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D-optimal designs for multi-response linear mixed models

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Abstract

Linear mixed models have become popular in many statistical applications during recent years. However design issues for multi-response linear mixed models are rarely discussed. The main purpose of this paper is to investigate D-optimal designs for multi-response linear mixed models. We provide two equivalence theorems to characterize the optimal designs for the estimation of the fixed effects and the prediction of random effects, respectively. Two examples of the D-optimal designs for multi-response linear mixed models are given for illustration.

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Acknowledgements

Dr. Yue and Dr. Liu were partially supported by the National Natural Science Foundation of China under Grants 11471216 and 11871143. Dr. Wong was partially supported by a Grant from the National Institute of General Medical Sciences of the National Institutes of Health under Award Number R01GM107639. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

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Authors and Affiliations

  1. College of Science, Donghua University, Shanghai, 201600, China

    Xin Liu

  2. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Rong-Xian Yue

  3. Department of Biostatistics, University of California, Los Angeles, CA, 90095-1772, USA

    Weng Kee Wong

Authors
  1. Xin Liu
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  2. Rong-Xian Yue
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  3. Weng Kee Wong
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Corresponding author

Correspondence to Rong-Xian Yue.

Appendix

Appendix

We provide justifications for Theorems 1 and 2 using the same notation from the main text.

1.1 A.1. Proof of Theorem 1 and 2

Here we establish the proof using similar arguments in Prus and Schwabe (2016) and invoking the general equivalence theorem (Kiefer 1959).

For the D-optimal criterion \(\psi _D(\xi )\) defined in (12), a direct calculation shows that the directional derivative at \(\varvec{M}_1\) in the direction of \(\varvec{M}_2\) is

$$\begin{aligned} -\text {tr}\left\{ (\varvec{M}_1^{-1}+{\varvec{\Delta }})^{-1}\varvec{M}^{-1}_1(\varvec{M}_2-\varvec{M}_1)\varvec{M}_1^{-1}\right\} . \end{aligned}$$

As in Silvey (1980), a standard argument based on convex analysis then establishes Theorem 1.

To determine the directional derivative for the modified D-criterion \(\psi _D^{pred}(\xi )\) we write the criterion in the form

$$\begin{aligned} \psi _D^{pred}(\xi )=\ln |{\varvec{M}}^{-1}(\xi )| +(n-1)\left( \ln |m\varvec{K}^T\varvec{K}|+\ln |(m\varvec{K}^T\varvec{M}(\xi )\varvec{K}+ \varvec{I}_q)^{-1}|\right) , \end{aligned}$$

where \(m\varvec{K}\varvec{K}^T={\varvec{\Delta }}.\) As a consequence, this criterion can be identified as a compound criterion with the directional derivative:

$$\begin{aligned} -\text {tr}\left\{ (n-1)({\varvec{\Delta }}-{\varvec{\Delta }}(\varvec{M}_1^{-1} +{\varvec{\Delta }})^{-1}{\varvec{\Delta }})(\varvec{M}_2-\varvec{M}_1)+\varvec{M}_1^{-1}(\varvec{M}_2-\varvec{M}_1)\right\} , \end{aligned}$$

which proves Theorem 2. \(\square \)

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Liu, X., Yue, RX. & Wong, W.K. D-optimal designs for multi-response linear mixed models. Metrika 82, 87–98 (2019). https://doi.org/10.1007/s00184-018-0679-7

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