Optimal Circular Balanced Repeated Measurements Designs for v = p through the Method of Cyclic Shift (MCS) Rule II

Abstract

Background. One of the common characteristics of preclinical genetic experimentation is the result of repeated measurements, and for this purpose, repeated measurements designs (RMDs) have gained much more significance. In the class of RMDs, balanced repeated measurements designs (BRMDs) are preferred as they balance out the residual effects since the experimentation is repeated over time on different subjects. This study provides the theoretical framework of universal optimal criteria proposed by Kiefer (1975) for the newly proposed circular balanced repeated measurements designs (CBRMDs) by Rajab et al. (2018). These universal optimality criteria were proved for the special class of designs where the number of treatments is equal to the number of periods. Universal optimality has been discussed considering all the possible effects in the models, i.e., units, subjects, treatments, and periods. Methodology. This study characterized CBRMDs, where several treatments and periods are equal in the contest of three separate models with their matrices of information in simplified form. We used these simplified matrices of information to ascertain the criteria for universally optimal CBRMDs under different conditions. These new CBRMDs have been constructed using the well-known method of cyclic shifts (MCS) rule II. Results. Universal optimality of the new proposed classes of designs has been discussed theoretically. Universally optimal CBRMDs were constructed for $v=p\left(\text{odd}\right)$ using the MCS rule II along with the confirmation of the universal optimality criteria proposed in the existing theory. Conclusions. The proposed class of new CBRMDs has been proven to have theoretically universally optimal designs, which have been constructed by the method of cyclic shifts rule II when the number of treatments is equal to the number of periods.