A design with non-zero D-efficiency can be used to estimate a model with all the main effects and all the two-factor interactions. The effect hierarchy assumption suggests that main effect estimation should be given more prominence than the estimation of two-factor interactions. The optimization functions in this package allow to user to create D-optimal designs that favor the main effect estimation over the estimation of two-factor interactions.
Doptimize. This function generates a single D-optimal design. The optimization function is specified using three parameters
$\alpha_1, \alpha_2, \alpha_3$. The following
function is then optimized:$$F = \alpha_1 D + \alpha_2 D_s + \alpha_3 D_1$$
Here $D$ is the $D$-efficiency of the design. The $Ds$- and $D1$-efficiency are defined further below.
When specifying $alpha_1=1$, $alpha_2=alpha_3=0$ the function generates design that optimizes the D-efficiency. For values of $alpha_2>0$ the main effects are given more weight. This allows the user to create designs with good estimation of the main effects, while still allowing to estimate all 2-factor interactions.
The $Ds$- and $D1$-efficiency are defined as follows. For a design $D$ the model matrix $X$ can be split into $[I X_1 X_2]$ with $I$ the intercept, $X_1$ the main effects and $X_2$ the second order effects. We let $X_{02}=[I X_2]$ and define $D1=(|X_1^T X_1|)^{1/(k+1)}$, $Ds=(|X^T X|/|X_{02}^T X_{02} |)^{1/k}$ with $k$ the number of factors of the design.
More details of the method and results of the the generation of optimal designs can be found the paper Eendebak (2015).
Doptimize and Defficiencies.For example code see the file example_Doptimize.R or example_design_usage.R included in the tests of the package.
p = Doptimize(N=32, k=7, nrestarts=20, alpha1=1, alpha2=1, alpha3=0)
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