Bergman-Hynen statistic

Although the Bergman-Hynen statistic provides a clever correction to some problems with the Box-Meyer statistic, it remains problematic in the face of multiple dispersion effects (see Brenneman and Nair, 2001 and McGrath and Lin, 2001). If factor j alone has a dispersion effect, the numerator and denominator of the statistic D H in (7) are unbiased estimators of the variances at the high and low levels of j. However, if several factors have dispersion effects, one has instead unbiased estimates of the average variances at these two levels, where the averaging includes the effects of all the other dispersion effects. This dependence of I)1-11 on additional dispersion effects can lead to inflated type I error probabilities and thus to spurious identification of dispersion effects. 

On the dyestuffs example, the Bergman-Hynen method also signals factor F as being related to dispersion. With the main effects of A and F in the location model, F has a Bergman-Hynen statistic of 3.27 (p-value = 0.001). The next strongest effects, as with the Box-Meyer method, are the ADEFinteraction, with a statistic of 2.24 (p-value = 0.017) and the CEF interaction, with a statistic of 0.45 (p-value = 0.035). 

Bergman and Hynen (1997) developed a method similar to that of Box and Meyer (1986), but with a simple and exact distribution theory for inference from the test statistic. The important observation of Bergman and Hynen was that the residuals from the fitted location model could complicate inference for the Box-Meyer statistic in two ways. First, the residuals in the two sums of squares could be correlated. Second, the residuals at the high (low) level of factor j typically depend on the actual variances at both levels of the factor, not just the level at which the run was made. 

Brenneman (2000) found that Harvey s method could underestimate the dispersion effect of factor j if that factor was left out of the location model. This result led Brenneman and Nair (2001) to propose a modified version of Harvey s method for two-level factorial experiments that is based on the results of Bergman and Hynen (1997). In the modified version, the dispersion statistic for factor j is computed from residuals from an expanded location model that includes the effect of factor j and all its interactions with other effects in the location model. For two-level designs, the modified Harvey s statistic for factor j is then... 

Arvidsson, M., Merlind, P. K., Hynen, A., and Bergman, B. (2001). Identification of factors influencing dispersion in split-plot experiments. Journal of Applied Statistics, 28, 269-283. 

Bartlett, M. S. (1937). Some examples of statistical methods of research in agriculture and applied biology. Supplement to the Journal of the Royal Statistical Society, 4, 137-183. Bergman, B. and Hynen, A. (1997). Dispersion effects from unreplicated designs in the... 

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