Multilevel models are needed for the analysis of data that have a hierarchical or clustered structure. Such data arise routinely in various fields, for instance, in educational research where pupils are nested within schools or in family studies with children nested within families. Clustered data may also arise as a result of the research design. For instance, repeated measures can be viewed as a series of measurements nested within individual subjects.
The models used in this chapter are multilevel regression models. The multilevel regression model assumes hierarchical data, with one response variable measured at the lowest level and explanatory variables at all existing levels. Conceptually the model is often viewed as a hierarchical system of regression equations. For example, assume we have data in J groups or contexts and a different number of individuals N. in each group. On the individual (lowest) level, we have the dependent variable Yf
Unless stated otherwise, the data used in the examples are artificial data. Data sets used in the examples are available from the authors.
and the explanatory variable Xy, and on the group level we have the explanatory variable Z.. Thus, we have a separate regression equation in each group:
The ¡i. are modeled by explanatory variables at the group level:
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