Exact Calculation of Power and Sample Size in Bioequivalence Studies Using Two One-Sided TestsŽ Meiyu Shen,A* Estelle Russek-Cohen,B and Eric V

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(wileyonlinelibrary.com) DOI: 10.1002/pst.1666 Published online 5 December 2014 in Wiley Online Library

Exact calculation of power and sample size in bioequivalence studies using two one-sided tests Meiyu Shen,a* Estelle Russek-Cohen,b and Eric V. Sludc,d

The number of subjects in a pharmacokinetic two-period two-treatment crossover bioequivalence study is typically small, most often less than 60. The most common approach to testing for bioequivalence is the two one-sided tests procedure. No explicit mathematical formula for the power function in the context of the two one-sided tests procedure exists in the statistical literature, although the exact power based on Owen’s special case of bivariate noncentral t-distribution has been tabulated and graphed. Several approximations have previously been published for the probability of rejection in the two one-sided tests procedure for crossover bioequivalence studies. These approximations and associated sample size formulas are reviewed in this article and compared for various parameter combinations with exact power formulas derived here, which are computed analytically as univariate integrals and which have been validated by Monte Carlo simulations. The exact formulas for power and sample size are shown to improve markedly in realistic parameter settings over the previous approximations. Copyright © 2014 John Wiley&Sons,Ltd.

Keywords: bioequivalence; single endpoint; sample size; power; crossover; pharmacokinetic study

1. INTRODUCTION The null hypothesis, H0,statesthatT and R are not equivalent. The alternative hypothesis representing equivalence, In a typical pharmacokinetic two-period two-treatment crossover Ha, is the intersection of the two one-sided parameter regions, bioequivalence study, subjects are randomized to one group f1 <T Rg and fT R <2g. receiving the reference drug (R) first or the other group receiv- No exact analytical formula has previously been published for ing the test drug (T) first. Each group receives the complementary the probability of rejection in the two one-sided tests procedure drug in the second period. Within each period for each sub- for crossover bioequivalence studies under general parameter ject a drug plasma concentration profile over time is obtained. settings, but there are several published approximations [3–6]. Then AUC (area under the concentration–time curve), Cmax (max- All of these approximations, as well as the analytical formu- imum concentration), and Tmax (time to reach maximum con- las derived in this paper, apply to the two one-sample tests centration) for each subject from each treatment are obtained procedure when the response variable of interest, log(AUC) from the observed concentration–time profiles. Because previous or log(Cmax), is normally distributed. Phillips [3] and Diletti, experience indicates that log(AUC) and log(Cmax) are approxi- Hauschke, and Steinijans [4] calculated the power and the mately normally distributed‡, we assume this throughout and sample size using Owen’s [7] special case of the bivariate non- denote by T and R themeanoftheresponsevariableof central t-distribution and presented some sample size tables interest used, respectively for the test product and reference product. The number of subjects in a pharmacokinetic two-period a two-treatment crossover bioequivalence study is typically small, Center for Drug Evaluation and Research, US Food and Drug Administration, Silver Spring, USA most often less than 60, with range from 12 to 170 according to b thesurveybyDavitet al. [1] of 12 years of bioequivalence studies Center for Biologics Evaluation and Research, US Food and Drug Administration, Silver Spring, USA submitted to the United States Food and Drug Administration. In order to conclude the bioequivalence of the test product and cMathematics Department, University of Maryland, College Park, USA the reference product, we should reject the null hypothesis in the dCenter for Statistical Research and Methodology, US Census Bureau, Maryland, following hypothesis test [2]: USA *Correspondence to: Meiyu Shen, DBVI, Office of Biostatistics, Office of Trans- H0 : T R 6 1 or T R > 2 lational Sciences, Center for Drug Evaluation and Research, Food and Drug (1) Administration, 10903 New Hampshire Ave, Silver Spring, MD 20993, USA. Ha : 1 <T R <2 E-mail: [email protected] †This article reflects the views of the authors and should not be construed to be Here, 1 and 2 are pre-specified constants, also called equiva- those of the US Food and Drug Administration. ‡ lence margins, and 1 <2. All logarithms used in this paper are natural, to base e. 95

