Biostatistics : One-Way Analysis of Variance : Decomposing the Variance and Its Meaning
DECOMPOSING THE VARIANCE AND ITS MEANING
Cochran’s theorem is the basis for the sums of
squares having independent chisquare distributions when Equation 13.1 holds
[see Rao (1997), page 4]. It can be deduced from Cochran’s theorem in the case
of the one-way ANOVA that Σ(Xij – )2 =
Σ(Xij – Xi)2 +
Σ(Xi –
)2, where the
following holds:
Xij is normally distributed with mean μi
The variance is σ2
Xij is the jth observation from the ith
group
Xi. is the average of all
observations in the ith group
X is the average over all the observations in all
groups
Let Q, Q1, and Q2 refer to total sum of squares, within-groups sum of
squares, and between-groups sum of squares, respectively. We have that Q = Q1
+ Q2 Q = S(Xij – )2 normalized
has a chi-square distribution with nw + nb – 1 degrees of freedom; Q1 = Σ(Xij – Xi.)2 has a
chi-square distribution with nw
degrees of free-dom; and Q2
= Σ(Xi. –
)2 has a chi-square
distribution with nb – 1
degrees of freedom. Q2 is
independent of Σ(Xij – Xi.)2. The symbol nb is the number of groups
and nw is the number of
degrees of freedom for error. The total sample size equals n – nb. For Q2 to have a chi-square
distribution when appropriately nor-malized, we need the null hypothesis that
all μi are
equal to be true. The F distri bution
is obtained by taking [Q2/(nb – 1)]/[Q1/nw]. When the alternative holds, the normalized Q2 has what is called a
noncentral chi-square distribution, and the ra-tio tends to be centered above
1. The distribution of [Q2/(nb – 1)]/[Q1/nw] is then called a noncentral F distribution.
The mathematical relationship between this F statistic and the sample multiple
correlation coefficient R2
(discussed in Chapter 12) is as follows: R2
= (nb – 1)F/ {(nb
– 1)F + nw} or F = {R2/(nb – 1)}/{(1 – R2)/nw}
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