Optimal designs in multiple group random coefficient regression models

Abstract

The subject of this work is multiple group random coefficients regression models with several treatments and one control group. Such models are often used for studies with cluster randomized trials. We investigate A-, D- and E-optimal designs for estimation and prediction of fixed and random treatment effects, respectively, and illustrate the obtained results by numerical examples.

Appendix

1.1 Proofs of Theorems 1–4

To make use of the available results for estimation and prediction, we recognize the model (7) as a special case of the linear mixed model (see e.g., Christensen 2002)

$$\begin{aligned} {\mathbf {Y}}={\mathbf {X}} {\varvec{\beta }} + {\mathbf {Z}} {\varvec{\gamma }} + {\varvec{\varepsilon }}, \end{aligned}$$

(31)

where ${\mathbf {X}}$ and ${\mathbf {Z}}$ are known fixed and random effects design matrices, ${\varvec{\beta }}$ and ${\varvec{\gamma }}$ are vectors of fixed and random effects, respectively. The random effects ${\varvec{\gamma }}$ and the observational errors ${\varvec{\varepsilon }}$ are assumed to be uncorrelated and to have zero means and non-singular covariance matrices ${\mathbf {G}}=\text{ Cov }\,({\varvec{\gamma }})$ and ${\mathbf {R}}=\text{ Cov }\,({\varvec{\varepsilon }})$.

According to Henderson et al. (1959), for full-column rank design matrix ${\mathbf {X}}$ the BLUE $\hat{{\varvec{\beta }}}$ for ${\varvec{\beta }}$ and the BLUP $\hat{{\varvec{\gamma }}}$ for ${\varvec{\gamma }}$ are provided by the mixed model equations

$$\begin{aligned} \left( \begin{array}{c} \hat{{\varvec{\beta }}} \\ \hat{{\varvec{\gamma }}} \end{array} \right) = {\left( \begin{array}{cc} {\mathbf {X}}^\top {\mathbf {R}}^{-1}{\mathbf {X}} &{} {\mathbf {X}}^\top {\mathbf {R}}^{-1}{\mathbf {Z}} \\ {\mathbf {Z}}^\top {\mathbf {R}}^{-1}{\mathbf {X}} &{} {\mathbf {Z}}^\top {\mathbf {R}}^{-1}{\mathbf {Z}}+{\mathbf {G}}^{-1} \end{array} \right) ^{-1}} \left( \begin{array}{c} {\mathbf {X}}^\top {\mathbf {R}}^{-1}{\mathbf {Y}} \\ {\mathbf {Z}}^\top {\mathbf {R}}^{-1}{\mathbf {Y}} \end{array} \right) \!, \end{aligned}$$

(32)

which can be rewritten in the alternative form

$$\begin{aligned}&\hat{{\varvec{\beta }}}=\left( {\mathbf {X}}^\top ({\mathbf {Z}} {\mathbf {G}}{\mathbf {Z}}^\top +{\mathbf {R}})^{-1}{\mathbf {X}}\right) ^{-1}{\mathbf {X}}^\top ({\mathbf {Z}} {\mathbf {G}}{\mathbf {Z}}^\top +{\mathbf {R}})^{-1}{\mathbf {Y}}, \end{aligned}$$

(33)

$$\begin{aligned}&\hat{{\varvec{\gamma }}}={\mathbf {G}}{\mathbf {Z}}^\top ({\mathbf {Z}}{\mathbf {G}}{\mathbf {Z}}^\top + {\mathbf {R}})^{-1}({\mathbf {Y}}-{\mathbf {X}}\hat{{\varvec{\beta }}})\!. \end{aligned}$$

(34)

The mean squared error matrix of the estimator and predictor $\left( \hat{{\varvec{\beta }}}^\top ,\, \hat{{\varvec{\gamma }}}^\top \right) ^\top$ is given by (see Henderson 1975)

$$\begin{aligned} \text{ Cov }\,\left( \begin{array}{c} \hat{{\varvec{\beta }}} \\ \hat{{\varvec{\gamma }}}-{\varvec{\gamma }} \end{array} \right) ={\left( \begin{array}{cc} {\mathbf {X}}^\top {\mathbf {R}}^{-1}{\mathbf {X}} &{} {\mathbf {X}}^\top {\mathbf {R}}^{-1}{\mathbf {Z}} \\ {\mathbf {Z}}^\top {\mathbf {R}}^{-1}{\mathbf {X}} &{} {\mathbf {Z}}^\top {\mathbf {R}}^{-1}{\mathbf {Z}}+{\mathbf {G}}^{-1} \end{array} \right) ^{-1}} \end{aligned}$$

