Multilevel Modeling

12. Multilevel Residual Mapping

While it is the variances that are estimated in a multilevel model at each of the specified levels, it is possible to estimate place-specific (posterior) residuals at each of the contextual levels. Residual mapping is an extremely useful application of multilevel models, especially when the interest lies in simultaneous multiple geographies, and when all the units at each of the geographic level can be observed in the analysis (e.g., the census) (Subramanian, Duncan et al., 2001). In order to appreciate this, Figure 9 unpacks the way in which residuals are constructed when there are two spatial levels.

The region-specific residuals (υ0k) at level-3 represent the difference from the fixed average line, β0. For example, REGION-A will have a negative residual given its lower rate of poor health compared to the overall average. REGION-B, in contrast, will have a positive residual, given its high rate compared to the average. Neighborhood-specific residuals (u0jk) at level-2, meanwhile, are measured as the difference from their respective regions to which they belong (hence, the subscript jk) and not as a difference from the fixed average.

Consider NEIGHBORHOOD-1 in REGION-A in Figure 9. From a conventional perspective, this neighborhood would be considered a ‘healthy place’ (given that it is below the average). From a multilevel perspective, however, this neighborhood would be considered an ‘unhealthy place’ given the healthy context of the region (low rate) to which it belongs, and as such, it would have a positive neighborhood residual. Such ideas are extremely useful in social policy (Goldstein and Spiegelhalter, 1996).

Figure 9

Figure describing the way in which residuals are constructed when there are two spatial levels as described in text.

Goldstein, H., Spiegelhalter, D. (1996) League tables and their limitations: Statistical issues in comparisons of institutional performance (with discussion). Journal of the Royal Statistical Society Series A 159: 385-443.
Subramanian, S. V., Duncan, C., et al. (2001) Multilevel perspectives on modeling census data. Environment and Planning A 33(3): 399-417.