Variance Components

Study guide: Exam MAS-II

Variance components are the parameters that split the total variance of a Linear Mixed Model into a piece for each source of randomness — one per random effect plus the residual.

$\mathrm{Var}(\mathbf{y}) = \mathbf{V} = \mathbf{Z}\mathbf{D}\mathbf{Z}^{\top} + \mathbf{R}$

$\text{random intercept: } \mathrm{Var}(y_{ij}) = \sigma_b^{2} + \sigma^{2}$

$\mathbf{D}$ collects the random-effect variances (and their covariances); $\mathbf{R}$ is the residual covariance structure, usually $\sigma^2 \mathbf{I}$
They are the mixed-model counterpart of the Variance of Hypothetical Means ( $\sigma_b^2$ ) and Expected Value of Process Variance ( $\sigma^2$ ) in credibility
Variances are non-negative, so a fitted component at the boundary $\hat\sigma_b^2 = 0$ is a signal that the grouping factor explains nothing — the mixed model has collapsed to ordinary regression
They drive everything downstream: the Intraclass Correlation, the shrinkage of each BLUP, and the standard errors of the fixed effects
REML is the default estimator because ML variance components are biased downward

Reading Variance Components from Software Output A random-intercept model of claim severity by claims adjuster reports: Random effects: adjuster (Intercept) Variance 250, Residual Variance 1,000. How much of the variation is between adjusters, and what does one adjuster's total variance look like?

Answer $\mathrm{Var}(y_{ij}) = \sigma_b^{2} + \sigma^{2} = 250 + 1{,}000 = 1{,}250$ $\mathrm{ICC} = \frac{250}{1{,}250} = 0.20$ 20% of the variance in severity sits between adjusters and 80% within. Two claims handled by the same adjuster have correlation 0.20; two handled by different adjusters are uncorrelated.