A Crossed Random Effects Model for Unbalanced Data With Applications in Cross-Sectional and Longitudinal Research

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Abstract
Hierarchical linear models have found widespread application when the data have a nested structure—for example, when students are nested within classrooms (a two-level nested structure) or students are nested within classrooms and classrooms are nested within schools (a three-level nested structure). Often, however, the data will have a more complex nested structure. In Example 1, students are nested within both neighborhoods and schools; however, a school can draw students from multiple neighborhoods, and a neighborhood can send students to multiple schools. In Example 2, children are nested within classrooms during the first year of the study; however, each child finds himself or herself with a new teacher and a new set of classmates during each subsequent year of the study. By combining Lindley and Smith’s (1972) concepts of exchangeability between and within regressions, this article formulates a crossed random effects model that applies to such data, provides maximum likelihood estimates via the EM algorithm, and illustrates application to study (a) neighborhood and school effects on educational attainment in Scotland and (b) classroom effects on mathematics learning during the primary years in the United States.