9. Specifying and Interpreting Models
Suppose we are interested in studying the variation in a poor health score, as a function of certain individual and neighborhood predictors. Let us assume that the researcher collected data on a sample of 50 neighborhoods and, for each of these neighborhoods, a random sample of individuals. We then have a two-level structure where the outcome is a poor health score, y, for individual i in neighborhood j. We will restrict this example to one individual-level categorical predictor, poverty, x1ij, coded as 0 if not poor and 1 poor, for every individual i in neighborhood j; and one neighborhood predictor, w1j, a socioeconomic deprivation index in neighborhood j.
Multilevel models operate by developing regression equations at each level of analysis. In the illustration considered here, models would have to be specified at two levels, level-1 and level-2. The model at level-1 can be formally expressed as:
1) yij=β0j+ β1χij+e0ij
In this level-1 model, β0j (associated with a constant, Χ0ij, which is a set of 1s, and therefore, not written) is the mean poor health score for the jth neighborhood for the non-poor group; β1 is the average differential in health score associated with individual poverty status (Χ1ij) across all neighborhoods. Meanwhile, eoij is the individual or the level-1 residual term. To make this a genuine two-level model we let β0j become a random variable, with an assumption that:
2) B0j = B0+u0j
where uoj is the random neighborhood-specific displacement associated with the overall mean poor health score (B0) for the non-poor group. Since we do not allow, at this stage, the average differential for the poor and non-poor group (B1) to vary across neighborhoods, uoj is assumed to be same for both groups. The equation in (2) is then the level-2 between-neighborhood model.