Abstract
Under the three-treatment three-period crossover design with simple carry-over effects, we derive the least-squares estimators for period effects, treatment effects and carry-over effects, as well as their covariance matrix in closed and explicit expressions. Using Monte Carlo simulation, we compare the test procedure adjusting carry-over with that ignoring carry-over with respect to Type I error and power. We further compare interval estimators adjusting carry-over with those ignoring carry-over with respect to the coverage probability and the average length. When the variation of responses within patients is small, the test procedure and interval estimators ignoring carry-over can lose accuracy in the presence of carry-over effects. When the variation of responses within patients is large, this loss of accuracy may become small or even minimal. We note that the loss of efficiency due to the adjustment of carry-over under the simple carry-over three-period crossover design is moderate, and is much less than that found for a two-period crossover design. We use the double-blind three-period crossover trial comparing formoterol solution aerosol and salbutamol suspension aerosol with a placebo for patients suffering from exercise-induced asthma on the forced expiratory volume in one second (FEV1) to illustrate the use of test procedures and interval estimators discussed here.
Acknowledgements
The author wishes to thank the two reviewers for many valuable comments and helpful suggestions to improve the clarity and contents of this paper.
A Appendix
On the basis of (2), we obtain the least-squares estimators of parameters under model (1) with assuming no carry-over effects as
and
The covariance matrix of these least-squares estimators (13)-(16) is given by
where
The inverse matrix
Using (19), we can estimate
where
B Appendix
On the basis of (6), we obtain the least-squares estimators
and
The covariance matrix under the simple carry-over model (1) is then given by
Under model (1) with carry-over effects, we can estimate
where
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