Generalized Linear Mixed Models (GLMMs) are a statistical modeling framework that extends Generalized Linear Models (GLMs) to account for the correlation and structure in the data due to hierarchical or nested factors. GLMMs are particularly useful when you have repeated measurements, data collected at multiple levels (e.g., individuals within groups), or other forms of clustered or hierarchical data.
Key features of GLMMs include:
- Generalization of GLMs: GLMMs extend the capabilities of GLMs, which are used for modeling relationships between a response variable and predictor variables, by allowing for the modeling of non-Gaussian distributions, like binomial or Poisson distributions, and by incorporating random effects.
- Random Effects: GLMMs include random effects to model the variability between groups or clusters in the data. These random effects account for the correlation and non-independence of observations within the same group.
- Fixed Effects: Like in GLMs, GLMMs also include fixed effects, which model the relationships between predictor variables and the response variable. Fixed effects are often of primary interest in statistical analysis.
- Link Function: Similar to GLMs, GLMMs use a link function to relate the linear combination of predictor variables and the response variable. Common link functions include the logit, probit, and log for binomial, Poisson, and Gaussian distributions, respectively.
- Likelihood Estimation: GLMMs typically use maximum likelihood estimation to estimate model parameters, including fixed and random effects.
Applications of GLMMs include analyzing data from various fields, such as epidemiology, ecology, psychology, and social sciences, where data often exhibit hierarchical or clustered structures. GLMMs are valuable for modeling the relationship between predictor variables and a response variable while accounting for the underlying correlation structure in the data, making them a versatile tool in statistical analysis.