Switch-back designs

SUMMARY Switch-back designs, originally proposed for cow lactation experiments, are repeated measurement designs appropriate for experiments in which the responses of experimental units vary with time according to different rates. A method for constructing such designs to compare an arbitrary number of treatments is presented, together with their analysis of variance. Robustness of the switch-back design when the treatments have residual as well as main effects is considered. Switch-back and balanced residual designs are compared when the treatment has three levels. In repeated measurement designs, also referred to as before-and-after, reversal, crossover, change-over, and multiple time series designs, each experimental unit is assigned more than once to a treatment. We consider a particular such design, the switch-back design, originally proposed by Brandt (1938) for comparing the effects of two different feeds on the milk yields of dairy cattle. This design adjusts for the natural decline at rates which vary widely from cow to cow in milk yield, during successive periods of lactation, by switching the feeds given to the cows in a balanced manner. Lucas (1956) extended this design to cover up to nine treatments, and computed the analysis of variance for treatment, and possibly block, effects. For ease of exposition our discussion is couched in terms of lactation experiments, although clearly our results are applicable in other contexts as well. In ? 2 we construct switch-back designs for an arbitrary number of treatments applied in three consecutive time periods. Section 3 provides the complete analysis of variance for treatment, period, lactation curve slope, cow and block effects. Lucas's (1956) F statistic for treatment effects is identical to ours if no block effects are assumed; however, when block effects are assumed, his error sum of squares has fewer degrees of freedom, since his analysis is based on a singular transformation of the data, so that the sum of squares for blocks must be partitioned from the error sum of squares. Also, he defines the block effect using the curvatures of the lactation curves of the cows in the block, while we, more appropriately we believe, use the cows' average milk yields. In ? 4 we consider the robustness of switch-back designs to residual effects, which occur when each treatment has a main effect on the experimental unit when it is applied, and a residual effect during the immediately following time period. In this case we derive the distribution of the switch-back test statistic for treatment effects, when residual effects are present.