Bioequivalence and Bioavailability Forum

Hi kimhuang,

❝ 1. For 4*4 William's design, if the subject only completes one period, should it be excluded from BE evaluations?

Yes – not informative in a fixed effect model.

❝ How many periods must each subject complete to include the data for BE evaluating?

Two relevant ones. ICH M13A recommends – without naming it as such – the ‘Two-at-a-Time’ approach in Section 2.2.3.1:

In studies with more than two treatment arms, e.g., a four-period study […], the analysis for each comparison should be conducted excluding the data from the treatment arms that are not relevant for the comparison in question.

In a 4-treatment Williams’ design you obtain Incomplete Block Designs (IBDs) for the pairwise comparisons.
As an example one of the six possible Williams’ designs (with the treatments \(\small{\text{A, B, C, D}}\)) gives for the comparisons \(\small{\text{A}}\,vs.\text{B}\) and \(\small{\text{C}}\,vs.\text{D}\) these IBDs (\(\small{\color{Red}\bullet}\) de­notes excluded data):

\(\small{\begin{array}{c|cccc}
s/p & \text{I} & \text{II} & \text{III} & \text{IV}\\\hline
1 & \text{A} & \text{C} & \text{B} & \text{D}\\
2 & \text{B} & \text{A} & \text{D} & \text{C}\\
3 & \text{C} & \text{D} & \text{A} & \text{B}\\
4 & \text{D} & \text{B} & \text{C} & \text{A}\\
\end{array}}{\color{Blue}\mapsto}
\begin{array}{c|cccc}
s/p & \text{I} & \text{II} & \text{III} & \text{IV}\\\hline
1 & \text{A} & {\color{Red}\bullet} & \text{B} & {\color{Red}\bullet}\\
2 & \text{B} & \text{A} & {\color{Red}\bullet} & {\color{Red}\bullet}\\
3 & {\color{Red}\bullet} & {\color{Red}\bullet} & \text{A} & \text{B}\\
4 & {\color{Red}\bullet} & \text{B} & {\color{Red}\bullet} & \text{A}\\
\end{array}{\color{Blue}\wedge}\small{\begin{array}{c|cccc}
s/p & \text{I} & \text{II} & \text{III} & \text{IV}\\\hline
1 & {\color{Red}\bullet} & \text{C} & {\color{Red}\bullet} & \text{D}\\
2 & {\color{Red}\bullet} & {\color{Red}\bullet} & \text{D} & \text{C}\\
3 & \text{C} & \text{D} & {\color{Red}\bullet} & {\color{Red}\bullet}\\
4 & \text{D} & {\color{Red}\bullet} & \text{C} & {\color{Red}\bullet}\\
\end{array}}\)

In each of the two resulting IBDs you exclude subjects with incomplete data (only one treatment).
Say, one subject in sequence \(\small{\text{B A D C}}\) dropped out after the second period. Fine for \(\small{\text{A}}\,vs.\text{B}\), but the subject has to be excluded for \(\small{\text{C}}\,vs.\text{D}\) (no relevant data). Another subject in sequence \(\small{\text{D B C A}}\) dropped out after the third period. Keep for \(\small{\text{C}}\,vs.\text{D}\), but the subject has to be excluded for \(\small{\text{A}}\,vs.\text{B}\) (only data of \(\small{\text{B}}\)).

❝ 2. For PK parameter analysis, can at least one period data to be included in PK parameter analysis?

Not sure what you mean. Can you reword / elaborate?
In a mixed effects model you could include the data but the result will be very similar (often identical) to a fixed effects model of complete data only (ANOVA).