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Bioequivalence trial with two generic drugs in 2 × 3 crossover design with missing data
Communications for Statistical Applications and Methods 2020;27:641-647
Published online November 30, 2020
© 2020 Korean Statistical Society.

Sang-Gue Park1,a, Seunghyo Kimb, Ikjoon Choic

aDepartment of Applied Statistics, Chung-Ang University, Korea;
bDepartment of Statistics, Graduate School of Chung-Ang University, Korea;
cDepartment of Pharmaceutical Industrial Management, Graduate School of Chung-Ang University, Korea
Correspondence to: 1Department of Applied Statistics, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul 06974, Korea. E-mail: spark@cau.ac.kr
Received July 27, 2020; Revised September 1, 2020; Accepted September 4, 2020.

The 2 × 3 crossover design, a modified version of the 3 × 3 crossover design, is considered to compare the bioavailability of two generic candidates with a reference drug. The 2 × 3 crossover design is more economically favorable due to decrease in the number of sequences, rather than conducting a 3×3 crossover trial or two separate 2 × 2 crossover trials. However, when using a higher-order crossover trial, the risk of drop-outs and withdrawals of subjects increases, so the suitable statistical inferences for missing data is needed. The bioequivalence model of a of 2×3 crossover trial with missing data is defined and the statistical procedures of assessing bioequivalence is proposed. An illustrated example of the 2 × 3 trial with missing data is also presented with discussion.

Keywords : average bioequivalence, 3 × 3 crossover design, 3 × 2 crossover design, missing at random, missing data
1. Introduction

A statistical procedure based on the 3 × 3 crossover design for assessing average bioequivalence of an original drug and two generic ones at once is proposed. This 3 × 3 bioequivalence trial has an advantage over conducting two separate 2×2 trials in terms of time and cost. The Korea Food and Drug Administration (KFDA, currently MFDS; Ministry of Food and Drug Safety) officially announced the partial acceptance for the regulation of the bioequivalence results from the 3×3 crossover design under the conditions that generic drugs contain the same active ingredients and are produced by the same company. Lee et al. (1998) proposed a model for assessing bioequivalence using a 3 × 3 crossover design which is a simple extension of the standard 2 × 2 design and Oh et al. (1999) discussed Lee’s work from a statistical point of view.

We believe that the 3 × 3 crossover design can be further modified to a 2 × 3 crossover design. The 2 × 3 crossover comprises two sequences with three periods, rather than three sequences in the case of the 3 × 3 crossover design. Such a deletion of a sequence would allow for a precise estimate of treatment effect while utilizing the same or less number of subjects; therefore, more ethically and economically favorable. Recently, Lim et al. (2005) proposed a statistical method of assessing average bioequivalence in 3 × 3 crossover design when some data are missing. Lim’s method is a modified version of Chow and Shao (1997) in 3 × 3 crossover design. Inferred from these statistical methods, the 2 × 3 crossover design could prove to be more efficient and is worthy to explore as a statistical method for assessing average bioequivalence.

In Section 2, we illustrate the statistical model for the 2 × 3 crossover design and develop simultaneous confidence intervals for drug effects. Next, we discuss the statistical method of constructing simultaneous confidence intervals for drug effects in Section 3 when dropouts occur in the later periods. In Section 4, we provide an example of this method and its results. Lastly in Section 5, concluding remarks are reported.

2. 2 × 3 crossover design

The standard form of 3 × 3 crossover trial is given in Table 1, where R, T1, and T2 stand for the reference drug and two generic drugs.

One weakness of the 3×3 crossover design are its difficulty to manage in trial due to the increased number of sequences and periods compared to the 2×2 crossover trial. One may want to reduce some sequence or period to conduct a trial more efficiently like Table 2. It may be more advantageous to reduce the sequence that can adjust the sample size. Let us say the second row is deleted.

The statistical model for the 2 × 3 crossover design can be written as

yijk=μ+Gk+Pj+F(j,k)+Sik+ɛijk,         (k=1,2;j=1,2,3;i=1,2,,nk),

where μ is the overall mean, Gk is the fixed effect of the kth sequence with ∑Gk = 0, Pj is the fixed effect of the jth period with ∑Pj = 0, F(j, k) is the fixed effect of the formulation in the kth sequence administered in the jth period such that


with ∑F(j, k) = 0, and the subject variability SikiidN(0,σs2), and the drug variability ɛijkiidN(0,σe2) and they are also assumed to be independent each other.