and power graphs. Phillips’ [3] calculations were for equivalence Clearly, the estimator DO is the average of the averages of the margins 2 D1 D 0.20, different from the values 2 D intra-subject difference between the test and the reference for 1 D log 1.25 D 0.2231 recommended in [8], the Food the two sequences, and S2 is a pooled estimate of the variance of and Drug Administration (FDA) Guidance for Industry. Kieser the intra-subject difference. For this two-period two-treatment and Hauschke [5] proposed approximate methods for sample crossover design, DO is a normally distributed unbiased estimate 2 2 2 size determination, and Chow and Wang [6] derived approxi- 2 R CT 1 1 .n1Cn22/S of with variance D 4 n C n ,and 2 2 is mate formulas for sample size calculation under crossover or DO 1 2 T CR 2 parallel designs. distributed as with D n1 C n2 2 degrees of freedom. In Section 2 of this article, we present the background, nota- tion, and standard assumptions in two-stage crossover designs for tests of bioequivalence, and in Section 3, we briefly derive the 3. EXACT POWER FUNCTION AND THE JOINT exact formula for the power of the two one-sided tests proce- PROBABILITY DENSITY FUNCTION OF T1 dure for testing bioequivalence based on a univariate normally AND T2 distributed response variable and indicate how numerical inte- qDO 1 gration easily provides accurate numerical values for power and First, we define test statistics T1 D and T2 D S 1 C 1 sample size. In Section 4, we compare the numerical results of the 2 n1 n2 O exact method with the tabulated sample size values of Diletti et qD2 as the basis for the hypothesis test (1). To correct the S 1 C 1 al. [4], the power values tabulated by Phillips [3] for 2 D1 D 2 n1 n2 0.20, and the sample size and power values generated by the misstatement of Liu and Li [9] that test statistics .T1, T2/ do not approximate method of Chow and Wang [6]. have a joint density (see first two lines after Equation (3) in [9]), 2 we simply observe that the function mapping DO , S to .T1, T2/, 2. LINEAR MIXED EFFECT MODEL FOR A with domain .1, 1/ .0, 1/ and range f.t1, t2/ : t1 > t2g, is both differentiable and differentially invertible, while the inde- TWO-PERIOD TWO-TREATMENT O 2 CROSSOVER BIOEQUIVALENCE STUDY pendent variables D and S have normal and gamma densities, respectively, as described in the previous text. For an exposition In a two-period two-treatment crossover design, there is a of the change-of-variable formula for differentiable one-to-one washout period between two periods of observation to limit transformations of random vectors with a joint density, see For- mula 4.3.2 on page 158 and Theorem 2.1.8 on page 53 of [10]. The or remove any carryover effect. Let Yijk be the response (e.g., log(AUC)) of the kth subject in the jth period of the ith sequence joint density for .T1, T2/ can also be obtained directly by differen- in the two-period two-treatment crossover study, where i D 1, 2, tiation with respect to t1 and t2 on the region t1 > t2 of the joint j D 1, 2, and k D 1, :::, ni. Then, the linear mixed effect model for cumulative distribution function derived and published by Owen Yijk is [7, Section 5]. Although the exact power function can easily be developed as Yijk D C Sik C Pj C Fij C "ijk (2) a double integral from the joint pdf of T1 and T2, we will derive a simpler form of the exact power function by integrating the con- where is the overall mean; Pj is the fixed effect of period j; Fij 2 is the fixed effect of the formulation administered in period j of ditional power given S . This alternative formula, in the same spirit sequence i; Sik is the random effect of subject k in sequence i; as Formulas (8) and (12) in Section 5 of [7], is simpler because and "ijk is the random error. From the treatment assignments, we it involves only a univariate integral over a bounded interval, 2 know that F11 D F22 D R and F12 D F21 D T . The param- after recognizing that the conditional power given S is a readily eters can be estimated only subject to restrictions: P1 C P2 D evaluated normal tail probability. T C R D 0, because otherwise the parameters P1, P2, T ,and The exact power function can be written, in terms of the ˛ and R are not separately identifiable. The Sik and the "ijk are all inde- 1 ˛ quantiles t.˛/ and t.1 ˛/ Dt.˛/ of the t distribution pendent normal random variables with mean 0. The variance of with degrees of freedom, as 2 2 Sik is and the variance of "ijk is if i ¤ j, 8i, j D 1, 2 for S T the test formulation and 2 if i D j, 8i, j D 1, 2 for the reference R P1 , n1, n2, T , R formulations. Let D T R,whereT and R are the true means of the D P RejectH0jT R D , 2 .1, 2/ , T , R test and reference formulations, respectively. An estimator of 0 is given as B DO 1 DO 2 D P @ q > t˛./ \ q 6 t1˛./ jT S 1 1 S 1 1 YN12 YN11 C YN21 YN22 C C DO D .(3)2 n1 n2 2 n1 n2 2 1

The pooled estimate of variance is C R D , 2 .1, 2/, T , RA " Xn1 2 s 2 1 * * S D Y12k Y11k Y 12 Y 11 S 1 1 n1 C n2 2 6 O 6 kD1 D P 1 C t˛./ C D 2 t˛./ # 2 n1 n2 n2 X * * 2 s ! C Y21k Y22k Y 21 Y 22 . S 1 1 kD1 C jT R D , 2 .1, 2/, T , R . 2 n1 n2

96 (4)

This last expression is the expectation over S of a conditional corresponding simulation standard errors less than 0.0005), for probability given S, which is expressed simply in terms of the a combination of equal to 0, 0.1, 0.2, and log(1.25), and of cumulative normal distribution function as D 0.2 and 0.3, that the simulated and exact values were always within 0.001 of one another, and that the numerical-integration 28 0 1 0 19 3 0 s ! 1 <ˆ => 1 6 B S C B S C 1 1 7 4 @ 2 q A @ 1 q A @ 6 2 1 A5 ES ˆˆ t˛./ ˆ C t˛./ > I S C , : DO 2 2 DO 2 2 ; t˛./ n1 n2 T C R T C R

error bounds were less than 0.0001 (usually by one or more orders where t˛./ is the 1 ˛ quantile of the t-distribution with of magnitude).˚ D n1 C n2 2 degrees of freedom, ˆ./ is the standard normal Because t2 ./ is an increasing function of (see the proof cumulative distribution function, I ./ is the indicator function, ˛ 2C 2 in Appendix B), both the upper limit of integration and the inte- and recall that 2 D R T 1 C 1 . Using the fact that DO 4 n1 n2 grand in the integral formula (6) are directly seen to be monotone 2 SP 2./ and DO N , 2 are independent, we write increasing as a function of n, so the integral (6) itself is also mono- 2C 2 O T R D tone increasing in n. Hence, we use a bisection search or other the previous expectation over S explicitly as numerical root-finder to find the required sample size n by first solving for the continuous value n D n at which the exact power P . , n , n , , / (6) with all parameters held fixed is equal to 1-ˇ and then round- 1 0 10 2 T R 1 ing it up to the smallest even number n > n. Code lines for Zc1 r B B C B B 2 x C doing this in R are also supplied in Appendix A. D @ˆ @ r t˛./ A 2 2 C 1 1 0 T R C 4 n1 n2 0 11 4. COMPARISON OF EXACT POWER WITH r B xCC 1 APPROXIMATE POWER ˆBr 1 Ct ./ CC x=21ex=2dx, @ ˛ AA =2 2C2 2 .=2/ T R 1 C 1 4.1. Comparison of exact power function with approximate 4 n1 n2 power function of Chow and Wang (5) 2 1 4.1.1. Approximate power formulas of Chow and Wang. Assuming .21/ 1 1 n o where c D C . 1 2 2 2 n n DO 1 t˛ ./.T CR / 1 2 P p 6 t˛ ./ C p 0, Chow and Wang [6] approxi- The gamma density and integrand written using the normal S 1=n S 1=n mated the power function P ., n, /for the two one-sided tests cumulative distribution function are readily evaluated, so the 1 procedure (1) when > 0and is relatively big to by integral (5) is easily evaluated in any good statistical computing ( package: R code for it is given in Appendix A. O CW D 2 In the two one-sided tests procedure for one single variable, the P D P p 6 p t˛ ./ jT S 1=n S 1=n power (P1/ is the probability of rejecting H0 when Ha in (1) is true. ) For sample size determination and power of two one-sided