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Comment on “Evidence for mesothermy in dinosaurs”

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Science  29 May 2015:
Vol. 348, Issue 6238, pp. 982
DOI: 10.1126/science.1260410


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 Gmax with maximum body mass M to estimate basal metabolic rate BMR. They determined Gmax for extant and extinct species by fitting Gompertz growth models Embedded Image to age-mass data; k is a growth rate parameter, t0 is the inflection point, and Gmax = kM/e. Log-log regressions of Gmax versus M show little scatter and high r2 [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.

Fig. 1 Data ranges for taxonomic groups.

The convex hulls of data points from Grady et al. (1) are plotted in terms of (A) k/e and (B) Gmax. The blue grid illustrates the shear transformation that relates k/e to Gmax. The solid black outline encompasses all data sets analyzed as dinosaurs by Grady et al. (1), whereas the dashed outline excludes Archaeopteryx. Both of the dinosaur data sets overlap every group except birds. (Not shown: monotremata and testudines.)

Grady et al. deviated from accepted statistical practice in estimating Gmax. 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 Gmax and k.

Regression of Gmax versus M is inappropriate because kM/e has Mas a factor. On a log-log scale, this relationship amounts to the geometric shear transformation yx + 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 Gmax = cMα, in which case Embedded Image. One should regress Gmax = 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, r2 = 0.798 for precocial birds when using Gmax, but 0.513 when using k/e.

Using k/e with the dinosaur data in (1) yields r2 = 0.549. I found 11 errors in these dinosaur data; when corrected, r2 = 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, r2 = 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 r2 = 0.386—commits a fundamental error in inference known as the “ecological fallacy” (7).

Grady et al. (1) argued that Gmax determines BMR [Case was skeptical (8)] and offered three lines of support. First, regressions of Gmax 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 Gmax scales similarly to B, where Gmax = G0Mα.” This suggests that Embedded Image 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 Embedded Image. Regression of BMR and Gmax is thus confounded because Gmax 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).

Fig. 2 Pairwise regressions between k/e, M, and BMR.

(A to D) reveal that correlation is nontransitive among growth rate, metabolism, and mass. The regression of k/e versus M is shown at the bottom in purple; Embedded Image for b = 0.649, 95% confidence interval (CI) (0.578, 0.719), r2 = 0.453. BMR versus M is shown at the left in green; Embedded Image for b = 0.672, 95% CI (0.577, 0.768), r2 = 0.622. BMR versus k/e is at the right in red; Embedded Image for d = –0.301, 95% CI (–0.593, –0.008), r2 = 0.034. The correlations are moderately strong for the first two pairs, with very similar exponents, but there is little correlation between k/e and BMR.

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.


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