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.
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This research has been supported by grant SCHW 531/16-1 of the German Research Foundation (DFG). The author thanks Radoslav Harman, Norbert Gaffke and Rainer Schwabe for fruitful discussions. The author is also grateful to two referees and the Editor-in-Chief, who significantly contributed to improving the presentation of the results.
Appendix
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)
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
which can be rewritten in the alternative form
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)
and can be represented as the partitioned matrix
where \({\mathbf {C}}_{11}=\mathrm {Cov}(\hat{{\varvec{\beta }}})\), \({\mathbf {C}}_{22}=\mathrm {Cov}\left( \hat{{\varvec{\gamma }}}-{\varvec{\gamma }}\right) \),
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 \({\varvec{\zeta }}\):
and
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):
where
and
where
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
which implies result (14).
For the vector \({\varvec{\varPsi }}\) of all individual treatment effects, it can be verified that
and the BLUP of \({\varvec{\varPsi }}\) is given by
Then the mean squared error matrix of \(\hat{{\varvec{\varPsi }}}\) is of general form (15) with
and
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
From (26), it follows that
where
for \(a_1=\frac{\sigma ^2\,N(Ku+1)}{K\,m}\) and \(a_2=\frac{\sigma ^2\,v\,(Ku+1)}{K(v+u)+1}\),
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
where
Then we obtain
which results in the following solutions of Eq. (40):
where \(s_{n,m}=K^2m^2v^2+2Km(m-n)(Ku+1)v+N^2(Ku+1)^2\), and
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.
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Prus, M. Optimal designs in multiple group random coefficient regression models. TEST 29, 233–254 (2020). https://doi.org/10.1007/s11749-019-00654-6
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DOI: https://doi.org/10.1007/s11749-019-00654-6
Keywords
- Optimal design
- Treatment and control
- Random effects
- Cluster randomization
- Mixed models
- Estimation and prediction