Comparison of Outlier Detection Methods in Crossover Design Bioequivalence Studies

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1 Journal of Pharmacy and Nutrton Scences, 01,, Comarson of Outler Detecton Methods n Crossover Desgn Boequvalence Studes A. Rasheed 1,*, T. Ahmad,# and J.S. Sddq,# 1 Deartment of Research, Dow Unversty of Health Scences, Karach, Pakstan Pharma Professonal Servces, Karach, Pakstan Deartment of Statstcs, Unversty of Karach, Pakstan Abstract: The sgnfcance of boequvalence (BE) studes s rsng due to large scale roducton and utlzaton of generc roducts all over the world. The correct dentfcaton of outlyng data n BE studes s substantal for decdng two roducts ether boequvalence or bonequvalent. For the detecton of outlers n BE studes wth the crossover desgns dfferent methods have been suggested n the lterature. In the resent work, we comared three outler detecton tests; () the Lkelhood dstance (LD) test () the estmated dstance (ED) test and the rncal comonent analyss (PCA) test. In ths work, the PCA test has been frst tme comared wth the LD and ED test. For the urose of comarson, we used two-way and three-way BE crossover data sets on lnear and logarthmc scales. Durng the course of work t was found nterestng and note-worthy that the erformances of the ED and PCA tests n the sense of outler detecton are better than the LD test and ths erformance erssts even for the log-transformed data. The results of our smulaton study also ndcated that the erformance of the ED test for outlers dentfcaton s better than the other two tests. Keywords: Boequvalence, Outlers, Lkelhood dstance, Estmated dstance, Prncal, Comonent. 1. INTRODUCTION 1.1. Boequvalence The term that consdered as the rate and extenson n whch an actve molecule s absorbed and becomes avalable at the drug acton ste s known as boavalablty. When comarson of harmacoknetc arameters related to boavalablty s made between two formulatons, the henomena referred as boequvalence. One of the two drugs s consdered as the reference. The major harmacoknetc arameters emloyed for the boavalablty assessment are: Area under the curve (AUC), Peak concentraton (C max ), Tme for achevng eak concentraton (T max ). Boequvalent drugs are harmaceutcal equvalents that, when admnstrated n the smlar molar dose, n dentcal condtons, does not reveal sgnfcant statstcal dfferences concernng boavalablty. The most broadly used measure of boequvalence s average boequvalence. Average boequvalence deends on the comarson of dfference between formulatons n the sense of mean and ncludng the fact that the dstrbutons of selected harmacoknetc arameters may dffer between two formulatons n other dstrbutonal characterstcs. Accordng to Food and Drug Admnstraton [1] gudelnes, two drugs or formulatons are declared to be boequvalent f the *Address corresondng to ths author at the Deartment of Research, Dow Unversty of Health Scences, Karach, Pakstan; Tel: ; E-mal: abdur.rasheed@duhs.edu.k # Co-Author E-mals: tasneem_ahmad001@yahoo.com, jssdr1@yahoo.com 90% confdence nterval of the rato of geometrc mean of harmacoknetc arameters such as, AUC and C max, les wthn the re-secfed range (80%, 15%). Sometme n regulatory submsson, for evaluaton of dfferent formulatons, the detecton of outlers becomes mandatory. Correct dentfcaton and treatment of outlyng data n BE studes s substantal for decdng two roducts ether boequvalence or bonequvalent. Durng the analyss of boequvalence the logarthmc transformaton of harmacoknetc arameters s recommended by the FDA [1] under the fact that logarthmc transformaton makes the dstrbuton much closer to the normal. 1.. Outlyng Observatons n Boequvalence Studes In boequvalence study, data may have some outlyng (extremely large or small) observatons. These outlyng observatons may have some rofound effect on the concluson of boequvalence studes. These extremely large or small values may be the results from dfferent mechansms, such as: (1) roduct falure (coated tablet broken; sngle tablet wth drug dosage) () Adverse event affectng drug absorton () Laboratory error/data transcrton error (4) unusual reacton of a sngle subject to one of the formulatons (so-called subject-by-formulaton nteracton). Mechansm 1 to can be vewed as outlers due to roduct or rocess falure and mechansm 4 can be ISSN: -806 / E-ISSN: /1 01 Lfescence Global

