A semi-parametric Bayesian approach to average bioequivalence

STATISTICS IN MEDICINEStatist. Med. 2007; 26:1224–1236Published online 11 July 2006 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/sim.2620

A semi-parametric Bayesian approach to average bioequivalence

Pulak Ghosh1,∗,† and Gary L. Rosner2

1Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303-3083, U.S.A.2Department of Biostatistics and Applied Mathematics, The University of Texas M. D. Anderson Cancer Center,

Houston, TX 77030, U.S.A.

SUMMARY

Bioequivalence assessment is an issue of great interest. Development of statistical methods for assessingbioequivalence is an important area of research for statisticians. Bioequivalence is usually determinedbased on the normal distribution. We relax this assumption and develop a semi-parametric mixed modelfor bioequivalence data. The proposed method is quite flexible and practically meaningful. Our proposedmethod is based on a mixture normal distribution and a non-parametric Bayesian approach using a Dirichletprocess mixture prior. A numerical example illustrates the use of our procedure. Copyright q 2006 JohnWiley & Sons, Ltd.

KEY WORDS: average bioequivalence; cross-over design; Gibbs sampling; mixture of Dirichlet processprior; mixture of normal; Markov chain Monte Carlo

1. INTRODUCTION

Bioequivalence assessment has become an issue of great interest to the biopharmaceutical industryduring the last few decades, especially after it became evident that having the same amounts of thedrug may exhibit marked differences in their therapeutic responses [1–5]. Nowadays, on the basisof simple pharmacokinetic concepts and parameters, bioavailability and bioequivalence studieshave been established as acceptable surrogates for expensive, complicated and lengthy clinicaltrials. Such trials are used worldwide to establish and ensure consistent product quality, as wellas reliable and therapeutically effective performance of marketed dosage form. Three situationshave thus been defined [6] in which bioequivalence studies are required: (i) when the proposedmarketed dosage form is different from that used in pivotal clinical trials, (ii) when significant

∗Correspondence to: Pulak Ghosh, Department of Mathematics and Statistics, Georgia State University, Atlanta, GA30303-3083, U.S.A.

†E-mail: [email protected]

Contract/grant sponsor: U.S. National Cancer Institute; contract/grant number: CA075981

Received 25 May 2005Copyright q 2006 John Wiley & Sons, Ltd. Accepted 8 May 2006

A SEMI-PARAMETRIC BAYESIAN APPROACH 1225

changes are made in the manufacture of the marketed formulation, and (iii) when a new genericformulation is tested against the innovator’s marketed product.

A bioequivalence study is an experiment to compare a test product (T) to a reference product (R).Bioequivalence studies compare both the rate and extent of absorption of various drug formulationswith the innovator (reference) product on the basis that if two formulations exhibit similar drugconcentration–time profiles in the blood/plasma, they should exhibit similar therapeutic effects.For an unapproved generic dosage form to be marketed and accepted as therapeutically effectivein relation to the innovator product, it must have established bioequivalence with the innovatorproduct, in vivo. The determination of bioequivalence is, thus, very important in the pharmaceuticalindustry because regulatory agencies like the U.S. Food and Drug Administration (FDA) allowa generic drug to be marketed only if its manufacturer can demonstrate that the generic productis bioequivalent to the innovator product. According to FDA regulations [7–9], a valid statisticalevaluation of bioequivalence trial is essential in order to guarantee the safety and efficacy of thegeneric drug products.

Bioequivalence studies usually proceed by administering dosages to subjects and measuring theconcentration of the drug in the blood just before and at set times after its administration. Theseconcentration-by-time measurements are often connected with a polygonal curve and measurementsof the drug’s pharmacokinetics, like AUC (area under curve), Cmax (maximum concentration) andTmax (time to maximum concentration) are calculated. For statistical analysis, these measures aretaken as the response variables. Often, these responses follow a lognormal distribution, and oneapplies a logarithmic transformation to the measurements. Following a log-transformation the useof ratios of AUCs is a more natural metric for measuring bioequivalence.

