Specifically, a Doptimal design maximizes D, where D is defined as follows:
Bayesian Doptimality is a modification of the Doptimality criterion. The Bayesian Doptimality criterion is useful when there are potentially active interactions or nonlinear effects. See DuMouchel and Jones (1994) and Jones et al (2008).
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X is the model matrix as defined in Simulate Responses

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K is a diagonal matrix with values as follows:

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k = 0 for Necessary terms

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k = 1 for If Possible main effects, powers, and interactions involving a categorical factor with more than two levels

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k = 4 for all other If Possible terms

The prior distribution imposed on the vector of If Possible parameters is multivariate normal, with mean vector 0 and diagonal covariance matrix with diagonal entries . Therefore, a value is the reciprocal of the prior variance of the corresponding parameter.
The values for k are empirically determined. If Possible main effects, powers, and interactions with more than one degree of freedom have a prior variance of 1. Other If Possible terms have a prior variance of 1/16. In the notation of DuMouchel and Jones, 1994, .
To control the weights for If Possible terms, select Advanced Options > Prior Parameter Variance from the red triangle menu. See Advanced Options > Prior Parameter Variance.
Ioptimal designs minimize the integral I of the prediction variance over the entire design space, where I is given as follows:
Here M is the moments matrix:
See Simulate Responses. For further details, see Goos and Jones (2011).
The moments matrix does not depend on the design and can be computed in advance. The row vector f (x)’ consists of a 1 followed by the effects corresponding to the assumed model. For example, for a full quadratic model in two continuous factors, f (x)’ is defined as follows:
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X is the model matrix, defined in Simulate Responses

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K is a diagonal matrix with values as follows:

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k = 0 for Necessary terms

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k = 1 for If Possible main effects, powers, and interactions involving a categorical factor with more than two levels

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k = 4 for all other If Possible terms

The prior distribution imposed on the vector of If Possible parameters is multivariate normal, with mean vector 0 and diagonal covariance matrix with diagonal entries . (See Bayesian DOptimality for more details about the values k.)
Alias optimality seeks to minimize the aliasing between effects that are in the assumed model and effects that are not in the model but are potentially active. Effects that are not in the model but that are of potential interest are called alias effects. For details about aliasoptimal designs, see Jones and Nachtsheim (2011).
Specifically, let X1 be the model matrix corresponding to the terms in the assumed model, as defined in Simulate Responses. The design defines the model that corresponds to the alias effects. Denote the matrix of model terms for the alias effects by X2.
The entries in the alias matrix represent the degree of bias associated with the estimates of model terms. See The Alias Matrix in Technical Details for the derivation of the alias matrix.
The sum of squares of the entries in A provides a summary measure of bias. This sum of squares can be represented in terms of a trace as follows:
Designs that reduce the trace criterion generally have lower Defficiency than the Doptimal design. Consequently, alias optimality seeks to minimize the trace of subject to a lower bound on Defficiency. For the definition of Defficiency, see Optimality Criteria. The lower bound on Defficiency is given by the Defficiency weight, which you can specify under Advanced Options. See Advanced Options > D Efficiency Weight.