On Identifying Dispersion Effects in Unreplicated Fractional Factorial Experiments

The analysis of dispersion effects is as important as the location effect analysis in the quality improvement. Unreplicated fractional factorial experiments are useful to analyse not only location effects but also dispersion effects. The statistics introduced by Box and Meyer (1986) to identify dispersion effects are based on residuals subtracting the estimates for large location effects from observations. The statistic is a simple form, however, the property is not completely discovered. In this article, the distribution of the statistic under the null hypothesis is derived in unreplicated fractional factorial experiments using an orthogonal array. The distribution under the null hypothesis cannot be expressed uniquely. The statistic has different null distributions depending on the combination of columns allocating factors. We concluded that the distributions can be classified into three types, i.e. the F distribution, unknown distributions close to the F distribution and the constant (not stochastic variable) which is one. Finally, the power of the test for detection of a single active dispersion effect is evaluated.