We consider an analysis-of-variance setting involving multiple variance components, one for each source of variation, arising from a $5 \times 5 \times 2$ split-plot latin square. The experiment contains five whole-plot treatments (A, B, C, D, E), with each plot divided into two subplots labeled 1 and 2. Each row of the variance-components table corresponds to a distinct source of variation, giving a total of nine variance components.

Let the nine variance parameters be $\theta_1^2, \ldots, \theta_9^2$. A simple noninformative analysis assigns a uniform prior:

$\displaystyle p(\theta_1, \ldots, \theta_9) \propto 1.$

A more structured approach places these parameters into a hierarchical model. Each $\theta_k$ is given a half-Cauchy prior with scale parameter $A$:

  • $\theta_k \sim \text{half-Cauchy}(A)$,
  • $A$ has a uniform prior.

This hierarchical model allows most variance components to be small while still permitting occasional large values, which matches typical Anova settings where most factors contribute little but a few may be influential.

Superpopulation vs. Finite-Population Standard Deviations

Inferences for the latin square example illustrate the contrast between uniform priors and hierarchical half-Cauchy priors for the variance parameters $\theta_k$.

Superpopulation interpretation

Each $\theta_k$ represents the standard deviation for an entire conceptual population of group effects, only a few of which were observed. With limited degrees of freedom and strong dependence among parameters in the latin-square structure, the uniform prior allows extremely large values for $\theta_k$. As shown in the left panel of Figure, this leads to very wide posterior intervals.

The hierarchical half-Cauchy model shrinks these intervals, especially at the upper end (e.g., the 75th and 97.5th percentiles), while leaving the medians largely unchanged. This demonstrates the benefits of partial pooling in situations where each variance component is only weakly identified.

Finite-population interpretation

For each variance component, the corresponding finite-population standard deviation is denoted $s_\alpha$. This is the standard deviation of the observed linear coefficients within that batch, after applying the necessary projection to the constraint space.

  • In simple hierarchical models, $s_\alpha$ is just the sample standard deviation of the $J$ values of $\alpha_j$.
  • In more complex structures, such as the split-plot latin square, $s_\alpha$ is the root mean square of the residual coefficients after enforcing the required constraints.

When degrees of freedom are small, $s_\alpha$ is typically better estimated than the superpopulation parameter $\theta_k$, because $s_\alpha$ depends directly on the observed coefficients rather than on a variance parameter with a highly uncertain posterior.

Figure shows posterior inferences for $s_\alpha$ under both the uniform and hierarchical half-Cauchy priors. The hierarchical model again yields sharper intervals, especially for variance components with very few degrees of freedom. In this example, the posterior for the scale parameter $A$ is centered near 1.8, with a 95% interval of $[0.5,, 5.1]$.

Key Points

  • Each source of variation in the split-plot latin square corresponds to its own variance component $\theta_k^2$.
  • A uniform prior allows very large variance estimates due to limited information and parameter entanglement.
  • A hierarchical half-Cauchy prior shrinks extreme values and produces more stable inferences.
  • Superpopulation standard deviations $\theta_k$ describe variability in hypothetical new groups.
  • Finite-population standard deviations $s_\alpha$ describe variability among the observed groups and are often more precisely estimated.
  • Hierarchical modeling is particularly effective when variance components are individually weakly identified.