have used this kind of analysis to describe hospital mortality rates where cluster-specific (hospital-specific) effects represent how much the death rates for patients at a particular hospital differ from the national rates for patients with the same covariate values (i.e., matched patients). Longford (1994) provides extensive references to applications involving empirical Bayes estimates of random effects.
An alternative approach to the analysis of suicide rate data is based on Generalized Estimating Equations (GEEs) models, which were introduced by Liang and Zeger (1986) and Zeger and Liang (1986). The GEE method models the marginal expectation (i.e., average response for observations having the same covariates) of outcomes as a function of the explanatory variables. In this approach, the coefficients measure differences in the average response for a unit change in the predictor; in contrast the mixed-effect model produces predictions that are cluster-specific. An important property of the GEE method is that the parameter estimates are consistent even if the working correlation matrix is misspecified as long as the model for the mean is correct. A disadvantage of GEE is that it does not provide cluster-specific (e.g., county-level) suicide rate estimates adjusted for case mix (i.e., covariate effects). Appendix A outlines the statistical foundations of both fixed-effects and mixed-effects Poisson regression models, as well as the alternative approach based on GEE.
To illustrate how Poisson regression models can be used to estimate the effects of age, race, and sex on clustered (i.e., within counties) suicide rate data, this example considers the effects of age divided into five categories (5–14, 15–24, 25–44, 45–64, and 65 and older), sex, and race (African American versus Other) in the prediction of suicide rates across the United States for the period of 1996–1998. These categories were used so that there would be sufficient sample sizes available to compare observed and expected annual suicide rates for both GEE and mixed-effects Poisson regression models. In general, the GEE and mixed effect parameter estimates were remarkably similar.
Table 10-1 displays observed and expected annual suicide rates for both methods of estimation, broken down by age, sex, and race calculated from the parameter estimates. Inspection of Table 10-1 reveals several interesting results. In general, suicide increases with age, is higher in males, and is lower in African Americans. Black females have the lowest suicide rates across the age range. In non-Black males, the suicide rate increases with age whereas in all other groups, the suicide rate either is constant or decreases after age 65. Comparison of the expected frequencies for the GEE and mixed-effects models reveal that they are quite similar and the GEE does a slightly better job of predicting the observed rates.
A special feature of the mixed-effects model is the ability of estimating county-specific rates using empirical Bayes estimates of the random