Optimal and efficient crossover designs when subject effects are random

Most studies on optimal crossover designs are based on models that assume subject effects to be fixed effects. In this article we identify and study optimal and efficient designs for a model with random subject effects. With the number of periods not exceeding the number of treatments, we find that totally balanced designs are universally optimal for treatment effects in a large subclass of competing designs. However, in the entire class of designs, totally balanced designs are in general not optimal, and their efficiency depends on the ratio of the subject effects variance and the error variance. We develop tools to study the efficiency of totally balanced designs and to identify designs with higher efficiency.