Until recently, regulatory guidelines have suggested the consideration of average bioequivalence(ABE) [10, 11]. ABE requires equivalence between the population means of the pharmacokineticmeasurements for the reference and test formulations. Over the past few years, the FDA [7, 8]supplemented ABE with two more criteria, viz., population bioequivalence (PBE) and individualbioequivalence (IBE). These new criteria have also been the subject of dispute, however, and ABEstill remains the main criterion for assessing bioequivalence between two formulations. The mainadvantage of ABE is its easier interpretation for the intended audiences, including regulators,prescribing physicians, pharmacists, and patients. The criterion of ABE has also found potentialapplications in several other areas such as, psychology [12], chemistry [13], and environmentalstatistics [14].

In this paper, we take a Bayesian approach to assessing ABE. The key advantage of using aBayesian approach for bioequivalence trials is the ability of the Bayesian inferential paradigm toincorporate background information thought pertinent to the clinical question being asked [15].Breslow [16] argued that bioequivalence is a perfectly natural concept to be subjected to Bayesiananalysis. Several authors have also advocated a Bayesian approach to average bioequivalenceinference [10, 17–21]. The main idea of all the above methods is to find the posterior distribution ofthe parameter of interest based on non-informative prior distributions for the parameters. Recently,Ghosh and Khattree [15] used an intrinsic Bayes factor approach to test ABE.

All of the existing literature on ABE, however, relies heavily on a normality assumption. Thenormality assumption in a bioequivalence trial may not always be true, however, and the inferencecan be misleading. Chow and Tse [22] and Bolton [23] discussed that the normality assumptionin a bioequivalence trial may lack robustness against outliers and skewness. Usually a bioequiv-alence trial is conducted with a small number of healthy subjects, and it is not always possibleto validate the normality assumption. Instead of following a normal distribution, the data from

Copyright q 2006 John Wiley & Sons, Ltd. Statist. Med. 2007; 26:1224–1236DOI: 10.1002/sim

1226 P. GHOSH AND G. L. ROSNER

a bioequivalence trial may have a mixture of normal distributions (e.g. diverse populations, suchas from pharmacogenetic variation), a skewed distribution or one with heavier tails, or some otherdistribution that cannot be easily specified. Thus, it is of practical interest to develop statisticalmodels in ABE that move beyond the traditional parametric model.

This paper addresses robust inference in bioequivalence studies by developing a robust Bayesiananalysis to assess ABE. We show how a robust Bayesian model can lead to better insights in abioequivalence study. Our method extends existing methods by allowing for possible heterogeneityof the subjects who are participating in the study. We propose a Bayesian non-parametric approachusing a Dirichlet process (DP) mixture [24–28]. Mixture distributions can characterize differentdistributional shapes and can describe different features of the bioequivalence data. We use theDP mixture prior to relax the distributional assumption and to accommodate possible populationheterogeneity. DP mixture models are, by far, the most widely used non-parametric Bayesian model,mainly because one can easily obtain posterior estimates using standard MCMC approaches, suchas Gibbs sampling [29, 30].

The plan of the paper is as follows. Section 2 introduces a parametric random effects model foraverage bioequivalence (ABE) trial which assumes a normal distribution for the random effects.In Section 3 we present the semi-parametric extension of the model which allows for a wide rangeof distributions for the random effects. In Section 4, we describe the data and the results of theempirical analysis are presented. Section 5 draws conclusions and provides an outlook on futureresearch.

