Foundations of Linear and Generalized Linear Models
Written by a highly-experienced author, Foundations of Linear and Generalized Linear Models is a clear and comprehensive guide to the key concepts and results of linear statistical models. The book presents a broad, in-depth overview of the most commonly used statistical models by discussing the theory underlying the models, R software applications, and examples with crafted models to elucidate key ideas and promote practical model building. The book begins by illustrating the fundamentals of linear models, such as how the model-fitting projects the data onto a model vector subspace and how orthogonal decompositions of the data yield information about effects of explanatory variables. Subsequently, the book covers the most popular generalized linear models, which include binomial and multinomial logistic regression for categorical data, and Poisson and negative binomial loglinear models for count data. Focusing on the theoretical underpinnings of these models, Foundations of Linear and Generalized Linear Models also features: An introduction to quasi-likelihood methods that require weaker distributional assumptions, such as generalized estimating equation methods An overview of linear mixed models and generalized linear mixed models with random effects for clustered correlated data, Bayesian modeling, and extensions to handle problematic cases such as high-dimensional problems Numerous examples that use R software for all text data analyses More than 400 exercises for readers to practice and extend the theory, methods, and data analysis A supplementary website with datasets for the examples and exercises An invaluable textbook for upper-undergraduate and graduate-level students in statistics and biostatistics courses, Foundations of Linear and Generalized Linear Models is also an excellent reference for practicing statisticians and biostatisticians, as well as anyone who is interested in learning about the most important statistical models for analyzing data.
Foundations of Linear and Generalized Linear Models
Introduction to Linear and Generalized Linear Models
This is a book about linear models and generalized linear models . As the names suggest, the linear model is a special case of the generalized linear model. In this first chapter, we define generalized linear models, and in doing so we also introduce the linear model.
Chapters 2 and 3 focus on the linear model. Chapter 2 introduces the least squares method for fitting the model, and Chapter 3 presents statistical inference under the assumption of a normal distribution for the response variable. Chapter 4 presents analogous model-fitting and inferential results for the generalized linear model. This generalization enables us to model non-normal responses, such as categorical data and count data.
The remainder of the book presents the most important generalized linear models. Chapter 5 focuses on models that assume a binomial distribution for the response variable. These apply to binary data, such as "success" and "failure" for possible outcomes in a medical trial or "favor" and "oppose" for possible responses in a sample survey. Chapter 6 extends the models to multicategory responses, assuming a multinomial distribution. Chapter 7 introduces models that assume a Poisson or negative binomial distribution for the response variable. These apply to count data, such as observations in a health survey on the number of respondent visits in the past year to a doctor. Chapter 8 presents ways of weakening distributional assumptions in generalized linear models, introducing quasi-likelihood methods that merely focus on the mean and variance of the response distribution. Chapters 1-8 assume independent observations. Chapter 9 generalizes the models further to permit correlated observations, such as in handling multivariate responses. Chapters 1-9 use the traditional frequentist approach to statistical inference, assuming probability distributions for the response variables but treating model parameters as fixed, unknown values. Chapter 10 presents the Bayesian approach for linear models and generalized linear models, which treats the model parameters as random variables having their own distributions. The final chapter introduces extensions of the models that handle more complex situations, such as high-dimensional settings in which models have enormous numbers of parameters.
1.1 COMPONENTS OF A GENERALIZED LINEAR MODEL
The ordinary linear regression model uses linearity to describe the relationship between the mean of the response variable and a set of explanatory variables, with inference assuming that the response distribution is normal. Generalized linear models (GLMs) extend standard linear regression models to encompass non-normal response distributions and possibly nonlinear functions of the mean. They have three components.
Random component : This specifies the response variable y and its probability distribution. The observations 1 on that distribution are treated as independent.
Linear predictor : For a parameter vector and a n × p model matrix X that contains values of p explanatory variables for the n observations, the linear predictor is X beta .
Link function : This is a function g applied to each component of that relates it to the linear predictor,
Next we present more detail about each component of a GLM.
1.1.1 Random Component of a GLM
The random component of a GLM consists of a response variable y with independent observations ( y 1, ..., yn ) having probability density or mass function for a distribution in the exponential family . In Chapter 4 we revie