Parameter estimation and inference in generalized linear mixed models

Generalized linear mixed models are models in statistical modeling that have a wide range of applications and can incorporate both fixed and random effects. These models provide flexibility beyond traditional linear models by accommodating various distribution types and link functions, allowing them to adapt to diverse types of data. They are particularly well-suited for analyzing complex data structures such as repeated measures, hierarchical data, and nested data sets. Widely used in the analysis of complex and hierarchical data, these models are crucial for accurate parameter estimation and the reliability of the obtained results. The aim of this study is to examine parameter estimation methods and inference in generalized linear mixed models. To this end, the fundamental concepts and applications of both linear models and generalized linear models have been initially reviewed. The structure of the models, various types of distributions, systematic components, and link functions are discussed in detail. Special emphasis is placed on commonly used distributions such as Poisson, Binomial, Normal, and Gamma distributions, and the likelihood and log-likelihood functions and properties of the exponential family of distributions are examined. Secondly, the structure of generalized linear mixed models and parameter estimation methods for these models are addressed. Additionally, various methods used to estimate the likelihood function in GLMMs are examined. In the parameter estimation section, methods such as Penalized Quasi-Likelihood and Pseudo Quasi-Likelihood, as well as the Laplace approximation, Gauss-Hermite Quadrature, and Adaptive Gauss-Hermite Quadrature methods, are highlighted. In the inference section, analyses and interpretations regarding the estimated parameters and model fit in GLMMs are shared. The findings and discussion part of the study includes applications conducted using the Bellbird and Owl datasets. These applications were examined using the R programming language.