A quantitative method for identifying active contrasts in unreplicated factorial designs based on the half-normal plot

Abstract

Normal or half-normal plots are often used to judge the significance of effect contrasts in unreplicated factorial or fractional factorial designs. Many quantitative procedures have also been proposed in recent literature to reduce the subjectivity of the graphical methods. In this paper we describe a new method for judging significance of effects that is both quantitative and graphical. The method consists of fitting a simple least-squares line and prediction limits to the half-normal probability plot. The rational for the statistic comes from recent papers by Lenth and Loh. This new method is a blend of Lenth's and Loh's method. It has the computational simplicity of Lenth's method with the graphical interpretation and increased power of Loh's method. We describe the calculations of this new statistic which is a supplement to the half-normal plot. We present a table of critical values for the statistic and show a simulation study to describe the power properties of the test. Finally, we present an example of the new technique using a data set from Box and Meyer. The new statistic developed in this paper is a powerful quantitative tool for reducing the subjectiveness of the half-normal plot in accessing the significance of effects in unreplicated designs. It can be computed simply with commands available in standard statistical program such as SAS, MINITAB or S-PLUS, and can be used graphically or numerically for more accuracy.