9.2. One-Way ANOVA 519To complete the test, we need to determine the distribution ofF underH 0.
First consider the sum of squares in the denominator,Q 3 ,whichwewriteas:
Q 3 /σ^2 =∑bj=1{
1
σ^2∑nji=1(Xij−X·j)^2}
.Notice, since we are discussing distributions, we are now using random variable
notation. By Part (c) of Theorem 3.6.1, forj=1,...,b, the term within the braces
has aχ^2 -distribution withnj−1 degrees of freedom. Further, the samples are
independent so theseχ^2 random variables are independent. Hence, by Corollary
3.3.1,Q 3 /σ^2 has aχ^2 -distribution with
∑b
j=1(nj−1) =n−bdegrees of freedom.
By Part (b) of Theorem 3.6.1, the random variableX·jis independent of the sum
of squares within the braces and further, by the independence of the samples, it
is independent ofQ 3 .Thus,allbsample means are independent ofQ 3. Because
X··=
∑b
j=1njX·j, the grand meanX··is a function of thebsample means, it
must be independent ofQ 3 , also. Therefore,Q 4 is independent ofQ 3 .Forthe
distribution of the numerator sum of squares, write the identity (9.2.10) as
Q/σ^2 =Q 3 /σ^2 +Q 4 /σ^2.For the left side, underH 0 ,Q/σ^2 has aχ^2 -distribution withn−1 degrees of freedom.
On the right sideQ 3 /σ^2 has aχ^2 -distribution withn−bdegrees of freedom and it
is also independent ofQ 4 /σ^2. By equating the mgfs of both sides, it follows that
Q 4 /σ^2 has aχ^2 -distribution with (n−1)−(n−b)=b−1 degrees of freedom.
Therefore, underH 0 ,theF test statistic, (9.2.11), has aF-distribution withb− 1
andn−bdegrees of freedom.
Suppose now that we wish to compute the power of the test ofH 0 againstH 1
whenH 0 is false, that is, when we do not haveμ 1 =μ 2 =···=μb. In Section
9.3 we show that underH 1 ,Q 4 /σ^2 no longer has aχ^2 (b−1) distribution. Thus we
cannot use anF-statistic to compute the power of the test whenH 1 is true. The
problem is discussed in Section 9.3.
Next, based on a simple example, we illustrate the computation of theF-test
using R.
Example 9.2.1. Devore (2012), page 412, presents a data set where the response
is the elastic modulus for an alloy that is cast by one of three different casting
processes. The null hypothesis is that the mean of the elastic modulus is not affected
by the casting process. The data are:
Cast Method Elastic Modulus
Permanent mold 45.5 45.3 45.4 44.4 44.6 43.9 44.6 44.0
Die cast 44.2 43.9 44.7 44.2 44.0 43.8 44.6 43.1
Plaster mold 46.0 45.9 44.8 46.2 45.1 45.5Thedataareinthefileelasticmod.rda.Thevariableelasticmodcontains the
response while the variableindcontains the casting method (1, 2, or 3). The R
code and results (test statisticF and thep-value) are: