Multilevel Analysis

Multilevel modeling is used in the analysis of data that have a clustered structure. Such data arise in various fields—for instance, in educational research, where pupils are nested in classes; in medical research, where patients are nested within hospitals; and also in political research, where individuals are nested in a social context. A crucial problem in the statistical analysis of clustered data is the dependencies between individual observations. For example, voters from the same city are not independent from one another, because they are all influenced by local policy. The statistical analysis performed on clustered data should account for this dependency. However, statistical analysis can also benefit from such dependency. The characteristics of different levels can be combined into one explanatory model and conclusions can be drawn about their effects, both at each level and in interaction. In the following sections, [Page 1638]the multilevel model is explained and illustrated by means of a political example.

The starting point for explaining the multilevel regression model is the idea that the dependent variable, allocated at the lowest level, is thought to be influenced by all distinguished levels. With respect to a two-level model, this principle can be seen in the following equation (the intercept-only model):

The dependent variable Y of an individual i in group j is decomposed into three different parts. Because there are no explanatory variables in the model yet, the explanatory part of the model consists only of the grand mean γ00. The unexplained part of the model consists of two parts: an error at the highest level (u0j) and an error at the lowest level (eij). This means that the total variance of the dependent variable is decomposed into two parts: the error variance at the lowest level (σ2e) and the error variance at the second level (σ2u0). A good explanation of the dependent variable is therefore based on both levels. Of course, one should keep in mind that the division over both levels can differ per variable. The more error variance there is at the lowest level, the less important the second level will be.

Adding explanatory variables of both levels gives the following equation:

In this equation, p explanatory variables of the lower level and q explanatory variables of the second level are added to the model. Interactions between variables of the lowest level are also called p, and interactions between variables of the second level are called q. Because these variables will explain a part of the variance at both levels, the errors will become smaller and the explained variance in the dependent variable can be calculated.

Interactions between variables of different levels take up a special position in the model. The influence of a lower-level variable on the dependent variable may depend on a second-level variable. This is called a cross-level interaction (or moderator effect). Adding cross-level interactions to the model, gives the final, and most elaborated, multilevel model:

Looking at this equation, one can see that not only are the cross-level interactions added to the model but there also are p extra error terms in the model: upjXpij. The errors upj, the unexplained parts of the regression coefficients of the lowest level, are multiplied with the lowest-level variables and are therefore different for different values of the X variables. This is called heteroskedasticity. In ordinary regression analysis, homoskedasticity is assumed, which means that the variance of the errors is independent of the values of the explanatory variables. Therefore, analyzing the model presented in Equation 3 requires a multilevel analysis. The variance of the errors upj is σ2up.

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