# Multilevel model

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Multilevel model

Multilevel models (also hierarchical linear models, nested models, mixed models, random coefficient, random-effects models, random parameter models, or split-plot designs) are statistical models of parameters that vary at more than one level. These models can be seen as generalizations of linear models, although they can also extend to non-linear models. Although not a new idea, they have been much more popular following the growth of computing power and availability of software.

For example, in educational research it may be necessary to assess the performance of schools teaching reading by one method against schools teaching reading by a different method. It would be a mistake to analyse the data as though the pupils were simple random samples from the population of pupils taught by a particular method. Pupils are taught in classes, which are in schools. The performance of pupils within the same class will be correlated, as will the performance of pupils within the same school. These correlations must be represented in the analysis for correct inference to be drawn from the experiment.

## Level

The concept of level is the keystone of this approach. In an educational research example, the levels might be:

• pupil
• class
• school
• district

The researcher must establish for each variable the level at which it was measured. In this example "test score" might be measured at pupil level, "teacher experience" at class level, "school funding" at school level, and "urban" at district level.

## Uses of multilevel models

Multilevel models have been used in education research or geographical research, to estimate separately the variance between pupils within the same school, and the variance between schools. In psychological applications, the multiple levels are items in an instrument, individuals, and families. In sociological applications, multi-level models are used to examine individuals embedded within regions or countries. Different covariables may be relevant on different levels. They can be used for longitudinal studies, as with growth studies, to separate changes within one individual and differences between individuals.

Cross-level interactions may also be of substantive interest; for example, when a slope is allowed to vary randomly, a level-2 predictor may be included in the slope formula for the level-1 covariate. For example, one may estimate the interaction of race and neighborhood so that an estimate of the interaction between an individual's characteristics and the context.

## Applications to longitudinal (repeated measures) data

Multilevel models can be used to model change over time in a variable of interest. An overall change function is fitted to the whole sample and the parameters can be allowed to vary. For example, in a study looking at income growth with age, individuals might be assumed to show linear improvement over time. The exact intercept and slope could be allowed to vary across individuals. The simplest models assume that the effect of time is linear. Polynomial models can be specified to allow for quadratic or cubic effects of time. Models that are nonlinear in their parameters may also be fitted in some software. Nonlinear models may be more appropriate in representing various growth functions where there may be various asymptotes that limit the range of possible values. Models may also incorporate time-constant or time-varying covariates as predictors.

## Other resources

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