Characterization of Optimal Saturated Main Effect Plans of the $2^n$ Factorial

Abstract

For the $2^n$ factorial a treatment design to estimate the $n$ main effects and the mean with $(n + 1)$ treatment combinations is known in the literature as a saturated main effect plan. Let the $(n \times 1) \times n$ matrix $D$ consisting of the 0's and 1's making up the subscripts of the observations, denote such a plan and let the $(n + 1) \times (n + 1)$ matrix $X$ stand for the corresponding design matrix of -1's and 1's, then optimal (in the sense of maximum absolute value of the determinant of $X'X$) designs have been characterized in terms of the information matrix $X'X$ by many authors, such as Plackett and Burman [6] and Raghavarao [7]. Williamson [9], Mood [3], and Banerjee [2], among others, have used (0, 1)-matrices to construct optimal and weighing designs. If the elements of the first row of $D$ are set equal to zero, then the $n \times n$ (0, 1)-matrix used in weighing designs is obtained from the last rows of $D$. However, $D$ is not restricted to always include the combination having all zero levels in this paper. For a summary concerning several aspects of optimal saturated main effect plans the reader is referred to Addelman's [1] paper. The aim of this paper is to characterize the optimal saturated main effect plans in terms of D'D rather than $X'X$. A major consequence of this is that all theory available for semi-normalized $(-1, 1)$-matrices is applicable to semi-normalized (0, 1)-matrices and vice versa. A second major consequence is that the normal equations for saturated main effect plans need not be obtained as they are readily derivable from the $D$ matrix.