Norm of alias matrices for balanced fractional 2 m factorial designs when interesting factorial effects are not aliased with effects not of interest in estimation

Abstract

Consider a balanced fractional 2 m factorial (2 m -BFF) design derived from a balanced array of full strength, which is called a simple array ( S-array). Let Θ 1, Θ 2 and Θ 3 be an n 1-vector of factorial effects to be of interest and to be estimated, an n 2-one not of interest in estimation and not assumed to be negligible, and an n 3-one assumed to be negligible, respectively, where n 1= m 0 + m 1 +⋯+ m ℓ , n 2= m ℓ+1 +⋯+ m ℓ+f andn 3=2 m−n 1−n 2. In this paper, by applying the multidimensional partially balanced association algebra, we give the norm of the alias matrix for a 2 m -BFF design derived from an S-array, when Θ 1 is not aliased with Θ 2. We also give best alias 2 6-BFF designs with 28⩽ N⩽41 for ℓ=1 and f=2, and with 32⩽ N⩽41 for ℓ=2 and f=1, where N is the number of assemblies.