On the equivalence of three estimators for dispersion effects in unreplicated two-level factorial designs

Box and Meyer [1986. Dispersion effects from fractional designs. Technometrics 28(1), 19–27] were the first to consider identifying both location and dispersion effects from unreplicated two-level fractional factorial designs. Since the publication of their paper a number of different procedures (both iterative and non-iterative) have been proposed for estimating the location and dispersion effects. An overview and a critical analysis of most of these procedures is given by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405]. Under a linear structure for the dispersion effects, non-iterative estimation methods for the dispersion effects were proposed by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405], Liao and Iyer [2000. Optimal 2n-p2n-p fractional factorial designs for dispersion effects under a location-dispersion model. Comm. Statist. Theory Methods 29(4), 823–835] and Wiklander [1998. A comparison of two estimators of dispersion effects. Comm. Statist. Theory Methods 27(4), 905–923] (see also Wiklander and Holm [2003. Dispersion effects in unreplicated factorial designs. Appl. Stochastic. Models Bus. Ind. 19(1), 13–30]). We prove that for two-level factorial designs the proposed estimators are different representations of a single estimator. The proof uses the framework of Seely [1970a. Linear spaces and unbiased estimation. Ann. Math. Statist. 41, 1725–1734], in which quadratic estimators are expressed as inner products of symmetric matrices.