where g is the link function. Common choice for the link function might be the identity link for continuous data, log link for count data, and logit link for binary data. For example, the link functions for the Poisson and logistic regression models are g(a) = log(a) and g(a) = log(a/(1 – a)), respectively.
In addition to the marginal model, the covariance structure of the correlated observations for a given unit of yi is modeled as
where Ai is a diagonal matrix of variance functions and R(a) is the working correlation matrix of yi specified by the vector of parameters a. Various types of working correlation structures such as exchangeable or autoregressive can be used in the model.
The maximum likelihood estimator can be obtained by solving the above estimating equations iteratively:
Returning to the national suicide data from the previous section, we now 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. For the purpose of illustration, we considered 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 U.S. 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 (maximum marginal likelihood - MMLE) Poisson regression models. To this end, we fit a Poisson regression model with all main effects and two-way interactions using both GEE and a full likelihood mixed-effects model. Given the large sample sizes almost all terms in the model were statistically significant although of widely varying effect sizes. Table 2 displays a comparison of parameter estimates and standard errors for the GEE and mixed-effects models. In general, the GEE and MMLE parameter estimates were remarkably similar. The only nonsignificant terms were two terms in the race by age interaction. The comparison of rates for ages 5-14 versus 15-24 and 5-14 versus 25-44, did not depend on race.