Animals undergo interrelated changes in size and shape on both developmental and evolutionary time scales. For example, the dramatic size changes from fetal development through adolescence occurs on a developmental time scale. Many evolutionary lineages also show a pronounced size increase through time, with early species being considerably smaller than later species. Accompanying these size changes are significant modifications of shape. These shape changes occur because specific body plans are not infinitely expandable; the physical constraints on body form are known to vary with size.
The fundamental problem faced by all organisms is called surface area  volume paradox, where the ratio between surface area and volume does not increase linearly. For example, consider two cubes, A = 1 cm on each side and B = 3 cm on each side. If L = length of a side, then the surface area L^{2} (= squared, or L^{2}, function) and volume L^{3} (= cubic, or L^{3} function). Consequently, cube B has 9 times the surface area and 27 times the volume of cube A. This creates severe problems at larger sizes, since nutrients and oxygen are consumed and waste and CO_{2} at produced at rates proportional to volume (i.e., they are cubic functions), while exchange rates at body surfaces are proportional to surface area and are squared functions. In all systems there occurs a critical size above which squared functions can not keep up with cubic functions.
The mathematical basis for analyzing the scaling relationships within organisms is described mathematically as:
If the slope (b) is equal to 1, then the variables exhibit equal proportional changes, and demonstrate isometry. In isometric relationships shape does not change as size increases. Instead, organisms exhibit geometric similarity for the variables being studied.
However, most variables in organisms do not scale
isometrically. Instead, they have unequal proportional changes, that
can
take a variety of forms:
• independence (b
= 0);
• positive allometry
(b > 1);
• negative allometry
(0 < b < 1);
• inverse allometry
(b < 0).
All allometric relationships are manifested as sizerelated changes
in shape, which are necessary to maintain functional efficiency.
Consequently,
shape differences between animals of unequal sizes must be evaluated
very
carefully when making paleobiological reconstructions.
Because allometric relationships are a power function (i.e., they have an exponent) they form a curved line when graphed on arithmetic axes (Figure 1 & 2). Since curved lines are relatively difficult to evaluate, allometric relationships are typically either graphed on logarithmic scales (Figure 3 & 4) or logtransformed values are plotted on arithmetic scales (Figure 5 & 6). Note that logarithmic scaling and logtransformed data give identical plots. We will work with base 10 logtransformed data in this course, because it simplifies calculations for data sets that cover huge size ranges. Unfortunately, humans do not have an easy familiarity with logarithms. Stating that the logarithm of the estimated body mass of the tyrannosaurid theropods Tyrannosaurus rex (in kg) is 3.756, while that of Tarbosaurus bataar is 3.322 means very little. But by converting back to actual masses (T. rex = 10^{3.756} and T. bataar = 10^{3.322}) we obtain 5700 and 2100 kg, respectively.
There are several important points to remember when working with logtransformed data. First, as illustrated by the tyrannosaurid data, even a small difference in logarithm values can reflect a very large difference in arithmetic values. Second, arithmetic plots which look superficially similar, may have very different equations. For example, the arithmetic plots in Figure 1 (a & d) look superficially similar, but have different causes. Figure 1 has the value of b held constant, while the value of a varies (y = 2x^{2} and y = x^{2}). Figure 2 has the converse, the value of a is held constant, while the value of b varies (y = x^{2.0} and y = x^{1.9}). Finally, a and b have different effects on logscaled and logtransformed data. Changing the value of a (Figures 3 & 5) produces parallel lines with identical slopes, but different origins. This is not unexpected, since a is the yintercept value. Changing the values of b (Figure 4 & 6) produces nonparallel lines with the same origin. Again, this is expected, since b represents the slope.
The purpose of this laboratory is to familiarize you with the analysis of allometric relationships. Much of your research this semester will involve the statistical analysis of allometry in living species, and the application of these relationships to extinct animals. Fundamental to an understanding of allometry is the ability to construct logtransformed plots of morphological data and derive a linear regression equation that describes the relationship between the variables.
