A Parametic Confidence Interval for a Moment-Based Scaled Criterion for Individualioequivalence

Transcription

1 Journal of Pharmcscokinetics and Biopharmaceutics, Vol. 25, No, 5, 1997 A Parametic Confidence Interval for a Moment-Based Scaled Criterion for Individualioequivalence Ebi K. Kimanani1 and Diane Potvin1 Received May 19, 1997 Final October 22, 1997 Confidence intervals of proposed individual bioequivalence metrics are difficult to determine in closed form because their stochastic distributions are unknown. In this article, it is shown that, with slightly modified weights, the Relative Individual Risk (RIR) moment-based scaled statistic for individual bioequivalence that was presented by Schall and Williams has an exact noncentral Fisher's F distribution with noncentrality parameter given by a scaled squared difference in formulations means. This can be approximated by a central F with adjusted degrees of freedom from which it follows that an upper (1-a) confidence bound for RIR is given by where RIR is the least square estimate of RIR; dfer is the degrees of freedom associated with the reference intrasubject variance estimate, v is the subject-by-formulation degrees of freedom adjusted for noncentrality and a is the significance level. Individual bioequivalence is concluded if UL does not exceed the regulatory limit. The performance of this confidence interval was investigated by comparing its experimental bioequivalence rate to that of the unweighted metric under known parameter situations through simulations of two formulations in a fully replicated study design. Results showed that the proposed metric is slightly less biased and more precise than the unweighted metric. KEY WORDS: bioequivalence; bootstrap; moment-based; noncentral F distribution; parametric confidence interval; relative individual risk; simulations; weights. INTRODUCTION The Food and Drug Administration (FDA) working group on individual bioequivalence (1) suggested a moment-based metric for individual bioequivalence and a mixed strategy for its implementation in which the 'Department of Biometrics and Pharmacokinetics R&D, Phoenix International Life Sciences, 2350 Cohen Street, Ville Saint-Laurent, Montreal, Quebec H4R 2N6, Canada X/97/lflflO-0595S12.50/0 < > 1997 Plenum Publishing Coiporalion

2 5% Kimanani and Potvin metric would be scaled for highly variable drugs and unsealed for drags with normal to low variability. A formula for setting the bioequivalenee range was also suggested. While the actual estimation and testing procedure was not stated, a 95% upper confidence bound was constructed using a bootstrap procedure in a data analysis example given in the article. The rationale for the suggested metric is to address the main limitations of the traditional average bioequivalence testing criteria, these being that highly variable drugs are penalized, the switchability of drugs is not addressed, and the formulation-specific variances are not compared. These issues are discussed extensively in the literature (2-5). The mixed strategy suggested is based on a scaled and unsealed approach of Schall and LuuS (4). Sheiner (5) developed a metric for individual bioequivalence that meets the above-mentioned requirements for bioequivalence. This metric was scaled by the intrasubject reference variance (the so-called "internalstandard scaling"). A point estimate was obtained by the substitution of maximum likelihood estimates of constituent parameters. Schall and Luus (4) used a bootstrap approach to estimate the upper 95% confidence bound for the unsealed analog of Sheiner's metric. This bootstrap approach was also used in Schall and Williams's (1) mixed strategy procedure. Parametric confidence intervals have been considered previously albeit for different measures of bioequivalence. Ekbohm and Melander (6) and Endrenyi (7) suggested a general statistical model that would be considered typical for at least a semireplicated design. In such a model, an estimate of the formulation by subject interaction variance term is a measure of the switchability of the formulation and is therefore extremely crucial to the assessment of individual bioequivalence. Their suggested metric was simply the interaction term scaled by the residual term and an upper 95% confidence interval was constructed using the central F distribution. This metric differs from Schall and Luus's (4) metric in that it does not explicitly account for differences in population means or variances of the formulations. Also, it is scaled using the residual variance, an average of both Test and Reference intrasubject variances. Holder and Hsuan (8) presented a metric and confidence interval that was similar to the above but explicitly accounting for differences in formulation means and scaled by a constant. In this article, an attempt is made to harmonize some of the key issues raised by the above authors. Motivated by Sheiner's criteria, a scaled metric that encompasses the three requirements for individual bioequivalence while accounting for design variables and variance components is proposed. The probability distribution of this metric is derived based on a linear mixedeffects model with a Gaussian error assumption and least squares parameter estimation procedure as did Ekbohm and Melander (6) and Endrenyi (7). The role of the subject by formulation interaction is shown to be crucial as

