Tsutakawa (1988) Mixed Model for Analyzing Geographic Variability in Mortality Rates

Summary

Tsutakawa (1988) proposes a mixed model for mortality data cross-classified by geographic region and demographic group (age $\times$ sex). The model combines a multiplicative Poisson structure for expected rates with two layers of random effects: a group-specific gamma rate $p_{ij}$ (capturing extra-Poisson variability within each geographic/demographic cell) and a common regional relative-risk $z_i$ (shared across all demographic groups in a county). Hyperparameters are estimated by maximum likelihood via a profile-likelihood approach, and rates are estimated by their posterior means — an empirical Bayes procedure that shrinks small-area estimates toward expected values while leaving large-county estimates data-driven.

Key Claims

• Model structure: Marginalize over $p_{ij}$ to obtain a negative binomial for $y_{ij}$ with variance = Poisson + extra-Poisson component $\sigma^2_j(n_{ij}\cdot\theta_j\cdot z_i)^2$. The $z_i$ is shared across all demographic groups in region $i$, inducing positive correlation of age-group rates within a county.
• Posterior mean (Eq. 3.1): $E(p_{ij} \mid y, \xi, z_i)$ = weighted average of observed rate $y_{ij}/n_{ij}$ and expected rate $\theta_j\cdot z_i$, with weights $n_{ij}\cdot\bar{p}_{ij}$ and $\sigma_j^{-2}$. Small counties shrink heavily toward the expected rate; large counties are data-driven.
• Empirical Bayes estimation: Hyperparameters $\xi = (\theta, \sigma^2, \gamma)$ estimated by profile likelihood (maximize over $\gamma$, integrate over $z_i$); rates then estimated by posterior means given $\hat\xi$. Avoids full Bayesian integration of hyperparameters.
• Missouri lung cancer application: 115 counties $\times$ 4 male age groups (45–54, 55–64, 65–74, 75+), 1972–1981. Maximum Likelihood (ML) estimate $\hat\gamma = 58$ (low geographic variability). Likelihood Ratio (LR) test vs. multiplicative Poisson (fixed effects): statistic 51.4, strongly favouring the random effects model. Empirical Bayes (EB) standard deviations up to 15% smaller than raw observed rates.
• Innovation over Tsutakawa (1985): The earlier paper handled a single demographic stratum. This extension introduces a cross-classification of $I$ regions $\times$ $J$ demographic groups with $z_i$ shared across all $j$, capturing the realistic feature that county-level environmental exposures affect all age groups simultaneously.

Concepts Introduced or Extended

• Bayesian Hierarchical Model

Entities Mentioned

• Robert K. Tsutakawa

Quotes

"Modeling the variability between and within units can improve estimates, as demonstrated frequently in empirical Bayes examples." (p. 37)

"The likelihood ratio statistic (minus twice the log of the likelihood ratio) is therefore 51.4, suggesting the presence of common geographic effects across age groups." (p. 41)

My Take

This is a clear and well-executed empirical Bayes paper. The two-level random effects structure (region-level $z_i$ plus cell-level $p_{ij}$) is a natural extension of the single-stratum model and maps cleanly to the gamma-Poisson conjugate pair. The profile-likelihood approach to hyperparameter estimation was state-of-the-art in 1988 before MCMC became practical; full Bayesian treatment with Markov Chain Monte Carlo (MCMC) would later be explored by Tsutakawa-Shoop-Marienfeld (1985) and successors. The Missouri lung cancer example is small but well-chosen — the LR test provides compelling evidence for geographic random effects. A limitation noted in the paper is the lack of formal goodness-of-fit diagnostics for the latent variable model; standardised residuals $r_{ij}$ are used informally.