Circular weakly balanced cross-over designs in periods of two and three different sizes

Cross-over designs (CODs) have applications in many branches of scientific inquiry. The use of CODs can lead to carryover effects, which are the main source of bias in the estimation of treatment effects. In order to reduce bias, minimal balanced, minimal partially balanced, and minimal weakly balanced CODs are preferred. In the literature, minimal weakly balanced CODs are reported for unequal period sizes when the first period is greater than the second period and so forth. In this article, the method of cyclic shifts is used to construct minimal partially balanced and minimal weakly balanced CODs for unequal period sizes of two and three, where the first period is less than the second period and so forth. All the proposed designs are efficient to control the carryover effects as well as to estimate the carryover effects and direct effects independently.