Read more about: Linear Models and their Extensions

In Regression Analysis, a linear model is used to describe the variability of a dependent (response) variable based on an assumed linear relationship with several independent variables (also called predictors, covariates, regressors). In Analysis-of-variance (ANOVA), linear models are used for the comparison of several group means. The simplicity of linear models makes them widely applicable in different disciplines including economics, agricultural, biological, engineering and social sciences. These models usually have as a basic assumption that the error terms are independently and identically distributed.

Linear mixed models can be viewed as extensions of a linear model, because one not only models the mean (fixed effects), but also the covariance (random effects and error term). Hence, linear mixed models provide a flexible framework for modeling a wide range of data types, for example clustered, longitudinal and spatial data. Including random effects in the model, provides an efficient way to model correlated data, especially when the dependence follows a certain pattern (compound symmetry, AR, circular symmetry). To determine the appropriate covariance structure for random can be a challenging task. The inference in linear mixed models is often about the estimation of the fixed effects (e.g. BLUE) and prediction of random effects (BLUP). In particular, the invariance of BLUEs and BLUPs under linear (mixed) models with different covariance matrices has been of interest.

Matrix algebra is nowadays an efficient tool for both model specification and inference. For many patterned covariance matrices, their spectral properties have been extensively studied, and the corresponding inverse matrices have been explicitly derived.

Research topics in the area

• Estimation and Prediction in Linear Mixed Models, equality of BLUEs and BLUPs under models with different covariance matrices.
• Identifying Influential Observations in Linear and Nonlinear Models.
• Estimation and Testing Block Covariance Structures in Multivariate Normal Models.
• Estimation in Multivariate Normal Mixture Models.
• Influence Analysis in Generalized Linear Mixed Models.