Biology Reference
In-Depth Information
Whether factors are fixed or random is important to the statistical design, as is whether
factors are crossed or nested. The distinction between fixed and random is important
because the question asked about fixed factors is whether the average shapes differ across
those specific levels. In this case, there is no experimental error associated with the selec-
tion or measurement of the levels. The levels are not randomly selected hence we do not
have to consider the random variation among them. But when the factor is random, the
levels we sample are random samples of the factor and it is therefore measured with error
(just like the dependent variable is). Hence there is an error term associated with the mea-
surement of the factor, just as there is with any random variable. That adds an additional
error term to the model. Even though the numerical values for each level do not depend
on whether the factor is fixed or random, there is an additional population parameter to
estimate when the factor is random. Whether factors are crossed or nested is important
not only because the nested terms are random but also because nesting affects the design
of the statistical test; as we discuss in more detail below, when we design a scheme for
permuting observations, the individuals being permuted must be equivalent to each other
in order to be exchangeable. Rather than exchanging individuals as if they were all equiva-
lent to each other, levels of the nested factor are the equivalent units (and therefore the
exchangeable ones). For example, rather than permuting all the deer mice as if they each
were equivalent to any other, we would permute whole litters.
Main Effects and Interaction Terms
The impact of each factor, considered one at a time, is known as a “main effect”. For
example, the impact of diet on shape is a main effect of diet. If we have just one factor,
that one main effect is all that concerns us (other than error). But when we have two or
more factors, both factors will have main effects and it is also possible that the two factors
interact. If they interact, the impact of one factor depends on the level of the other. For
example, consider those male and female flies that we collected at different altitudes. We
might find that females differ from males in their response to altitude. That differential
response is the interaction effect
the effect of one factor (altitude) depends on the level
of the other (sex). When the interaction term is significant, then we cannot generalize
about the effect of the main factor because its effect is conditional on another factor. We
could say that altitude's effect on female shape is general (so long as that does not also
depend on the impact of time) but we cannot say that altitude's effect on shape is general
because it is not
it depends on sex. In general, we would not interpret main effects in
the presence of interactions. When the design is unbalanced, meaning that the sample
sizes are not equal for all the levels in the analysis, it can be difficult to partition main
effects and interactions.
Decomposition of Variance
We often need to decompose the variation of the dependent variable (shape) into the
contributions made by the factors, covariate(s), interaction terms and error. This decompo-
sition is done by determining the variation explained by the combinations or subsets of
