Linear mixed-effects models

In Chapter 3, linear mixed models are introduced and specified. Linear mixed modeling is a statistical approach with widespread applications in longitudinal data analysis. Given a considerable body of linear mixed modeling techniques, this chapter is focused on the general specifications, basic inferences, and estimating procedures of the fixed effects in the presence of the specified random effects. For analytic convenience, three specific cases of linear mixed models are delineated first, based on which statistical inferences of linear mixed models are then formalized. Next, the maximum likelihood estimator is specified with a focus on the estimation of the fixed-effects. It is emphasized that when the sample size is small, the maximum likelihood estimator yields a variance estimate that is biased downward because a penalty term is missed with unknown population means. Some other statistical procedures in linear mixed models are also presented in this chapter. Lastly, an illustration is provided to display how to apply linear mixed models in empirical research.