3 Parametric Confidence Interval for a Criterion for Individual Bioequivalence 597 a measure of switchability within the framework of such a model (3,6,7). The resulting metric is similar to that suggested by the FDA group, with slightly different weights. The main objective of this article is to present this metric and show that the suggested weights, probability distribution, and parameter estimation are all consequences of the statistical model used. Other objectives are to suggest a "working" bioequivalence range; compare the bootstrap and parametric estimation procedures for the proposed metric; and compare the performance of the proposed metric to that of Schall and Luus (4). STATISTICAL MODEL FOR BIOEQUIVALENCE The model presented is based on any crossover design that is at least semireplicated in all the formulations. It is assumed that the design is appropriately optimized to measure the formulation effect. For illustrative purposes only, the parameters in the model below are estimated based on a two-formulation [Test (T) and Reference (R)], four-period, two-sequence, balanced and fully replicated study design with Sequence 1 as T, R, R, T and Sequence 2 as R, T, T, R. Balancedness in this case refers to period balancedness, that is, each subject completes all four periods of the study. In general, parameter estimation is not restricted to this condition. However, period imbalancedness will cause parameter estimates to be biased whatever the method of estimation may be (least squares, maximum likelihood, or restricted maximum likelihood). Let YIJ/C denote the bioavailability response on the kth subject in Sequence / and period j. Then the assumed linear mixed-effects statistical model is given by The fixed effects are g1, g/, nit nj the effects of Sequence / and period j, respectively; itfj Kjj is the effect of formulation i in period j and will sometimes be denoted by rt or rr when the formulation is T or R, respectively. &</) Sk(t) and (rs)*ko (rs)ik(t) are the random effects due to Subject k in Sequence / and the interaction between formulation i and subject k in Sequence /, respectively while Sijk is the residual term. It is further assumed that the mixed effects are all zero mean Gaussian distributed and mutually independent. The subject within sequence term has variancect (which may be referred to as the pure interindividual variability). The residual variances are cr or erl, OR, if the observation is on the T or R formulation, respectively. The interaction term has variance (>)\, OR for the T and R formulations, respectively, with a covariance of wrt. The full

4 S9S Kimanani and Potvin variance-covariance matrix for the model in Eq. (1) is given by where and Ej is the same as above but with T and R reversed. The least squares method was used to estimate parameters. The corresponding analysis of variance table (ANOVA) and variance components estimates are given in Table I. Note that OR, OT were estimated separately based on a submodel of Eq. (1) with terms for sequence, period(sequence), and subject(seqiience) using data only from formulation R or T to estimate OR with MSB, or OT with MSET, respectively. This procedure is equivalent to carrying out an ANOVA based on Eq. (1) and then estimating OR by taking the sum of squares of the residuals over the cells in which formulation R was observed. Least square estimates of the parameters may be obtained in SAS PROC GLM, for example, specifying the subject and interaction term as random. Alternatively, restricted maximum likelihood estimates may be obtained in SAS PROC MIXED with a heterogenous compound symmetry randomcomponent structure and a residual-error variance structure that is stratified by formulation. A METRIC FOR INDIVIDUAL BIOEQUIVALENCE A generally weighted form of the moment-based scaled metric proposed by the FDA working group for bioequivalence is The specific metric used by the FDA group and denoted in this article by RIR11, was unweighted (Cj = C2=C3=l). It is remarked that using this rationale, when adjusted to account for design factors but conditional on nonsignificant period effects, Sheiner's metric (5) would have weights Ci = C3=2, C2=l which are equivalent to Ci = C3 = l, C2=l/2 (see derivation in the Appendix). The metric proposed here 5s denoted by RIR15 and has

