Quantitative Analysis of the Balance Property in Factorial Experimental Designs 2⁴ to 2⁸

Abstract

Experimental designs are built by using orthogonal balanced matrices. Balance is a desirable property that allows for the correct estimation of factorial effects and prevents the identity column from aliasing with factorial effects. Although the balance property is well known by most researchers, the adverse effects caused by the lack or balance have not been extensively studied or quantified. This research proposes to quantify the effect of the lack of balance on model term estimation errors: type I error, type II error, and type I and II error as well as R2, R2adj, and R2pred statistics under four balance conditions and four noise conditions. The designs considered in this research include 24–28 factorial experiments. An algorithm was developed to unbalance these matrices while maintaining orthogonality for main effects, and the general balance metric was used to determine four balance levels. True models were generated, and a MATLAB program was developed; then a Monte Carlo simulation process was carried out. For each true model, 50,000 replications were performed, and percentages for model estimation errors and average values for statistics of interest were computed.

Keywords

• design of experiments
• design matrix
• balance
• orthogonality
• general balance metric