The SAGE Handbook of Multilevel Modeling Edited by Marc A. Scott, Jeffrey S. Simonoff and Brian D. Marx “Handbook_Sample.tex” — 2013/7/23 — 9:39 — page 52 1 The Multilevel Model Framework J e f f G i l l Washington University, USA A n d r e w J. W o m a c k University of Florida, USA 1.1 OVERVIEW then nested in clinics or hospitals, which are then nested in healthcare management sys- Multilevel models account for different lev- tems, which are nested in states, and so on. els of aggregation that may be present in In the classic example, students are nested data. Sometimes researchers are confronted in classrooms, which are nested in schools, with data that are collected at different lev- which are nested in districts, which are then els such that attributes about individual cases nested in states, which again are nested in the are provided as well as the attributes of nation. In another familiar context, it is often groupings of these individual cases. In addi- the case that survey respondents are nested in tion, these groupings can also have higher areas such as rural versus urban, then these groupings with associated data characteris- areas are nested by nation, and the nations in tics. This hierarchical structure is common regions. Famous studies such as the American in data across the sciences, ranging from National Election Studies, Latinobarometer, the social, behavioral, health, and economic Eurobarameter, and Afrobarometer are obvi- sciences to the biological, engineering, and ous cases. Often in population biology a physical sciences, yet is commonly ignored hierarchy is built using ancestral informa- by researchers performing statistical analyses. tion, and phenotypic variation is used to Unfortunately, neglecting hierarchies in data estimate the heritability of certain traits, in can have damaging consequences to subse- what is commonly referred to as the “animal quent statistical inferences. model.” In image processing, spatial relation- The frequency of nested data structures in ships emerge between the intensity and hue of the data-analytic sciences is startling. In the pixels.There are many hierarchies that emerge United States and elsewhere, individual vot- in language processing, such as topic of dis- ers are nested in precincts which are, in turn, cussion, document type, region of origin, or nested in districts, which are nested in states, intended audience. In longitudinal studies, which are nested in the nation. In health- more complex hierarchies emerge. Units or care, patients are nested in wards, which are groups of units are repeatedly observed over “Handbook_Sample.tex” — 2013/7/25 — 11:25 — page 3 4 THE MULTILEVEL MODEL FRAMEWORK a period of time. In addition to group hier- like a linear model or generalized linear archies, observations are also grouped by the model. The departure comes from the treat- unit being measured. These models are exten- ment of some of the coefficients assigned to sively used in the medical/health sciences to the explanatory variables. What can be done model the effect of a stimulus or treatment to modify a model when a point estimate is regime conditional on measures of interest, inadequate to describe the variation due to such as socioeconomic status, disease preva- a measured variable? An obvious modifica- lence in the environment, drug use, or other tion is to treat this coefficient as having a dis- demographic information. Furthermore, the tribution as opposed to being a fixed point. frequency of data at different levels of aggre- A regression equation can be introduced to gation is increasing as more data are generated model the coefficient itself, using information from geocoding, biometric monitoring, Inter- at the group level to describe the heterogeneity net traffic, social networks, an amplification in the coefficient. This is the heart of the mul- of government and corporate reporting, and tilevel model. Any right-hand side effect can high-resolution imaging. get its own regression expression with its own Multilevel models are a powerful and assumptions about functional form, linearity, flexible extension to conventional regression independence, variance, distribution of errors, frameworks.They extend the linear model and and so on. Such models are often referred to the generalized linear model by incorporating as “mixed,” meaning some of the coefficients levels directly into the model statement, thus are modeled while others are unmodeled. accounting for aggregation present in the data. What this strategy produces is a method of As a result, all of the familiar model forms for accounting for structured data through utiliz- linear, dichotomous, count, restricted range, ing regression equations at different hierar- ordered categorical, and unordered categor- chical levels in the data. The key linkage is ical outcomes are supplemented by adding that these higher-level models are describing a structural component. This structure clas- distributions at the level just beneath them for sifies cases into known groups, which may the