A Note on Fractions of 34n+1Plans

Abstract

Consider a 3m factorial experiment where the 3 levels of each factor are denoted by -1, 0 and 1. Every treatment combination in such an experiment can be represented by an m-dimensional vector with elements -1, 0 and 1. Every treatment combination may be called a design point. The orthogonal main effect plans for symmetrical factorial experiments involving (sn l)/(s 1) factors, each having s levels, with Sn design points are given by Addelman [1] and for experiments involving {2(sn 1)/(s 1) 1} factors, each having s levels, with 2s design points are given by Addelman and Kempthrone [2]. In both these cases s is taken to be a prime or a prime power. Thus for a symmetrical factorial experiment where each factor has 3 levels, orthogonal main effect plans are available involving 4, 7, 13, 25 etc., factors. When an orthogonal main effect plan is not available for a particular number of factors, say m, we select an orthogonal main effect plan where the number of factors involved is greater than m and omit the components in each treatment combination corresponding to the additional factors. When there is a large difference between the number of factors, m, and the next largest number in either of the series (31 1)/2 or 3m 2, where n = 2, 3, . . ., the cost of taking each observation may, at times, prohibit the selection of an orthogonal plan of the type described. If we relax the condition that all main effects should be orthogonal, and impose the weaker conditions that