On balancing in repeated measurements designs

Abstract

An introspection of various criterions to define balanced Repeated Measurements Designs (RMD's) in literature was carried out. It is proved that the VIIth condition in the definition of Patterson (4) is implied by the other six conditions, hence VIIth condition is not required and the remaining six conditions are sufficient conditions for defining the balanced RMD; wherein all the elementary contrasts of direct effects are estimated with equal variance, say(v1) and all the elementary contrasts of first order residual effects are estimated with equal variance, say(v2). It is shown that reduction in these six conditions by Hedayat and Afsarinejad (3) to define balanced RMD does not estimate the elementary contrasts of direct effect and first order residual effects with equal variance. Similarly, use of four plus one i.e. five conditions by Dey et al. (2) are not sufficient to define variance balanced RMD. Hence, the definition of Patterson (4) can be modified to specify only first six conditions to define, balanced Repeated Measurements Designs. Also the conditions specified by Chawla and Dey (1) for defining balanced ternary change over designs (Repeated Measurements Designs) are not necessary conditions; they are sufficient conditions.