On properties of Toeplitz-type covariance matrices in models with nested random effects
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On properties of Toeplitz-type covariance matrices in models with nested random effects Yuli Liang1
· Dietrich von Rosen2,3 · Tatjana von Rosen4
Received: 28 September 2019 / Revised: 3 August 2020 © The Author(s) 2020
Abstract Models that capture symmetries present in the data have been widely used in different applications, with early examples from psychometric and medical research. The aim of this article is to study a random effects model focusing on the covariance structure that is block circular symmetric. Useful results are obtained for the spectra of these structured matrices. Keywords Covariance matrix · Circular block symmetry · Random effects model · Symmetry model · Eigenvalue · Eigenvector
1 Introduction Real populations which are of interests in various research areas such as medicine, biology, social studies, often exhibit hierarchical structures. For instance, in educational research, students are grouped within classes and classes are grouped within schools; in medical studies, patients are nested within doctors and doctors are nested within hospitals; in breeding studies, offsprings are grouped by sire and sires are grouped within some spatial factors (region); in political studies, voters are grouped within districts and districts are grouped within cities; in demographic studies, chil-
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Yuli Liang [email protected] Dietrich von Rosen [email protected] Tatjana von Rosen [email protected]
1
Department of Statistics, Örebro University School of Business, Örebro, Sweden
2
Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden
3
Department of Mathematics, Linköping University, Linköping, Sweden
4
Department of Statistics, Stockholm University, Stockholm, Sweden
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Y. Liang et al.
dren are grouped within families and families are grouped within a macro-context such as neighborhoods or ethnic communities. It has been recognized that such grouping induces certain dependence between population units within different clusters and, hence statistical models based upon independence assumption become invalid. Mixed linear models are routinely used for data analysis when the data exhibit dependence and/or various sources of variation can be identified, e.g., repeated measures, longitudinal and hierarchical data. In general, the mixed linear model has the following form Y = Xβ + Zγ + ,
(1)
where Y : n × 1 is a response vector, X: n × p is a known design matrix, β : p × 1 is a vector of fixed effects, γ : k × 1 is a vector of random-effects with a known incidence matrix Z : n × k, : n × 1 is a vector of random errors. It is assumed that γ ∼ N (0, G), ∼ N (0, R), and Cov(γ , ) = 0. Hence, Y is normally distributed with expectation Xβ and covariance matrix = ZG Z + R. In this article, we consider a two factor nested model. Let γ : n 2 × 1 and ξ : n 2 n 1 × 1 be two vectors of random effects, where ξ is nested within factor γ , and : n 2 n 1 ×1 be the vector of random errors. Further, it is assumed that γ ∼ N (0, 1 ), ξ
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