Bergman-Hynen method

Table 1. Expanded location models for the Bergman-Hynen method for several potential dispersion effects when the location model includes the main effects of factors A and B. The location model always includes main effects for A and B and also includes the main effect of the dispersion candidate and its interactions with A and with B...

Blomkvist et al. (1997) extended the Bergman-Hynen method to identify dispersion effects from unreplicated multi-level experiments. Arvidsson et al. (2001) showed how the method could be applied to split-plot experiments. 

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). 

Use the Bergman-Hynen (1997) method as a quick screen for dominant dispersion effects. 

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... 

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... 

Blomkvist, O., Hynen, A., and Bergman, B. (1997). A method to identify dispersion effects from unreplicated multilevel experiments. Quality and Reliability Engineering... 

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