Biostatistics

314 CHAPTER 8 ANALYSIS OF VARIANCE
The Among Groups Sum of Squares To obtain the second component of
the total sum of squares, we compute for each group the squared deviation of the group
mean from the grand mean and multiply the result by the size of the group. Finally, we add
these results over all groups. This quantity is a measure of the variation among groups and
is referred to as the sum of squares among groups or SSA. The formula for calculating this
quantity is as follows:
SSA ¼ Xk
j¼1
n j x :j x ::
2
(8.2.4)
In summary, then, we have found that the total sum of squares is equal to the sum of
the among and the within sum of squares. We express this relationship as follows:
SST ¼ SSA þ SSW
From the sums of squares that we have now learned to compute, it is possible to obtain two
estimates of the common population variance, s 2 . It can be shown that when the
assumptions are met and the population means are all equal, both the among sum of
squares and the within sum of squares, when divided by their respective degrees of
freedom, yield independent and unbiased estimates of s 2 .
The First Estimate of s 2
Within any sample,
X n j
i¼1
x ij x :j
2
n j 1
provides an unbiased estimate of the true variance of the population from which the sample
came. Under the assumption that the population variances are all equal, we may pool the k
estimates to obtain
MSW ¼
X k X n j
j¼1 i¼1
X k
j¼1
x ij x :j
2
n j 1 (8.2.5)
This is our first estimate of s 2 and may be called the within groups variance, since it is
the within groups sum of squares of Equation 8.2.3 divided by the appropriate degrees of
freedom. The student will recognize this as an extension to k samples of the pooling of
variances procedure encountered in Chapters 6 and 7 when the variances from two
samples were pooled in order to use the t distribution. The quantity in Equation 8.2.5
is customarily referred to as the within groups mean square rather than the within
groups variance.