2 164 Journal of Pharmacy and Nutrton Scences, 01 Vol., No. Rasheed et al. vewed as outlers due to subject-by-formulaton nteracton []. 1.. Outlyng Observatons n Crossover Boequvalence Studes In a non-relcated crossover desgn comarng f formulatons of a drug, the model s gven as y jl = μ + S + P l + F j + e jl =1,...,n j,l =1,... f (1) Where y jl s the resonse of the th subject n lth erod for jth formulaton, μ s the overall mean, P l s the fxed effect of the lth erod wth P l =0, F j s the fxed effect for the jth formulaton wth F j =0, S s the random effect for the th subject, and e jl s the random error n observng y jl. It s assumed that the { S } and { e jl } are ndeendently and normally dstrbuted wth means zero and varances s and e, resectvely. For the model (1) we can dfferentate two tyes of outlers 1. Sngle data ont outler: Unusual subjects who reveal extremely low or hgh boavalablty relatve to reference treatment, ths tye of outler also termed as wthn-subject outlers.. Subject outler: Unusual subjects who reveal an extreme boavalablty for both test and reference treatments, ths tye of outlers termed as between-subject outlers. FDA [1] advocates that roduct or rocess falure and subject-by-formulaton nteracton are two reasons of outlers n crossover boequvalence studes. These mechansms for outler detecton, the roduct or rocess falure and the subject-by-formulaton nteracton can show clearly as a sngle data ont (wthn-subject) outler. In crossover desgn for two treatments wth two erods and two sequences, t s not ossble to searate outlers occurred due to roduct or rocess falure from the outlers occurred due to subject-by-formulaton nteracton []. The crossover desgn for three treatments wth three erods and three sequences has also the smlar ssue. Consequently regulatory authortes do not allow the excluson of outlers from the statstcal analyss of x crossover boequvalence studes only based on statstcal crtera, but f a crossover BE data set contan outlyng observaton then t mght be of nterest to resent the results wth and wthout outlyng observatons. Lund [] has develoed a method for outler detecton n the lnear model, use of ths method has also been suggested by the FDA but Chow and Lu [4] onted out and roved that ths method no longer arorate for the crossover desgn due to correlated harmacoknetc resonses from the same subject. For detecton of outlers n boequvalence studes, two rocedures based on Cook s lkelhood dstance and the estmated dstance, were roosed by Chow and Tse [5]. Enachescu and Enachescu [6] used rncal comonents to ntroduce a test for outler detecton n boequvalence studes wth crossover desgn. Ramsay and Elkum [7] comared dfferent outler detecton methods roosed by Chow and Tse and Lu and Weng [5, 8] and wth the hel of the smulaton study, he resented that the estmated dstance test erformed better than other tests. Enachescu and Enachescu [6] has ntally used rncal comonents for the dentfcaton of outlyng observatons n crossover BE studes. In ths work we frst comared the outlers dentfcaton test based on rncal comonents wth other two tests based on Cook s lkelhood dstance and the estmated dstance. We erformed these three tests on non-relcated X and X crossover BE data sets and observed the numbers of subjects were detected as outlers. Ths work was carred on the lnear and logarthmc scale as recommended by the FDA [1]. The erformance of these test were also observed through a smulaton study.. THREE OUTLIER DETECTION TEST FOR CROSSOVER DESIGN.1. Prncal Comonent Analyss (PCA) The objectve of PCA s to dscover or to reduce the dmensonalty of the data set and dentfy new meanngful underlyng varables. [6] has mentoned that for normally dstrbuted observaton U are ndeendent varables, where s called Egen 1, j value denotes the varance of the -th rncal comonent. He also Consdered 1, j the weghted sum of square dstance to zero of the

