Biology Reference
In-Depth Information
depends on what hypothesis one intends to test, and exactly how data collection is struc-
tured. It is not a feature of the variable itself.
Typically, the statistical significance of random factors is not the hypothesis that really
interests us. Those factors are measured simply to determine how much variation can be
attributed to them. We might also want to quantify the variance explained by fixed factors,
and doing so may seem to blur the distinction between the two kinds of factors. The dis-
tinction, however, remains important, especially when a study contains both fixed and
random factors because the denominator of the F-ratio depends on whether the factor is
random or fixed.
Crossed and Nested Factors
In the examples presented above, we measured males and females at all altitudes in all
years. At least in principle, any individual could have been allocated to any “treatment”
(by nature if not by us). In some cases, all the individuals at one level on one factor are
also all at the same level on another. For example, in an experiment on the impact of die-
tary consistency on mandible morphology, infant deer mice could not be removed from
their mothers and raised by different mothers (
Myers et al., 1996
). Consequently, all sib-
lings ate the same food. That is important because we would expect siblings to be more
similar to each other than to unrelated mice not only because they eat the same diet but
also because they are genetically related to each other and share the same uterine and nes-
tling environment. There are many families that eat the same diet, but no families eat
more than one diet. Thus, within any single level of the diet factor there are many families,
but at any single level of the family factor there is only one diet. In this case, family is
nested within the diet factor. Because variation among families may contribute substan-
tially to the variation in shape, this source of variation needs to be taken into account
when assessing the impact of diet on shape. As may be obvious, we are not particularly
interested in whether family A differs from family B, or from any other family. All that we
care about is whether variation among families contributes to variation in shape, and
whether all families respond similarly to diet, making it safe to generalize about the
impact of dietary consistency on shape.
To take another sample, suppose that we are rearing fish in a large number of tanks to
test for the impact of water temperature on body shape. For each temperature, we have
multiple tanks but (obviously) all the fish within any given tank are reared at the same
temperature. We want to assess the impact of water temperature on body shape and to
ensure that our result is general rather than depending on the particular tank in which the
fish were reared. Because the fish within the same tank may have something in common
aside from the temperature of rearing, and that common feature might vary across tanks
even if water temperature does not vary, that other (unmeasured) factor could contribute
to the variation among fish. It might contribute both to the variation among fish reared at
the same temperature and to the response of the fish to water temperature. Although we
do not care whether the fish in one tank differ, on average, from the fish in any other spe-
cific tank, we do care whether variation among tanks contributes to our experimental error
and, even more importantly, whether the response to the factor of interest depends on the
nested term. Again, the nested factor is also a random factor.