tests procedure in the bioequivalence literature, Schuirmann [2], Diletti R D , 2 .1, 2/, . et al. [4], and Chow and Wang [6] all assumed that n1 D n2 D n=2 2 2 2 and D D . Following this convention, we will compare CW CW T R Clearly, from the right sides of Pn and P1 . , n, /, P over-o the exact power with the approximate power of Chow and Wang DO 1 estimates P1 . , n, / because P p 6 t˛ ./ C p > 2 2 2 S 1=n S 1=n under the assumption that n1 D n2 D n=2andT D R D . p The exact power function becomes 0. Replacing S by 2 in the right-hand expression for PCW , [6] further approximated P1 . , n, / for n2 D1 D CW DO and relatively large > 0byP P p 6 p P1 , n, o S 1=n 2=n ! Zc2 r t˛ .n 2/ j D , 2 . , /, . T R 1 2 2 x D ˆ p t˛./ With this replacement, for 2 D1 D and D 2=n CW 0 0,nˇ Chow andˇ Wang approximatedo P1 . , n, / by P D r !! ˇ DO ˇ P ˇ p ˇ 6 p t ./ . Hence, the sample size formulas x 1 = = ˛ 1 p =21 x=2 S 1 n 2 n ˆ C t˛./ x e dx, of Chow and Wang are 2=n 2=2.=2/ (6) 8 2 ˆ 2 . /C . / n. /2 < 2 .t˛ n 2 tˇ n 2 / 2 1 2 if ¤ 0 where c2 D 2 2 . .j j/ 8t˛ ./ n > (7) ˆ 2 2 The power values computed numerically from (5) using a stan- : 2 .t˛ .n2/Ctˇ=2.n2// if D 0 dard numerical-integration routine, integrate in R [11], were 2 carefully checked both in terms of their own estimated error bounds and by comparing them with Monte Carlo simulations 4.1.2. Comparison of exact power and approximate power of of rejections in the two one-sided tests procedure. As examples Chow and Wang. The approximate power function of Chow and 6 of the results, we found using 10 Monte Carlo replications (with Wang [6] tends to overestimate power by removing the upper-tail 97

O bound on Dp , but because of the further replacement of sam- The product is considered a highly variable drug when CV for S 1=n untransformed AUC or Cmax is greater than 0.3 [12,13]. Regard- O ple standard deviation by true in the inequality for Dp , their S 1=n less of the magnitude of the standard deviation from exam- approximation does not overestimate for all possible parameter ination of many numerical results, the approximate power of combinations. The differences between exact power and approx- Chow and Wang [6] seems to overestimate the exact power imate powers are illustrated in Figures 1–4 for wide ranges of when both n and are very small. The power curve of Chow standard deviations and true mean differences. and Wang has a peak because their calculation used differ- As deﬁned in Section 2, 2 is the variance of log-transformed ent formulas for power when is zero and nonzero. The dif- data from the reference product and thep coefﬁcient of variation ference between the approximate power of [6] and the exact (CV) in the untransformed data is CV D e 2 1. power decreases as the total sample size increases for the

Figure 1. Approximate power of Chow and Wang versus exact power numerically calculated from Equation (6) against true mean difference when D 0.2, at different total sample sizes, n.