3 Comarson of Outler Detecton Methods n Crossover Desgn Journal of Pharmacy and Nutrton Scences, 01 Vol., No. 165 rojected data nto rncal factoral lane, wth E 1, j = = and Var 1, j =. Now the Observatons wth a square dstance greater than m (the rule of ) may be consdered as outlers where m = +... Lkelhood Dstance (LD) Test Chow and Tse [5] ntroduced the lkelhood dstance test for dentfyng outler n a boequvalence study where the null hyothess assumes that there are no erod and formulaton effects. Now the model becomes y j = μ + S + e j =1,...,n j=1,... f () The arameters of nterest are μ, s, and e. Let =( 1 ) t, where 1 = μ, = e, and = e + f s. The maxmum lkelhood functon s The maxmum lkelhood estmators of the arameters are then 1 =Y = 1 Y nf, j = 1 (Y n( f 1) j Y ) = f (Y n Y ) j j The lkelhood dstance statstc for the th subject s twce the dfference between the model log lkelhood evaluated by usng estmates from all the subjects and from estmates obtaned after deletng the th subject. LD ()= [L() L ( ) Where s the maxmum lkelhood estmator of obtaned by deletng the th subject from the data. Asymtotcally, as n, LD () s dstrbuted as a ch-square statstc wth three degrees of freedom. The th subject s called an outler f LD ()> () j L()= Nf log N log( f 1 ) 1 (Y j 1 ) = f 1 1 ( Y 1 ) N N f j=1 ().. Estmates Dstance (ED) Test Estmated dstance statstcs deends on the dfference n the arameter estmates arsng from the deleton of th observaton, rather than on the dfference n the log-lkelhood. The estmated dstance statstc s Table 1: X BE Crossover Data on Lnear and Logarthmc Scale Perod I Perod II Perod III Perod I Perod II Perod III Sequence Subject no. AUC on lnear scale AUC on logarthmc scale ACB ACB ACB ACB ACB ACB ACB ACB BAC BAC BAC BAC BAC BAC BAC CBA CBA CBA CBA CBA CBA

4 166 Journal of Pharmacy and Nutrton Scences, 01 Vol., No. Rasheed et al. ED () = f =( ) t 1 ( ) Where 1 s the maxmum lkelhood estmator of the varance matrx / n 0 0 = 0 /(n1) [5] roved that, ED () s asymtotcally dstrbuted as a ch-square varable wth three degrees of freedom. Hence, the estmated dstance test declares the th subject as an outler f ED ()> ().. APPLICATION AND RESULTS.1. x Crossover Desgn In order to aly above defned test rocedures for detectng outlers n BE studes, we found a BE data set on FDA webste, where t s mentoned as data set 8. In that BE study there were formulatons (A, B and C), erods and sequences (ACB, BAC and CBA); t was a crossover desgn wth an equal number of sequence, erods and formulatons. 1st Sequence contaned 8 subjects, nd sequence contaned 7 and rd had only 6 subjects. The data set on lnear and logarthmc scales are gven n Table 1. As only 1 subjects out of 4 subjects comleted the study, now for the sake of our convenence we coded them as subject number from 1 to 1. In Fgure 1, the lkelhood and estmated dstances for the LD and ED tests and the squared dstances for observatons n the PCA test are resented for the lnear scale data. For the LD and ED tests, subject s to be consdered as outler f LD ()> ()= and ED ()> () = resectvely and for the PCA test, subject wth the squared dstance greater than threshold value ( m = + ) s Fgure 1: For x crossover BE data on lnear scale, the lkelhood and estmated dstances for the LD and ED tests and the squared dstances for observatons n the PCA test resectvely. Fgure : For x crossover BE data on the logarthmc scale, the lkelhood and the estmated dstances for the LD and ED tests and the squared dstances for observatons n the PCA test resectvely.