5 Parametric Confidence Interval for a Criterion for Individual Bioquivalence 599

6 em Kimanani and Potvin weights where K1, K2 are the number of subjects in Sequences 1 and 2, respectively. As will be demonstrated using least square parameter estimates for the model in Eq. (1) (see Table I), the method-of-moments estimate of the proposed RIR1S metric is denoted by RIR1S and given by where with LSMT(R) denoting the least square mean of the Test (or Reference) formulation, MSER denoting the Reference intrasubject mean sum of squares and SS NT is the sum of squares due to the subject by formulation interaction effect (see Table I). The main advantage of this choice of weights is that it is a natural consequence of the statistical linear mixed-effects model in Eq. (1), its distributional assumptions and its least square parameter estimates and it facilitates the parametric distribution of the resulting statistic as shown in the proposition below. Proposition; Parametric Csnldence Interval for RIR Let Fobs be defined by Eq. (5). Then the following distributional properties are claimed: 1. (oi/02) Fobs follows a noncentral F distribution with K, + K2 - I and dff.9. degrees of freedom and noncentrality parameter S2/ Hence a parametric (I a) upper confidence interval for RIR15 is given by

7 Parametric Confidence Interval for a Criterion for Individual Bioequivalence mi where dfer = the reference intrasubject degrees of freedom Fa.vj&ix ~ the «critical value of a centra! F distribution with v, dfer degrees of freedom Proof of Claim (I) The proof of the proposition is based on the statistical model and assumptions described in the Statistical Model for Bioequivalence section. The main part of the proof lies in showing that F0bS has a noncentral F distribution with numerator degrees of freedom Kt + K2 l, denominator degrees of freedom dfer and with noncentrality parameter given by 2/92. To prove this claim, the definitions of noncentral F and #2 as given below are used (9). Noncentral x2 Distribution. A random variable X follows a noacentral X2 distribution with v degrees of freedom and noncentrality parameter &2, denoted by %2&\v if it can be written as X = Y2+%2V _ i where F is a Gaussian variable with mean A and variance 1, #2-i is a chi-square variable with v-1 degrees of freedom and Y and xl-\ are statistically independent. The expected value ofx is A2 + v. Noncentral F Distribution. A random variable follows the noncentral F distribution with noncentrality parameter A2 and degrees of freedom V, Va if it can be written as a ratio of two quantities, (%\/v\)/(%\/v where %\ follows a noncentral j2 with vt degrees of freedom and noncentrality parameter A2 and is statistically independent from %\ which follows a %2 with v2 degrees of freedom. Also necessary for the proof are the following consequences of the distributional assumptions of the model being used. 1. SSiN-r/02 is a x2 variable with K^+K2-2 degrees of freedom (see Table I). 2. SSI?R/OR is a %2 variable with degrees of freedom ^TER (see Table I).

