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(Comparative Anatomy and Physiology Brought Up to Date--continued, Appendix 2)

APPENDIX 2: Encephalization quotients, Kleiber's Law, and statistical methods

Derivation of Encephalization Quotients

In order to measure encephalization, a statistical model is needed to relate body and brain size across species. The use of a model allows the statistical estimation of "expected" brain size, given body size. Foley and Lee [1991], citing Martin [1981, 1983], use the equation:

BM(e) = 1.77 * (W0.76)

[Note: * indicates multiplication]

where BM(e) = predicted brain mass, in grams, and W = body mass, in grams. A similar equation is given in Martin [1989], as cited in Foley and Lee [1991], specifically for primates.

The above equation is an example of scaling, as it defines a relationship between a dependent variable (in the above, BM(e) or expected brain mass) and a measure of body size or scale. Further, the mathematical relationship above is non-linear, or allometric. A short, easily readable introduction to the topic of scaling is provided by Martin [1992].

Returning to the topic of encephalization, once an estimate of brain mass is obtained using a credible statistical model (based on data from a range of species), encephalization can be measured. Foley and Lee [1991] provide a good explanation of encephalization quotients (p. 223):

Encephalization quotients (EQ) represent the positive or negative residual value of brain mass, calculated by:

EQ = BM(0) ÷ BM(e)

where BM(0) = actual brain mass, and BM(e) = predicted brain mass for body size.

The coefficient of the allometric equation [for BM(e), above] is close to 0.75, similar to the relationship between body size and basal metabolic rate (BMR, Kleiber 1961). This implies that brain size and BMR are isometrically related, from which the further inference may be drawn that the size of an individual's brain is closely linked to the amount of energy available to sustain it (Milton 1988, Parker 1990).

Note that encephalization quotients are of the form:

actual brain mass ÷ predicted brain mass

Thus, a quotient greater than 1 indicates actual brain mass is greater than predicted, while quotients less than 1 indicate actual brain mass is less than the predicted value.

Kleiber's Law and Statistical Primer

Recall that Kleiber's Law expresses the relationship between body size--specifically body mass--and body metabolic energy requirements, i.e., RMR or BMR. The general equation is:

RMR = 70 * (W0.75)

where RMR is measured in kcal/day, and W = weight in kg. Note the similarity in form between Kleiber's Law and the equation for BM(e) above from Foley and Lee [1991].

Kleiber's Law and similar exponential equations are usually fitted in logarithmic form. (Exponential equations are commonly used for scaling in comparative anatomy analysis.) That is, given you want to estimate an equation of the form,

Y = K * (Xz)

where X and Y are the (given) independent and dependent variables, K is a constant, and z the exponent. (K and z are to be estimated, i.e., are the numbers of interest.). However, it is easier to estimate K and z by fitting the equivalent (linear) logarithmic form, i.e.:

log(Y) = K + (z * log(X))

In the case of Kleiber's Law, the fitted form is:

log(RMR) = log(70) + (0.75 * log(W))

The above equations are fitted by regression--ordinary least squares, a standard statistical technique for solving linear equations to find the linear equation that minimizes total "squared error"--the differences (squared) between the independent variable and the estimate of the independent variable calculated using the linear equation. The term "model fit" used herein usually refers to the process of doing a regression on an independent and dependent variable, which yields estimates of the coefficients of the linear equation (in the case of the equations above: K, z), and an estimate of errors and goodness of fit.

One measure of how good a model fit is, is provided by the r-square statistic, which can be interpreted as the proportion (or percentage) of variability in the independent variable, explained by the estimated linear model (that uses the dependent variable(s)). There are no hard-and-fast rules on what r-square "must" be, but values above 0.85 are considered "good" by most criteria.

The term "confidence interval" is also mentioned herein. A confidence interval is a range of values such that the probability the underlying true parameter (coefficient, estimated value, or other quantity) lies within the specified range of values is X%, where X% is usually chosen as 95% or 99%. Confidence intervals are calculated using the results of the model fit and the underlying assumptions of the model (as regards probability of errors). (Note to statistically adept readers: the preceding is for non-technical readers. Certain technical fine points have been omitted.)

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