Estimates, Pseudo-Components in Mixture Experiments

Both the Quick tab and the ANOVA/Effects tab of the Analysis of a Mixture Experiment dialog box contain a Summary: Estimates, pseudo-components button. Click this button to produce a spreadsheet with parameter (coefficient) estimates for the current model, based on the transformed pseudo-components. The effects that will be computed depend on the current model, as specified via the options in the Model group box on the Model tab.

All estimates in this spreadsheet pertain to the rescaled factor settings, that is, to pseudo-components. To see the results for the original (untransformed) factor settings, click the Estimates, original comps button on the Quick tab or the ANOVA/Effects tab. Note that if your original components were already scaled to the standard 0-1 range (and the mixture total is equal to 1.0), the Estimates, original comps button is dimmed, and the results shown for this option are identical to those for the original components (because the pseudo-component transformation will not change the factor values in that case; see the formula below).

Pseudo-components

The computation of pseudo-components is discussed in the Introductory Overview. In short, when analyzing standard mixture designs, we would like to rescale the original factor values so that the low and high factor settings for each factor (in a simplex design) are transformed to 0 and +1, respectively. Specifically, during the analysis, the component settings are customarily recoded to so-called pseudo-components so that (see also Cornell, 1990a, Chapter 3; or the Introductory Overview):

x'i = (xi -Li )/(Total-L)

Here, x'i stands for the i'th pseudo-component, xi stands for the original component value, Li stands for the lower constraint (limit) for the i'th component, L stands for the sum of all lower constraints (limits) for all components in the design, and Total stands for the mixture total. This transformation makes the coefficients for different factors comparable in size. Also, since this is a linear transformation of the variables, the conclusions from the experiment will not be affected.

Main effects

The values shown in the column labeled Effect, and in the rows labeled by the main effects, are the linear effect coefficients. Note that the statistical significance of these estimates (if computed) must be interpreted with caution. Because of the general mixture constraint (sum of component values must be constant), there are only q-1 degrees of freedom associated with the q components.

Other effects

The other rows of the spreadsheet will show the coefficient estimates for the different terms in the respective polynomial model (as selected in the Model group box on the Model tab). Note that these are the estimates for the model in the canonical form (see Introductory Overview).

Standard error of coefficients

The standard errors for the parameter estimates (ANOVA estimates and coefficients) are computed from the residual sums-of-squares for the current model.