Box-Meyer statistic

Box and Meyer also derived a useful result (which is applied in some of the subsequent methods in this chapter) that relates dispersion effects to location effects in regular 2k p designs. We present the result first for 2k designs and then explain how to extend it to fractional factorial designs. First, fit a fully saturated regression model, which includes all main effects and all possible interactions. Let /3, denote the estimated regression coefficient associated with contrast i in the saturated model. Based on the results, determine a location model for the data that is, decide which of the are needed to describe real location effects. We now compute the Box-Meyer statistic associated with contrast j from the coefficients 0, that are not in the location model. Let i o u denote the contrast obtained by elementwise multiplication of the columns of +1 s and—1 s for contrasts i and u. The n regression coefficients from the saturated model can be decomposed into n/2 pairs such that for each pair, the associated contrasts satisfy i o u = j that is, contrast i o u is identical to contrast j . Then Box and Meyer proved that equivalent expressions for the sums of squares SS(j+) and SS(j-) in their dispersion statistic are... 

When applied to the data on hue in the 26 dyestuffs experiment, the Box-Meyer statistic (3) points to factor F as having the most potential for a dispersion effect, with a statistic of 0.59. The effect for factor F stands out, though not dramatically, on a normal plot of the Box-Meyer statistics (Figure 5). The next strongest... 

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. 

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. 

The statistic D is similar to the Box-Meyer statistic D , but uses the geometric averages of the squared residuals rather than the arithmetic averages. 

Figure 5. Normal probability plot of the Box-Meyer dispersion statistics for the 2s experiment on hue, with A and F in the location model.

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). 

The indices / (+) and / (-) in the above definitions are used to denote summation over the design points with factor j at its high and low levels, respectively. The test statistic proposed by Box and Meyer was one-half the log of the ratio of these sums of squares,... 

Box and Meyer did not present any formal inference procedures for using this statistic to identify dispersion effects. The use seems to be informal for screening factors with large effects from those with no or small effects on dispersion, for example, by making a normal probability plot of the statistics see Montgomery (1990) for an application of this idea. 

Holm and Wiklander (1999) presented test statistics for dispersion effects derived from quadratic functions of the location effects. These test statistics are equivalent to the Box and Meyer (1986) statistics. The Holm and Wiklander version emphasizes how they can be seen as correlation coefficients among the null location effects, which can be used as a basis for making statistical inferences about possible dispersion effects. 

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