On Estimation and Construction in Fractional Replication

Abstract

It was shown by the authors [1], [2] how to adjust the treatment design matrix $X$ to furnish estimates of effects as orthogonal linear functions of observations for any irregular fractional replicate from an $N$ treatment factorial. The fractional replicate considered earlier was such that the design matrix $X$ was of dimensions $p \times p$ implying that $p$ effect parameters be estimated from $p$ observations. The method consisted in finding a matrix $\lambda$ such that the design matrix $X$ and the observation vector $Y$ were augmented to become $X_1 = \lbrack X'\vdots X'\lambda\rbrack'$ of dimensions $(p + m) \times p$ and $Y_1 = \lbrack Y'\vdots Y'\lambda\rbrack'$ of dimensions $(p + m) \times 1$ with $p + m = N$ in such a way that $\lbrack X_1'X_1\rbrack$ reduced to a diagonal matrix. In the present note, the earlier results have been generalized in the sense that the design matrix $X$ need not be square, that is, of dimensions $p \times p$, but is of dimensions $(p + m_1) \times p, p + m_1 < N$, implying that $p$ effect parameters be estimated from $(p + m_1)$ observations. Besides this generalization the following additional results were obtained: (i) the structural relationship between the effect parameters retained and the observations omitted was derived, (ii) a working rule was developed for constituting the irregular fractional replicate with observations that are internally consistent making it possible to estimate the effect parameters, and (iii) a desirable procedure of designing the fractional replicate to obtain maximum efficiency was set forth.