Economic Trend Resistant2n-(n-k) Designs of Resolutions III and IV Based on Hadamard Matrices

Abstract

This article utilizes the Normalized Sylvester-Hadamard Matrices  of size 2^kx2^kand their associated  saturated orthogonal arrays OA(2^k, 2^k - 1, 2, 2) topropose analgorithmbased on factor projection (Backward/Forward) for the construction of three systematic run-after-run2^n-(n-k) fractional factorial designs: (i) minimum cost  trend free 2^n-(n-k)designsof resolution III (2^k-1≤n≤2^k– 1 – k)by backward factor deletion (ii) minimum cost trend free 2^n-(n-k) designsof resolution III (k+1≤n≤ 2^k-1– 2+k ) by forward factor addition (iii) minimum costtrend free  2^n-(n-k) designsof resolution IV (2^k-2≤n≤2^k-1-2) ,where each 2^n-(n-k)design is economic minimizing the number of factor level changes between the 2^ksuccessive runs and allows for the estimation of all factor main effects unbiased by the linear time trend,which might be present in the 2^ksequentially generated responses. The article gives for each 2^n-(n-k)design: (i) the defining contrast displaying the design’s alias structure(ii) the k independent generators for sequencingthe design’s  2^n-(n-k) runs  by the Generalized Fold over Scheme and (ii) the minimum total cost of factor level changes between the 2^n-(n-k) runs of the design. Proposed designs compete well with existing systematic2^n-(n-k)designs (of either resolution) in minimizing the experimental costandin securing factors’ resistance to the non-negligible time trend.

Keywords: Sequential fractional factorial experimentation; Time trend free systematic run orders; Generalizedfoldover scheme for sequencing experimentalruns; The total cost of factor level changes between successive runs; The Normalized Sylvester –Hadamard Matrices; Orthogonal Arrays and factor projection; Design resolution and the alias structure.

DOI: 10.7176/JEP/11-25-05

Publication date:September 30^th 2020