8 «2 Kimanani and Potvin 3. S/d = (4KlK2/(Kl+K2))l/2(LSMT-LSMR)/0 is normally distributed with mean 6/e = (4KlK2/(K, +K2))1/2(n rr)/0 and variance 1 based on the assumptions going along with Eq. (1) for a fully replicated four-period two-sequence design since var(lsmt-lsmr) = (Kl + K2/(4KlK2))62. Hence 52/02 = («,^2/ Ki + K2)(LSMT-LSM«.)2/d2 is a noncentral x2 variable with noncentrality parameter S /d2 and 1 degree of freedom. The weight C3~4KiK2/((Kt + K2)(Ki + K2-- 1)) enters merely as a normalizing constant for the difference in means and will be close to 1 for moderate Kt = K2. If 8, SSER, SSint are pairwise independent, then according to the definitions of the noncentral F above, (S2 + SSWT)/B2 is a noncentral x2 variable with noncentrality parameter S2/d2 and Ki + K2-l degrees of freedom. Hence (<jr/d2)fobs follows a noncentral F distribution with Kt + K2 l, dfer. degrees of freedom and noncentrality parameter S2/@2. To complete the proof, it remains to prove the pairwise stochastic independence of the three quantities S, SSER, SSim. This is done by first rewriting the sums of squares in terms of the parameters of the model in Eq. (1). Pairwise stochastic independence is then shown using the fact that two sums of squares are stochastically independent if their summands are pairwise independent (9). The notation used is defined below: yfj.t (sj.) emphasises that the mean of observations (residuals) in cell (/,/) (Sequence /, Period j) is with respect to formulation T while y'l.k, (s'i.*) denotes the observation (residual) mean of the Mi subject in Sequence / over those periods in which Formulation i (i=t, R) was administered. E\J. is the residual cell mean in Period j in the sequences in which Formulation i was administered. Si.k sf.k + sj.k since the design is period-balanced. (In the ease of imbalancedness, /jt = (./R/i /jt+/tvtslo/4 where Jfu,,Jrt would be the number of periods in which Subject k receives Formulation R or T, respectively.) «,.-,./, denotes multiple summation over sequence (/= 1,2), subject (k = 1,2,.., KI), formulation (i = R, T) and the periods over which formulation / is observed in Sequence / for Subject k(ji). The least square estimates of the fixed parameters are given by

9 Parametric Confidence Interval for a Criterion for Individual Bioequivalence 603 Using this notation, the three parameters can be rewritten in terms of the model parameters as follows. The marginal residual sum of squares due to the reference formulation is obtained from Table I, denoted by SSER and given by To show stochastic independence ofssintand SSER, note that the covariance of their summand is given by Similarly, to show stochastic independence of 2 and MSSINT, the covariance of their summands is given by

10 604 Kimanani ami Potvin This completes the proof that (0i./82)Fobs has a noncentral F distribution (or equivalently, Fob5 ha a central ((82/(^.)F) with noncentrality parameter given by 82/&2 and K, + K2-l, dfer degrees of freedom. Proof of Claim (2) To derive the upper confidence interval for MR IS, the following result is used: Central Approximation to the Noncentral F Distribution. A noncentral cr2f with degrees of freedom vt, v2, and noncentrality parameter A2 can be well approximated with a central <72(1 + A2/Vj)Fv,V2 distribution (10), where Thus I70bs has a central where Based on this, an upper (I a) confidence bound for RIR15 is given by which completes the proof of the proposition. Corollary: An Approximate Confidence Bound for the Ratio of Variances An upper confidence bound for a /02 is given by Since S2/02 is not known, an approximate (1 - a) confidence bound could be constructed by replacing 82/62 with its method of moments estimate

11 Parametric Confidence Interval for a Criterion for Individual Bioequivalence 605 which is 2 = 4KiK2/(K, + K2) (LMSr-LSMRf and 6>2 = MSS1NT (see Table I). This result is useful since it provides a quick comparison of variances and switchability whatever the relative formulation means are. Application of the Proposition to the Evaluation of Individual Bioequvalence The individual bioinequivalence null hypothesis is and the alternative is where AMS is the regulatory bioequivalence range. If the necessary conditions for the proposition hold, then the above null hypothesis is rejected (or individual bioequivalence concluded) if the upper (1 a) confidence bound for RIR1S is smaller than AMS. Thus the proposition provides a very convenient way to evaluate an upper bound for the estimated RIR parameter that can be obtained in any statistical software program. This would provide an alternative to the nonparametric bootstrap method that has been suggested so far. One of the main objectives of the simulation experiment described below is to compare the performance of this parametric estimate to its bootstrap counterpart and also to bootstrap estimates that would be obtained using the unweighted metric, under various known parameter combinations. This comparison can only be made with respect to some bioequivalence range. Since a regulatory limit has not been set yet, a simulation experiment was performed to suggest a "working bioequivalence range." Using this working bioequivalence range, more simulations were done to investigate the impact of the approximation on the nominal significance level and Type II error (producer risk) and also the performance of RIR 15 metric for different parameter combinations in comparison to RIR 11. The simulation studies are discussed further in the next section. SIMULATION STUDIES Simulation Parameters To be within a realistic sample size for a fully replicated two-formulations study, 24-subject studies were simulated. The natural logarithm of each

