Dispersion effects in unreplicated fractional factorials

Unreplicated fractional factorial designs (specifically, $2k-p$ designs) are often used in industry as screening experiments. In these studies, the conventional goal is to find the factors that have the largest impact on the mean (location) of a response or responses. The Pareto principle is applied, separating the vital few from the trivial many. The assumption of constant variance is commonly made and ordinary least squares (OLS) analysis is used to estimate the location effects.; In addition to having a desirable mean response, a consistent product/process will also have little variability, i.e. dispersion. When the variance of the response differs between the two levels of a column in the effect matrix, that column produces a dispersion effect. The widely-used unreplicated $2k-p$ designs can play an important role in detecting dispersion effects with a minimum expenditure of resources. Dispersion effects are identified by analyzing the residuals from the location model.; After providing an overview of $2k-p$ designs, we show that if there is a dispersion effect, a correlation among pairs of location effect estimates is created. We recommend a joint confidence region approach to identifying location effects in the presence of a dispersion effect. We also show that if two location effects are mistakenly left out of the model in the above procedure, then a spurious dispersion effect is created in their interaction column. Most existing methods for dispersion effect testing in fractional factorial designs are subject to these spurious effects. We propose a method of dispersion effect testing based on geometric means of residual sample variances. To our knowledge, this is the only method that works as expected in the presence of multiple dispersion effects. Additionally, we discuss nonparametric dispersion effect tests as existing tests are quite sensitive to their normality assumptions.