Multilevel Modeling

10. Modeling Contextual Effects

The random-effects and the fixed-effects estimates for each neighborhood, meanwhile, are related (Jones and Bullen, 1994). The neighborhood-specific random intercept (β_0j) in a multilevel model is a weighted combination of the specific neighborhood coefficient in a fixed effects model (β^*_0j) and the overall multilevel intercept (β₀), in the following way: β_0j= w_jβ^*_0j+(1– w_j) β₀, with the overall multilevel intercept being a weighted average of all the fixed intercepts: β₀=(Σ w_jβ^*_0j)/Σ w_j.

Each neighborhood weight is the ratio of the true between-neighborhood parameter variance to the total variance, which additionally includes sampling variance resulting from observing a sample from the neighborhood. Consequently, the weights represent the reliability or precision of the fixed terms:

               σ²_uo
w_j=   ————
      υ²_{j +} σ²_uo

, where the random sampling variance of the fixed parameter is: 

          σ²_e
υ²_j=   ———
          n_j,

with n_j being the number of observations within each neighborhood.

When there are genuine differences between the neighborhoods and the sample sizes within a neighborhood are large, the sampling variance will be small in comparison to the total variance. As a result, the associated weight will be close to 1, with the fixed neighborhood effect being reliably estimated, and the random-effect neighborhood estimate close to the fixed-neighborhood effect. As the sampling variance increases, however, the weight will be less than 1 and the multilevel estimate will increasingly be influenced by the overall intercept based on pooling across neighborhoods. Shrinkage estimates allow the data to determine an appropriate compromise between specific estimates for different neighborhoods and the overall fixed estimate that pools information across places over the entire sample (Jones and Bullen, 1994).

Importantly, the fixed effects approach to modeling neighborhood differences using cross-sectional data is not a choice for a typical multilevel research question. This is especially true where there is an intrinsic interest in an exposure measured at the level of neighborhood such as the one specified in model (2); in such instances, a multilevel modeling approach is a necessity. This is because the dummy variables associated with the neighborhoods (measuring the fixed effects of each neighborhood) and the neighborhood exposure is perfectly confounded, and, as such, the latter is not identifiable (Fielding, 2004). Thus, the fixed effects specification to understand neighborhood differences is unsuitable for the sort of complex questions which multilevel modeling can address.