Generalized linear mixed model
Encyclopedia
In statistics
, a generalized linear mixed model (GLMM) is a particular type of mixed model
. It is an extension to the generalized linear model
in which the linear predictor contains random effects in addition to the usual fixed effects. These random effects are usually assumed to have a normal distribution.
Fitting such models by maximum likelihood
involves integrating
over these random effects. In general, these integrals cannot be expressed in analytical form. Various approximate methods have been developed, but none has good properties for all possible models and data set
s (ungrouped binary data being particularly problematic). For this reason, methods involving numerical quadrature or Markov chain Monte Carlo
have increased in use as increasing computing power and advances in methods have made them more practical.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a generalized linear mixed model (GLMM) is a particular type of mixed model
Mixed model
A mixed model is a statistical model containing both fixed effects and random effects, that is mixed effects. These models are useful in a wide variety of disciplines in the physical, biological and social sciences....
. It is an extension to the generalized linear model
Generalized linear model
In statistics, the generalized linear model is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to...
in which the linear predictor contains random effects in addition to the usual fixed effects. These random effects are usually assumed to have a normal distribution.
Fitting such models by maximum likelihood
Maximum likelihood
In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....
involves integrating
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
over these random effects. In general, these integrals cannot be expressed in analytical form. Various approximate methods have been developed, but none has good properties for all possible models and data set
Data set
A data set is a collection of data, usually presented in tabular form. Each column represents a particular variable. Each row corresponds to a given member of the data set in question. Its values for each of the variables, such as height and weight of an object or values of random numbers. Each...
s (ungrouped binary data being particularly problematic). For this reason, methods involving numerical quadrature or Markov chain Monte Carlo
Markov chain Monte Carlo
Markov chain Monte Carlo methods are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its equilibrium distribution. The state of the chain after a large number of steps is then used as a sample of the...
have increased in use as increasing computing power and advances in methods have made them more practical.