Figure 2. Approximate power of Chow and Wang versus exact power numerically calculated from Equation (6) against true mean difference when D 0.3, at different

98 total sample sizes, n.

Figure 3. Approximate power of Chow and Wang versus exact power numerically calculated from Equation (6) against true mean difference when D 0.4, at different total sample sizes, n.

Figure 4. Approximate power of Chow and Wang versus exact power numerically calculated from Equation (6) against true mean difference when D 0.7, at different total sample sizes, n. same and n, and also decreases as the true difference () the true power slightly, which results in having a few more sub- increases. When increases from 0.2 to 0.7, there are more cases jects than necessary. when the approximate power of Chow and Wang exceeds the Previous authors [3,4,6] often focused on CV<0.3. However, exact power. errors of approximation may also be important for the large Thus, sample sizes are underestimated using the approximate values of highly variable drugs. power of Chow and Wang for the combination of small n and From Figure 1, it is seen that the approximate power of Chow small . When studies are underpowered, they may fail to meet and Wang [6] is very close to the exact power for n >18 and the study objectives. For a combination of large and small n, *>0.04 when D 0.2. From Figures 2–4, it is seen that the

the approximate power of Chow and Wang can underestimate approximate power of Chow and Wang overestimates the exact 99

Table I. Comparison of sample size from exact power given by Equation (6) and approximate power of Chow and Wang for achieving 80% power at different combinations of and . D 0.01 D 0.02 D 0.03 D 0.04 Exact Chow and Wang Exact Chow and Wang Exact Chow and Wang Exact Chow and Wang 0.166666666 0.21614161418161818 0.33428343036323836 0.45846605062566662 0.5 90 70 92 78 94 86 100 94 0.6 128 100 130 110 136 122 144 136 0.7 172 136 176 150 184 164 194 184

Table II. Comparison of the exact power given by Equation (6) and power values from Figure 1(c) in Diletti and colleagues’ paper when CV D 20% and n D 24. D T R Method 0 log(1.05)D0.04879 log(1.1)D0.09531 Diletti and colleagues’ power 0.97 0.9 0.7 Exact power, P1 . , n, / 0.9679 0.902 0.696

power for many more combinations of and n when > 0.3 than when D 0.2. For example, for n D 50, D 0.02 and Table III. Comparison of the exact power given D 0.4, the approximate power of Chow and Wang is 80.35%, by Equation (6) and power values from Figure 3 [3] while the exact power is 71.63%. when CV D 20% and n D 24. D T R 4.1.3. Comparison of exact sample size with approximation of Chow Method 0.015 0.05 0.1 and Wang. In Table I, we compare the total sample sizes from the exact power as in Equation (6) and the approximate power of Phillips’ power 0.9 0.8 0.5 Chow and Wang [6] for achieving 80% power. The total sample Exact power, P1 . , n, / 0.909 0.809 0.517 size is rounded up to the next even number. Table I shows that the difference in the total sample sizes increases as increases, and decreases as increases, for each given . For example, compared the exact power values by Equation (6) and the the total sample size from the Chow–Wang approximate power power values from Figure 3 [3] for these three sets of param- is about 10% less than that from exact power when D 0.03 eters and list these in Table III. In order to compare the exact and D 0.2, about 15% less when D 0.02 and D 0.4, and power with Phillips’ values, the exact values were computed using about 20% less than that from exact power when D 0.01 and 2 D1 D 0.20. D 0.3. From Table III, Phillips’ power is quite close to the exact power. However, the graphs and tables present a limited number of cases 4.2. Comparison of the exact power and Diletti and as Phillips [3] did. Exact and explicit power formulas, such as P1 in colleagues’ power Equations (5) and (6), allow any situation to be considered.