(35)

and can be represented as the partitioned matrix

$$\begin{aligned} \mathrm {Cov}\,\left( \begin{array}{c} \hat{{\varvec{\beta }}} \\ \hat{{\varvec{\gamma }}}-{\varvec{\gamma }} \end{array} \right) =\left( \begin{array}{cc} {\mathbf {C}}_{11} &{} {\mathbf {C}}_{12} \\ {\mathbf {C}}_{12}^\top &{} {\mathbf {C}}_{22} \end{array} \right) \!, \end{aligned}$$

(36)

where ${\mathbf {C}}_{11}=\mathrm {Cov}(\hat{{\varvec{\beta }}})$, ${\mathbf {C}}_{22}=\mathrm {Cov}\left( \hat{{\varvec{\gamma }}}-{\varvec{\gamma }}\right)$,

$$\begin{aligned} {\mathbf {C}}_{11}= & {} \left( {\mathbf {X}}^\top \left( {\mathbf {Z}} {\mathbf {G}}{\mathbf {Z}}^\top +{\mathbf {R}}\right) ^{-1}{\mathbf {X}}\right) ^{-1},\\ {\mathbf {C}}_{22}= & {} \left( {\mathbf {Z}}^\top {\mathbf {R}}^{-1}{\mathbf {Z}} +{\mathbf {G}}^{-1}-{\mathbf {Z}}^\top {\mathbf {R}}^{-1}{\mathbf {X}} ({\mathbf {X}}^\top {\mathbf {R}}^{-1}{\mathbf {X}})^{-1}{\mathbf {X}}^\top {\mathbf {R}}^{-1}{\mathbf {Z}}\right) ^{-1},\\ {\mathbf {C}}_{12}= & {} -{\mathbf {C}}_{11}\, {\mathbf {X}}^\top {\mathbf {R}}^{-1}{\mathbf {Z}}\left( {\mathbf {Z}}^\top {\mathbf {R}}^{-1}{\mathbf {Z}} +{\mathbf {G}}^{-1}\right) ^{-1}. \end{aligned}$$

For ${\varvec{\beta }}={\varvec{\theta }}_0$, ${\varvec{\gamma }}={\varvec{\zeta }}$, ${\mathbf {X}}=\mathrm {Vec}_{j=1}^J\left( {\mathbf {1}}_{r_j}\otimes \left( {\mathbf {1}}_K\,{\mathbf {f}}(j)^\top \right) \right)$, ${\mathbf {Z}}=\mathrm {Diag}_{j=1}^J\big ({\mathbf {I}}_{r_j}\otimes \big ({\mathbf {1}}_K\,{\mathbf {f}}(j)^\top \big )\big )$, ${\mathbf {G}}=\sigma ^2\,{\mathbf {I}}_N\otimes \text {block-diag}(u,\, v\,{\mathbf {I}}_{J-1})$ and ${\mathbf {R}}=\mathrm {Cov}({\varvec{\varepsilon }})=\sigma ^2\,{\mathbf {I}}_{NK}$ our model (7) is of form (31).

Using formulas (33) and (34) and employing some linear algebra, we obtain the following BLUE and BLUP for the fixed and random effects ${\varvec{\theta }}_0$ and ${\varvec{\zeta }}$:

$$\begin{aligned} \hat{{\varvec{\theta }}}_0=\left( \begin{array}{c}\bar{{\mathbf {Y}}}_J \\ \mathrm {Vec}_{j=1}^{J-1}\left( \bar{{\mathbf {Y}}}_j-\bar{{\mathbf {Y}}}_J\right) \end{array}\right) \!, \end{aligned}$$

