On Estimates for Fractions of a Complete Factorial Experiment as Orthogonal Linear Combinations of the Observations

Abstract

A procedure for adjusting the treatment design matrix to furnish estimates as orthogonal linear functions of stochastic variates has been given for fractional replicates of factorial experiments composed of $ks^{-m}s^n$ treatments for $m < n$, for $s$ a prime number, and $k$ not a multiple of $s$. These are irregular fractional replicates. Ordinarily irregular fractional replicates do not lead to estimates of effects which are orthogonal linear combinations of the observations. In this paper a new relationship and some generalizations on the theory of irregular fractional replicates of complete factorials have been developed. En route to these developments two theorems in matrix transformations were proved. In addition, the relationship between the method utilized here and ordinary missing plot techniques is pointed out.