Theory of Linear Models

Bent Jorgensen

• Publisher: Chapman and Hall/CRC
• Year: 2019
• ISBN:
  9781351408615
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Providing a self-contained exposition of the theory of linear models, this treatise strikes a compromise between theory and practice, providing a sound theoretical basis while putting the theory to work in important cases.

CHAPTER 1

Simple Linear Regression

Linear models are used for studying the explanation of a given variable in terms of a linear combination of given explanatory variables. In the present chapter, we discuss the case of one explanatory variable, as a preparation for the general case, which is treated in Chapter 2 and onwards. Readers already familiar with simple linear regression may want to go directly to Chapter 2, with occasional reference to Chapter 1 as necessary.

1.1 The linear regression model

Consider an experiment in which we make simultaneous measurements of two variables x and y for a range of different experimental conditions. If we make n measurements, let

$(x_1,y_1),\dots ,(x_n,y_n)$

denote the corresponding n pairs of observations. We construct a statistical model for the situation where the relationship between x and y is thought to be linear or approximately so.

Often x represents the experimental conditions, and y represents the outcome of the experiment. The variable y is called the response variable or the dependent variable. We assume that y₁, …, y_n are realizations of independent random variables Y₁, …, Y_n. In contrast, the values x_l, …, x_n are considered constant (non-random), and x is called the explanatory variable or the independent variable. We must hence make a clear distinction between the response and explanatory varia...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Chapter 1–Simple Linear Regression
• Chapter 2–The General Linear Model
• Chapter 3–One-Sample and One-Factor Analysis of Variance
• Chapter 4–Multiple Regression Models
• Chapter 5–Analysis of Residuals
• Chapter 6–Analysis of Variance With Two or Three Factors
• References and Bibliography
• Index