The main difference between a linear mixed model and a generalized linear mixed model is that a linear mixed model is used to analyze continuous data, while a generalized linear mixed model is used to analyze data that has a non-normal distribution, such as count data. A linear mixed model uses the linear predictor equation, while a generalized linear mixed model uses a link function to transform the linear predictor equation. Additionally, a generalized linear mixed model can include categorical predictor variables, while a linear mixed model cannot.

## Generalized linear mixed model

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At the Statisticsan generalized mixed linear model (GLMM) is an extension of generalized linear model (GLM) where the linear predictor contains random effects beyond the usual fixed effects. They also inherit from the GLMs the idea of ​​extending mixed linear models do not stop-normal Dice.

GLMMs provide a wide range of models for analyzing pooled data, as differences between groups can be modeled as a random effect. These models are useful in analyzing many types of data, including longitudinal data.

## Model

GLMMs are generally defined so that, subject to random effects

$\displaystyle u$

you

\displaystyle you

the dependent variable

$\displaystyle y$

y

\displaystyle y

is distributed according to the exponential family
with its expectation related to the linear predictor

$\textstyle X\beta +Zu$

x
β
+
Z
you

\textstyle X\beta +Zu

$\textstyle g$

g

\textstyle g

:

$\displaystyle g(E[y\vert u])=X\beta +Zu$

g
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\displaystyle g(E[y\vert u])=X\beta +Zu

.

On here

$\textstyle X$

x

\text style X

and

$\textstyle \beta$

β

\textstyle\beta

are the design matrix of fixed effects and fixed effects, respectively;

$\textstyle Z$

Z

\textstyle Z

and

$\textstyle u$

you

\textstyle you

are the random effects design matrix and the random effects, respectively. To understand this brief definition, you first need to understand the definition of a generalized linear model and from one mixed model.

Generalized linear mixed models are special cases of hierarchical generalized linear models where random effects are normally distributed.

The complete probability

$\displaystyle \ln p(y)=\ln \int p(y\vert u)p(u)du$

ln

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\displaystyle \ln p(y)=\ln \int p(y\vert u)p(u)du

it has no general closed form, and integration over random effects is usually extremely computationally intensive. In addition to numerically approximating this integral (for example, via Gauss-Hermite square), methods motivated by the Laplace approximation were proposed. For example, the penalized near-likelihood method, which essentially involves repeated (ie, doubly iterative) fitting of a weighted normal mixed model with a working variable, is implemented by several commercial and open source statistical programs.

## Fitting a model

Assembly of GLMMs via maximum probability (how see AIC) involves integrating about random effects. In general, these integrals cannot be expressed in analytical way. Several approximate methods have been developed, but none have good properties for all possible models and data sets (e.g. ungrouped binary data are particularly problematic). For this reason, methods involving numerical quadrature or Markov Chain Monte Carlo increased in use as increasing computing power and advances in methods made them more practical.

The Akaike Information Criterion (AIC) is a common criterion for model selection. AIC estimates for GLMMs based on certain exponential family distributions were recently obtained.

## Software

• Several packages contributed to R provide GLMM functionality, including lme4 and glmm.
• GLMM can be installed using SAS and SPSS
• MATLAB also provides a function called “fitglme” to fit GLMM models.
• The Python Statsmodels package supports binomial and poisson implementation
• Julia’s MixedModels.jl package provides a function called GeneralizedLinearMixedModel that fits a GLMM to the given data.
• DHARMa: residual diagnosis for hierarchical regression models (multilevel/mixed) (utk.edu)

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