Biology Reference
In-Depth Information
analyzing unbalanced designs can be disappointingly uncertain (
Searle, 2006
).
Although permutation methods will be helpful in many applications involving shape
data, they share many of the limitations of classical analytic methods. Thus, the impact
of unbalanced designs should be considered when designing the study to avoid being
forced to confront the problems they pose after the data are collected. It can be very
disheartening to contemplate leaving out measured individuals or even factors to bal-
ance the data.
Fixed and Random Factors
The distinction between fixed and random factors can be both subtle and difficult to
explain because the same factor can be either fixed or random depending on context. In
general, when we are interested in the specific factors and can collect data on all relevant
levels of the factor, then the factor can be treated as fixed. In contrast, when the factors are
not of particular interest in their own right, and the levels that are measured are a random
sample of the levels that could have been measured, the factor is treated as random. In
general, in the case of a random factor, the factor itself might not be of any special interest
but it is one that needs to be taken into account when testing hypotheses about the fixed
factors. Random factors are sometimes viewed as “nuisance terms” because they are both
uninteresting and contribute to variation in the sample.
To make this distinction more concrete, we can consider two cases in which the same
factor is either fixed or random depending on the specifics of the question being asked. In
the first case, suppose that we want to test the hypothesis that the shapes of fly wings
depend on altitude and sex. We have sampled both sexes at three altitudes, and we have
sampled them at different times (although we made sure to sample both sexes, and flies at
all altitudes, at the same times). We are not interested in seasonality of shape, but we
might nonetheless suspect that the season in which the flies were collected might affect
wing shape, and that flies at higher altitudes might be differently affected by season than
those at lower altitudes. Thus, even though we are not interested in time, we need to con-
sider the possibility that some of the variation within the sample is temporal. In that case,
habitat and sex would be fixed factors, but time would be random. The dependence of
shape on time is not a hypothesis that we wish to test, nor have we exhaustively sampled
(or controlled) for it. We have merely taken a sample of shapes at several times. We do
not particularly care when any sample was collected
we only care that they were col-
lected at different times. As a result, we want to know how much of the variation is due
to collecting date, but we do not care whether the flies collected at one particular time dif-
fer significantly from those collected at another specific time. In this context, time is
merely a nuisance variable because it could explain some of the variance in the sample;
including it as a term in the model is important both to avoid ascribing the variation
explained by it to one of the fixed factors of interest, and to lessen the unexplained varia-
tion in the data (by explaining it). In contrast, say that we do want to know whether time
has an impact on wing shape. We may want to know whether high and low altitude flies,
of either sex, respond to climate change, and if that response depends on altitude or sex.
In this case, time would be a fixed factor. Thus, whether a factor is fixed or random
