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.
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
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
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.
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
where
with ∑
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
Now we can assess average bioequivalence by Dunnett’s (1–2
where
One can claim the bioequivalence among drugs if the calculated (1–
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
Without loss of generality, we assume that in the
One can express the model (
here
Consider the linear transformation
the conditional distribution of
Under model (
Let
where
and
Where ⊗ is the Kronecker product,
Under model (
The estimator of the covariance matrix of
where
By using
has a
From this we can calculate Dunnett’s test (1–2
Table 6 shows Ondansetron example data given by Lim
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
The 90% confidence intervals
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.
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
3 × 3 crossover design
Sequence | Period | ||
---|---|---|---|
1 | 2 | 3 | |
1 | R | T2 | T1 |
2 | T2 | T1 | R |
3 | T1 | R | T2 |
2 × 3 crossover design
Sequence | Period | ||
---|---|---|---|
1 | 2 | 3 | |
1 | R | T2 | T1 |
2 | T1 | R | T2 |
ANOVA table for 2 × 3 crossover design
Sources | Degrees of freedom | Expected mean squares |
---|---|---|
Between | 2 | |
Sequence | 1 | |
Residual | 2 | |
Within | ||
Period | 2 | |
Drug | 2 | |
Residual | 4 | |
Total | 6 |
Coefficients for estimates of pairwise formulation effects in 2 × 3 crossover design
Sequence | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Period | ∑ | Period | ∑ | Period | ∑ | |||||||
1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | ||||
1 | −1 | 2 | −1 | 6/9 | −2 | 1 | 1 | 6/9 | −1 | −1 | 2 | 6/9 |
2 | 1 | −2 | 1 | 6/9 | 2 | −1 | −1 | 6/9 | 1 | 1 | −2 | 6/9 |
Variance |
Coefficients are multiplied by 3; Variance when
2 × 3 Crossover design with dropouts
Sequence | Period | ||
---|---|---|---|
1 | 2 | 3 | |
1 | R | T1 | T2 |
( | ( | ||
2 | T2 | R | T1 |
( | ( |
AUC value for Ondansetron example adopted from Lim
Sequence Period | Sequence | Period 1 | Period 2 | Period 3 |
---|---|---|---|---|
1 | 1 | 12176 | 11424 | 14319 |
2 | 10913 | 12114 | 11640 | |
3 | 11004 | 11802 | 11234 | |
4 | 14377 | 15322 | 14700 | |
5 | 15110 | 18308 | 18598 | |
6 | 21644 | 23917 | (24176) | |
7 | 11367 | (10524) | (13224) | |
8 | 2 | 12153 | 9771 | 12794 |
9 | 14121 | 12292 | 18396 | |
10 | 6339 | 7860 | 7907 | |
11 | 20062 | 17667 | 23253 | |
12 | 12306 | 17170 | (15114) | |
13 | 19123 | (15472) | (17058) | |
14 | 20043 | (15816) | (19540) |
( ): missing data