2. MODEL

In most bioequivalence trials, a test formulation is compared with the innovator reference for-mulation in a group of normal, healthy subjects, as recommended by the US FDA [8, 9]. Eachparticipant receives the treatments alternatively in a cross-over study. The most commonly usedstatistical design for comparing average bioequivalence between a test formulation (T ) and a ref-erence formulation (R) of a drug is a two-sequence, two-period, cross-over design [6]. We referto this design as a standard 2 × 2 cross-over design. The following statistical model is usuallyconsidered for a 2 × 2 cross-over design.

yi jk =mi,k + Si + Pk + �i j + ei jk (1)

In this model, we consider yi jk to be the logarithm of response in the i th sequence from the kthperiod for the j th subject, (i = 1, 2; j = 1, 2, . . . , ni ; k = 1, 2). We use the logarithm, becauseoften the response measures in a bioequivalence study appear to follow a lognormal distributionbecause of skewness. In (1), mi,k is the direct effect of the formulation in the i th sequence thatis administered at the kth period, Si is the fixed effect of the i th sequence (S1 + S2 = 0), Pk isthe fixed effect of period k (P1 + P2 = 0), �i j is the random effect of the j th subject in the i thsequence, and ei jk is the within subject random error in observing yi jk .

If we assume, without loss of generality, that the first period in the first sequence is the referenceformulation (R), then

mi,k ={mR if k = i

mT if k �= i; mR + mT = 0



The random variables �i j are assumed to be i.i.d normal with mean 0 and variance �2. The ei jk arei.i.d normal with mean 0 and variance �2l , where l = R if k = i and l = T otherwise. We assumethat �i j and ei jk are mutually independent.

2.1. Average Bioequivalence Criteria

Two drugs are called average bioequivalent if the population means of the drug-specific AUCsare sufficiently close. In statistical terms, the problem of ABE is to decide if the difference of twoparameters �=mT − mR is close to zero. Formally, the hypothesis of average bioequivalence isformulated as:

H0 : ��L or ��U versus Ha : �L<�<�U (2)

where the lower and upper tolerance limits �L and �U are known constants specified by the FDA.The limits �L = log(0.8) and �U = log(1.25) are widely accepted by drug authorities for testingbioequivalence in terms of AUCs.

The hypothesis testing set up in (2) is the reverse of the ordinary view of testing. Whereas a nullhypothesis is usually a hypothesis of equivalence, we now consider the lack of equivalence the nullhypothesis that we seek to disprove. This formulation makes a great deal of sense for bioequivalencetrials. Here the type I error is the probability of declaring the drugs to be bioequivalent when theyare not. Therefore, by setting up the hypothesis as in (2), the consumer’s risk is protected. Once theconsumer’s risk is restricted to, say, a level 5 per cent error, the agency leaves the pharmaceuticalindustry to determine the extent of manufacturer’s risk via the type II error.

In a Bayesian hypothesis test, the construction might be based on the posterior probabilities ofthe hypothesis. In general, the hypothesis with higher posterior probability is accepted. Thus, in ourcase, if the posterior probability of Ha is greater than 0.5 then ABE will be established. One couldalso incorporate utilities or losses and apply decision-theoretic criteria to make decisions aboutABE. We do not follow that extension in this paper. One can, however, give a decision-theoreticargument for retaining the hypothesis with the greater posterior probability. The decision to declareaverage bioequivalence is based on whether the difference of the means (or ratio of AUCs) is inthe prescribed interval or not. If the investigator uses a 0–1 loss function for deciding between thetwo hypotheses, then the Bayes rule is to take the decision corresponding to the hypothesis withhigher posterior probability.

3. ROBUST DISTRIBUTION OF THE RANDOM EFFECT

The most common choice for the distribution of the random effect �i j in the cross-over design in(1) is the normal distribution. Almost all inferential procedures currently in use for assessing ABEare based on this assumption. The choice of the normal distribution for the random effect is quitearbitrary, however, and it may well happen that this distribution does not fit the data at hand. Thisassumption will be violated, for example, if the data are skewed, include outliers, or consist ofdiverse populations. Thus, it might be desirable to consider the distribution of the random effectsa priori to belong to a sufficiently large class to capture such possibilities.

