D- and A-Optimal Designs - Options Tab

Select the Options tab of the D- and A-Optimal Designs dialog box to access options to control various technical parameters of the optimization process.

Maximum number of iterations

Enter a value in the Maximum number of iterations box to specify the maximum number of iterations that are to be performed. Note that if the Sequential (Dykstra) method is selected on the Optimization Methods tab, this parameter is ignored (since that method is not iterative in nature). If you choose the DETMAX method, one iteration is defined as one complete excursion with improvement.

Recode factor values

By default, STATISTICA optimizes the design (i.e., computes X'X, where X stands for the design matrix) after recoding the factor settings found in the list of candidate points. If you clear the Recode factor values check box, no recoding will be performed. If this check box is selected, then, if you select a Response surface (with intercept) model on the Model tab, the design matrix is computed from the factor settings recoded to the ±1 range (i.e., 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, 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 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. For additional discussion of these transformations refer also to the results dialogs for Analysis of a Central Composite (Response Surface) Experiment and Analysis of a Mixture Experiment..

Initial design

The option button selected in the Initial design group box specifies the method for selecting the points for the initial design. All optimization algorithms available on the Optimization methods tab, except for the Sequential (Dykstra) method, require an initial design as a starting point (see the Introductory Overview for details). Remember that you can "force" design points into the final design via the Display and modify candidate points option on the Candidate points tab.

Select first N data points from candidate list

If this option button is selected, the first N data points from the candidate list will be taken as the initial design.

Select via sequential (Dykstra) algorithm

If this option button is selected, then the initial design is selected from the candidate list via the Sequential (Dykstra) algorithm.

Select randomly

If the Select randomly option button is selected, the design points for the initial design are selected randomly from the candidate list. Note that the random seed (the value in the Seed box) automatically changes (i.e., the random number generator is automatically reseeded) every time that you enter the D- and A-Optimal Designs dialog box.

Alpha value for avoiding singularity

It may happen that the original choice of design points (for the initial design) yields an X'X matrix (where X stands for the design matrix) that is singular. The determinant is equal to 0 (zero) for singular matrices, and they cannot be inverted. Hence, the search algorithm will terminate because no initial valid X'X matrix can be computed (and updated). Mitchell (1974b) proposed to use during the optimization not the actual X'X matrix, but rather the augmented matrix:

(X'X)augmented = X'X + a *((X'0X0)/N0)

Here X0 stands for the design matrix computed for all candidate points, and N0 stands for the number of candidate points.

In the Alpha value for avoiding singularity box, specify the Alpha (a) value that is to be used for computing the X'X matrix for the initial design.

Note: Avoiding singularity during DETMAX excursions. An additional improvement to the original DETMAX algorithm, that is also used in the implementation in the Experimental Design module, was proposed by Nachtsheim (1979). In general, if during an excursion a singular design matrix occurs, then the respective design is ostracized by placing it into the list of "failure designs." If a singular X'X matrix occurs during a negative excursion, then from that point on, only positive excursions are allowed. For details for these procedures, refer to Mitchell (1974b) and Nachtsheim (1979).

Tolerance value for D/A improvement

The value in the Tolerance value for D/A improvement box is used by the various search algorithms to determine whether successive iterations have yielded an improvement in the selected optimality criterion. Specifically, the criterion for improvement is that:

Criterionold +Tolerance < Criterionnew

where Criterionold and Criterionnew stand for the chosen optimality criterion for the old (prior to the last iteration) and the new design (after the most recent iteration).

Tolerance value for checking constant mixture total

STATISTICA uses the value in the Tolerance value for checking constant mixture total box to verify that the component values for the points in the candidate list sum to a constant ± tolerance (the mixture total, plus or minus the tolerance value). This value is ignored if a Response surface design model is selected on the Model tab. If you entered approximate component values into the candidate list (e.g., 0.33 in order to denote the component value 1/3, i.e., one-third) then you may have to adjust the Tolerance value.