5 Comarson of Outler Detecton Methods n Crossover Desgn Journal of Pharmacy and Nutrton Scences, 01 Vol., No. 167 Table : The Result of Outler Detecton from all Three Methods for x Crossover Data on the Lnear and Logarthmc Scale BE data on lnear scale BE data on logarthmc scale LD test ED test PCA test LD test ED test PCA test No outler Subj. 0 s outler Subj. 0 s outler No outler Subj. 0 s outler No outler consdered as outler. Fgure dects the results on the logarthmc scale and for the PCA test threshold value was From above fgures and table t s evdent that none of the subject was detected as an outler from the lkelhood method, subject 0 was detected as an outler from the estmated dstance method on both scales whereas the rncal comonent method detected t only on the lnear scale... x Crossover Desgn We consdered a BE data set from [4], ths study was conducted wth 4 healthy volunteers. The desgn of the BE study was x crossover.e., there were two sequences (RT and TR) and two erods. Each of the two sequences contaned 1 subjects. Fve-50-mg tablets as test formulatons or 5-mL of an oral susenson as a reference formulaton were gven to each of the subjects durng each dosng erod. The data set on lnear and logarthmc scales are gven n the Table. In Fgure, the lkelhood and estmated dstances for the LD and ED tests and the squared dstances for observatons n the PCA test are resented for the lnear scale data. For the lkelhood and estmated dstance tests, the crteron for declarng a subject as an outler s same as defned above and for the PCA test, subject wth squared dstance greater than threshold value ( m = + ) s Table : X BE Crossover Data on Lnear and Logarthmc Scale Perod I Perod II Perod I Perod II Sequence Subject no. AUC on lnear scale AUC on logarthmc scale RT RT RT RT RT RT RT RT RT RT RT RT TR TR TR TR TR TR TR TR TR TR TR TR

6 168 Journal of Pharmacy and Nutrton Scences, 01 Vol., No. Rasheed et al. Fgure : For x crossover BE data on the lnear scale, the lkelhood and the estmated dstances for the LD and ED tests and the squared dstances for observatons n the PCA test resectvely. Fgure 4: For x crossover BE data on the logarthmc scale, the lkelhood and estmated dstances for the LD and ED tests and the squared dstances for observatons n the PCA test resectvely. consdered as outler. Fgure 4 shows the results on the logarthmc scale. For the PCA test threshold value was From above Fgures, 4 and Table 4 t s evdent that none of the subject was detected as an outler from the lkelhood method, subjects 1 and 19 were detected as outlers from the estmated dstance method on the lnear scales and subjects 1 and were detected as outler on the logarthmc scale, moreover, on the logarthmc, subject 19 was not found as an outler but t was very close to the threshold value. Whereas from the rncal comonent analyss method subjects 19 and 1 were found as outlers on lnear and logarthmc scales resectvely. Subjects 19 and were very close to the threshold value on the logarthmc but were not found as outlers. 4. SIMULATION STUDY For each combnaton of samle sze=18, 4 and ntrasubject varabltes= 0, 40, a total of 100 data sets of AUC values were generated from the statstcal model for X crossover desgn under the normalty assumton. For the sake of our convenence we assumed that there were no erod and carryover effects. The true test and reference means were both chosen to be 100. The results of smulatons study are gven n the Tables 5 and 6 on the lnear and logarthmc scales resectvely. From Tables 5 and 6 t s evdent that the rate of outlers dentfcaton of ED test s hgher than LD and PCA tests, lkewse the ercentage of the smulaton (n arenthess) n whch ED test rate was at least as good as each of the alternatve s also hgh. For examle n Table 5 for the lnear scale, for samle sze 4 and ntrasubject Table 4: The Result of Outler Detecton from all Three Methods for x Crossover Data on the Lnear and Logarthmc Scale BE data on lnear scale BE data on logarthmc scale LD test ED test PCA test LD test ED test PCA test No outler Subj.1 and 19 are outlers Subj. s 19 outler No outler Subjs. 1 and are outler Subj. 1 s outler

