A new class of designs for mixture-of-mixture experiments

Abstract

This paper deals with the implementation of an additive linear model for mixture of mixtures including major and minor components. Experimental designs, derived from designs for qualitative factors, are built for the two classical cases of type A or type B mixtures. With such designs the determination of the least square estimators of the model parameters or the determination of the D-efficiency can be achieved in an easy way.

Appendix A: Proof of proposition 3

Using Proposition $2$, the associated MoM design is balanced and its information matrix under constraints (5) is given by (see proposition $1$):

$$\begin{aligned} X^{*t}X^{*}=\hbox {diag}\left[ n,a_{1}\left( I_{q_{1}-1}+J_{q_{1}-1}\right) , \, \ldots \, ,a_{p}\left( I_{q_{p}-1}+J_{q_{p}-1}\right) \right] . \end{aligned}$$

It is well known that $\left( aI_{n}+bJ_{n}\right) ^{-1}=a^{-1}\left( I_{n}-\left( b/a+nb\right) J_{n}\right)$ so:

$$\begin{aligned} \left( X^{*t}X^{*}\right) ^{-1}=\hbox {diag}\left[ \frac{1}{n},\frac{1}{a_{1}}\left( I_{q_{1}-1}-\frac{1}{q_{1}}J_{q_{1}-1}\right) , \, \ldots \, ,\frac{1}{a_{p}}\left( I_{q_{p}-1}-\frac{1}{q_{p} }J_{q_{p}-1}\right) \right] . \end{aligned}$$

Thus ($\forall$ $i=1,\ldots ,p$ and $\forall$ $j=1,\ldots ,q_{i}-1$):

$$\begin{aligned} \hbox {Var}\left( \widehat{\beta }_{0}\right) =\frac{\sigma ^{2}}{n}\text { and }\hbox {Var}\left( \widehat{\beta }_{ij}\right) =\frac{\sigma ^{2}}{a_{i}}\left( 1-\frac{1}{q_{i}}\right) =\frac{\sigma ^{2}\left( q_{i}-1\right) }{q_{i}a_{i}} \end{aligned}$$

with $a_{i}=n\left( 1-q_{i}\alpha _{i}\right) ^{2}/q_{i}.$ For the least squares estimators we immediately obtain $\widehat{\beta }_{0}=1/n\left( \mathbb {I}_{n}^{t}Y\right) =\overline{Y}.$ Concerning $b_{ij}$ the block diagonal structure of $X^{*t}X^{*}$ implies that the parameters in $\gamma _{i}=\left( \beta _{i1},\beta _{12},\ldots ,\beta _{i\left( q_{i}-1\right) }\right) ^{t}$, associated to the major component $i$ ($i=1,\ldots ,p$), can be estimated by:

$$\begin{aligned} \begin{array}{ll} \widehat{\gamma }_{i} &{} =\dfrac{1}{a_{i}}\left( I_{q_{i}-1}-\dfrac{1}{q_{i} }J_{q_{i}-1}\right) A^{\left[ q_{i}\right] t}D_{i,\alpha _{i}}^{t}Y\\ &{} =\overset{}{\dfrac{1}{a_{i}}\left( I_{q_{i}-1}-\dfrac{1}{q_{i}}J_{q_{i} -1}\right) \left[ \left( 1-q_{i}\alpha _{i}\right) A^{\left[ q_{i}\right] t}X_{i}^{t}Y+\alpha _{i}A^{\left[ q_{i}\right] t}J_{q_{i}n}Y\right] }\\ &{} =\overset{}{\dfrac{q_{i}}{n\left( 1-q_{i}\alpha _{i}\right) }\left( I_{q_{i}-1}-\dfrac{1}{q_{i}}J_{q_{i}-1}\right) A^{\left[ q_{i}\right] t}X_{i}^{t}Y}\text { because }A^{\left[ q_{i}\right] t}\mathbb {I}_{q_{i}}=0. \end{array} \end{aligned}$$

Remember that $X_{i}$ is a binary matrix of the qualitative factors design so (denoting by $S_{ij}$ the sum of the responses for which minor component $j$ of major component $i$ is used):

$$\begin{aligned} X_{i}^{t}Y=\left( \begin{array}{c} S_{i1}\\ \vdots \\ S_{iq_{i}} \end{array} \right) \text { and }A^{\left[ q_{i}\right] t}X_{i}^{t}Y=\left( \begin{array}{c} S_{i1}-S_{iq_{i}}\\ \vdots \\ S_{i\left( q_{i}-1\right) }-S_{iq_{i}} \end{array} \right) . \end{aligned}$$

Applying the operator $\left( I_{q_{i}-1}-\left( 1/q_{i}\right) J_{q_{i} -1}\right)$ to the vector $A^{\left[ q_{i}\right] t}X_{i}^{t}Y$ gives (for $j=1,\ldots ,q_{i}-1$):

$$\begin{aligned} \begin{array}{ll} \hat{\beta }_{ij} &{} =\dfrac{q_{i}}{n\left( 1-q_{i}\alpha _{i}\right) }\left[ S_{ij}-S_{iq_{i}}-\left( \dfrac{q_{i}-1}{q_{i}}\right) \left( \dfrac{1}{q_{i}-1} {\displaystyle \sum \limits _{j=1}^{q_{i}-1}} S_{ij}-S_{iq_{i}}\right) \right] \\ &{} =\overset{}{\dfrac{q_{i}}{n\left( 1-q_{i}\alpha _{i}\right) }\left[ S_{ij}-\left( \dfrac{1}{q_{i}}S_{iq_{i}}+\dfrac{1}{q_{i}} {\displaystyle \sum \limits _{j=1}^{q_{i}-1}} S_{ij}\right) \right] }\\ &{} =\dfrac{q_{i}}{n\left( 1-q_{i}\alpha _{i}\right) }\left( S_{ij}-\dfrac{1}{q_{i}} {\displaystyle \sum \limits _{j=1}^{q_{i}}} S_{ij}\right) . \end{array} \end{aligned}$$

The design for qualitative factors is orthogonal, so $X_{i}^{t}\mathbb {I} _{n}=\left( n/q_{i}\right) \mathbb {I}_{q_{i}}$ and $S_{ij}$ is a sum involving $n/q_{i}$ terms. This remark lead us to $S_{ij}=\left( n/q_{i}\right) \overline{Y}_{ij}$ and then:

$$\begin{aligned} \hat{\beta }_{ij}=\dfrac{1}{\left( 1-q_{i}\alpha _{i}\right) }\left( \overline{Y}_{ij}-\dfrac{1}{q_{i}} {\displaystyle \sum \limits _{j=1}^{q_{i}}} \overline{Y}_{ij}\right) =\dfrac{1}{\left( 1-q_{i}\alpha _{i}\right) }\left( \overline{Y}_{ij}-\overline{Y}\right) . \end{aligned}$$

Note that all these results are true under the constraints (5), that is without the removed parameters $\beta _{iq_{i}}$ ($i=1,\ldots ,p$). We easily verify that the results are still true for these parameters using the relation:

$$\begin{aligned} \forall \, i=1,\ldots ,p, \beta _{iq_{i}}=-\sum _{j=1}^{q_{i}-1} \beta _{ij} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \square \end{aligned}$$