Multilevel Modeling

Any substantive interpretation of σ²_u0or α₁ is entirely dependent on the ‘appropriate’ specification of the other parts of the model: specifically, the fixed part of the model (i.e., individual-level and ecological-level confounders) as well as the random part specification of level-1.

Returning to our hypothetical example above and a substantive interpretation of the fixed-effect of neighborhood deprivation, the model that controlled just for personal income was probably inadequate. Personal socio-economic position is a complex and multidimensional construct that is often viewed as including income, educational attainment, social class, and (more recently) some measure of personal deprivation of hardship. Therefore, controlling for just one socio-economic factor is unlikely to fully adjust for personal socio-economic position. That is, residual confounding would remain. Nevertheless, the practice of adjusting for just one socio-economic factor before declaring neighborhood effects is not uncommon.

Even the model that fully adjusts for all the parallel individual-level variables that went into the construction of the neighborhood deprivation index may still be prone to residual confounding. First, for a fluctuating variable such as income, the average neighborhood income may be a better measure of your likely average income than personally declared (including reporting errors) income for the last year. Second, a measure or a personal socio-economic factor at one point in time does not capture dynamics over the lifecourse. While just one study, it is interesting to note that there was no association of neighborhood deprivation with mortality among a cohort of Scottish men after adjustment for social class at multiple points in the lifecourse (Davey Smith, Hart et al., 1997). Third, even if a satisfactorily full set of socio-economic factors across the life-course can be included in the analysis, the issue of (inevitable) measurement error of these covariates and resulting ‘resonant confounding’ of the neighborhood deprivation-mortality association remains (Marshall and Hastrup, 1996).

It is a common finding that such variation is small in social epidemiology studies, and often not statistically significant – be it before or after the adjustment for individual-level covariates. However, direct interpretation of these variations for neighborhood-health research must be done cautiously, especially when the outcome is not linear (e.g., binary). Rather, the major utility of σ²u0 lies statistically in allowing robust and reliable fixed effect estimates of level-2 exposures on health (i.e., estimating the size and precision of direct ecological effects and cross-level effect modification). Within the framework of the ‘context versus composition’ question, undue focus on the random variation may be misleading, especially when we have only observational data. Importantly, moderately strong and statistically significant ecological effects are often found in the absence of statistically significant between-neighborhood variation (Merlo, 2003).

This is a difficult, if not unanswerable, question for any observational study. By including further covariates, we gain from adjusting for further potential confounding. But, on the other hand, we risk including variables that are also on the causal pathway from neighborhood deprivation to health. For example, smoking may be patterned by one's context, meaning that adjusting for smoking is actually an attempt at quantifying the indirect ecological effect of neighborhood deprivation mediated by smoking.

This problem of variables that are both likely to be confounders and mediating variables in the association of an exposure with outcome is a perplexing problem in all observational studies. Short of conducting intervention studies on neighborhood deprivation, longitudinal studies with repeated measures of individual histories of changing neighborhood deprivation is one study design that may assist. That said, data-sets with both repeated measures and ecological-level variables are uncommon.

This chapter has attempted to illustrate that multilevel statistical modeling meshes well with multilevel thinking. But there are other approaches to modeling clustered data (e.g., Generalized Estimating Equation, or GEE). However, it is beyond the scope of this chapter to canvass these options in detail. Briefly though, the GEE approach will often deliver the same result. The key difference between the GEE and the multilevel approach is that the latter models the random variation as being of intrinsic interest, rather than a nuisance to overcome. As such, the choice of strategy is really dependent upon the conceptual motivation of the researcher (Heagerty and Zeger, 2000).

Since individuals do, to some extent, choose where to live, 'unobserved' individual or family factors can be mistaken for neighborhood effects. Similar 'unobserved' factors may also characterize neighborhoods and other spatial levels of analysis. This problem of endogeneity ---whereby an unobserved variable is related to a set of predictors and the response--- is only beginning to be addressed within the context of multilevel methods. The issue of endogeneity is even more complex in multilevel analysis because the unmeasured influences of omitted variables in the fixed part gets incorporated in the random part of the model, thereby violating the assumption of the independence of regressors and model disturbances (Rice, Jones et al., 1998). Three ways of dealing with this issue have been suggested in the multilevel literature. The first is to include data that actually "measure the crucial omitted variable" (Duncan and Raudenbush, 1999). The second is to apply specially developed multilevel instrumental variable estimation techniques (Spencer, 1998; Spencer and Fielding, 2000), i.e., the standard solution to endogeneity problems in single-level regression now extended to multilevel regression models. The third is to use a repeated measures, cross-classified structure, longitudinal fixed-effects model based on the nesting of panel observations for those who change neighborhoods within a cross-classified structure, with time varying covariates at each level of the analysis (Rasbash and Goldstein, 1994). This strategy, of course, is extremely data intensive and involves intensive computational demands. It is recommended that future applications be sensitive to the critical implication that the issue of endogeneity poses for multilevel research.

Even when one is not making uninformed predictions for specific contextual units, there are issues that need to be considered while making inferences for the population of contextual units. Simply because multilevel models treat the higher-level units as a sample drawn from a common population does not automatically mean that unconditional inferences are possible. There are crucial exchangeability judgments that are often neglected (Morris, 1995). Specifically, researchers need to ensure that the sample of neighborhoods comes from/can be exchanged with/is similar to the population that they wish to make inference about, with this being true for each specific neighborhood for which they have data. If there are reasons to believe that certain neighborhoods are truly independent or that they come from different populations they should not be regarded as exchangeable with the remaining random sample of neighborhoods and as such should be treated as fixed effects. While one option is to perform diagnostics after the model is fitted and/or conducting multilevel analysis with and without those neighborhoods that are believed to share exchangeable properties, a conceptually sound approach is to carefully plan the selection of neighborhoods at the design stage, analogous to the sampling of individuals in a survey (Draper, 1995).