## Hierarchical (multilevel) models for survey data

The basic idea of hierarchical modeling (also known as multilevel modeling, empirical Bayes, random coefficient modeling, or growth curve modeling) is to think of the lowest-level units (smallest and most numerous) as organized into a hierarchy of successively higher-level units. For example, students are in classes, classes are in schools, schools are in school districts, school districts are in states. We can then describe outcomes for an individual student as a sum of effects for the individual student, for her/his class, for the school, for the district and for the state. Each of these effects can often be regarded as one of an exchangeable collection of effects (e.g. all school-level effects) drawn from a distribution described by a variance component. There may also be regression coefficients at some or all of the levels.

Once a model of this type is specified, inferences can be drawn from available data for the population means at any level (school, class, district, etc.). These estimators, which can be regarded from a Bayesian perspective as posterior means or from a frequentist perspective as "Best Linear Unbiased Predictors" (BLUPs: see Robinson, Statistical Science, 1991, pp. 15-32), often have better properties than simple sample-based estimators using only data from the unit in question. This makes them useful in the problem of "small-area estimation," i.e. making estimates for units or domains for which there is a very limited amount of information (see Ghosh and Rao, Statistical Science, 1994, pp. 55-76).

Hierarchical models are often applicable to modeling of data from complex surveys, because usually a clustered or multistage sample design is used when the population has a hierarchical structure in the sense described above. Hierarchical modeling describes populations with complex structures in a way that differs fundamentally from the survey analysis software listed on our main page. Typically, classical survey data analysis estimates "finite population parameters" that describe a population of units at a given level (such as the mean test score of all students, or the regression of mean test scores on per-pupil expenditure across schools). The hierarchical structure of the population is considered because it affects the sample design and therefore the properties of the estimator used; for example, the variance of the mean student score is different when students are drawn as a cluster sample (where schools are clusters) than it would be if students are drawn by simple random sampling from a national population.

Hierarchical modeling, on the other hand, takes variation at different levels of the hierarchy as an object of inference, as described in the first paragraph of this page. A hierarchical model can involve estimands (separate regression coefficients at the school and student level, for example) that have no meaning without recognition of the hierarchical structure of the population. Obviously, the choice between these approaches must depend on the descriptive and scientific objectives of the study.

### Bibliography and further information

For more discussion of multilevel models, including principles, software, and applications, see the Centre for Multilevel Modeling at the University of Bristol. In particular, there are links to books and other information resources, a collection of software reviews, and some more introductory discussion.

The UCLA Academic Computing Services maintains a collection of links and software examples at its Multilevel Modeling Portal.

The following are tutorial introductions to hierarchical modeling for practitioners:

• Raudenbush and Bryk, Hierarchical Linear Models, Sage Publications, 2002 (first edition was 1992).