D- and A-Optimal Designs - Optimization Methods Tab

Select the Optimization methods tab of the D- and A-Optimal Designs dialog box to access the options described here.

Note: Search algorithms. The search procedures or algorithms that have been proposed are described below (for a review and detailed comparison, see Cook and Nachtsheim, 1980; see also the Introductory Overview). They are described here in the order of speed, that is, the Sequential or Dykstra method is the fastest method, but often most likely to fail, that is, to yield a design that is not optimal (e.g., only locally optimal; see the Introductory Overview).

Sequential (Dykstra). Click the Sequential (Dykstra) button to initiate the method described here. This algorithm is due to Dykstra (1971). Starting with an empty design, Statistica searches through the candidate list of points and chooses in each step the one that maximizes the selected criterion. There are no iterations involved; Statistica simply picks the requested number of points sequentially. Thus, this method is the fastest of the ones discussed. Also, by default, this method is used to construct the initial designs for the remaining methods.

Simple exchange (Wynn-Mitchell). Click this button to initiate the method described here. This algorithm is usually attributed to Mitchell and Miller (1970) and Wynn (1972). The method starts with an initial design of the requested size (by default constructed via the sequential search algorithm described above; see also the Options tab). In each iteration, one point (run) in the design is dropped from the design and another added from the list of candidate points. The choice of points to be dropped or added is sequential, that is, at each step the point that contributes least with respect to the selected optimality criterion (D or A) is dropped from the design; then the algorithm chooses a point from the candidate list so as to optimize the respective criterion. The algorithm stops when no further improvement is achieved with additional exchanges.

DETMAX. Click the DETMAX button to initiate the method described here. This algorithm, due to Mitchell (1974b), is probably the best known and most widely used optimal design search algorithm. Like the simple exchange method, the search begins with an initial design (by default constructed via the sequential search algorithm described above). The search begins with a simple exchange as described above. However, if the respective criterion (D or A) does not improve, the algorithm undertakes excursions. Specifically, the algorithm adds or subtracts more than one point at a time, so that, during the search, the number of points in the design may vary between ND+ Nexcursion and ND- Nexcursion, where ND is the requested design size, and Nexcursion refers to the maximum allowable excursion, as specified in the Max excursion box. You can enter a new value in the Max excursion box, or adjust the value using the microscrolls. The search stops when the selected criterion (D or A) no longer improves within the maximum excursion.

Modified Fedorov (simult. switching). Click this button to initiate the method described here. This algorithm represents a modification (Cook and Nachtsheim, 1980) of the basic Fedorov algorithm described below. It also begins with an initial design of the requested size (by default constructed via the sequential search algorithm). In each iteration, the algorithm exchanges each point in the design with one chosen from the candidate list, so as to optimize the design according to the selected criterion (D or A). Unlike the simple exchange algorithm described above, the exchange is not sequential, but simultaneous. Thus, in each iteration each point in the design is compared with each point in the candidate list, and the exchange is made for the pair that optimizes the design. The algorithm terminates when there are no further improvements in the respective optimality criterion.

Fedorov (simultaneous switching). Click this button to initiate the method described here. This is the original simultaneous switching method proposed by Fedorov (see Cook and Nachtsheim, 1980). The difference between this procedure and the one described above (Modified Fedorov) is that in each iteration only a single exchange is performed; that is, in each iteration all possible pairs of points in the design and those in the candidate list are evaluated. The algorithm then exchanges the pair that optimizes the design (with regard to the selected criterion). Thus, it is easy to see that this algorithm potentially can be somewhat slow, since in each iteration ND*NC comparisons are performed, in order to exchange a single point.

Optimization criterion
The option selected in the Optimization criterion group box specifies whether to use D-optimal or A (or T) optimal techniques. The optimal design criteria are discussed in the Introductory Overview.
D-optimal (maximize determinant)
Select this option button to maximize the determinant of the cross-product matrix X'X, where X stands for the design matrix. The larger the determinant of the design matrix, the less redundant are the columns of the design matrix, and, hence, the more information will they extract from the dependent variable. Because the determinant of the X'X matrix can be updated (as points are added or dropped from the design) without having to recompute the matrix from scratch (see Galil and Kiefer, 1980), the computations are usually noticeably faster when this criterion is selected, as compared to the A criterion.
A- (or T-) optimal (minimize trace of X'X inv.).
Select this option button to minimize the trace of the inverse of X'X, where X stands for the design matrix. Updating the inverse of X'X requires more computations than updating the determinant of X'X. Hence, if you select this option, the computations may require more time.
Number of points in final design
Enter the number of points for the final design in the Number of points in final design box.