Polish Statistical Association

Central Statistical Office of Poland

**Subject:** Economics , Statistics & Probability

**ISSN:** 1234-7655

**eISSN:** 2450-0291

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Anasu Rabe / D. K. Shangodoyin / K. Thaga

**Keywords : **
correlated random effects,
covariance matrix,
linear Cholesky decomposition,
linear mixed models

**Citation Information : **
Statistics in Transition New Series. Volume 20,
Issue 4,
Pages 59-70,
DOI: https://doi.org/10.21307/stattrans-2019-034

**License : **
(CC BY-NC-ND 4.0)

**Received Date : **03-October-2017
/
**Published Online: ** 13-December-2019

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Modelling the covariance matrix in linear mixed models provides an additional advantage in making inference about subject-speciﬁc effects, particularly in the analysis of repeated measurement data, where time-ordering of the responses induces signiﬁcant correlation. Some difﬁculties encountered in these modelling procedures include high dimensionality and statistical interpretability of parameters, positive definiteness constraint and violation of model assumptions. One key assumption in linear mixed models is that random errors and random effects are independent, and its violation leads to biased and inefﬁcient parameter estimates. To minimize these drawbacks, we developed a procedure that accounts for correlations induced by violation of this key assumption. In recent literature, variants of Cholesky decomposition were employed to circumvent the positive deﬁniteness constraint, with parsimony achieved by joint modelling of mean and covariance parameters using covariates. In this article, we developed a linear Cholesky decomposition of the random effects covariance matrix, providing a framework for inference that accounts for correlations induced by covariate(s) shared by both ﬁxed and random effects design matrices, a circumstance leading to lack of independence between random errors and random effects. The proposed decomposition is particularly useful in parameter estimation using the maximum likelihood and restricted/residual maximum likelihood procedures.

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