From the usual ANOVA construction given by Chow and Liu (2008) and Park (2014) assuming the equal sample sizes, we can obtain Table 3.

Table 4 provides the coefficients for estimates of pairwise formulation effects in order to draw a statistical inference on the drug effects.

Denoting θ1R = FT1FR, θ2R = FT2FR, and θ21 = FT2FT1, the unbiased estimators of these parameters can be obtained through:


Now we can assess average bioequivalence by Dunnett’s (1–2α)×100% simultaneous confidence intervals of θ1R = FT1FR and θ2R = FT2FR


where d(α, a, b) is the critical values for Dunnett’s test (a is number of treatment means and b is the degrees of freedom) and MSIntra is obtained from Table 3.

One can claim the bioequivalence among drugs if the calculated (1–α)×100% simultaneous confidence intervals based on log transformed scale belong to the pre-assigned limit (log(0.8), log(1.25)).

3. 2 × 3 crossover design with missing data

The 2 × 3 crossover design consists of three periods of testing periods with washouts between the periods. Responses are often not obtained properly for various reasons, such as protocol violations, failure of assay methods, or missed visits. The unobserved responses are considered as dropouts. In this design, the subjects are likely to drop out in the second or the third period. Dropouts in the second period would result in missing data in both the second and third period; whereas dropouts in the third period would result in missing data only in the third period.

When we have some dropouts, we might try Grizzle (1965)’s idea after deleting subjects with dropouts. But it may cause some significant loss of information in statistical inference, if subjects with missing data are deleted and statistical analysis is performed. Chow and Shao (1997) proposed a general statistical method to compare a generic drug to the reference one in a two sequence, three period crossover design with unbalanced or incomplete data. Lim et al. (2005) extended their method in a 3 × 3 crossover design and this paper also modifies Lim et al.’s way in a 2 × 3 crossover design.

Without loss of generality, we assume that in the kth sequence, the first mk2 subjects have the data for all three periods; the next mk1mk2 subjects have data for period 1 and period 2 like Table 5. The sample sizes mkl(l = 1, 2) may be random and assume whether or not Yi jk is independent of the measurement error εi jk (Chow and Shao, 1997).

One can express the model (2.1) as


here X is a design matrix, β′ = (μ, G1, P1, P2, FR, FT1) is a fixed parameter vector, S′ = (S11, . . . , S n11; S12, . . . , S n22) is a vector of random subject effects, Z is the corresponding subject effect design matrix, and ε is a error vector. S and ε are assumed to be mutually independent with N(0, σ2sI) and N(0, σ2eI), respectively.

Consider the linear transformation HY with HZ, where H is some suitably defined matrix. Since


the conditional distribution of HY, given mkl(l = 1, 2), is still normal if ε is normal and independent of mkls. This kind of transformation was used by Fuller and Batesse (1973).

Under model (3.2), we obtain the unbiased estimator P1, P2, FR, FT by considering a special transformation H which is equivalent to taking within-subject differences (between two periods). The within-subject differences di1k(1 ≤ imk1) and di2k(1 ≤ imk2) are defined as follows


Let d be the vector of these differences and then d is independent of S and d = HY for some H satisfying HZ = 0. Assuming that mkl’s are independent of ε, we obtain






Where ⊗ is the Kronecker product, 1v is the v-vector of ones, Iv is the identity matrix of order v, and 0 is the matrix of 0s of an appropriate order.

Under model (3.3), the maximum likelihood estimator of θ̂,


The estimator of the covariance matrix of θ̂ can be obtained by the least square methods:




By using equations (3.4), (3.5), and (3.6) we can then construct an exact confidence interval for lθ with a fixed vector l because


has a t-distribution with m11 + m21 + m12 + m22 − 4 degrees of freedom.

From this we can calculate Dunnett’s test (1–2α)×100% confidence intervals for FRFT1, FRFT2 for two drug formulations with the reference one. The bioequivalence among drugs if the calculated 90% simultaneous confidence intervals based on log transformed scale belong to pre-assigned limit (log 0.8, log 1.25) .

