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Multilevel models (also known as hierarchical linear models, linear mixed-effect models, mixed models, nested data 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. [1]
A mixed model, mixed-effects model or mixed error-component model is a statistical model containing both fixed effects and random effects. [ 1 ] [ 2 ] These models are useful in a wide variety of disciplines in the physical, biological and social sciences.
In statistics, a generalized linear mixed model (GLMM) is an extension to the generalized linear model (GLM) in which the linear predictor contains random effects in addition to the usual fixed effects. [1] [2] [3] They also inherit from generalized linear models the idea of extending linear mixed models to non-normal data.
Best linear unbiased predictions" (BLUPs) of random effects are similar to best linear unbiased estimates (BLUEs) (see Gauss–Markov theorem) of fixed effects. The distinction arises because it is conventional to talk about estimating fixed effects but about predicting random effects, but the two terms are otherwise equivalent. (This is a bit ...
Moreover, the generalized linear mixed model (GLMM) is a special case of the hierarchical generalized linear model. In hierarchical generalized linear models, the distributions of random effect do not necessarily follow normal distribution. If the distribution of is normal and the link function of is the identity function, then hierarchical ...
Models for discrete longitudinal data. New York: Springer Science+Business Media, Inc. ISBN 978-0387251448. Covers non-linear models. Has SAS code. Pinheiro, Jose; Bates, Douglas M. (2000). Mixed-effects models in S and S-PLUS. New York, NY u.a: Springer. ISBN 978-1441903174. Uses S and S-plus but will be useful for R users as well.
Thus, in a mixed-design ANOVA model, one factor (a fixed effects factor) is a between-subjects variable and the other (a random effects factor) is a within-subjects variable. Thus, overall, the model is a type of mixed-effects model.
REML estimation is implemented in Surfstat, a Matlab toolbox for the statistical analysis of univariate and multivariate surface and volumetric neuroimaging data using linear mixed effects models and random field theory, [6] [7] but more generally in the fitlme package for modeling linear mixed effects models in a domain-general way. [8]