This section provides information about I-, D-, Bayesian I-, Bayesian D-, and Alias-Optimal designs.
is the default design type produced by the custom designer except when the RSM button has been clicked to create a full quadratic model.
has runs whose purpose is to lower the variability of the coefficients of this pre-stated model. By focusing on minimizing the standard errors of coefficients, a D-Optimal design may not allow for checking that the model is correct. It will not include center points when investigating a first-order model. In the extreme, a D-Optimal design may have just p distinct runs with no degrees of freedom for lack of fit.
maximizes D when
D-optimal split plot designs maximize D when
where V -1is the block diagonal variance matrix of the responses (Goos 2002).
Bayesian D-Optimality
is a modification of the D-Optimality criterion that effectively estimates the coefficients in a model, and at the same time has the ability to detect and estimate some higher-order terms. If there are interactions or curvature, the Bayesian D-Optimality criterion is advantageous.
works best when the sample size is larger than the number of Necessary terms but smaller than the sum of the Necessary and If Possible terms. That is, p + q > n > p. The Bayesian D-Optimal design is an approach that allows the precise estimation of all of the Necessary terms while providing omnibus detectability (and some estimability) for the If Possible terms.
uses the If Possible terms to force in runs that allow for detecting any inadequacy in the model containing only the Necessary terms. Let K be the (p + q) by (p + q) diagonal matrix whose first p diagonal elements are equal to 0 and whose last q diagonal elements are the constant, k. If there are 2-factor interactions then k = 4. Otherwise k = 1. The Bayesian D-Optimal design maximizes the determinant of (X'X + K2). The difference between the criterion for D-Optimality and Bayesian D-Optimality is this constant added to the diagonal elements corresponding to the If Possible terms in the X'X matrix.
is more appropriate than D-Optimality if your goal is to predict the response rather than the coefficients, such as in response surface design problems. Using the I-Optimality criterion is more appropriate because you can predict the response anywhere inside the region of data and therefore find the factor settings that produce the most desirable response value. It is more appropriate when your objective is to determine optimum operating conditions, and also is appropriate to determine regions in the design space where the response falls within an acceptable range. Precise estimation of the response therefore takes precedence over precise estimation of the parameters.
minimizes this criterion: If f '(x) denotes a row of the X matrix corresponding to factor combinations x, then
Bayesian I-Optimality
Bayesian I-Optimality has a different objective function to optimize than the Bayesian D-optimal design, so the designs that result are different. The variance matrix of the coefficients for Bayesian I-optimality is X'X + K where K is a matrix having zeros for the Necessary model terms and some constant value for the If Possible model terms.
The Bayesian I-Optimal design minimizes the average prediction variance over the design region:
where M is defined as before.
Specifically, let X1 be the design matrix corresponding to the model effects, and let X2 be the matrix of alias effects, and let