Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

516 Inferences About Normal Linear Models

X 1 ,X 2 ,...,Xnfrom an arbitrary distribution. Thus

(n−1)S^2 =

∑n

i=1

(Xi−X)^2 =

∑n

i=1

Xi^2 −nX

2

=

∑n

i=1

Xi^2 −

n

n^2

(n

∑

i=1

Xi

) 2

=

∑n

i=1

Xi^2 −

1

n

⎛

⎝

∑n

i=1

Xi

∑n

j=1

Xj

⎞

⎠

=

∑n

i=1

Xi^2 −

1

n

⎛

⎝

∑n

i=1

Xi^2 +2

∑

i<j

XiXj

⎞

⎠

=

n− 1

n

∑n

i=1

Xi^2 −

2

n

∑

i<j

XiXj.

So the sample variance is a quadratic form in the variablesX 1 ,...,Xn.

9.2 One-WayANOVA

Considerbindependent random variables that have normal distributions with un-
known meansμ 1 ,μ 2 ,...,μb, respectively, and unknown but common varianceσ^2.
For eachj=1, 2 ,...,b,letX 1 j,X 2 j,...,Xnjjrepresent a random sample of size
njfrom the normal distribution with meanμjand varianceσ^2. The appropriate
model for the observations is

Xij=μj+eij; i=1,...,nj,j=1,...,b, (9.2.1)

whereeijare iidN(0,σ^2 ). Letn=

∑b

j=1njdenote the total sample size. Suppose

that it is desired to test the composite hypothesis

H 0 :μ 1 =μ 2 =···=μbversusH 1 :μj =μj′,forsomej =j′. (9.2.2)

We derive the likelihood ratio test for these hypotheses.
Such problems often arise in practice. For example, suppose for a certain type
of disease there arebdrugs that can be used to treat it and we are interested
in determining which drug is best in terms of a certain response. LetXjdenote
this response when drugjis applied and letμj=E(Xj). If we assume thatXj
isN(μj,σ^2 ), then the above null hypothesis says that all the drugs are equally
effective; see Exercise 9.2.6 for a numerical illustration of this situation involving
drugs that are intended to lower cholesterol. In general, we often summarize this
problem by saying that we have one factor atblevels. In this case the factor is the
treatment of the disease and each level corresponds to one of the treatment drugs.
Model (9.2.1) is called aone-waymodel. As shown, the likelihood ratio test
can be thought of in terms of estimates of variance. Hence, this is an example of an