## Abstract

Grady *et al*. (Reports, 13 June 2014, p. 1268) studied dinosaur metabolism by comparison of maximum somatic growth rate allometry with groups of known metabolism. They concluded that dinosaurs exhibited mesothermy, a metabolic rate intermediate between endothermy and ectothermy. Multiple statistical and methodological issues call into question the evidence for dinosaur mesothermy.

Grady and co-workers (*1*) followed the method of Case (*2*) to investigate dinosaur metabolism by using allometric scaling of maximum growth rate *G*_{max} with maximum body mass *M* to estimate basal metabolic rate *BMR*. They determined *G*_{max } for extant and extinct species by fitting Gompertz growth models to age-mass data; *k* is a growth rate parameter, *t*_{0} is the inflection point, and *G*_{max} = *kM*/*e*. Log-log regressions of *G*_{max} versus *M* show little scatter and high *r*^{2} [Fig. 1 and fig. S1 in (*1*)], classifying extant groups by endothermic or ectothermic metabolism. The dinosaur regression falls between these two groups, which they call mesothermy.

Grady *et al*. deviated from accepted statistical practice in estimating *G*_{max}. Model selection should be done on a data set (species) basis (*3*); instead they chose a model by the mean Akaike information criterion across all species. For many species, they assume a value for *M* rather than using regression to estimate it, and/or they add hypothetical data points for neonates. Either practice could distort estimates of *G*_{max } and *k*.

Regression of *G*_{max } versus *M* is inappropriate because *kM*/*e* has *M*_{}as a factor. On a log-log scale, this relationship amounts to the geometric shear transformation *y* → *x* + *y*, which compresses data points along the line *y* = *x* (Fig. 1). High correlation and low scatter is a geometric artifact of their chosen regression variable.

That choice is unnecessary. Grady *et al*. propose *G*_{max} = *cM*^{α}, in which case . One should regress *G*_{max} = *k*/*e* (i.e., mass-specific rate) versus *M* to estimate *c* and α and to test this hypothesis. This method, which predates Case (*4*) and is common in growth rate studies (*5*), yields identical values for *c* and α but with much weaker correlations. For example, *r*^{2} = 0.798 for precocial birds when using *G*_{max}, but 0.513 when using *k*/*e*.

Using *k*/*e* with the dinosaur data in (*1*) yields *r*^{2} = 0.549. I found 11 errors in these dinosaur data; when corrected, *r*^{2} = 0.514. Grady *et al*. inappropriately included *Archaeopteryx* as a dinosaur; it is taxonomically a bird, and its age-mass data require different treatment (*6*). It also has a disproportionate impact: minus *Archaeopteryx*, *r*^{2} = 0.386 for dinosaurs.

The analyses of Grady *et al*. (*1*) and Case (*2*) were based entirely on robust correlation in these regressions. However, the shear transformation creates an illusion; in reality, dinosaur data overlap every group except birds. Many extant groups also overlap, including endotherms and ectotherms (Fig. 1).

Growth and metabolism are properties of individual species. Classifying them by group averages (i.e., regression)—especially when *r*^{2} = 0.386—commits a fundamental error in inference known as the “ecological fallacy” (*7*).

Grady *et al*. (*1*) argued that *G*_{max } determines *BMR* [Case was skeptical (*8*)] and offered three lines of support. First, regressions of *G*_{max }versus *M* seemed to classify endotherms and ectotherms; this is a fallacy, as discussed above. Second, metabolic scaling theory predicts it; empirical evidence contradicts this theory, however (*9*). Third, “[e]mpirical evidence (*13*) indicates that *G*_{max} scales similarly to *B*, where *G*_{max} = *G*_{0}*M*^{α}.” This suggests that and thus that metabolic rate may be inferred from growth (*1*). Statistical correlation, however, is not transitive (*10*), as a simple test reveals.

It is well known (*9*) that . Regression of *BMR* and *G*_{max} is thus confounded because *G*_{max} has *M* as a factor. Instead, we must pair *BMR* with *k*/*e*. I performed pairwise regressions of *k*/*e*, *M*, and *BMR*, using data sets from Grady *et al*. (*1*). The regressions of *k*/*e* and *M*, and of *BMR* and *M*, yield moderately strong correlations, but effectively no correlation appears between *k*/*e* and *BMR* (Fig. 2).

In conclusion, one cannot classify dinosaurs as mesotherms on the basis of growth rate allometry. The growth rates of the dinosaur taxa studied by Grady *et al*. (*1*) match those of both endotherms and ectotherms. The data actually show that growth rate does not predict metabolism.