and

$$\begin{aligned} \hat{{\varvec{\zeta }}}=\left( \begin{array}{c}\frac{K}{K(v+u)+1}\,\mathrm {Vec}_{j=1}^{J-1}\mathrm {Vec}_{i=N_{J-1}+1}^{N_j}\left( \left( \begin{array}{c}u \\ v\,{\mathbf {e}}_j\end{array}\right) \left( \bar{{\mathbf {Y}}}_{j,i}-\bar{{\mathbf {Y}}}_j\right) \right) \\ \frac{K}{Ku+1}\,\mathrm {Vec}_{i=N_{J-1}+1}^{N_j}\left( \left( \begin{array}{c}u \\ {\mathbf {0}}_{J-1}\end{array}\right) \left( \bar{{\mathbf {Y}}}_{J,i}-\bar{{\mathbf {Y}}}_J\right) \right) \end{array}\right) \!. \end{aligned}$$

Now the results (8)–(13) of Theorems 1 and 2 are straightforward to verify.

To prove Theorems 3 and 4, we firstly compute the blocks ${\mathbf {C}}_{11}$, ${\mathbf {C}}_{12}$ and ${\mathbf {C}}_{22}$ of the mean squared error matrix (36):

$$\begin{aligned} {\mathbf {C}}_{11}= & {} \frac{\sigma ^2(Ku+1)}{Km}\left( \begin{array}{cc} 1 &{} -{\mathbf {1}}_{J-1}^\top \\ -{\mathbf {1}}_{J-1} &{} \frac{(K(u+v)+1)m}{(Ku+1)n}{\mathbf {I}}_{J-1}+{\mathbf {1}}_{J-1}{\mathbf {1}}_{J-1}^\top \end{array}\right) \!, \end{aligned}$$

(37)

$$\begin{aligned} {\mathbf {C}}_{22}= & {} \sigma ^2\left( \begin{array}{cc}{\mathbf {C}}_{221} &{} {\mathbf {0}} \\ {\mathbf {0}} &{} {\mathbf {C}}_{222} \end{array}\right) \!, \end{aligned}$$

(38)

where

$$\begin{aligned} {\mathbf {C}}_{221}= & {} {\mathbf {I}}_{n(J-1)}\otimes \text {block-diag}(u,\, v\,{\mathbf {I}}_{J-1})\nonumber \\&-\,\frac{K}{K(v+u)+1}\mathrm {Diag}_{j=1}^{J-1}\left( \left( {\mathbf {I}}_n-\frac{1}{n}{\mathbf {1}}_n{\mathbf {1}}_n^\top \right) \otimes \left( \left( \begin{array}{c}u \\ v\,{\mathbf {e}}_j\end{array}\right) \left( \begin{array}{c}u \\ v\,{\mathbf {e}}_j\end{array}\right) ^\top \right) \right) \!,\\ {\mathbf {C}}_{222}= & {} {\mathbf {I}}_{m}\otimes \text {block-diag}(u,\, v\,{\mathbf {I}}_{J-1})-\frac{k}{Ku+1}\left( {\mathbf {I}}_m-\frac{1}{m}{\mathbf {1}}_m{\mathbf {1}}_m^\top \right) \\&\otimes \left( \left( \begin{array}{c}u \\ {\mathbf {0}}_{J-1}\end{array}\right) \left( \begin{array}{c}u \\ {\mathbf {0}}_{J-1}\end{array}\right) ^\top \right) \!, \end{aligned}$$

and

$$\begin{aligned} {\mathbf {C}}_{12}=-\sigma ^2\left( {\mathbf {C}}_{121}\,\, \vdots \,\, {\mathbf {C}}_{122}\right) \!, \end{aligned}$$

(39)

where

$$\begin{aligned} {\mathbf {C}}_{121}= & {} \mathrm {tVec}_{j=1}^{J-1}\left( \frac{1}{n}{\mathbf {1}}_n^\top \otimes \left( \left( \begin{array}{c} 0 \\ {\mathbf {e}}_j\end{array}\right) \left( \begin{array}{c}u \\ v{\mathbf {e}}_j\end{array}\right) ^\top \right) \right) \!,\\ {\mathbf {C}}_{122}= & {} \frac{1}{m}{\mathbf {1}}_m^\top \otimes \left( \left( \begin{array}{c} 1 \\ -{\mathbf {1}}_{J-1}\end{array}\right) \left( \begin{array}{c}u \\ {\mathbf {0}}_{J-1}\end{array}\right) ^\top \right) \!. \end{aligned}$$