You will be performing
three
analyses in this exercise. The first analysis is simply to familiarize
you with the techniques for analyzing shape data and uses data from
common
carpentry nails. The second analysis concerns the relationship between
body length and body mass in theropod dinosaurs. The third analysis
examines
the lengthwidth relationship in the extinct sand dollar, Encope
tamiamiensis,
from the Pliocene of Florida. This species occurs in two different body
forms (i.e., broad and narrow), and we will use allometric analysis to
determine whether one or two species are actually present.
Scaling in Carpentry Nails
Common sense would suggest that carpentry nails should exhibit simple isometry; a nail that is twice as long as second one, should also have twice the diameter. But is this actually the case?
Carpentry nails ranging
in
size from three penny common (abbreviated 3D) to thirty penny common
(30D)
are available in the laboratory for analysis. Using calipers, measure
the
length and midshaft diameter in millimeters for one nail of each size
and record your data in a spreadsheet for analysis. As with most
allometric
analyses, the data will need to be logtransformed before it is
graphed.
Once the data has been entered and analyzed, plot a graph of log LENGTH
vs. log DIAMETER.
Scaling of Theropod Body Length and Body Mass
In the second portion of this exercise, you will be evaluating the relationship between body length and body mass for theropod dinosaurs. Data has been provided from published information for all theropod species for which reasonable accurate length and mass estimates are available (data from G. S. Paul. 1988. Predatory dinosaurs of the world. A complete illustrated guide. Simon and Schuster, New York, NY, 464 pp.). NOTE: Paul used an idiosyncratic system of taxonomic nomenclature for his book. Standard taxon names, as of 2007, are provided below along with Paul's 1988 names.
Table. Estimated body length and body mass for theropod dinosaurs
known from relatively complete skeletons.

(m) 
(kg) 








































































Part 3. Allometry of the Echinoid, Encope tamiamiensis
Because of the inherent variability in organisms it is frequently unclear whether a collection of specimens represents a single, highly variable species, or two (or more) closely related species. The evolutionary pathways within these species swarms must be clearly delineated, if more than a cursory understanding of the paleobiology of the group is to be obtained. Problems of biological variability are compounded by the prevalence of heterochronic evolution within groups of closelyrelated species. Consequently, similar body forms may occur at different sizes in related species.
One of the most important uses of allometric analysis is the evaluation of such species swarms by analyzing sizerelated changes in shape. Evaluating developmental changes in shape are fundamental for separating sibling species within a swarm.
Unfortunately, a detailed allometric analysis would be too complex for this course, since it would require independent assessments of numerous sizeshape relationships. However, it is possible to make a preliminary evaluation based on only simple measurements. For example, if we were to make a loglog plot of length and width for a large sample of specimens with similar body form, we would expect one of two possible results. If a single, variable species was present, data points should cluster about the regression and there should be a relatively high correlation coefficient. If two species were present, relatively few data points would reside along the regression line; there would be two distant clusters of data points and a low correlation coefficient.
In this portion of the laboratory, we will be evaluating sizerelated changes in body form for the extinct echinoid, Encope tamiamiensis (Tamiami Formation; Pliocene; Charlotte Co., Florida). This species occurs in two different body forms; broad and narrow. Whether these forms represent separate species, or simply extreme variants of a single species is to be evaluated with an allometric analysis of body lengthbody width relationships.
There are six sets of E. tamiamiensis specimens available in the laboratory. Each research group should obtain one set of specimens for study. Using calipers, measure maximum length and maximum width for each specimen. Record this data in a spreadsheet. When all groups have obtained their data, it will be pooled for analysis. Analyze the data as in the previous studies obtaining:
Questions
Do nails exhibit isometry?
Why might nails change shape with increasing size?
The relationship between body length and body mass in theropods
correlates
a linear measurement (= length) with a cubic measurement (= mass).
Theoretically,
what would you predict as the value for b?
Does your value of b deviate from this prediction, and
if so, why?
Does the E. tamiamiensis sample we evaluated support the
hypothesis
that there two closelyrelated species present?
Suggest three other allometric analyses that could be performed with
these specimens to either confirm or refute your conclusions.