Circular First- and Second-Order Balanced Repeated Measurements Designs

Abstract

Sharma (1977) and Aggarwal et al. (2006) considered non circular construction of first- and second-order balanced repeated measurements designs. Sharma et al. (2002) constructed circular first- and second-order balanced repeated measurements designs only for a class with parameters (v, p = 3n, n = v ²) and also showed its universal optimality. In this article, we consider circular construction of first- and second-order balanced repeated measurements designs and strongly balanced repeated measurements designs by using the method of cyclic shifts. Some new circular designs with parameters (v, p, n) for cases p = v, p < v and p > v are given.

Keywords:

• A-optimal
• Carry-over effects
• Circular balanced designs
• Cyclic shifts
• Second-order designs
• Strongly balanced designs

Mathematics Subject Classification:

• 05B05
• 62K05
• 62K10

Acknowledgments

The authors wish to thank the referee for valuable comments which led to the improvement in the first version of this article. We are grateful to Dr. Seema Jaggi, Indian Agricultural Statistics Research Institute India, for providing the Sharma et al. (2002) article.

Notes

Note: We can construct smaller designs which are not uniform on periods but which are still balanced for first- and second-order residual treatment effects by taking the smaller set of blocks which is (1/v)th of each of full set of blocks given in Table 1. For example, first- and second-order CUBRMD for v = p = 5 can by constructed by using the following sets of shifts , which is given below.

Note: In shift [2, 4]t, t means add (v − 1) treatments in the last row.