Estimates of Effects for Fractional Replicates

Abstract

Given any fraction of a factorial experiment in which the treatments either occur zero or one time, previous results were obtained on augmentation of the treatment design matrix, $X$, such that the product of the transpose and of the augmented matrix, $X_1 = \lbrack X'\vdots X'\lambda\rbrack'$, resulted in a diagonal matrix, and on a transformation of $X_1$ to another matrix $X_2 = FX_1$. In the present paper results are obtained on the evaluation of the variances of estimated effects under augmentation, on the existence and evaluation of $F$ and $\lambda$, on the determination of aliases of effects, and on the calculation of inverses for $\lbrack X'X\rbrack$ and for the information matrix $\lbrack X'_{22}X_{22}\rbrack$ for the deleted treatments.