Model-Discrimination Designs
This webpage collects the two-level balanced and unbalanced optimal model-discrimination designs generated by the Particle Swarm Exchange (PSE) algorithm. The model-discrimination design aims to identify the underlying model that associates the experiment outcome with the control variables in the early stage of the experiments. It provides maximal estimation capability and discrimination capability for a set of competing models. On this webpage, we show the resulting model-discrimination designs investigated in our submitted paper below. In addition to the proposed PSE algorithm, we also developed and tested the codes of CP (Li and Wu, 1997) and SIBSSD (Phoa et al., 2016) algorithms for the balanced designs and CE algorithm (Meyer and Nachtsheim, 1995) for the unbalance designs.
Chen P.-Y., Chen, R.-B., Li, J.-P. and Li, W. (2021+). Particle Swarm Exchange Algorithms with Applications in Generating Optimal Model-Discrimination Designs. Preprint.
$\blacktriangleright$ CLICK HERE to the example codes for finding to model-discrimination designs.
$\blacktriangleright$ CLICK HERE to jump to the lists of model-discrimination design collections.
Model Space
Let $\mathcal{F}$ be the set of linear models with $m$ main effects and some two-factor interactions that are possibly useful for the experiment data. There are two types of model spaces, namely, $\mathcal{F}=MEPI_g$ and $\mathcal{F}=PMS_q$, considered in this website.
The “Main Effects Plus $g$ Interactions,” $MEPI_g$, model space (Agboto et al., 2010; Jones et al., 2007; Li and Nachtsheim, 2000; Sun, 1994) consists of linear models with all ($m$) main effects and $g$ active two-factor interactions. The higher-order interactions are assumed to be negligible. The size of the $MEPI_g$ model space is ${m \choose 2} \choose g$.
The “Projective Model Space of dimension $q$,” $PMS_q$, model space (Loeppky et al., 2007) consists of linear models with only $q$ out of $m$ main effects and all corresponding two-factor interactions of the $q$ main effects. The higher-order interactions are assumed to be negligible. The size of the $PMS_q$ model space is ${m\choose q}$.
Model-Discrimination Design Criteria
Suppose there are $n$ design points. Given model space $\mathcal{F}$ and two models $f_i$ and $f_j$ in $\mathcal{F}$, without loss of generality, let $f_i$ to be the reference model and $f_j$ to be the competing model. Define Let $f_i^{(j)}$ be the terms that are in $f_i$ but not in $f_j$, and $\mathbf{X}_i^{(j)}$ is the corresponding $n\times p_i^{(j)}$ model matrix of $f_i^{(j)}$. Then, the Fisher information matrix (FIM) for model $f_i^{(j)}$ is $\mathbf{M}_i^{(j)} = {\mathbf{X}_i^{(j)}}’(\mathbf{I} - \mathbf{H}_j)\mathbf{X}_i^{(j)}$ where $\mathbf{H}_j = \mathbf{X}_j\left(\mathbf{X}’_j\mathbf{X}_j\right)^{-1}\mathbf{X}’_j$.
There are three types of model-discrimination design criteria based on the FIM (Agboto et al., 2010) considered on this webpage.
The $\overline{AF}$-optimal design maximizes the average log-determinant values of the nonsingular FIMs for all pairs of $f_i$ and $f_j$ in $\mathcal{F}$, that is,
$\overline{AF} = \frac{1}{|\mathcal{F}|\left(|\mathcal{F}| - 1\right)}
\sum_{i=1}^{|\mathcal{F}|}\sum_{j \neq i}AF_{ij}
~\text{ where }~
AF_{ij} = \frac{1}{p_i^{(j)}}
\log{\det{\left(\mathbf{M}_i^{(j)}\right)}}$
The $\overline{A^S}$-optimal design minimizes the average trace values of the inverse matrix of the nonsingular FIMs for all pairs of $f_i$ and $f_j$ in $\mathcal{F}$, that is,
$\overline{A^S} = \frac{1}{|\mathcal{F}|\left(|\mathcal{F}| - 1\right)}
\sum_{i=1}^{|\mathcal{F}|}\sum_{j \neq i}A^S_{ij}
~\text{ where }~
A^S_{ij} = \frac{1}{p_i^{(j)}}
\mbox{trace}{\left(\mathbf{M}_i^{(j)}\right)^{-1}}$
The averaged Expected Noncentrality Parameter, $\overline{ENCP}$, -optimal design maximizes the average trace values of the nonsingular FIMs for all pairs of $f_i$ and $f_j$ in $\mathcal{F}$, that is,
$\overline{ENCP} = \frac{1}{|\mathcal{F}|\left(|\mathcal{F}| - 1\right)}
\sum_{i=1}^{|\mathcal{F}|}\sum_{j \neq i}ENCP_{ij}
~\text{ where }~
ENCP_{ij} = \frac{1}{p_i^{(j)}}
\mbox{trace}{\left(\mathbf{M}_i^{(j)}\right)}$