Linear Mixed Model

Study guide: Exam MAS-II

A Linear Mixed Model (LMM) extends ordinary linear regression by including both fixed effects (parameters that apply to the entire population) and random effects (parameters that vary across groups or subjects). This accommodates correlated observations, repeated measures, and grouped data.

$\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{u} + \boldsymbol{\varepsilon}$

$\mathbf{u} \sim N(\mathbf{0}, \mathbf{G})$

$\boldsymbol{\varepsilon} \sim N(\mathbf{0}, \mathbf{R})$

$\mathbf{u} \perp \boldsymbol{\varepsilon}$

Model components:

Symbol	Name	Description
$\boldsymbol{\beta}$	Fixed effects	Population-level coefficients
$\mathbf{u}$	Random effects	Group-level deviations
$\mathbf{X}$ , $\mathbf{Z}$	Design matrices	For fixed and random effects
$\mathbf{G}$	Random effect covariance	Variance of group-level deviations
$\mathbf{R}$	Residual covariance	Within-group error structure

Marginal mean: $E[\mathbf{y}] = \mathbf{X}\boldsymbol{\beta}$ ; marginal covariance: $\text{Cov}(\mathbf{y}) = \mathbf{Z}\mathbf{G}\mathbf{Z}^\top + \mathbf{R}$
Parameters are estimated by ML or REML (Restricted Maximum Likelihood). Two covariance structures with the same fixed effects can be compared by a likelihood ratio test on either fit, but models that differ in their fixed effects must be compared with ML fits — the REML likelihood depends on the fixed-effects specification
Hierarchical Models are a special case of LMMs with nested random effects
LMMs connect to Bühlmann-Straub Credibility: the BLUP (Best Linear Unbiased Predictor) for a random effect corresponds to the credibility estimate

Random Intercept Model for Repeated Claims Claims from $i = 1, \ldots, 50$ policyholders are observed over $t = 1, \ldots, 4$ years. An actuary fits: $Y_{it} = \beta_0 + \beta_1 \text{age}_i + u_i + \varepsilon_{it}$ , where $u_i \sim N(0, \sigma_u^2)$ is a policyholder random intercept. Interpret $\sigma_u^2$ .

Answer $\sigma_u^2$ is the between-policyholder variance — how much policyholders differ in their baseline claim levels after accounting for age. A large $\sigma_u^2$ relative to $\sigma_\varepsilon^2$ indicates substantial policyholder heterogeneity not explained by age.