12 606 Kimanani and Potvin subject's AUC was directly simulated using a MultiNormal (ft, E) distribution, where ft was the mean vector for Periods 1 to 4 and E was the corresponding variance-covariance matrix [see Eq. (2)]. For each parameter combination, 1000 studies were generated with 12 subjects in each of two seqraences according to the design in the Statistical Model for Bioequivalence section. Differences in means were controlled through 8' = Tr TR while switcn&bility was controlled by varying the subject by formulation interaction given by oi>=g>t + rar~2cbrt. The intrasubject Reference and Test variances were controlled through their ratio denoted by j and assumed to be equal to the interindividual variance ratio, thus 7 = Or/OR=: (o- +<s>2r)/(ff +c4). An a value of 5% was used. Different bioequivalence ranges were generated by varying oio and s in where cr,^ is a prespecified threshold value for scaling and is some allowance for change variability in the variances. MR Evaluation Methods In each simulation study, least square estimates of model parameters were obtained as detailed in the Statistical Model for Bioequivalence section. Method of moments point estimates of the two RIR metrics were calculated. The parametric confidence interval for RIR 15 was then evaluated according to the proposition and denoted by RIR15para. Nonparametric bootstrap confidence intervals for both metrics are denoted by RIRl5boot and RIR 11boot and were obtained as the 95th percentiles from 2000 bootstrap samples of the simulated study. Efron and Tibshirani's (10) bootstrap algorithm and program (uncorrected for possible bias) was used for the latter purpose. Working Bioeqnivalence Range The producer risk p, which is the probability of falsely rejecting bioequivalence, is controlled by designing the study to achieve a certain power given a threshold consumer risk, a, and using the regulatory limit stated for the test. For purposes of suggesting a working bioequivalence range, simulations were designed to identify at which bioequivalence range a minimal satisfactory power (1 P) would be achieved for a fixed (realistic) sample size. For this purpose, only perfectly bioequivalent studies were simulated. These were

13 Parametric Confidence Interval for a Criterion for lndividual Biuequivalence 607 studies in which R and T formulations were simulated as perfectly switchable (0i» = 0), with equal variances (y = l) and equal means (S' = 0). Two variance scenarios were simulated, typical and highly variable with <T =0.025, 0.15 or Reference intrasubject CV=16 and 40%, respectively. The corresponding inter-subject variances were oi+«r = 0.1,1.0, respectively, while an equal formulation mean value of 4.5 was used. A trial set of 48 working bioequivalence ranges (AMS) was determined as a combination of crwo=0.15, 0.20, 0,25, 0.30, and e=0, 0.4, 9.8, 1, 1.2, 1.4,1.6,1.8,2,2.2,2.4,2.6 using Eq. (10). Using each value of bioequivalence range in the trial set, individual bioequivalence was concluded for each RIR metric and estimation method if the estimated upper bound for RIR was less than the given bioequivalence range. This process was repeated for 1000 simulated studies, p was estimated as the percentage of studies for which individual bioequivalence was rejected for each combination of RIR metric, estimation method and trial value for bioequivalence range. The results are presented in Fig. 1 for both typical and highly variable scenarios. Performance of RIR15 To investigate the impact of using a parametric upper bound on the experimental or achieved producer risk, the performance of the confidence bound for RIR15para was compared to its bootstrap version, RIRlS^oot and also to RIR 11 boot- The parameters varied were y, ', OR while G\> was fixed at 0 and a%+ OR was fixed at 0.1 for oi = and 1.0 for oi=0.15, 0.30, The results for the 1000 studies were summarized by taking the median estimation bias, precision, and upper confidence bound using RIR15boot', RIR15boot, and RIR I lboot. The frequencies of individual bioequivalence conclusions were recorded for each metric. RESULTS Bioeqivalence Range Results from the 1000 studies are plotted in Fig. 1. The y-axis, labeled Producer Risk corresponds to the percentage of studies in which individual bioequivalence was rejected at each bioequivalence range (indicated on the x-axis) using each of the RIR metrics. From these results, the following observations were made: (i) that rejection rates were very similar for both CVs of 16 and 40%; (ii) that a bioequivalence range of 2 insures a producer risk of around 20% (corresponding to a power of 80%) for a 24-subject replicated study using either the parametric or bootstrap method to estimate RIR15; (iii) that a bioequivalence range for RIR 11 is around 2.75, this

