Multilevel Modeling

10. Modeling Contextual Effects

It is worth drawing parallels between a simple multilevel or a random-intercepts model (3) and the conventional OLS or fixed-effects regression model. Consider the fixed-effects model, whereby the neighborhood effect is estimated by including a dummy for each neighborhood, as shown below:

12) yij = β0 + βxij+βNj+(e0ij)

where Nj is a vector of dummy variables for N – 1. neighborhoods. The key conceptual difference between the fixed and the random-effects approach to modeling contexts is that while the fixed part coefficients are estimated separately, the random part differentials (u0j) are conceptualized as coming from a distribution (Goldstein, 2003).

This conceptualization results in three practical benefits (Jones and Bullen, 1994):

  1. Pooling information between neighborhoods, with all the information in the data being used in the combined estimation of the fixed and random part; in particular, the overall regression terms are based on the information for all neighborhoods;
  2. Borrowing strength, whereby neighborhood-specific relations that are imprecisely estimated benefit from the information for other neighborhoods; and
  3. Precision-weighted estimation, whereby unreliable neighborhood-specific fixed estimates are differentially down-weighted or shrunk toward the overall city-wide estimate. A reliably estimated within-neighborhood relation will be largely immune to this shrinkage.
Jones, K., Bullen, N. (1994) Contextual models of urban house prices: A comparison of fixed- and random-coefficient models developed by expansion. Economic Geography 70: 252-272.
Goldstein, H. (2003) Multilevel statistical models. London: Edward Arnold.