Chapter 3 - Linear Models, Generalized Linear Models (GLMs), and Random Effects Models: The Components of Hierarchical Models

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Hierarchical models (HMs) represent a sequence of probability models for dependent random variables, of which one is typically observed (this is the data), and one or more random variables are unobserved and thus latent. Typical examples of the latter are the occurrence or abundance state of a local population. To describe patterns in each random variable, we typically use linear models, and for parameters representing rates or probabilities, we use link transformations just like in generalized linear models (GLMs). To fully exploit the power of HMs, you therefore need to have a good practical understanding of both linear models and GLMs. This chapter provides an applied introduction to both topics, which are essential in all of applied statistical modeling. HMs naturally contain random effects—i.e., sets of parameters assumed to be drawn from some statistical distribution—hence, HMs can also be called mixed models. Mixed models are a confusing topic to many ecologists, and hence we briefly review random-effects, or mixed, models as well. Linear models express the effects of covariates on a response as a simple sum. The covariates may be continuous, as in a regression model, or categorical (factors), as in analysis of variance (ANOVA). Linear models are an extremely powerful and yet simple way to describe patterns in the expected response from some stochastic system—i.e., the thing we want to describe and understand with our statistical model. The combination of different covariates or factors, interactions, and polynomial terms of continuous covariates offer tremendous power to describe such patterns. GLMs extend the principle of linear models to responses that need not be normal, but may be Poisson, Bernoulli, or binomial, and hence to the modeling of counts, proportions, or probabilities. Random effects are sets of parameters, or latent variables, that are described as the outcome of a stochastic system, and hence are given a probability distribution in the statistical model. They are typically invoked for a variety of reasons that include the following: extending the scope of inference of an analysis, quantifying and partitioning variability in one set of parameters, quantifying and modeling covariability (correlations) between two or more sets of parameters, modeling hidden structure in the data, borrowing strength, and combining information from different studies or data sets. We give an example of a normal-normal mixed model and a Poisson-normal generalized linear mixed model (GLMM). This chapter is meant to provide an applied review of linear models, GLMs, and random effects regardless of the inference paradigm (frequentist or Bayesian), and we give R and BUGS code to fit all the models.