Confounding of location and dispersion effects in unreplicated fractional factorials

When studying both location and dispersion effects in unreplicated fraction al factorial designs, a "standard" procedure is to identify location effect s using ordinary least squares analysis, fit a model, and then identify dis persion effects by analyzing the residuals. In this paper, we show that if the model in the above procedure does not include all active location effec ts, then null dispersion effects may be mistakenly identified as active. We derive an exact relationship between location and dispersion effects, and we show that without information in addition to the unreplicated fractional factorial (such as replication) we can not determine whether a dispersion effect or two location effects are active.