Biology Reference
In-Depth Information
and the Type III sums of squares for the interaction term AB (SS
AB(III)
) is given by
SS
AB
ð
III
Þ
5
SS
A
1
B
1
A
3
B
2
SS
A
1
B
(9.39)
When computing the F-ratio, the Type III sums of squares are substituted for the sums
of squares computed for a balanced design. The Type III sums of squares do not depend
on the order in which the factors are entered and do not yield an additive partitioning of
variance.
Which Sums of Squares to Use?
There is considerable controversy among statisticians regarding the merits of these vari-
ous types of sums of squares. Some authors contend that Type III sums of squares are the
appropriate ones to use, as a general rule, because the hypotheses being tested are more
straightforward, not being functions of the sample sizes for the cells (
Quinn and Keogh,
2002
). It is certainly true that we do not usually frame our hypotheses in terms of the num-
ber of observations per group, and hypotheses framed in those terms may indeed seem
nearly nonsensical. But, on the other hand, we usually rely on the additivity of sums of
squares when defining the main effects of the factor. That property of additivity is unique
to Type I sums of squares. An important consideration favoring Type I sums of squares
is that, if factors are confounded (or if interaction terms are statistically significant)
we would not normally interpret the main effects as if they do not depend on the design.
We will not tell you which sums of squares to use, but we suggest that, whenever possible,
examine both Type I and III sums of squares and, when using Type I sums of squares,
which are sensitive to the order in which factors are entered, enter the factors in different
orders to develop your understanding of your data.
WO
RKING WITH MULTIVARIATE SUM OF SQUAR
ES
The sums of squares for multivariate data are sum of squares and cross products
(SSCP) matrices, rather than scalars. In this section, we first consider multivariate statisti-
cal approaches to testing hypotheses based on classical analytic multivariate methods and
then permutation-based methods based on inter-specimen (Procrustes) distances.
Classical Analytic Approaches to Significance Testing of GLM Models
Given a model, we can calculate the SSCP of the hypothesis that we wish to test (SS
H
),
a denominator SSCP (SS
denominator
), which may be the residuals or some other SSCP
depending on the structure of the desired F-ratio test, as well as the total SSCP (SS
Total
).
There are a range of different analytic multivariate tests based on either the relationship
between the SSCP of the hypothesis and the total, SS
H
(SS
total
)
2
1
or on the relationship
between the SSCP of the hypothesis and the denominator, SS
H
(SS
denominator
)
2
1
(the nega-
tive exponent indicates a matrix inversion). The SSCP matrices are assumed to follow the
multivariate distributions of partitioned Wishart matrices (
Mardia et al., 1979
). There are