7 Comarson of Outler Detecton Methods n Crossover Desgn Journal of Pharmacy and Nutrton Scences, 01 Vol., No. 169 Table 5: The Smulatons Percentage where the Estmated Dstance Test Performed Better than other Tests on the Lnear Scale, n the Sense of Detectng Outlers More Frequently, and the Percentage (n Parenthess) n whch t was at Least as Good Lnear scale Samle sze Intrasubject varabltes ED>LD (EDLD) ED>PCA (EDPCA) (100) 8(100) 18 94(100) 71(100) (100) 77(100) 18 97(100) 78(100) Table 6: The Smulatons Percentage where the Estmated Dstance Test Performed Better than other Tests on the Logarthmc Scale, n the Sense of Detectng Outlers More Frequently, and the Percentage (n Parenthess) n whch t was at Least as Good Log scale Samle sze Intrasubject varabltes ED>LD (EDLD) ED>PCA (EDPCA) (100) 6(99) 18 86(100) 61(98) (100) 61(9) 18 84(100) 6(99) varabltes 0, the estmated dstance test erformed better than the lkelhood dstance test n 97% of the smulatons and better than the PCA test n 8% of the smulatons. Under the same condton the estmated dstance test erformed at least as well as the lkelhood dstance test n 100% of the smulatons and at least as well as the PCA test n 100% of the smulatons. In Table 6 for the logarthmc scale, for samle sze 4 and ntrasubject varabltes 0, the estmated dstance test erformed better than the lkelhood dstance test n 90% of the smulatons and better than the PCA test n 6% of the smulatons. Under the same condton the estmated dstance test erformed at least as well as the lkelhood dstance test n 100% of the smulatons and at least as well as the PCA test n 99% of the smulatons. 5. CONCLUSION Fndngs of our work roose that the estmated dstance test s sueror to the lkelhood dstance and the rncal comonent analyss test. The erformance of the estmated dstance test erssts even for logtransformed x and x crossover BE data sets. The rncal comonent analyss test has been frst tme comared n ths research wth any of the outler detecton test. It s obvous from above fndngs that the rncal comonent analyss test s much sueror to the lkelhood dstance test as lkelhood dstance test dd not detect any subject as an outler on any of the scales. Moreover, the results of our smulaton study advocate that, the erformance of outlers dentfcaton of the estmated dstance test s sueror to the lkelhood dstance test and the PCA test. These smulatons results also suort the erformance of the estmated dstance test over the lkelhood dstance test and the rncal comonent analyss test when tested on real crossover BE data sets. Excluson of the outlyng observatons n BE studes are allowed only when they are caused by the roduct or rocess falure; when the reason of outlyng observaton s subject-by-formulaton nteracton, the excluson of such outlyng observaton may not be allowed. In x or x crossover BE t s not ossble to searate outlers aeared due to the roduct or rocess falure from the outlers aeared due to the subject-by-formulaton nteracton, only bass of statstcal crtera. Although t s substantal to resent the statstcal results of crossover BE study wth and wthout outler observatons. REFERENCES [1] FDA. Gudance for ndustry on statstcal aroaches to establshng boequvalence.center for Drug Evaluaton and

8 170 Journal of Pharmacy and Nutrton Scences, 01 Vol., No. Rasheed et al. Research, Food and Drug Admnstraton, Rockvlle, Maryland 001. [] Schall R, Endreny L, Rng A. Resduals and outlers n relcate desgn crossover studes. J Boharm Stat 010; 0(4): htt://dx.do.org/ / [] Lund RE. Tables for an aroxmate test for outlers n lnear models. Technometrcs 1975; 17: htt://dx.do.org/ / [4] Chow S-C, Lu J-P. Desgn and analyss of boavalablty and boequvalence studes ( rd ed.). New York: Dekker 009. htt://dx.do.org/10.100/sm [5] Chow SC, Tse SK. Outler detecton n boavalablty/ boequvalence studes. Stat Med 1990; 9(5): [6] Enachescu D, Enachescu C. A new aroach for outlyng records n boequvalence trals. Paer resented at the The XIII Internatonal Conference "Aled Stochastc Models and Data Analyss" 009. [7] Ramsay T, Elkum N. A comarson of four dfferent methods for outler detecton n boequvalence studes. J Boharm Stat 005; 15(1): 4-5. htt://dx.do.org/ /bip [8] Lu JP, Weng CS. Detecton of outlyng data n boavalablty/boequvalence studes. Statst Med 1991; 10(9): htt://dx.do.org/10.100/sm Receved on Acceted on Publshed on DOI: htt://dx.do.org/ /