Centre for Multilevel Modelling

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What are multilevel models and why should I use them?

What are multilevel models?

Many kinds of data, including observational data collected in the human and biological sciences, have a hierarchical or clustered structure. For example, children with the same parents tend to be more alike in their physical and mental characteristics than individuals chosen at random from the population at large. Individuals may be further nested within geographical areas or institutions such as schools or employers. Multilevel data structures also arise in longitudinal studies where an individual’s responses over time are correlated with each other.

Multilevel models recognise the existence of such data hierarchies by allowing for residual components at each level in the hierarchy. For example, a two-level model which allows for grouping of child outcomes within schools would include residuals at the child and school level. Thus the residual variance is partitioned into a between-school component (the variance of the school-level residuals) and a within-school component (the variance of the child-level residuals). The school residuals, often called ‘school effects’, represent unobserved school characteristics that affect child outcomes. It is these unobserved variables which lead to correlation between outcomes for children from the same school.

Multilevel models can also be fitted to non-hierarchical structures. For instance, children might be nested within a cross-classification of neighbourhoods of residence and schools.

Why use multilevel models?

There are a number of reasons for using multilevel models:

  1. Correct inferences: Traditional multiple regression techniques treat the units of analysis as independent observations. One consequence of failing to recognise hierarchical structures is that standard errors of regression coefficients will be underestimated, leading to an overstatement of statistical significance. Standard errors for the coefficients of higher-level predictor variables will be the most affected by ignoring grouping.
  2. Substantive interest in group effects: In many situations a key research question concerns the extent of grouping in individual outcomes, and the identification of ‘outlying’ groups. In evaluations of school performance, for example, interest centres on obtaining ‘value-added’ school effects on pupil attainment. Such effects correspond to school-level residuals in a multilevel model which adjusts for prior attainment.
  3. Estimating group effects simultaneously with the effects of group-level predictors: An alternative way to allow for group effects is to include dummy variables for groups in a traditional (ordinary least squares) regression model. Such a model is called an analysis of variance or fixed effects model. In many cases there will be predictors defined at the group level, eg type of school (mixed vs. single sex). In a fixed effects model, the effects of group-level predictors are confounded with the effects of the group dummies, ie it is not possible to separate out effects due to observed and unobserved group characteristics. In a multilevel (random effects) model, the effects of both types of variable can be estimated.
  4. Inference to a population of groups: In a multilevel model the groups in the sample are treated as a random sample from a population of groups. Using a fixed effects model, inferences cannot be made beyond the groups in the sample.

An example

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