Because Diletti et al. [4] did not provide the explicit mathematical formula for their power calculation, we read three power values 5. DISCUSSION AND CONCLUSIONS from Figure 1(c) of [4]. The exact power values versus those read from Figure 1(c) of [4] for these parameters are listed in Table II. The exact power has been derived from the joint density func- From Table II, the power in [4] is very close to the exact value. tion of the two correlated test statistics (T1 and T2)usedin However, the graphs and tables in [4] present a limited num- the two one-sided tests procedure. The exact power numerically ber of cases. For instance, [4] did not provide any power value integrated from Equation (5) is corroborated by the results of the for CV>30%, as would be the case with highly variable drugs. Monte Carlo simulations and is readily available using numerical Exact and explicit power functions will allow such situations to integration, and R code for this is provided in Appendix A. Exact be considered. sample size calculation is then easy using the exact power for any parameter combinations in bioequivalence studies based on two 4.3. Comparison of the exact power and PhillipsŠ power one-sided tests. The Chow–Wang values are often used in standard software packages, for example, nQuery [14]. The fact that Because Phillips [3] did not provide the explicit mathematical the approximate power of Chow and Wang [6] markedly overes- formula for his power calculation, we also read power values timates the exact power for many combinations of and n as 100 for three sets of parameter values from Figure 3 of [3]. We then shown in Figures 1–4 and Table I demonstrates the preferability