We can observe that ${\varvec{\varPsi }}_0=({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1}){\varvec{\theta }}_0$ and then $\hat{{\varvec{\varPsi }}}_0=({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1})\hat{{\varvec{\theta }}}_0$ is the BLUE of ${\varvec{\varPsi }}_0$. Consequently, the covariance matrix of $\hat{{\varvec{\varPsi }}}_0$ can be determined using the formula

$$\begin{aligned} \mathrm {Cov}\left( \hat{{\varvec{\varPsi }}}_{0}\right) =({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1}){\mathbf {C}}_{11}({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1})^\top , \end{aligned}$$

which implies result (14).

For the vector ${\varvec{\varPsi }}$ of all individual treatment effects, it can be verified that

$$\begin{aligned} {\varvec{\varPsi }}=\left( {\mathbf {1}}_N\otimes ({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1})\right) {\varvec{\theta }}_0+\left( {\mathbf {I}}_N\otimes ({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1})\right) {\varvec{\zeta }} \end{aligned}$$

and the BLUP of ${\varvec{\varPsi }}$ is given by

$$\begin{aligned} \hat{{\varvec{\varPsi }}}=\left( {\mathbf {1}}_N\otimes ({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1})\right) \hat{{\varvec{\theta }}}_0+\left( {\mathbf {I}}_N\otimes ({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1})\right) \hat{{\varvec{\zeta }}}. \end{aligned}$$

Then the mean squared error matrix of $\hat{{\varvec{\varPsi }}}$ is of general form (15) with

$$\begin{aligned} {\mathbf {B}}_{1}= & {} \left( {\mathbf {1}}_N\otimes ({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1})\right) {\mathbf {C}}_{11}\left( {\mathbf {1}}_N\otimes ({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1})\right) ^\top ,\\ {\mathbf {B}}_{2}= & {} \left( {\mathbf {1}}_N\otimes ({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1})\right) {\mathbf {C}}_{12}\left( {\mathbf {I}}_N\otimes ({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1})\right) ^\top \end{aligned}$$

and

$$\begin{aligned} {\mathbf {B}}_{3}=\left( {\mathbf {I}}_N\otimes ({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1})\right) {\mathbf {C}}_{22}\left( {\mathbf {I}}_N\otimes ({\mathbf {0}}_{J-1}\, \vdots \, {\mathbf {I}}_{J-1})\right) ^\top . \end{aligned}$$

After applying (37)–(39), we obtain the result of Theorem 4.

1.2 Proof of Lemma 2

To determine the eigenvalues of $\text {Cov}\left( \hat{{\varvec{\varPsi }}}-{\varvec{\varPsi }}\right)$, we have to solve the equation

$$\begin{aligned} \text {det}\left( \text {Cov}\left( \hat{{\varvec{\varPsi }}}-{\varvec{\varPsi }}\right) -\lambda \,{\mathbf {I}}_{N} \right) =0. \end{aligned}$$

(40)

From (26), it follows that

$$\begin{aligned} \text {Cov}\left( \hat{{\varvec{\varPsi }}}-{\varvec{\varPsi }}\right) -\lambda \,{\mathbf {I}}_{N} =:\left( \begin{array}{cc} \tilde{{\mathbf {H}}}_{11} &{} {\mathbf {H}}_{12}\\ {\mathbf {H}}_{12}^\top &{} \tilde{{\mathbf {H}}}_{22} \end{array}\right) \!, \end{aligned}$$

where

$$\begin{aligned} \tilde{{\mathbf {H}}}_{11}=(a_1-\lambda )\,\frac{1}{n}{\mathbf {1}}_n{\mathbf {1}}_n^\top +(a_2-\lambda )\left( {\mathbf {I}}_{n}-\frac{1}{n}{\mathbf {1}}_n{\mathbf {1}}_n^\top \right) \end{aligned}$$

for $a_1=\frac{\sigma ^2\,N(Ku+1)}{K\,m}$ and $a_2=\frac{\sigma ^2\,v\,(Ku+1)}{K(v+u)+1}$,

$$\begin{aligned} \tilde{{\mathbf {H}}}_{22}=a_3\,\frac{1}{m}{\mathbf {1}}_m{\mathbf {1}}_m^\top +(\sigma ^2v-\lambda ){\mathbf {I}}_{m} \end{aligned}$$

for $a_3=\frac{\sigma ^2(K(v+u)+1)m}{K\,n}+\frac{a_1m}{N}$, and ${\mathbf {H}}_{12}$ is the same as in (26).

