Generalized Linear Mixed Models (GLMMs) combine aspects of Generalized Linear Models (GLMs) and mixed effects models, offering a versatile statistical approach. They are particularly valuable for handling non-normally distributed data with correlations and hierarchical structures, making them ideal for analyzing complex datasets like police fatal shootings. In this context, GLMMs serve to uncover demographic disparities, identify temporal patterns, analyze geographic distributions, quantify risk factors, and evaluate the impact of policy changes, contributing to a deeper understanding of this critical issue in law enforcement.
- 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.
5. Likelihood Estimation: GLMMs typically use maximum likelihood estimation to estimate model parameters, including fixed and random effects.