Hierarchical Model

Study guide: Exam MAS-II

A Hierarchical Model (also called a multilevel model or nested model) organizes data and parameters into levels, where units at a lower level are nested within units at a higher level. It is a special case of the Linear Mixed Model with nested random effects.

Two-level hierarchical model: Y_{ij} = \beta_0 + \beta_1 x_{ij} + u_j + \varepsilon_{ij}

u_j \sim N(0, \sigma_u^2)

\varepsilon_{ij} \sim N(0, \sigma_\varepsilon^2)

• Level 1: individual observations i within group j (e.g., claims within a policyholder)
• Level 2: groups j (e.g., policyholders within a territory)
• u_j is the random intercept for group j — the group-level deviation from the population mean
• The intraclass correlation coefficient (ICC) measures how much of the total variance is between groups: \text{ICC} = \frac{\sigma_u^2}{\sigma_u^2 + \sigma_\varepsilon^2}
• A high ICC indicates that knowing one observation's group provides substantial information about other observations in the same group
• Hierarchical models can also include random slopes, where the effect of a predictor x_{ij} varies across groups j

ICC in a Claims Hierarchical Model An actuary fits a hierarchical model to claims data from 200 territories, each with multiple policyholders. The estimated variance components are \hat{\sigma}_u^2 = 0.04 (between territories) and \hat{\sigma}_\varepsilon^2 = 0.16 (within territories). Calculate the ICC.

Answer \text{ICC} = \frac{0.04}{0.04 + 0.16} = \frac{0.04}{0.20} = 0.20 20% of the total variance in claims is attributable to territory-level differences. Territory membership explains a meaningful but not dominant share of claim variability.