Introductory Biostatistics

2. The di¤erence between the two sums of squares above,

SSB¼SST
SSW

¼

X

i;j

ðxi
xÞ^2

¼

X

i

niðxi
xÞ^2

is called thebetween sum of squares. SSB represents the variation or dif-

ferences between the sample means, a measure very similar to the nu-

merator of a sample variance; thenivalues serve asweights.

Corresponding to the partitioning of the total sum of squares SST, there is
partitioning of the associated degrees of freedom (df). We haveðn
1 Þdegrees
of freedom associated with SST, the denominator of the variance of the com-
bined sample. SSB hasðk
1 Þdegrees of freedom, representing the di¤erences
betweenkgroups; the remaining½n
k¼

P

ðni
1 ފdegrees of freedom are
associated with SSW. These results lead to the usual presentation of the
ANOVA process:

1. Thewithin mean square

MSW¼

SSW

n
k

¼

P

iðni
^1 Þs

2

P i

ðni
 1 Þ

serves as an estimate of the common variances^2 as stipulated by the one-

way ANOVAmodel. In fact, it can be seen that MSW is a natural

extension of the pooled estimatesp^2 as used in the two-samplettest; It is a

measure of the average variation within theksamples.

2. Thebetween mean square

MSB¼

SSB

k
 1

represents theaveragevariation (or di¤erences) between theksample

means.

3. The breakdowns of the total sum of squares and its associated degrees of
   freedom aredisplayedin the form of ananalysis of variance table(Table
   7.6). The test statisticFfor the one-way analysis of variance above com-
   pares MSB (theaveragevariation—or di¤erences—between theksample
   means) and MSE (the average variation within theksamples). A value

ONE-WAY ANALYSIS OF VARIANCE 265