Repeated measurements designs for a model with self and simple mixed carryover effects

Abstract

We study the problem of design and analysis of two-period repeated measurements (crossover or changeover) designs based on two or more treatments. In our model, we allow the appearance of two types of carryovers (residuals) in the observations collected in the second period. It is assumed that each treatment has three types of effects: direct effect, self-carryover effect, and simple mixed carryover effect. The direct effect will manifest itself no matter where and when the treatment is applied. Carryover effects will only appear in the second period. The nature and the magnitude of this carryover effect from a treatment in the first period which will appear in the second period depends on the treatment into which it carries this carryover effect. It is called a self-carryover effect if it carries to itself. Otherwise, it is called a simple mixed carryover effect. It is proved that if the study is properly designed, we can efficiently estimate any contrast in direct treatment effects even if there are self and simple mixed carryover effects in the data. Further, it is shown that the most efficient way of constructing a two-period repeated measurements design is equivalent to constructing an efficient block design in a special class of block designs based on t(⩾2) treatments utilizing in total a fixed number of experimental units. In addition, if we are interested in contrasts involving self-carryover effects only, then it is easy to design the study so that we can estimate these contrasts. The problem of identifying small designs that allow the unbiased estimation of independent contrasts in simple mixed carryover effects is addressed.