Understand how to use a hierarchical model.

A pricing analyst has five years of loss ratios for each of 40 territories. Territory means vary, sample sizes vary, and the grand mean is informative. A hierarchical model gives you per-territory estimates that borrow strength from the rest of the portfolio.

Why hierarchy. Observations inside the same group are correlated. OLS treats them as independent and understates standard errors. A hierarchical model puts the correlation in the model where it belongs and yields a posterior-style estimate for each group that pulls noisy small-sample groups toward the grand mean.

Index Level-1 observations $i=1,\dots ,n_j$ within Level-2 group $j=1,\dots ,J$. The simplest hierarchical model is:

$$y_{ij}={\beta }_0+u_j+{\epsilon }_{ij},u_j\sim N(0,{\tau }^2),{\epsilon }_{ij}\sim N(0,{\sigma }^2)$$

The fixed part is ${\beta }_0$; the random part is $u_j+{\epsilon }_{ij}$. Two variance components: between-group ${\tau }^2$ and within-group ${\sigma }^2$.

Intraclass correlation. The correlation between any two observations in the same group is:

$$\rho =\frac{{\tau }^2}{{\tau }^2+{\sigma }^2}$$

Common mistakes

• Reporting ${\bar{y}}_j$ as the hierarchical estimate. The shrunk estimate is ${\hat{\beta }}_0+{\hat{u}}_j$, not ${\bar{y}}_j$. Quoting 0.820 for territory C overstates expected loss ratio by about 5 points.
• Using ML log-likelihoods to compare variance structures. Variance-component comparisons require REML log-likelihoods (with identical fixed effects). ML biases ${\hat{\tau }}^2$ downward and gives misleading LRTs across random structures.
• Treating the Level-2 group identifier as a fixed effect with 200 dummies. That is the no-pooling OLS approach. Group means have no shrinkage, no borrowing of strength, and inflated MSE for thinly observed groups.

Bottom line

• Hierarchical means grouped data with random effects. Level-1 units (claims, years, students) nest inside Level-2 groups (territories, schools, drivers), and random effects capture group-to-group variation.
• Random intercept model: $y_{ij}={\beta }_0+u_j+{\epsilon }_{ij}$, with $u_j\sim N(0,{\tau }^2)$, ${\epsilon }_{ij}\sim N(0,{\sigma }^2)$. ICC $={\tau }^2/({\tau }^2+{\sigma }^2)$ is the share of total variance between groups.
• BLUP shrinkage: the predicted group effect ${\hat{u}}_j={\lambda }_j({\bar{y}}_j-{\hat{\beta }}_0)$, with ${\lambda }_j={\tau }^2/({\tau }^2+{\sigma }^2/n_j)$. Larger $n_j$ shrinks less.
• Shrinkage is toward the fixed-effect intercept, not zero; the group rate is ${\hat{\beta }}_0+{\hat{u}}_j$, not the raw mean ${\bar{y}}_j$.

Exam shortcut

If the question gives ${\tau }^2$, ${\sigma }^2$, $n_j$, and ${\bar{y}}_j$, compute the BLUP via ${\lambda }_j({\bar{y}}_j-{\hat{\beta }}_0)$ and add to ${\hat{\beta }}_0$; the same skeleton answers most random-intercept items. If two groups share ${\bar{y}}_j$ but differ in $n_j$, the larger group gets the larger ${\lambda }_j$ and lands farther from the grand mean; rank by $n_j$ without recomputing.

The full lesson (about 1,941 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B2

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