This section considers a model for random effects that generalizes the normality assumption of�i j to a larger class of distributions. The aim of this generalization is to protect the inference frompossible misspecification of the random effects’ distribution. This generalization has the potential



to be robust to departures from a normal distribution while having good performance if the actualdistribution is normal.

One possible extension would assume a finite mixture of normals for the random effects’ distri-bution. A further and more flexible extension for specifying the random effects’ prior distribution,one that goes beyond a finite mixture, is to take a Bayesian non-parametric approach [24, 31, 32].

That is, one models the random effects’ prior distribution as a flexible class via a Bayesiannon-parametric prior distribution. Specifically, we propose a Bayesian non-parametric priorusing a Dirichlet process (DP) mixture. This proposal relaxes the distributional assumption andaccommodates possible population heterogeneity. DP mixtures are by far the most widely usednon-parametric Bayesian model, mainly because of the following desirable properties of DirichletProcess mixtures: (1) ease of computation, (2) ability to characterize different shapes, and (3) con-sistency. For more technical details about the properties of DP mixture see, References [33, 32].We consider this approach in the following section.

3.1. Mixture of Dirichlet process

In this section, we model set the prior distribution of �i j using a Dirichlet process mixture priorgiven by

�i j ∼N(� j , �2�) (3)

� j ∼ G (4)

G ∼DP(�G0) (5)

G0 ∼N(0, �2G) (6)

�2� ∼ IG(c, d) (7)

The above prior is a mixture of normals with respect to a mixing measure G. The mixing measure,G is a Dirichlet process. The parameters of a Dirichlet process are G0 a probability measure, and�, a positive scalar assigning mass to the real line. The parameter G0 is often called the basemeasure and is our best prior guess at G. The concentration parameter � reflects our prior beliefabout how similar G is to G0. Large values of � lead to a G that is very close to G0. Small valuesof � allow G to deviate more from G0 and put most of its probability mass on just a few atoms.The prior for � is discussed in the next section. The hyperparameters (�2G, c, d) are assumed tobe known in our example. One can also consider putting prior distributions on these parametersto allow learning (see end of Section 3.1.1).

3.1.1. A finite approximation. The above representation (equations (3)–(7)) provides a formaldefinition of the Dirichlet process mixture prior. There are several ways to implement a DPmixture prior. Recent research has focussed on using the constructive definition of the Dirichletprocess to produce MCMC computational algorithms [34–37]. Following Sethuraman [38], oneway to generate the DP mixture prior is to regard the infinite dimensional parameter G as aninfinite mixture. Thus, the Dirichlet process DP(�G0) can be written as

G =∞∑l=1

vl�Zl = V1�Z1 +∞∑l=2

(1 − V1)(1 − V2) · · · (1 − Vl−1)Vl�Zl (8)



where V1, V2, . . . are i.i.d. Beta(1, �) random variables. Since the infinite series (8) is almost surelyconvergent, as l increases the random vectors (Vl , Zl) will have diminishing effect on the priordistribution and thus on the posterior distribution of �i j . Thus, an approximation to the Dirichletprocess can be obtained by truncating the higher order terms in the stick-breaking representations(12). This results in an approximating random probability measure [39, 40] of the form

G =L∑

l=1vl�Zl = V1�Z1 +

L∑l=2

(1 − V1)(1 − V2) · · · (1 − Vl−1)Vl�Zl (9)

where V1, V2, . . . , VL−1 are i.i.d. Beta(1, �) random variables, and VL is set to one to ensure thatthe random weights sum to unity.

