Biology Reference
In-Depth Information
the last chapter. Because we are considering only univariate models at this point, Y would
be “size”. We have two factors; the first is “sex”, the second is “region” because the chip-
munks were sampled in two regions, the Yosemite region and the southern Sierras. So factor
A would be “sex” and factor B would be “region”. For this case, the full model is:
Y
A
B
A
B
1
ε
(9.22)
5
1
1
3
A is the effect of sex on size, B is the effect of region on size and A
B is the interaction term
3
σ
e
2
and mean zero (we
between sex and region, and
ε
is again the error term with variance
again center Y). Note that the significance of the interaction term, A
B is tested according
to the contribution that it makes to the total variance in addition to that made by the two
main effects (i.e. the contributions made by A and B, treated individually). There are a num-
ber of different possible designs for this two factor case because either A or B (or both) can
be fixed or random. In this example of sex and region, both factors are fixed. But there are
models containing two factors with one fixed and the other random; this type of model is
said to be mixed. A two-factor mixed model is also used for studies of fluctuating asymme-
try (i.e. random deviations from bilateral symmetry). There are two main effects,
“Individual” and “Side”. The first is the variation among individuals, the second is the dif-
ference between right and left sides, and the interaction term, “Individual
3
Side” is the var-
iation among individuals in “sidedness”. In this case, “Individual” is a random term
3
we
are randomly sampling individuals and we do not particularly care whether any one indi-
vidual differs from any other in shape. “Side” is a fixed factor (because we do care about
the difference between right and left sides). In this case, it is actually the interaction term
that most interests us because our objective is to quantify fluctuating asymmetry. In some
cases, both factors are random, a common design used in quantitative-genetic studies where
the objective is to decompose the variance into the part explained by variation in genotype
and that explained by variation in environment.
When one factor is fixed and the other is random, it becomes possible to consider two dif-
ferent types of constraints on the interaction terms A
B.RememberthatY is centered, and
so the sums of all group factors (for both A and B,asin
Equation 9.13
) are also required to be
zero. However, there are two possibilities for the interaction terms A
3
B, either that the
summed interaction terms be zero (as in
Equation 9.13
, but summed over the interaction
terms), which is called the restricted interaction model, or alternatively that there is no restric-
tion on the interaction terms. The restricted model does imply a correlation of the interaction
terms, any two interaction terms will not be independent within a given level of B for exam-
ple (
Quinn and Keough, 2002
). We follow the recommendations of Quinn and Keough (2003)
and
Searle et al. (1992)
in presenting and using the unrestricted interaction model.
Table 9.2
shows the sums of squares for each of the terms in the model. Again, the sums
of squares are shown in a simple summation notation. The F-ratios used to test the signifi-
cance of each term in the model, for the various combinations of random and fixed factors,
are shown in
Table 9.3
(note the different denominators for A and B depending on whether
the factors are fixed or random, and that the restricted model of interactions is used here),
and the calculation of the variance components for each combination is shown in
Table 9.4
.
Results for the analysis of the impact of sex and region on chipmunk jaw size are shown in
Table 9.5
. Sex does have a significant impact on chipmunk jaw size; females' jaws are larger
3
