Biology Reference
In-Depth Information
The notation SS A(II) means a Type II estimate of the sum of square for A (SS A refers to a
SS for A calculated in the same manner as for a balanced design).
Similarly:
SS B ð II Þ 5
SS A 1 B
SS A
:
(9.34)
2
The interaction term is computed as:
SS A 3 B ð II Þ 5
SS A 1 B 1 A 3 B 2
SS A 1 B
(9.35)
When computing the F-ratio, the Type II sums of squares are substituted for the sums
of squares computed for a balanced design. Unlike the Type I sums of squares, Type II
sums of squares do not depend on the order in which terms at the same or lower level are
entered. In this context, “lowest” means the main effects of the factors, pairwise interaction
terms are at a higher level than the main effects, and three-way interactions are at a higher
level than pairwise interactions, etc. The contribution made by a term is assessed by com-
paring the model that contains that term to a model that lacks it. When comparing the
model for a factor to a model that lacks it, the interaction terms are omitted from the
model. Another important distinction between Type II and Type I sums of squares is that,
in the case of Type II sums of squares, the sums of squares are not additive
the sum of
all the sums of squares need not equal the total sum of squares.
Type III Sums of Squares
Type III sums of squares are also called “marginal sums of squares” because the
method is based on the marginal means: the grand mean (i.e. the overall mean) is calcu-
lated from the means of the means. To see the distinction, look again at Table 9.1 where
we tabulated the sample sizes for handedness by sex. We could compute the mean for
right-handed individuals, summing the values for the 50 right-handed females and 70
males and dividing by 120, and doing the same for the left-handed individuals: summing
the values for the 50 females and 30 males and dividing by 80. Alternatively, we could
compute the mean value for right-handed females, and the mean for right-handed males
and then compute the mean of those two means, doing the same for the left-handed
means. That second approach is the one used in computing the Type III sums of squares.
When testing hypotheses, the method based on Type III sums of squares compares the
sum of squares explained by the full model to the sum of squares explained by the model
without the factor of interest (the reduced model) so, in this case, the Type III sums of
squares for A (SS A(III) ) is given by:
SS A ð III Þ 5
SS A 1 B 1 A 3 B
2
SS B 1 AB
(9.36)
Where SS A 1 AB is obtained by fitting
Y
B
A
B
1 ε
(9.37)
5
1
3
Similarly, the Type III sums of squares for B (SS B(III) ) is given by:
SS B ð III Þ 5
SS A 1 B 1 A 3 B
SS A 1 A 3 B
(9.38)
2
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