14 608 Kimanano and Povtin

15 Parametric Confidence Interval for a Criterion for Individual Bioequivalence 609 higher value reflecting the greater weight on inequality of variance in the latter; and (5v) that the bioequivalence range stays constant with respect to the scaling factor. The negligible effect of intraindividual variation on rejection rates suggests that this variability is not an issue for perfectly bioequivalent studies. This is seen later in Table III, where the rejection rate and upper confidence limit do not depend on the intraindividual variability when there is no mean difference. However, it is evident from Table III that rejection rates increase more steeply for increasing mean differences for lower intra-individual variation. The constancy of the bioequivalence range is attained by allowing the s variation in Eq. (10). It suggests that by not allowing this flexibility, that is setting s=0 in all cases, one risks incurring a much greater producer risk than the nominal value. These results suggest that the bioequivalenee range does not depend on the Reference intraindividual threshold value «r«o per se, but on its combination with e such that for a fixed power, the resulting upper bound is constant across both <?l,o and CTR. This coastant value strongly depends on the minimal power that is desired. However for the purpose of further simulations it seemed reasonable to assume a power of 80% in a 24-subject, two-way fully replicated study. Hence, a "working bioequivalenee range" of AMS = 2 was used in the rest of the simulations studies for RIR15 and 2.75 for RIR11. Performance of RIR15 Point Estimate RIR11 and RIR1S differ only in their values of (C2, C3) which are respectively, C2=l, C,= l and C2-0.5, C3=4KlK2/(Ki+K2)(Kl + K2~l). Their method of moments point estimates for a given simulated study were obtained by substituting their constituent parameter estimates into the respective metric formulas. Estimates from 1000 simulated studies were summarized by the median value. True and estimated values of RIR were obtained by substituting the true and estimated values of constituent parameters in the respective formulas. The true RIR 15 and RIR 11 values differ because of the different weights. For the same variance and switchability parameters, both metrics seem to lie in about the same range for varying S' [4.72 = (4.51, -0.20) for RIR15 and 4.52 = (4.12, -0.40) for RIR 11 when CFR = and 7=0.60 for example] (see Table II). However, when <fo,or,s' are fixed, RIR 11 has a wider range over a varying ratio of variances (r) than RIR 15. For example for 5'=0, <TR =0.025, the range for RIR1I is 0.90=(0.50, -0.40) compared to that of RIR1S which is 0.45= (0.25, -0.20) indicating the greater sensitivity of RIR 11 to variances.

