Professor Geraert and I agree on the importance of examining arithmetic data instead of logarithmic transformations but we disagree on other issues pertaining to the fitting of statistical models in bivariate allometry. The most important of our differences concerns the utility of quadratic equations, which sometimes provide better fits than power functions to observations in bivariate displays. However, a quadratic equation is an unrealistic representation of allometric variation because the quadratic term in the model causes the fitted curve to assume the shape of a parabola (see Finney, 1989) (http://www.mathopenref.com/quadraticexplorer.html). When the coefficient in the quadratic term is positive, the curve has a minimum and both ends point upwards. When the coefficient is negative, the curve has a maximum and both ends point downwards. The problem with a parabolic curve may not be immediately apparent when the tracing is limited to the range of data in the sample (as in the case of Uca pugnax), but coefficients in the statistical model are biologically uninterpretable in any case (Gould, 1966; Finney, 1989).

Moreover, a quadratic model is not as good a fit to data for chela mass versus body mass in Uca pugnax as the two- and three-parameter power functions that were reported in my essay (Packard, 2012a). All three models satisfy tests for normality and constant variance (Table 1), and all three of the mean functions closely follow the path of the observations [see fig. 4A in my study (Packard, 2012a) and fig. 1 in the accompanying Correspondence (Geraert, 2012)]. However, PRESS statistics indicate that both the power functions are substantially better fits than the quadratic model (Table 1). Thus, the quadratic equation in this case is not favored on statistical grounds any more than it is favored on biological grounds.

Table 1.

Diagnostics for three statistical models fitted to arithmetic values for chela mass versus body mass in fiddler crabs

Professor Geraert also believes that a three-parameter power function has no biological meaning, owing presumably to the term for an intercept. It is worth remembering here that Huxley himself argued in favor of using a three-parameter function as the theoretical starting point in allometric analyses [see p. 241 of Huxley (Huxley, 1932)]. Huxley went on to suggest that the intercept might be relatively unimportant in the scheme of things, so that an investigator might revert to a two-parameter model with minimal loss of fit or relevance. Of course, this argument may have been a convenience for Huxley because he did not have ready access at that time to a procedure for fitting a three-parameter power model to bivariate data.

Professor Geraert is correct when he suggests that I have no biological interpretation to attach to the negative intercept for the three-parameter power function describing the scaling of chela mass to body mass in fiddler crabs. Thus, it might be advisable to revert to the simpler, two-parameter power function (which still shows that the allometric exponent does not change with body size). However, Professor Geraert's assertion that a three-parameter power model generally is meaningless is incorrect, because a non-zero intercept frequently has biological as well as statistical significance when enough is known about the system under study (Bales, 1996; Sartori and Ball, 2009; Packard, 2012b).

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