Multilevel model
Encyclopedia
Multilevel models are statistical models of parameters that vary at more than one level. These models can be seen as generalizations of linear model
s, 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 sample
s 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
.
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.
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.
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.
Linear model
In statistics, the term linear model is used in different ways according to the context. The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However the term is also used in time series analysis with a different...
s, 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 sample
Simple random sample
In statistics, a simple random sample is a subset of individuals chosen from a larger set . Each individual is chosen randomly and entirely by chance, such that each individual has the same probability of being chosen at any stage during the sampling process, and each subset of k individuals has...
s 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
Inference
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...
to be drawn from the experiment
Design of experiments
In general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments...
.
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
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
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.
See also
- HyperparameterHyperparameterIn Bayesian statistics, a hyperparameter is a parameter of a prior distribution; the term is used to distinguish them from parameters of the model for the underlying system under analysis...
- Mixed-design analysis of varianceMixed-design analysis of varianceIn statistics, a mixed-design analysis of variance model is used to test for differences between two or more independent groups whilst subjecting participants to repeated measures...
- Restricted randomizationRestricted randomizationMany processes have more than one source of variation in them. In order to reduce variation in processes, these multiple sources must be understood, and that often leads to the concept of nested or hierarchical data structures. For example, in the semiconductor industry, a batch process may operate...
Software
- Mplus
- HLM
- SASSAS SystemSAS is an integrated system of software products provided by SAS Institute Inc. that enables programmers to perform:* retrieval, management, and mining* report writing and graphics* statistical analysis...
using the MIXED procedure - MLwiN
- aML
- RR (programming language)R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....
- Latent Gold
- xtmixed in STATA (version 9 and after)
- SPSSSPSSSPSS is a computer program used for survey authoring and deployment , data mining , text analytics, statistical analysis, and collaboration and deployment ....
using the Advanced Statistics product