Design and analysis of two treatments in five periods cross-over designs

Abstract

A crossover design is a repeated measurements design such that each experimental unit receives different treatments during the different time periods. A cross-over design with 𝑡 treatments, 𝑝 periods, and 𝑠 sequences is denoted by C (𝑡,𝑝,𝑠). In a majority of bioequivalence studies, design and analysis of lower order cross-over designs are normally associated with erroneous results. Higher order crossover designs are desirable in the analysis of crossover designs to eliminate carryover effects. The purpose of the study was to design and analyze two treatments in five periods crossover designs. The specific objectives of the study were to: Estimate treatments and residual effects of the designs; evaluate the design’s optimality criteria; evaluate the design’s robustness for missing data; and compare the Bayesian and the 𝑡- test analysis methods on treatments and carryover effects. The treatments and residual estimates were obtained using the Best Linear Unbiased Estimation (BLUE) method while the optimality criteria of the designs were determined by the variances of the treatments and carry-over effects, where the designs with minimum variance were considered to be optimum. In addition, the covariance of the two effects was used to evaluate the optimality of designs which estimate treatment effects in the presence of carry-over effects. Break down numbers were used to rank the designs according to their robustness against missing data. In the Bayesian method of analysis, the posterior quantities were obtained for the mean intervals of treatments and carry-over effects and the highest posterior density (HPD) graphs were plotted and interpreted using conditional probability statements. For validation purposes, the 𝑡-tests were performed and their results were compared with the Bayesian results. The C(2,5,2) in this study comprised of fifteen designs (𝐷1−𝐷15) while the C(2,5,4) comprised of twelve designs(𝐷16−𝐷27) . The findings of the study indicated that a majority of the designs considered gave estimates for treatments and carry-over effects . Additionally, two designs were optimal in estimating treatment effects for C (2×5×2) cross-over designs. Moreover, one design was found to be optimal and robust for missing data for C (2×5×4), and it was hence used in the analysis of a hypothetical example. From the Bayesian analysis, the probability of significant treatment difference in the presence of carryover effects was 1, while from the 𝑡-test, the calculated 𝑡−value of 11.73 was greater than the two sided tabulated value at 5% level of significance. The two analysis methods implied significant differences in the treatment effects. Finally, the mean subject profiles for a majority of periods and their respective sequences implied a direct treatment effect in favor of treatment B. In conclusion, it was established that variance-balance plays a major role in determining a suitable design. This is due to the fact that the optimal and robust for missing data in the study was more variance-balanced as compared to the other designs whose optimality and robustness for missing data were relatively lower. The study recommends that the optimal and robust for missing data design in this study be applied in bioequivalence experiments in assessment of efficacy of new treatments against standard ones. For further research the BLUE method should be used in estimation of effects for designs with more than two treatments.