Then we compute the determinant of $\text {Cov}\left( \hat{{\varvec{\varPsi }}}-{\varvec{\varPsi }}\right) -\lambda \,{\mathbf {I}}_{N}$ as

$$\begin{aligned} \mathrm {det}\left( \text {Cov}\left( \hat{{\varvec{\varPsi }}}-{\varvec{\varPsi }}\right) -\lambda \,{\mathbf {I}}_{N} \right) =\mathrm {det}\left( \tilde{{\mathbf {H}}}_{11}\right) \mathrm {det}\left( \tilde{{\mathbf {H}}}_{22}-{\mathbf {H}}_{12}^\top \,\tilde{{\mathbf {H}}}_{11}^{-1}{\mathbf {H}}_{12}\right) \!, \end{aligned}$$

where

$$\begin{aligned}&\mathrm {det}\left( \tilde{{\mathbf {H}}}_{11}\right) =(a_1-\lambda )(a_2-\lambda )^{n-1},\\&\tilde{{\mathbf {H}}}_{11}^{-1}=\frac{1}{a_1-\lambda }\,\frac{1}{n}{\mathbf {1}}_n{\mathbf {1}}_n^\top +\frac{1}{a_2-\lambda }\left( {\mathbf {I}}_{n}-\frac{1}{n}{\mathbf {1}}_n{\mathbf {1}}_n^\top \right) \!,\\&\tilde{{\mathbf {H}}}_{22}-{\mathbf {H}}_{12}^\top \,\tilde{{\mathbf {H}}}_{11}^{-1}{\mathbf {H}}_{12}=\left( a_3-\frac{a_1^2\,m}{(a_1-\lambda ) n}\right) \frac{1}{m}{\mathbf {1}}_m{\mathbf {1}}_m^\top +\left( \sigma ^2v-\lambda \right) {\mathbf {I}}_{m},\\&\mathrm {det}\left( \tilde{{\mathbf {H}}}_{22}-{\mathbf {H}}_{12}^\top \,\tilde{{\mathbf {H}}}_{11}^{-1}{\mathbf {H}}_{12}\right) =\left( \sigma ^2v-\lambda \right) ^{m-1}\left( a_3-\frac{a_1^2\,m}{(a_1-\lambda ) n}+\sigma ^2v-\lambda \right) \!. \end{aligned}$$

Then we obtain

$$\begin{aligned}&\mathrm {det}\left( \text {Cov}\left( \hat{{\varvec{\varPsi }}}-{\varvec{\varPsi }}\right) -\lambda \,{\mathbf {I}}_{N} \right) \\&\quad =(a_2-\lambda )^{n-1}\left( \sigma ^2v-\lambda \right) ^{m-1}\left( (a_3+\sigma ^2v-\lambda )(a_1-\lambda )-\frac{a_1^2\,m}{n}\right) \!, \end{aligned}$$

which results in the following solutions of Eq. (40):

$$\begin{aligned} \lambda _1= & {} \frac{\sigma ^2\,v\,(Ku+1)}{K(v+u)+1},\\ \lambda _2= & {} \sigma ^2v,\\ \lambda _3= & {} \frac{\sigma ^2N}{2\,Kn\,m}(Km\,v+N(Ku+1)+\sqrt{s_{n,m}})\!, \end{aligned}$$

where $s_{n,m}=K^2m^2v^2+2Km(m-n)(Ku+1)v+N^2(Ku+1)^2$, and

$$\begin{aligned} \lambda _4=\frac{\sigma ^2N}{2\,Kn\,m}(Km\,v+N(Ku+1)-\sqrt{s_{n,m}})\!. \end{aligned}$$

After applying $n=N\,w$ and $m=N(1-w)$, we can see that $s_{n,m}=N^2s_w$ and we obtain the result of Lemma 2.