On nonorthogonality and nonoptimality of Addelman's main-effect plans satisfying the condition of proportional frequencies

The condition of proportional frequencies for main-effect plans is stated in Addelman (1962, 1963). The condition of equal frequencies is given in Rao (1946, 1947, 1973) and also in Plackett (1946) as a necessary and sufficient condition for orthogonality of the main-effect plans. The condition of equal frequencies implies the condition of proportional frequencies, but not conversely. The paper examines the main-effect plans which satisfy only the condition of proportional frequencies. The fact is that the orthogonality of such plans is not correct. The paper studies the difference between Addelman's and Fisher's definitions of the general mean and main effects. The paper demonstrates a few undesirable properties for the Addelman's definition. The paper also shows with four illustrative examples that the Addelman plans are not necessarily optimal.