedure for ioequivalece ig a tadard wo reatmet Croover Deig, Diviio of ioequivalece, Office of Geeric Drug, Cetre for Drug Evaluatio ad eearch,.. Food ad Drug Admiitratio, ockville, MD. Food ad Drug Admiitratio (FDA) (988a). tatitical Method for Obtaiig Cofidece Itervall for Idividual ad Populatio ioequivalece Criteria. Publihed o Iteret : tatproc.htm Food ad Drug Admiitratio (FDA) (988b).Data et of ioequivalece for Idividual ad Populatio ioequivalece. Publihed o Iteret: idex.htm Weerahadi,. (995) Exact tatitical Method for Data Aalyi. priger, New York. 585