On the Structure and Analysis of Singular Fractional Replicates

Abstract

In a recent paper [1], the authors presented a procedure showing how the treatment design matrix for an irregular fractional replicate of an $N$ treatment factorial might be adjusted to furnish estimates of the effect parameters as orthogonal linear functions of the observations.The procedure may be summarizes as follows: If the $N \times N$ design matrix $X$ of the complete factorial is partitioned into four submatrices $X{11}, X_{12}, X_{22}$ having dimension $(p + m_1) \times p, (p + m_1) \times m, m_2 \times p$ and $m_2 \times m$ respectively, with $m_1 + m_2 = m$ and $p + m = N, X_{11}$ would correspond to the design matrix of an irregular fractional replicate consisting of $p$ effect parameters. With the help of an auxiliary design matrix $\lambda$, the design matrix $X_{11}$ and the corresponding observation vector $Y_{p+m_1}$ of $(p + m_1)$ components were then augmented to become $X_1 = \lbrack X'_{11} : X'_{11} \lambda\rbrack'$ of dimensions $(p + m_1 + m_2) \times p$, and $Y_1 = \lbrack Y' : Y' \lambda\rbrack'$ of dimensions $(p + m_1 + m_2) \times 1$ in such a way that $\lbrack X_1'X_1\rbrack$ reduced to a diagonal matrix. The success of the procedure depended on being able to find least squares estimates for each of $m_2$ omitted observations, and this, in turn, was possible because it was possible to have least squares estimates for each of the effect parameters retained. In other words, this meant that both $(X'_{11} X_{11})^{-1}$ and $(X_{22}X'_{22})^{-1}$ existed. The structural relationship between the effect parameters retained and the observations omitted was such that existence of one of the inverses implied the existence of the other. If the rank of $(X'_{11}S_{11})$ is not full, either as a result of defective or intentional construction of the fractional replicate, the rank of $(X_{22}X'_{22})$ will be less than full, and, as a result, it will not be possible to have unique least squares estimates for each of the omitted observations. When this is the case, the corresponding fractional replicate would be what might be characterized as a singular fractional replicate. A question then arises as to how the results presented in [1] would be affected by this singuarlity. The question has been resolved and analogous results have been obtained through use of generalized inverses (referred to as $g$-inverses) in the present paper. By way of an aid to the derivation of the analogues, a few additional results on the ranks of the associated submatrices have also been obtained.