1 Analysis of Variance versus Experimental Design
Not all data analyzed by ANOVA are from a designed experiment. On the other hand, some designed
experiments lead to data for which ANOVA methods are inappropriate. However, there is a strong historical
connection between ANOVA and Experimental Design.
2 Analysis of Variance
The key idea behind an analysis of variance involves a decomposition of the total sum of squares. For a
one-way ANOVA (possibly arising from a completely randomized design) the decomposition is TSS = SSB
+ SSW (Our text uses SST instead of SSB, and SSE instead of SSW). This expression not only looks a lot
like TSS = SSR + SSE from regression, but we will see later that we can set up a regression model using
dummy variables for which these expressions are identical, where SSB = SSR and SSW = SSE.
If Yij is the jth observation in group i, then Yij −Y.. = (Yij −Yi.)+(Yi. −Y..), which leads to:
k
X
i=1
n
X
j=1
(Yij −Y..)2=
k
X
i=1
n
X
j=1
(Yij −Yi.)2+
k
X
i=1
n
X
j=1
(Yi. −Y..)2,or TSS = SSW +SSB.
As an example, consider three groups with the following data: Group 1 has Y1jvalues of 1, 2, and 3,
Group 2 has Y2jvalues of 5, 3, and 4, and Group 3 has Y3jvalues of 6, 7, and 5. The overall sample mean
is Y.. = (Pk
i=1 Pn
j=1 Yij)/kn = 36/9 = 4.Then TSS is
k
X
i=1
n
X
j=1
(Yij −Y..)2= (1 −4)2+ (2 −4)2+... + (5 −4)2= 30.
The group means are Y1.= 2, Y 2.= 4,and Y3.= 6,so SSW and SSB are
SSW =
k
X
i=1
n
X
j=1
(Yij −Yi.)2= (1 −2)2+ (2 −2)2+ (3 −2)2+ (5 −4)2+... + (5 −6)2= 6,and
SSB =
k
X
i=1
n
X
j=1
(Yi. −Y..)2=
k
X
i=1
n(Yi. −Y..)2= 3(2 −4)2+ 3(4 −4)2+ 3(6 −4)2= 24.
Thus TSS = SSW +SSB or 30 = 6 + 24 partitions the total sum of squares about the overall mean into
two parts, one within groups (due to error, or effects not accounted by the model) and one between groups
(measuring the difference between sample means). Since each group here has nobservations, each group
contributes n−1 degrees of freedom for the within group sum of squares, for a total of k(n−1) degrees of
freedom for SSW. SSB is calculating the sum of squares of ksample means about their (overall) mean, so
it has k−1 degrees of freedom. For the example data above, k(n−1) = 3(2) = 6, and k−1=3-1=2.
We can summarize this information in an analysis of variance table:
Source SS df MS F
Between groups 24 2 12 12
Within groups 6 6 1
Total sum of squares 30 8
1
