H-Likelihood Approach to Random-Effect Models

Abstract

In this chapter, we introduce an h-likelihood approach to the general class of statistical models with random effects. Consider a linear mixed model (LMM), for $i=1, \ldots , q$ and $j=1, \ldots , n_{i}$, $y_{ij}=x_{ij}^{T}{\varvec{\beta }}+v_{i}+e_{ij}$, where $y_{ij}$ is an observed random variable (response), $x_{ij}=(x_{ij1}, \ldots , x_{ijp})^{T}$ is a vector of covariates, ${\varvec{ \beta }}$ is a vector of fixed effects, $v_{i}\sim N(0,\alpha )$ is an i.i.d. random variable for the random effects, $e_{ij}\sim N(0,\phi )$ is an i.i.d random error or measurement error, and $v_{i}$ and $e_{ij}$ are independent. Parameters $\phi$ and ${\varvec{\alpha }}$ are the variance components. In this model, there are two types of unknowns; the fixed unknowns $\theta =(\beta ,\phi ,\alpha )^{T}$ and the random unknowns $v=(v_{1}, \ldots , v_{q})^{T}$.