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Erschienen in:
2017 | OriginalPaper | Buchkapitel
3. H-Likelihood Approach to Random-Effect Models
verfasst von : Il Do Ha, Jong-Hyeon Jeong, Youngjo Lee
Erschienen in: Statistical Modelling of Survival Data with Random Effects
Verlag: Springer Singapore
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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 \(v=(v_{1}, \ldots , v_{q})^{T}\).