Quasi-Orthogonal Arrays And Optimal Fractional Factorial Plans

Abstract

In this talk, further results on the optimality of fractional factorial plans for arbitrary factorials are presented. A new class of arrays, analogous to orthogonal arrays, are introduced. These arrays are called quasi-orthogonal arrays. We show that fractional factorial plans represented by quasi-orthogonal arrays are universally optimal (and hence, in particular A-, D- and E-optimal) under a specified model. We also give a general method of construction of quasi-orthogonal arrays. Fractional factorial plans represented by quasi- orthogonal arrays can be sometimes saturated in the sense that the number of parameters to be estimated equals the number of runs in the plan. In such a situation, one cannot obtain an internal estimate of the error variance and hence, the standard F-test for testing the significance of factorial effects cannot be used. It is shown that the `plus one run plans' obtained by adding a single run to a plan represented by a quasi-orthogonal array is optimal with respect to every generalized criterion of type 1, which includes in particular the commonly used optimality criteria, like the A-, D- and E-criteria.