16 610 Kimana

17 Parametric Confidence Interval for a Criterion for Individual Bioequivalence 611 Estimation accuracy (Bias) and precision (Interquartile Range, IQR) are given in Table II. From these values, it is evident that the estimates for both metrics are positively biased. However, the bias is slightly greater in RIR11 (overall mean bias = 0.21) than in RIR15 (overall mean bias=0.14). Dispersion, as estimated by the interquartile range, is also higher in the former (overall mean IQR =1.20) than in the latter (overall mean IQR-1.08). Thus, in the actual estimation process, RIR1S seems to be slightly less biased and more precise than RIR11. Type I and Type II Error The effect of the two RIR metrics (RIR11 and RIR1S) and their estimation procedures was investigated by comparing the rejection rates in 1000 simulated studies. The results are presented in Table III. The UL and %Rej columns correspond to the Upper Confidence Limit and the Rejection rate, Table III. Effect of the Estimation Procedure (Parametric for RIR15para and Bootstrap for RIRilboot and RIR15booL) on the Upper Confidence Limit for RIR and the Rejection Rates in 1000 Simulated Studies O7> = 0) *2R r Metric 8' = 0.00 UL RIR15para 0.63 RIR15b<K)l 0.66 RIRllboo, RIR15para 1.06 RIR15boo,.11 RIR I Ibooi RIR15pora 1.52 RIR15bool 1.58 RIRllboo, RIR15para 0.65 RIR!5boo, 0.65 RIRl!bM RIRI5p3ra RIRlSboot RIRlltao, RIR15para.52 RIRlSboo,.57 RIR!1 bool RIR15pala 0.65 RIR15boo, 0.67 RIR 1! boo, RIR15p.ra.06 RSR15b0o,.05 RIR 11 booi K-ilx. I -)para.52 RIRI5,.56 RIR I i boo, 2.49 %Rej UL 5' = 0.10 %Rej UL ' = 0.22 %Rej ' =0.34 UL ! %Rej

18 612 Ktmanani and Potvin respectively. From this table, it is observed that the bootstrap and parametric RIR1S confidence intervals (taken as the median of the simulated samples and denoted by RIR]5b0ot and RIR15para, respectively) are within the same magnitude. It is also observed in Table III that in a "perfect" bioequivalence case (5' = 0, OB=O, y=l), the producer risk is approximately 20% (which is expected since the bioequivalence range was set to reproduce this power). This value is constant for increasing CFR, implying an effective scaling. For fixed variance parameters, the rejection rate increases with increasing 8', and extremely rapidly for small <TR values. For example, for RIR15para when OR = 0.025, 7 = 1 the rejection rate increases from about to 40.60% for ' = 0.00 to 0.10, respectively. For the same increase in ' the rejection rate increases from to 21.00% for OR = This illustrates the issue raised by Schall and Williams (!) that scaling to an internal standard (i.e., the estimated intraindividual reference variance) may penalize drugs with low variability by inflating the producer risk. On the other hand, when the true ratio of means is 140% (8' = 0.34), crr = 0.30 (CK=55%), and the ratio Test to Reference variances is 1.50, the internal standard scaling allows for a rejection rate close to 60%, for either of the metrics RIR11 or RIR15. Thus, 40% of the formulations in this category would pass individual bioequivalence, a rather high Type I error, demonstrating that internal scaling may be too permissive for highly variable drugs. DISCUSSION The metric proposed in this article was initially motivated by an effort to derive the probability distribution of Sheiner's scaled metric modified to account for typical design variables. The implication of a noncentral F was glaring given the distributional assumptions of the statistical model suggested. The main result is that the proposed metric (RIR15), Sheiner's metric (RIRSHEINER), and Schall and Luus's metric (RIR11) are all of the general form given in Eq. (3). Following the order (C3, C\, C2), the weights for the three metrics are given by (1,1,1) for Schall and Luus's metric, (2,2,1) (or 1,1,0.5) for Sheiner's, and (C3> 1,0.5) for RIR15. Note that in the latter case, C3 = 4KiK2/(Kl + K2)(K, + K2-l) is a normalizing constant for the noncentrality parameter and is close to one for moderate sample sizes. Thus it is claimed that RIR15 is equivalent to WRSHEINER when design factors are accounted for. All three metrics account for the three main components of individual bioequivalence criteria which are equality of means, equality of intrasubject variances through the terms (/IT A*R) and (of OR), and similarity of the average mean differences between individuals (switchability measure) through the term <BT+fi>R~2fljRT.

