Biology Reference
In-Depth Information
and third is 2 mm. Because 2 mm is equal to 2 mm, we can say that the difference between
the first and second organisms is equal to that between the second and third. We might
choose a scale that takes proportions into account, so that 2 mm counts for more when
organisms are near 1 mm than when they are near 100 mm, but still the scale is unambigu-
ous and measurements are mathematically commensurable. In contrast, if we classify
morphologies into three types
“one” and “two” are taken
to be one unit apart, as are “two” and “three”, but we cannot say that the difference
between “one” and “two” is equal to the difference between “two” and “three”. Perhaps
the first two types differ by the presence or absence of a notochord, whereas the second
two differ by the presence or absence of a tubercle on the tibia. The problem faced here
does not arise when coding discrete classes for phylogenetic analyses because the charac-
ters may be equally informative in that context. However, weighting them equally in
studies of disparity implies that they contribute equally to morphological variety.
Fortunately, size and shape data are continuously valued variables, so we will concentrate
on metrics of disparity suited to continuously valued variables.
The metrics for continuously valued variables can be either Euclidean or non-Euclidean
distances, although most workers use Euclidean distances. We can also distinguish among
metrics by whether the measures are of: (1) linear distances between forms (corresponding
to a standard deviation); (2) squared distances between forms (corresponding to a vari-
ance); or (3) volumes. Measures of volume might seem most desirable because they could
appear to capture the most information about the size of the occupied morphospace.
Unfortunately, no satisfactory measure of volumes is available yet, because measuring
them involves multiplication rather than addition. When distances along dimensions are
multiplied, a trivial distance along one deflates the size of the space. For example, if we
multiply distances along several dimensions, such as 0.4, 0.3 and 0.2, we get a volume of
0.024 and, if we multiply that product by 0.002, we get 0.000048
therefore, adding infor-
mation about that fourth dimension reduces the size of the space to nearly zero. Logically,
we would expect that the additional information would only increase the size of the space.
Another disturbing feature of this volume-based approach to disparity is that the volume
of several slightly disparate variables can be far larger than the volume of three very
disparate variables and one nearly invariant variable. In the example above we had three
disparate variables and one that is nearly invariant. We might have another case in which
there are also four variables, each with a disparity of 0.1; the product of (0.1)(0.1) (0.1)
(0.1)
“one”, “two” and “three”
0.0001, which is more than twice the volume of the first case (0.000048). In contrast,
if we restrict our analysis to only the first three variables, the disparity would be (0.1)(0.1)
(0.1)
5
0.001
substantially less than that of the first case (0.024).
If we had an objective and non-arbitrary method for ignoring some dimensions
(so that their low levels of disparity do not deflate the space), we could circumvent these
problems. However, all methods for deciding whether to exclude a variable depend on
subjective arguments, and the decision about whether to exclude a variable can have an
enormous impact on the results. For that reason, we prefer metrics based on standard
deviations and variances. Both standard deviations and variances are equally useful
metrics, and there is no reason to debate which of them is preferable because one is easily
derived from the other. The major reason for using a variance is that variances are
additive. Because of that property, we can calculate the overall disparity of a group, then
5
