Multilevel Analysis

Most of statistical inference is based on replicated observations of UNITS OF ANALYSIS of one type (e.g., a sample of individuals, countries, or schools). The analysis of such observations usually is based on the [Page 674]assumption that either the sampled units themselves or the corresponding RESIDUALS in some statistical model are independent and identically distributed. However, the complexity of social reality and social science theories often calls for more COMPLEX DATA SETS, which include units of analysis of more than one type. Examples are studies on educational achievement, in which pupils, teachers, classrooms, and schools might all be important units of analysis; organizational studies, with employees, departments, and firms as units of analysis; cross-national comparative research, with individuals and countries (perhaps also regions) as units of analysis; studies in GENERALIZABILITY THEORY, in which each factor defines a type of unit of analysis; and META-ANALYSIS, in which the collected research studies, the research groups that produced them, and the subjects or respondents in these studies are units of analysis and sources of unexplained variation. Frequently, but by no means always, units of analysis of different types are hierarchically nested (e.g., pupils are nested in classrooms, which, in turn, are nested in schools). Multilevel analysis is a general term referring to statistical methods appropriate for the analysis of data sets comprising several types of unit of analysis. The levels in the multilevel analysis are another name for the different types of unit of analysis. Each level of analysis will correspond to a POPULATION, so that multilevel studies will refer to several populations—in the first example, there are four populations: of pupils, teachers, classrooms, and schools. In a strictly nested data structure, the most detailed level is called the first, or the lowest, level. For example, in a data set with pupils nested in classrooms nested in schools, the pupils constitute Level 1, the classrooms Level 2, and the schools Level 3.

Hierarchical Linear Model

The most important methods of multilevel analysis are variants of REGRESSION analysis designed for hierarchically nested data sets. The main model is the HIERARCHICAL LINEAR MODEL (HLM), an extension of the GENERAL LINEAR MODEL in which the probability model for the errors, or residuals, has a structure reflecting the hierarchical structure of the data. For this reason, multilevel analysis often is called hierarchical linear modeling. As an example, suppose that a researcher is studying how annual earnings of college graduates well after graduation depend on academic achievement in college. Let us assume that the researcher collected data for a reasonable number of colleges—say, more than 30 colleges that can be regarded as a sample from a specific population of colleges, with this population being further specified to one or a few college programs—and, for each of these colleges, a random sample of the students who graduated 15 years ago. For each student, information was collected on the current income (variable Y) and the grade point average in college (denoted by the variable X in a metric where X = 0 is the minimum passing grade). Graduates are denoted by the letter i and colleges by j. Because graduates are nested in colleges, the numbering of graduates i may start from 1 for each college separately, and the variables are denoted by Yij and Xij. The analysis could be based for college j on the model

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