Optimality of block designs with variable block sizes and random block effects

Abstract

We consider the simple block model with random block effects, the block effects having variance σ ²_b =λσ²;, with σ² the variance of the errors. It is assumed that the experimenter can vary the sizes of the blocks. The universal optimality of certain designs for all λ over all designs with the same number of blocks and the same number of observations is shown. It is of interest to note that if Balanced Incomplete Block Designs compete, then they perform equally well for λ=0 and for λ=∞, i.e. in the one way classification model and in the simple block model with fixed block effects, but they perform worse for every λ∈(0, ∞).

The result is, however, theoretical in nature. It treats a situation which is not very likely to happen in practice. The interest lies in the fact that it provides a counterexample to a conjecture on optimality of designs in mixed models.