of the exact power function P1 . , n, /in planning of bioequiv- [12] Blume H, Midha KK. Report of Consensus Meeting: Bio-international alence studies, especially when is small relative to .Mean ’92. Conference on Bioavailability, Bioequivalence and Pharmacoki- differences in log(AUC)orlog(C ) as small as 0.03 are of practical netic Studies. Bad Homburg, Germany, 20–22 May, 1992. Eur. J. max Pharm. Sci. 1993; 1: 165–171. interest based on the review by Davit et al. [1]of12yearsofbioe- [13] Blume H, McGilveray I, Midha K. Report of Consensus Meeting: quivalence studies from the FDA, which found that more than Bio-international ’94, Conf. on Bioavailability, Bioequivalence and 50% of studies have mean differences of less than 0.05 between Pharmacokinetic Studies. Munich, 14–18 June, 1994. Eur. J. Pharm. generic product and innovator (these are mean differences in Sci. 1995; 3: 113–124. [14] Elashoff JD. nQuery advisor, version 5.0 user’s guide. Statistical Solu- log(AUC)orlog(Cmax), when the corresponding mean differences tions Ltd: Los Angeles, 2002. in AUC and Cmax between generic and innovator products are respectively 3.56% and 4.35%). While the exact power agrees closely with [3,4] in the few cases displayed in those papers, it is important to have exact powers APPENDIX A: R CODE FOR EXACT POWER AND and sample sizes in all bioequivalence study settings, including SAMPLE SIZE those for highly variable drugs with CV > 30%. The power P1 . , n1, n2, T , R/ in (5) is calculated by the following lines of R code. Here, the arguments n1 and n2 respectively Acknowledgements represent the sample sizes n1, n2of sequences 1 and 2; sigT , sigR represent T and R; th1 and th2 respectively represent 1 and This research forms part of the PhD dissertation of the first author 2; and tstar represents .Infact,.n1, n2, T , R/ enters formula in the Statistics Program at the University of Maryland, under (5) only through the quantities 2 2 2 1 n 1 n 4 O D R C T . = 1 C = 2/= the joint guidance of the other authors. The first author is grate- D and D n1 Cn2 2, and arguments sigD and nu represent DO and ful for numerous discussions with Dr. Yi Tsong and thanks Dr. Yi . Pow1$value is the power in formula (5) (with Pow1$abs.error Tsong and Dr. Stella Machado for their constant encouragement the estimated absolute error of integration), and SSiz denotes and support during her graduate studies while she has worked the sample size obtained by equating formula (6) to 1-beta. as a full time statistician at the FDA. Finally, the authors thank the We use two standard R functions: integrate to perform univari- anonymous reviewers for their comments. ate numerical integration, and uniroot to perform root-finding or inversion. REFERENCES Pow1 = function(sigD, nu, th2,th1,tstar, alpha=.05){ [1] Davit B, Nwakama P, Buehler G, Conner D, Haidar S, Patel D, Yang Y, Integrand = function(x) { Lawrence Y, Woodcock J. Comparing generic and innovator drugs: (pnorm((th2-tstar)/sigD-qt(1-alpha, nu)*sqrt(x/nu))- a review of 12 years of bioequivalence data from the United States Food and Drug Administration. Annals of Pharmacotherapy 2009; pnorm((th1-tstar)/sigD+qt(1-alpha, 43:1583–1597. nu)*sqrt(x/nu)))*dchisq(x,nu) [2] Schuirmann D. A comparison of the two one-sided tests procedure } and the power approach for assessing the equivalence of average integrate(Integrand, lower = 0, upper = bioavailability. Journal of Pharmacokinetics and Biopharmaceutics nu*((th2-th1)/(2*qt(1-alpha, nu)*sigD))^2)[1:2] 1987; 15(6):657–680. [3] Phillips KE. Power of the two one-sided tests procedure in bioe- } quivalence. Journal of Pharmacokinetics and Biopharmaceutics 1990; SSiz = function(beta, sigT,sigR,th2,th1,tstar, alpha=.05, 18:137–143. uppern=300){ [4] Diletti E, Hauschke D, Steinijans VW. Sample size determination for Pow2 = function(n) { bioequivalence assessment by means of confidence intervals. Inter- national Journal of Clinical Pharmacology, Therapy, and Toxicology Pow1(sqrt((sigT^2+sigR^2)/n), n-2, th2, th1, tstar, 1991; 29:1–8. alpha)$value-1+beta } [5] Kieser M, Hauschke D. Statistical methods for demonstrating equiv- uniroot(Pow2,c(3,uppern))$root alence in crossover trials based on the ratio of two location param- } eters. Drug Information Journal 2000; 34:563–568. [6] Chow SC, Wang H. On sample size calculation in bioequivalence trials. Journal of Pharmacokinetics and Pharmacodynamics 2001; 28:155–169. APPENDIX B: PROOF OF MONOTONICITY OF [7] Owen DB. A special case of a bivariate non-central t-distribution. p Biometrika 1965; 52:437–446. =t˛ ./ [8] U.S. Food and Drug Administration Center for Drug Evaluation and . ıp p Research. Guidance for Industry: Statistical Approaches to Estab- Define t Z Y ,whereZ is the standard lishing Bioequivalence, January 2001. Available at: http://www. ./ D ./ fda.gov/downloads/Drugs/Guidances/ucm070244.pdf (accessed normal random variable and Y ./ is the chi-squaredP random 25.11.2014). 2 variable with degrees of freedom. Because Y./ D iD1 Zi , [9] Liu F, Li Q. Exact Sequential Test of Equivalence Hypothesis Based on where Z , Z , :::, Z are independent and identical standard nor- Bivariate Non-central T-statistics. Computational Statistics and Data 1 2 Analysis 2014; 77:14–24, Available at: http://dx.doi.org/10.1016/j. mal random variables, then obviously Y. C 1/ is stochastically > csda.2014.02.007. larger than Y./, which means for any real value x, P.Y.ıpC 1/ [10] Casella G, Berger RL. Statistical inference (2nd edn). The Wadsworth x P Y > x . From this, it follows that t 1 1 /> . ./ / ıp . C / C Group, 2002. is stochastically smaller than t ./ , from which it follows [11] R Core Team. R: A Language and Environment for Statistical Com- ıp ı p p puting. R Foundation for Statistical Computing. Vienna, Austria, that t˛ . C 1/ C 1 < t˛ ./ . Therefore, =t˛ ./ is a 2013. Available at: http://www.R-project.org/ (accessed 01.10.2014). monotonic increasing function of . 101