19 Parametric Confidence Interval for a Criterion for Individual Bioequivalence 613 An advantage of the proposed metric RIR 15 is that, as shown in this article, its method of moments estimate has a noncentrai F distribution. This can be approximated by a central F and hence a parametric confidence interval for it exists [Eq. (6)] and can be easily evaluated using most statistical software. This provides a parametric alternative to the bootstrap estimation procedure. To facilitate parameter estimation, at least a partially replicated experimental design should be used. The estimation procedure depends on the statistical model used because it will adjust for all the effects included. In particular, design parameters such as period or sequence effects can be estimated and their effect on bioequlvalence assessed. Hence the results in this article are based on, but not restricted to, the design and model described in the Statistical Model for Bioequivalence section. SUMMARY OF results With slightly modified weights, the Relative Individual Risk momentbased scaled statistic for individual bioequivalence proposed by the FDA Working Group on Individual Bioequivalence (1) has an exact noncentrai Fisher's F distribution with noncentrality parameter given by the square of the standardized difference in formulation means. This noncentrai F distribution can be approximated by a central F with adjusted degrees of freedom and a scale factor. The parametric confidence bound for RIR is based on this central F and can be easily evaluated in any statistical software and provides a quick comparison and/or alternative to the bootstrap procedure that has been used so far. The Type I error rate in bioequivalence conclusions seems high for highly variable drugs (to the order of 40%) implying a possibly too permissive scaling procedure. This suggests that a threshold value for scaling may be more accurate than the internal standard scaling that was used here. The probability distribution of a generally weighted metric is yet to be evaluated. ACKNOWLEDGMENTS The authors acknowledge John Hooper and Richard Lalonde for motivating this research and, Walter Hauck, Robert Schall, and Laszlo Endrenyi for many interesting discussions on the subject and useful comments on the first drafts of this article. Mr Jean Lavigne provided essential programming support.

20 614 Kimanani and Potvin APPENDIX Sheiner's Metric Modified for Design parameters Using the notation in this article, Sheiner's (4) scaled RIR metric was based on the ratio where y/.* is the mean of the observations on Subject k on Formulation i; E( } is the mathematical expectation and E(-\k) is the conditional mathematical expectation on Subject k. When design parameters are included, then the above ratio evaluates to where n' is the mean of the period effects in the periods in which Formulation / was administered in Sequence /. Under the assumption of no period effects, the metric becomes It follows that RIRSHEINER -1 is in the form of the general weighted metric in Eq. (3), with weights given by (2, 2,1). REFE1ENCES 1. R. Schall and R. L. Williams. Towards a practical strategy for assessing individual bioequivalence. J. Pharmacokin. Biopharm. 34: (1996). 2. S. Anderson and W. W. Hauck. Consideration of individual bioequivalenee. J. Pharmacokin. Biopharm. 18: (1990). 3. S. Anderson and W. W. Hauck. Measuring switchability and prescribability: when is average bioequivalenee sufficient? J. Pharmacokin. Biopharm. 22: (1994). 4. R. Schall and H. G. Luus. On population and individual bioequivalenee. Stat. Med. 12: (1993). 5. L, B. Sheiner. Bioequivalenee revisited. Statist. Med. 11: (1992). 6. G. Ekbohm and H. Melander. The subject-by-formulation interaction as a criterion of interchangeability of drugs. Biometrics 45: (1990). 7. L. Endrenyi. A simple approach for the evaluation of individual bioequivalenee. Drug Inform. J. 29: (1995). 8. D. J. Holder and F. Hsuan. Moment-based criteria for determining bioequivalence. Biometrika 80: (1993). 9. H. Scheffe. The Analysis of Variance, Wiley, New York (1959). 10. B. Efron and R. J. Tibshirani. An Introduction to the Bootstrap, Chapman and Hall, New York, 1993.