Data Analysis Using Regression and Multilevel/Hierarchical Models

As with linear and logistic regressions, generalized linear models can be fit to multilevel structures by including coefficients for group indicators and then adding group-level models. We illustrate in this chapter with three examples from our recent applied research: an overdispersed Poisson model for police stops, a multinomial logistic model for storable voting, and an overdispersed Poisson model for social networks.

Overdispersed Poisson regression: police stops and ethnicity

We return to the New York City police example introduced in Sections 1.2 and 6.2, where we formulated the problem as an overdispersed Poisson regression, and here we generalize to a multilevel model. In order to compare ethnic groups while controlling for precinct-level variation, we perform multilevel analyses using the city's 75 precincts. Allowing precinct-level effects is consistent with theories of policing such as the “broken windows” model that emphasize local, neighborhood-level strategies. Because it is possible that the patterns are systematically different in neighborhoods with different ethnic compositions, we divide the precincts into three categories in terms of their black population: precincts that were less than 10% black, 10%–40% black, and more than 40% black. We also account for variation in stop rates between the precincts within each group. Each of the three categories represents roughly one-third of the precincts in the city, and we perform separate analyses for each set.

Overdispersion as a variance component

As discussed in Chapter 6, data that are fit by a generalized linear model are overdispersed if the data-level variance is higher than would be predicted by the model.