Data Analysis Using Regression and Multilevel/Hierarchical Models

Introduction

Generalized linear modeling is a framework for statistical analysis that includes linear and logistic regression as special cases. Linear regression directly predicts continuous data y from a linear predictor Xβ = β0 + X1β1 + ⋯ + Xkβk. Logistic regression predicts Pr(y = 1) for binary data from a linear predictor with an inverselogit transformation. A generalized linear model involves:

1. A data vector y = (y1, …, yn)

2. Predictors X and coefficients β, forming a linear predictor Xβ

3. A link function g, yielding a vector of transformed data ŷ = g−1(Xβ) that are used to model the data

4. A data distribution, p(y|ŷ)

5. Possibly other parameters, such as variances, overdispersions, and cutpoints, involved in the predictors, link function, and data distribution.

The options in a generalized linear model are the transformation g and the data distribution p.

• In linear regression, the transformation is the identity (that is, g(u) ≡ u) and the data distribution is normal, with standard deviation σ estimated from data.

• In logistic regression, the transformation is the inverse-logit, g−1(u) = logit−1(u) (see Figure 5.2a on page 80) and the data distribution is defined by the probability for binary data: Pr(y = 1) = ŷ.