D- and A-Optimal Designs - Model Tab
Select the Model tab of the D- and A-Optimal Designs dialog box to access options to specify the model that you want to fit to the data (after you have completed the experiment). Statistica selects points from the candidate list to optimize the chosen criterion given the current model, that is, given the current design matrix (see also the Introductory Overview).
Use the Experimental Design module to construct optimal designs for standard response surfaces, where the model includes an intercept (or grand mean), and for mixtures where the models (in their standard canonical form) do not include an intercept. To learn more about these different models refer to the to the information on central composite designs and mixture designs in the Introductory Overview.
- Resp. surface (with intercept).
- Select the Resp. surface (with intercept) option button to specify a response surface design. The common second-order (quadratic) model for response surface designs can be written as:
y = b0 + b1*x1 +...+ bk*xk + b12*x1*x2 + b13*x1*x3 +...+ bk-1,k*xk-1*xk + b11*x12 +...+ bkk *xk2
where b0 is the intercept, b1 is the coefficient for the main effect for factor 1 (b2 for factor 2, and bk for factor k), b12 (bjk) is the coefficient for the interaction effect for factors 1 and 2, and b11 (bkk) is the coefficient for the quadratic effect.
- Response surface model
- The option button selected in the Response surface model group box specifies the order of the model, that is, which types of effects (e.g., quadratic main effects, interactions, etc.) to retain in the model. You can choose from the following models:
Linear main effects
Lin./quad. main effects
Linear main eff. + 2-ways
Lin/quad main eff. + 2-ways
- Mixture (no intercept)
- Select the Mixture (no intercept) option button to specify a mixture design.
- Mixture model
- The option button selected in the Mixture model group box specifies the model for the design. The common models (in their canonical form, see Introductory Overview) for mixture experiments, for the example case of three factors, can be written as:
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.)
Notes: Recoding of factor effects. By default, Statistica optimizes the design (i.e., compute X'X, where X stands for the design matrix) after recoding the factor settings found in the list of candidate points. This recoding can be turned off on the Options tab. Specifically, if you select a Response surface (with intercept) model (see above), the design matrix is computed from the factor settings recoded to the ±1 range (e.g., the smallest value for each factor is recoded to -1, the largest value is recoded to +1). If you select a Mixture (no intercept) model (see above), the design matrix is computed from factor settings transformed to pseudo-components (see also 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 ith component, L stands for the sum of all lower constraints (limits) for all components in the design, and Total stands for the mixture total. For additional discussion of these transformations, refer also to the results dialogs of Analysis of a Central Composite (Response Surface) Experiment and Analysis of a Mixture Experiment.
- Non-standard models
- If you want to design a response surface experiment that includes non-standard terms, you can proceed as follows. First create via spreadsheet transformations the desired terms from the factors; for example, you could add a linear-by-quadratic three-way interaction by creating a new variable in the data file, and then specifying the formula =x1*x22*x3 to compute the values for the new variable. After you have computed all terms for the model, select all variables, including the new variables you computed, as the variables for the experiment (i.e., for the candidate list). Then 1) turn off the recoding option via the Recode factor values check box on the Options tab, and 2) select the Linear main effects model in the Response surface model group box (see above). The points for the optimal design will now be determined from the untransformed values, as you computed them in your data file. Thus, you can use the optimal design facilities to find optimal designs for any type of model.
- Ignore some effects
- Select the Ignore some effects check box or click the Effects to ignore button to display the Customized (Pooled) Error Term dialog box. Select individual effects that you want to be ignored when generating the optimal design.