The finite approximation (9) for DP can be used in WinBUGS to implement the Gibbs samplingfor fitting a DP mixture model. This can be done by introducing latent variables J= (J1, J2, . . . , Jn)that indicate group membership for the unobserved variables � j , along with a probability vectorw= (w1, w2, . . . , wL)T. Thus, model (7)–(11) can be written as

�i j ∼N(�J j , �2�) (10)

J j |w∼Multinomial({1, 2, . . . , L},w) (11)

�l ∼ G, l = 1, 2, . . . , L (12)

G ∼DP(�G0) (13)

G0 ∼N(0, �2G) (14)

w∼Dirichlet( �

L,

�

L, . . . ,

�

L

)(15)

�2� ∼ IG(c, d) (16)

Note that the value of L in (9) is chosen to control the size of the tail probability

∞∑k=L+1

vk

The truncation point L needs to be chosen appropriately. The effect of truncation on the distributionof functionals of a Dirichlet process has been studied by Muliere and Tardella [36, 40, 41]. Ishwaranand Zarepour [36] suggested L = √

n for large n and L = n for small n. Since there is a approximatelinear relationship between � and L , we adopt a uniform prior for �. We did try adding anotherlayer of prior uncertainty by putting a prior distribution on �2G , the variance in the prior mean ofthe Dirichlet process. The results were remarkably close for both formulations of the model, sowe only present results for the model with fixed �2G .

3.2. Prior distribution

Parameters in model (1) are S1 = − S2, P1 = − P2, mR = − mT , �2T , �2R, and �. For simplicitywe assume S = S1, P = − P1, m =mR . To complete the Bayesian specification of the model,



we assign weakly informative priors to the unknown fixed effect parameters. Specifically, we useconjugate priors for the overall mean, m ∼ N(m0, �2m), the sequence effect S ∼ N(S0, �2s ), and theperiod effect P ∼ N(P0, �2P). For the error variance, we specify, �2l ∼ IG(al , bl), l = R, T , whereIG(al , bl) denotes the inverse gamma distribution with shape parameter al and scale parameter bl .The hyperparameters (�2m, c, d, m0, S0, �2s , P0, �2P , aR, aT , bR, bT ) are assumed to be known.We also fit a model that included a prior on the hyperparameters of the model, but the results werevery similar, so we present only the results with fixed hyperparameters.

3.3. Gibbs sampling

The posterior distributions are analytically intractable and thus computations are done via MonteCarlo approximations with the help of the MCMC method. The Gibbs sampler is probably themost widely used MCMC method and is implemented in the software package WinBUGS [42]. Ingeneral, MCMC works by drawing samples from distributions that converge to the correct posteriordistribution of the parameters. In Gibbs sampling, one draws samples from the conditional fullposterior distributions of each of the parameters given the most recent draws of the other parameters.Thus, what is required for the Gibbs sampler to work is the ability to sample from the full conditionalposterior distribution of the parameters. The conditional distribution of all the parameters areobtained from the joint distribution of all the parameters. We skip the explicit expression of theconditional distribution as WinBUGS calculates the conditional distribution automatically. Themain code for implementing the model in WinBUGS is included in the Appendix A.

4. DATA ANALYSIS

We illustrate the usefulness of the above method by analysing a real data set [43] in this section.We compare the DP mixture prior to a model with a single Normal prior for the subject-specificrandom effects in model (1).

4.1. Illustration

A two-by-two cross-over study randomized twenty-six healthy male subjects to one of two treatmentsequences. The objective of the trial was to determine if the pharmacokinetic characteristics of one40mg capsule of a drug made by Company A are the same as the concurrent administration of two20mg capsules of the same drug made by Company B. The two treatment sequences were eithertreatment A in the first period followed by treatment B in the second period or vice versa. A five-dayor seven-day washout period separated the treatment periods. The pharmacokinetic parameter AUCwas calculated for each subject in each treatment period from drug levels assayed from plasmasamples taken at 0, 0.33, 0.66, 1, 1.5, 2, 3, 4, 5, 6, and 8 h post dose. The data are skewed andinclude a few outliers. See the histogram plots (Figure 1). The first row of panels in Figure 1 showshistograms of the period differences in the first sequence of the log transformed data. There is an out-lier in both sequences. The second and third row of Figure 1 gives the exploratory plots for the treat-ment effects, assuming no period differences. The histograms indicate strong skewness in the data.

