Optimality Status of Incomplete Layout Three-Way Balanced Designs and Optimal Designs Under Heteroscedastic Errors in Linear Regression

The setup is that of a row-column design (also called a three-way design) and the model is that of fixed effects additive model with homoscedastic errors. We say that the row and column classifications represent known heterogeneity directions and the treatment effects correspond to the third component of variation. Our primary task is to provide efficient comparisons of the treatment effects. Further to this, we have two sets of differential effects, each within a specific heterogeneity direction, to be compared (or at least eliminated). Technically, these are referred to as row effects and column effects. A three-way design involving R rows, C columns and v treatments is said to possess a complete (incomplete) layout if the number of experimental units (eu's) in the design is equal to (less than, respectively) RC. The row-column layout is generally made available to the experimenter who has to design treatment allocation over the eu's. The statistical analysis of data arising out of such a design is fairly straightforward and we refer to Shah and Sinha (1996) for this. There are three (assignable) sources of variation: rows, columns and treatments. These correspond to three classifications and the resulting data is often referred to as three-way classified data in the literature. A three-way design is said to be three-way balanced (or, totally balanced) whenever the variances of estimates of normalized effects contrasts are the same for each classification. We readily verify that the simplest example of a three-way balanced design is a Latin square which is known to possess a complete layout involving the same number of rows, columns and treatments. Incomplete three-way balanced designs are not easy to come across. Agrawal (1966) was the first to provide some series of such designs. Afterwards, Hedayat and Raghavarao (1975) also provided one series of such designs. Note that a three-way balanced design in an incomplete three-way layout provides complete symmetry (c.s.) of the C-matrices involving comparisons of row effects, or the column effects, or the treatment effects. In view of this kind of strong symmetry possessed by these designs, it is reasonable to expect that such designs will be optimal for inference on contrasts of the parameters for each of the three sources of variation: rows, columns and treatments !!