GLMM

Misc

• Packages
  • {glmmTMB}
    • Beta, Beta-Binomial, Beta-Ordinal, various extensions for Negative Binomial and Poisson, Log-Normal, Tweedie, Student-T, Gamma, Zero-Inflated, Truncated (See Manual >> nbinom2 for details)
    • Currently has an experimental implementation of fixed effect priors which can be a solution to complete separation errors, singular fits, and instability (Link).
    • Supports parallelization through OpenMP but might not be supported by Mac and has significant overhead on Windows.
  • {GLMMadaptive} - For binary/dichotomous data and count data with small counts and few repeated measurements, the accuracy of Laplace approximations (i.e. {glmmTMB}) is low. This package uses adaptive Gaussian quadrature rule which is the recommended, albeit more computationally intensive, approximation method for MLE in this situation.
    • Supported Distribution Families: Student’s-t, Beta, Zero-Inflated and Hurdle Poisson and Negative Binomial, Hurdle Log-Normal, Hurdle Beta, Gamma, and Censored Normal
    • Currently no nested or crossed random effects designs
    • Can penalize fixed effects with a student-t prior
    • Parallelization through {optimParallel} which doesn’t seem to have any OS limitations.
  • {glmmPen} (Vignette)
    • Fits high dimensional penalized generalized linear mixed models using the Monte Carlo Expectation Conditional Minimization (MCECM) algorithm.
    • Performs variable selection on both the fixed and random effects simultaneously through MCP, SCAD, or LASSO penalties.
    • Supports Binomial, Gaussian, and Poisson data with canonical links
• Papers
  • High performance implementation of the hierarchical likelihood for generalized linear mixed models: an application to estimate the potassium reference range in massive (codelectronic health records datasets
    • A high-performance, direct implementation of the hierarchical-likelihood for GLMMs in the R package TMB
    • The hierarchical likelihood approach to GLMMs is a methodologically rigorous framework for the estimation of GLMMs that is based on the Laplace Approximation (LA), which replaces integration with numerical optimization, and thus scales very well with dimensionality.
    • Code (+ parallelization option)
      • SingleRE: analyzes a Poisson GLM with a single random effect
      • MultipleRE: analyzes a Poisson GLM with multiple random effects

Model Equation

• Equation
  \[ g\{E(y_i \mid b_i)\} = \beta X_i + b_i Z_i, \]
  
  • \(g(⋅)\): A monotonic link function
  • \(X_i\): A design matrix for the fixed effects
  • \(\beta\): Fixed effects coefficients
  • \(Z_i\): A design matrix for the random effects that’s assumed to be a subset of \(X_i\)
  • \(b_i\): Random effects coefficients