Experimental Design - Box-Cox Plots for Selecting Transformations

It is assumed in analysis of variance that the variances in the different groups (experimental conditions) are homogeneous, and that they are uncorrelated with the means. If the distribution of values within each experimental condition is skewed, and the means are correlated with the standard deviations, then one can often apply an appropriate power transformation to stabilize the variances, and to reduce or eliminate the correlation between the means and standard deviations. In addition to the Box-Cox transformation button available from the Box-Cox tab of most results dialogs, Statistica contains the example Statistica Visual Basic program Boxcoxp.stb, which computes the so-called Box-Cox diagnostic plot, useful for selecting a (power) transformation of the dependent variable.

The Boxcoxp.stb program can be run after you click the Summary: Display (design) button in the results dialog box for analyzing 2(k-p) designs, screening designs, 3(k-p), or central composite (response surface) designs. Ensure that the respective Design:... spreadsheet must be the one that is currently active or in the foreground, and then run the Statistica Visual Basic program Boxcoxp.stb.

Statistica produces a scatterplot of 1) the means versus the standard deviations, and 2) the log of means versus the log of standard deviations (if all means and standard deviations are greater than zero).

For the latter plot, the slope (b) of the regression line suggests a transformation for the dependent variable (y), so that:

y' = y(1-b) b¹1

y' = natural log(y) b=1

In practice, it is not important that you use the exact value of b as estimated for (and displayed in) the second graph. Rather, as a rule of thumb, you should consider the following transformations:

For additional information regarding this family of transformations, see Box and Cox (1964), Box and Draper (1987), and Maddala (1977). Statistica also contains the related Statistica Visual Basic programs Boxcox.stb and Boxtid.stb. These programs provide additional methods for selecting transformation of the variables in an ANOVA.