Foundations of Linear and Generalized Linear Models

CHAPTER 2 Linear Models: Least Squares Theory

The next two chapters consider fitting and inference for the ordinary linear model. For n independent observations with μ_i = E(y_i) and , denote the covariance matrix by

Let denote the n × p model matrix, where x_ij is the value of explanatory variable j for observation i. In this chapter we will learn about model fitting when

where is a p × 1 parameter vector with p ≤ n and I is the n × n identity matrix. The covariance matrix is a diagonal matrix with common value σ² for the variance. With the additional assumption of a normal random component, this is the normal linear model, which is a generalized linear model (GLM) with identity link function. We will add the normality assumption in the next chapter. Here, though, we will obtain many results about fitting linear models and comparing models that do not require distributional assumptions. ...