Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

Chapter 9

Inferences About Normal

Linear Models

9.1 Introduction................................

In this chapter, we consider analyses of some of the most widely used linear mod-

els. These models include one- and two-way analysis of variance (ANOVA) models

and regression and correlation models. We generally assume normally distributed

random errors for these models. The inference procedures that we discuss are, for

the most part, based on maximum likelihood procedures. The theory requires some

discussion of quadratic forms which we briefly introduce next.

Consider polynomials of degree 2 innvariables,X 1 ,...,Xn,oftheform

q(X 1 ,...,Xn)=

∑n

i=1

∑n

j=1

XiaijXj,

forn^2 constantsaij. We call this form aquadratic formin the variablesX 1 ,...,Xn.
If both the variables and the coefficients are real, it is called areal quadratic
form. Only real quadratic forms are considered in this book. To illustrate, the form
X 12 +X 1 X 2 +X 22 is a quadratic form in the two variablesX 1 andX 2 ;theform
X 12 +X^22 +X 32 − 2 X 1 X 2 is a quadratic form in the three variablesX 1 ,X 2 ,andX 3 ;
but the form (X 1 −1)^2 +(X 2 −2)^2 =X 12 +X 22 − 2 X 1 − 4 X 2 +5 is not a quadratic
form inX 1 andX 2 , although it is a quadratic form in the variablesX 1 −1and
X 2 −2.

LetXandS^2 denote, respectively, the mean and variance of a random sample

515