Constrained Mixture Design Results - Confounding Tab

Click the Correlation matrix (no intercept) button to display a correlation matrix of the columns of the current design matrix. The number of effects displayed in this matrix depend on the choice of the model in the Model group box (see below). Note that these correlations are computed from the transformed pseudo-component values (see below; see also the Introductory Overview, for a discussion of pseudo-components). Also, the entries in this table are not the standard Pearson product-moment correlations; instead they are the standardized cross-products for the pseudo-component values. To estimate the parameters for mixture models, usually the polynomial model is rewritten into the so-called canonical form. In short, given the overall mixture constraint (that all component values must add to a constant), the standard linear, quadratic, and cubic models can be rewritten into equivalent no-intercept regression models. For more details, refer to the Introductory Overview.

Note: Pseudo-components. The computation of pseudo-components is discussed in the Introductory Overview. In short, when analyzing standard mixture designs, you should rescale the original factor values so that the low and high factor settings for each factor are transformed to 0 and +1, respectively. Specifically, during the analysis, the component settings are customarily recoded to so-called pseudo-components so that:

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

(See also Cornell, 1990a, Chapter 3.) 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.

Click the Inverse button to display (1) the inverse of the correlation matrix, and (2) the standardized inverse of the correlation matrix. This matrix can be interpreted as the correlation matrix of effects; that is, it is the standardized variance/covariance matrix of the parameter estimates for the current model. The greater the absolute value of a correlation between effects in this matrix, the more redundant are the respective effects.

The option selected in the Model group box determines which terms are included in the model. The common mixture models are shown below, for the example case of a 3-component design (see Cornell, 1990b, for additional details).

Linear.

y = b1*x1 + b2*x2 + b3*x3

Quadratic.

y = b1*x1 + b2*x2 + b3*x3 + b12 *x1*x2 + b13 *x1*x3 + b23*x2*x3

Special cubic.

y = b1*x1 + b2*x2 + b3*x3 + b12 *x1*x2 + b13 *x1*x3 + b23*x2*x3 + b123*x1*x2*x3

Full cubic.

 y = b1*x1 + b2*x2 + b3*x3 + b12 *x1*x2 + b13 *x1*x3 + b23*x2*x3 + d12*x1*x2*(x1-x2) +   d13*x1*x3*(x1-x3) + d23*x2*x3*(x2-x3) + b123*x1*x2*x3

(Note that the dij's are also parameters of the model.)