For the Bayesian analysis, we choose relatively diffuse priors. Specifically, we assume inde-pendent diffuse prior distribution N(0, 103) for the parameters m, S, P and assume a weaklyinformative gamma prior distribution �(0.001, 0.001) for the �−2

l . We also assume a N(0, 103)prior for the random effect �i j . Because we assume a normal prior for �i j , we call this model



−0.5 0.0 0.5 1.00

12

34

Period differences (sequence 1)

Fre

quen

cy

6 7 8 9

6 7 8 9

02

46

8

Formulation A

Fre

quen

cy0

24

68

Formulation B

Fre

quen

cy

Figure 1. Histogram of period differences and treatment formulation.

‘Normal model’. We chose these priors for illustrative purposes only. A formal analysis shouldinclude an evaluation of the sensitivity of the inference to prior assumptions.

For the DP model (7), we assume G0 ∼ N(0, 1000). A gamma prior distribution �(0.001, 0.001)is assumed for �2�. A Uniform(0.5, 4) covers a sufficiently wide range of values of �. The upperbound 4 is essentially arbitrary and some sensitivity analysis on this may be useful. We triedvarious values of L and found that L = 30 works very well.

We investigated what effect a prior on L would have in the following way. With our finiteapproximation, there is a multinomial distribution on the number of unique values of �l in model(10)–(16). For each iteration of the MCMC, we computed the maximum number of multinomialprobabilities (for classes 1 . . . L) that were non-negligible. By examining the histogram of themaximum effective number of clusters in the mixture, we determined that there would be no morethan 16 unique means in the mixture. From this exercise, we infer that a prior on L would likelyyield the same inference as our finite approximation, as long as this prior is not too precise. Also,with 26 subjects in this data set, a truncation point 30 is sufficiently high.

The initial values for the fixed parameters were selected by starting with the prior mean andcovering ±3 standard deviations. The initial values for the precision were arbitrarily selected. Inthe analysis, we used 5000 burn-in iterations and 10 000 updates. The posterior estimates of theparameters are presented in Table I.

The estimates of the parameters across models agree broadly. In Table I we present the posteriormeans. The treatment effect is quite high in all the models, and it is significant, in the sense that the95 per cent credible interval does not contain zero. The negative estimates for the sequence showsthat the AUC at the second sequence seems larger than at the first sequence. Negative estimatesfor the period effect bear a similar interpretation. The variance estimates for formulation B aregreater than the corresponding estimates for formulation A in all the models.



Table I. Posterior mean of the parameters.

Parameter Normal model Dirichlet model

Sequence −0.1919 −0.2251Period −0.08439 −0.03874Treatment 1.942 1.07�2eA 0.2575 0.339

�2eB 0.6515 0.6858PABE 0.131 0.435

The Bayesian hypothesis test requires calculating the posterior probability of the hypothesesdescribed in (2). Thus, the posterior probability of average bioequivalence is computed using thefollowing equation with the random draws from the posterior distributions produced by the Gibbssampler:

P[ABE |data] = Pr[log(0.8)<mT − mR<log(1.25)|Data]

∼= 1

m

M∑p=1

I [log(0.8)<mTp − mRp< log(1.25)]

Here, (mTp−mRP : p= 1, . . . , M) is a sample from the observed posterior density of (mT −mR),

I (.) denotes the indicator function, and M = 10 000 is the number of iterations. If the posteriorprobability defined by the above equation is greater than 0.5, then average bioequivalence isaccepted. In Table I, PABE is the posterior probability of ABE. ABE was rejected in all the modelssince the posterior probability of ABE is less than 0.5. We note that the frequentist threshold of0.05 plays no role in interpreting posterior probabilities and we think the relevant threshold is 0.5,suggesting that one should retain the hypothesis with the higher posterior probability.