4. An illustrative example

Table 6 shows Ondansetron example data given by Lim et al. (2005), but with one sequence deleted. Ondansetron is used to prevent nausea and vomiting caused by cancer chemotherapy, radiation therapy, anesthesia, and surgery. Ondansetron comes as the brand name drug Zofran and a company developed generic drugs Vominon 4 mg and Vomonon 8 mg for Zofran. The 21 healthy Korean male subjects selected from the well defined protocol were randomized and received each formulation. The plasma concentrations of Ondansetron were monitored by the high performance liquid chromatography for over a 12 hour period after administration. In this experiment, all planed-for data are actually observed and both test drugs were found to be bioequivalent to Zofran. We will use first and second sequences data. However, pretend that some of the subjects of period 3 in each sequence are dropped out for the purpose of demonstrating the results.

According to MFDS’s standard on pharmaceutical equivalence test (2018), bioequivalence analysis should be based on the log-transformed data rather than original one. When the dropouts occur, the common way is to exclude the corresponding subjects’ data and analyze the bioequivalence study with the rest of the data. In this case the deletion of subjects with dropouts leads m11 = 6, m12 = 5, m21 = 5, and m22 = 4. The study loses significant amount of information due to the dropouts and it may cause to declare bioinequivalence.

The 90% confidence intervals FRFT1, FRFT2 based on the proposed method are


The confidence intervals are compared to preassigned limit (log 0.8 = −0.22314, log 1.25 = 0.22314) and the tested two drugs are claimed to be bioequivalent with the reference drug since both intervals given by the proposed method are within this limit. It is noted that the lengths of intervals based on the proposed method are shorter than ones based on the complete case. Consequently, there is a minor loss of information due to missing data when the proposed method is applied.

5. Conclusion

When determining the bioequivalence of multiple test drugs and a reference drug, performing several separate 2 × 2 crossover trials proves less efficient than performing one higher-order crossover trial (Lim et al., 2005). We proposed the statistical inference of 2 × 3 crossover design with missing data to assess the bioequivalence between two generic drugs with the reference drug. The 2 × 3 crossover design may grant pharmaceutical companies some economical advantages such as benefits from a decreased trial duration and a financial expenditure from possible decreased number of subjects required to demonstrate bioequivalence. This study may be more meaningful and practical as it includes the statistical procedure and an illustrated example to assess the bioequivalence, especially when missing data occur during the trial. However, it may be needed to show the efficiencies of the 2 × 3 design and the method of handling missing data under more general circumstances like the well-designed simulation studies in further studies.

This research was supported by the Chung-Ang University Research Scholarship Grants(A) of the second author in 2018.

Table 1

3 × 3 crossover design



Table 2

2 × 3 crossover design



Table 3

ANOVA table for 2 × 3 crossover design

SourcesDegrees of freedomExpected mean squares
Between SS2n − 1
 Sequence SS1σe2 + σs2 + 3n (G12 + G22)
 Residual SS (Inter)2n − 2σe2 + 3σs2

Within SS
 Period SS2σe2+n2{(P1-P2)2+(P1-P3)2+(P2-P3)2}+n12{(FR-FT1)2+(FR-FT2)2+(FT2-FT1)2}-n{P1FT2+P2FT1+P3FR}
 Drug SS2σe2+n4{(FR-FT1)2+(FT1-FT2)2+(FR-FT2)2}
 Residual SS (Intra)4n − 4σe2

Total SS6n − 1

Table 4

Coefficients for estimates of pairwise formulation effects in 2 × 3 crossover design

Sequenceθ1R = FT1FRθ2R = FT2FRθ21 = FT2FT1

Periodc jk2Periodc jk2Periodc jk2



Coefficients are multiplied by 3; Variance when n = nk for k = 1, 2.

Table 5

2 × 3 Crossover design with dropouts


Y111, Y211, . . . Yn111Y121, Y221, . . . , Ym1121Y131, Y231, . . . , Ym1231
(n1m11) missing(n1m12) missing

Y112, Y212, . . . Yn212Y122, Y222, . . . , Ym2132Y132, Y232, . . . , Ym2232
(n2m21) missing(n2m22) missing

Table 6

AUC value for Ondansetron example adopted from Lim et al. (2005)

Sequence PeriodSequencePeriod 1Period 2Period 3


( ): missing data

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