Mathematical Analysis of a Five Periods Crossover Design for Two Treatments
Authors: Nyakundi Cornelious Omwando , Koske Joseph Kipsigei , Mutiso John Muindi , Tum Isaac Kipkosgei 
Published: June 2021

Introduction: A cross-over design is a repeated measurements design such that each experimental unit receives different treatments during different time periods. Lower order cross-over designs such as the two treatments, two periods and two sequences C (2, 2, 2) design have been discovered to be inefficient and erroneous in their analysis of treatments efficacy. In this regard, higher order cross-over designs have been recommended and developed like: the two treatments, three periods and four sequence C (2, 3, 4) design; and the two treatments, four periods and four sequence C (2, 4, 4) designs. However, there still exists more efficient higher order cross-over designs for two treatments which can be used in bioequivalence experiments. This study gives a new design and analysis for two treatments, five periods and four sequence C (2, 5, 4) cross-over design that gives more precise estimates and provides estimates for intra subject variability.
Method: A hypothetical case study was considered on 160 experimental units which are assumed to be randomly selected from a given population. A cross over design of two treatments (A, B) in five periods whose sequences are given by BABAA, ABABB, BAABA and ABBAB were used. Each of the experimental units was used as its own control. The estimates for both direct treatments and treatments carry-over effects were obtained using best linear unbiased estimation method (BLUE). We simulated data for two treatments in five periods and four sequences and used it to test the null hypotheses of no significant differences in both the direct treatments and treatments carry-over effects using the  test. The subject profiles plots were used to determine the general trend so as to enable an experimenter make a decision on which of the two treatments under consideration was more efficacious.
Results: In testing the null hypothesis of no significant difference in carry-over effects for the two treatments (A&B), the calculated value was found to be 0.55 which was less than the tabular value at 156 degrees of freedom at 95 % confidence level, hence the null hypothesis was not rejected. Similarly, In testing the null hypothesis of no significant difference in treatment effects for A&B, the calculated value was found to be 11.73 which was higher than the tabular value at 156 degrees of freedom at 95% confidence level hence the null hypothesis was rejected, and it was concluded that there was indeed a significant difference in the treatment effects. The mean subject profiles plots for a majority of periods and their respective sequences indicated that the general trend implied that treatment B was more effective as compared to treatment A.
Conclusion: In cross-over designs, the presence carry-over effects affect the precision of treatments effects estimates in an experiment. Apart from increasing the washout periods, increasing the number of periods in cross-over designs can help in eliminating the carry-over effects. The C (2, 5, 4) design in this study gives more precise estimates and can provide estimates for intra subject variability. The simulated data indicated that there was significant difference in the treatment effects, and in comparison of the two treatments, treatment B was more effective as compared to treatment A.