Rejection of ABE can be described by the high difference between the two treatments. Note,however, that the treatment difference reduces from the normal model to the Dirichlet model. Thishappens because of the effect of outliers and skewness of the data in the Dirichlet model. Thus,the posterior probability of ABE also increases in the Dirichlet model compared with the normalmodel. This example thus clearly indicates the usefulness of the mixture model, especially whenthe bioequivalence data are skewed and contain outliers. The advantage of the DP mixture modelcan also be justified from the residual plot in Figure 2. Residuals from the DP mixture modelappear to have better behaviour in the figure.

We compare the two models informally by computing the effective number of parameters pDand the deviance information criterion (DIC) as presented by Spiegelhalter et al. [44]. DIC canbe implemented in WinBUGS and can be used to compare complex models. Large differences inthe criterion can be attributed to real predictive differences in the models. The smaller the DIC,the better the fit, and a difference larger than 10 is overwhelming evidence in favour of the bettermodel [45].

Using DIC values in Table II, we see that the MDP model gives improved model fit overthe other two models. Spiegelhalter et al. [44] mention that pD roughly indicates the number ofparameters in the model. We see that DP model has maximum pD.



Quintiles of Standard Normal

Res

idua

l

-2 -1 0 1 2-1.0

0.0

0.5

1.0

1.5

Quintiles of Standard Normal

Res

idua

l und

er M

DP

mod

el

-2 -1 0 1 2

-1.0

0.0

0.5

1.0

Figure 2. Residual plots.

Table II. Effective number of parameters, pD and DICfor the three fitted models for the first data.

pD DIC

Normal model 9.393 58.410DP model 12.811 45.063

5. CONCLUSION

We have provided an easily implemented robust Bayesian model for studying the effect of anassumption of normality for the random effects’ distribution in bioequivalence trials. Our modelaffords the flexible use of informative priors. The flexibility stems from the fact that it allowsfor accommodation of the uncertainty in the distribution. The method yields flexible data-driveninference for bioequivalence. We have discussed how such inference can be obtained and illustratedour method with an example. We found that the models with the normal distribution, as expected,lead to different inference than with the DP mixture model.

We have illustrated our method of analysis in the context of a single parameter in the 2× 2cross-over design with an equal number of subjects in each sequence and no drop-outs. In practice,our Bayesian method could be extended to more parameters and to other criteria, such as individualbioequivalence. Such extensions are an area of ongoing research.



APPENDIX A: IMPLEMENTATION USING WinBUGS

We describe the MDP prior WinBUGS code to implement the methods described in this paper.The full code is available from the authors upon request.

∗ T denotes the treatment effect∗ S denotes the sequence effect∗ P denotes the period effect∗ delta[g[i]] denotes random subject effect∗ g[i] is a variable that assigns a common subject number to each set of two observations taken

from the same subject

Model{

A.1. 2× 2 cross-over design

for ( i in 1:N){

y[i]∼ dnorm(mu[i],tau)mu[i]<-T * x[i,1]+S * x[i,2]+P * x[i,3]+delta[g[i]]

}tau∼dgamma(0.001,0.001)

A.2. MDP model distribution of random effect

for (j in 1:K){

delta[j]∼ dnorm(beta[group[i]],tau1)group[i]∼ dcat(p[])

}* Constructive DPPp[1]<-r[1]for (j in 2:L){

p[j]<-r[j]*(1-r[j-1])*p[j-1]/r[j-1]}p.sum<-sum(p[])for (j in 1:L){

beta[j]∼dnorm (0,tau2)r[j]∼dbeta(1,alpha)

* scaling to ensure sum to 1pi[j]<-p[j]/p.sum}alpha∼dunif(0.5,4)a∼ dnorm(0,0.001)



tau1 ∼ dgamma(0.001,0.001)tau2 ∼ dgamma (0.001,0.001)}

ACKNOWLEDGEMENTS

We thank Prof. Peter Muller for many useful discussion and helpful comments. We also thank Dr ThomasE. Bradstreet for providing the bioequivalence data. This research was partially supported by Grant numberCA075981 from the U.S. National Cancer Institute.

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A semi-parametric Bayesian approach to average bioequivalence