coefficient that they model as if it were have their own set of explanatory variables itself an outcome variable. This means that at the group level. So a hierarchy is estab- multilevel models are highly symbiotic with lished such that some explanatory variables Bayesian specifications because the focus in are assigned to explain differences at the indi- both cases is on making supportable distribu- vidual level and some explanatory variables tional assumptions. are assigned to explain differences at the Allowing multiple levels in the same group level. This is powerful because it takes model actually provides an immense amount into account correlations between subjects of flexibility. First, the researcher is not within the same group as distinct from cor- restricted to a particular number of levels. The relations between groups. Thus, with nested coefficients at the second grouping level can data structures the multilevel approach imme- also be assigned a regression equation, thus diately provides a set of critical advantages adding another level to the hierarchy, although over conventional, flat modeling where these it has been shown that there is diminishing structures emerge as unaccounted-for hetero- return as the number of levels goes up, and geneity and correlation. it is rarely efficient to go past three levels What does a multilevel model look like? At from the individual level (Goel and DeGroot the core, there is a regression equation that 1981, Goel 1983). This is because the effects relates an outcome variable on the left-hand of the parameterizations at these super-high side to a set of explanatory variables on the levels gets washed out as it comes down the right-hand side. This is the basic individual- hierarchy. Second, as stated, any coefficient level specification, and looks immediately at these levels can be chosen to be modeled “Handbook_Sample.tex” — 2013/7/25 — 11:25 — page 4 1.2 BACKGROUND 5 or unmodeled and in this way the mixture of Lee and Bryk (1989). These applications con- these decisions at any level gives a combina- tinue today as education policy remains an torially large set of choices. Third, the form of important empirical challenge. Work in this the link function can differ for any level of the literature was accelerated by the development model. In this way the researcher may mix lin- of the standalone software packages HLM, ear, logit/probit, count, constrained, and other ML2, VARCL, as well as incorporation into forms throughout the total specification. the SAS procedure MIXED, and others. Addi- tional work by Goldstein (notably 1985) took the two-level model and extended it to sit- 1.2 BACKGROUND uations with further nested groupings, non- nested groupings, time series cross-sectional It is often the case that fundamental ideas in data, and more. At roughly the same time, statistics hide for a while in some applied a series of influential papers and applica- area before scholars realize that these are tions grew out of Laird and Ware (1982), generalizable and broadly applicable princi- where a random effects model for Gaussian ples. For instance, the well-known EM algo- longitudinal data is established. This Laird– rithm of Dempster, Laird, and Rubin (1977) Ware model was extended to binary out- was pre-dated in less fully articulated forms comes by Stiratelli, Laird, and Ware (1984) by Newcomb (1886), McKendrick (1926), and GEE estimation was established by Zeger Healy andWestmacott(1956), Hartley (1958), and Liang (1986). An important extension to Baum and Petrie (1966), Baum and Eagon non-linear mixed effects models is presented (1967), and Zangwill (1969), who gives the in Lindstrom and Bates (1988). In addition, critical conditions for monotonic conver- Breslow and Clayton (1993) developed quasi- gence. In another famous example, the core likelihood methods to analyze generalized lin- Markov chain Monte Carlo (MCMC) algo- ear mixed models (GLMMs). rithm (Metropolis et al. 1953) slept quietly Beginning around the 1990s, hierarchical in the Journal of Chemical Physics before modeling took on a much more Bayesian com- emerging in the 1990s to revolutionize the plexion now that stochastic simulation tools entire discipline of statistics. It turns out that (e.g. MCMC) had arrived to solve the result- hierarchical modeling follows this same sto- ing estimation challenges. Since the Bayesian ryline, roughly originating with the statistical paradigm and the hierarchical reliance on dis- analysis of agricultural data around the 1950s tributional relationships between levels have (Eisenhart 1947, Henderson 1950, Scheffé a natural affinity, many papers were produced 1956, Henderson et al. 1959). A big step and continue to be produced in the inter- forward came in the 1980s when education section of the two. Computational advances researchers realized that their data fit this during this period centered around customiz- structure perfectly (students nested in classes, ing MCMC solutions for particular problems classes nested in schools, schools nested (Carlin et al.