## Abstract

D’Emic and Myhrvold raise a number of statistical and methodological issues with our recent analysis of dinosaur growth and energetics. However, their critiques and suggested improvements lack biological and statistical justification.

We presented evidence linking ontogenetic growth to metabolic rate and thermoregulation in dinosaurs (*1*). D’Emic (*2*) and Myhrvold (*3*) raise issues regarding our analysis, which we address here.

In his Comment, D’Emic criticizes our use of continuous models to estimate growth rate, arguing that they are inappropriate for many ectotherms and, in particular, underestimate maximum rates in dinosaurs. D’Emic suggests that we double dinosaur growth rates, yielding rates approaching extant endotherms. However, there are several serious flaws with this approach.

As D’Emic states, our calculations of maximum growth rates are derived from models of continuous sigmoidal growth over ontogeny, which are commonly employed for ectotherms (*4*, *5*). Even for large ectotherms without strict determinate growth [as in (*6*)], the amount of new biomass added annually generally declines toward zero over ontogeny, consistent with a sigmoidal relationship between mass and age (*6*–*9*). For continuous models, maximum growth rate represents the average maximum rate for the organisms in question. The use of averages has analytic and theoretical utility, yielding typical, rather than extreme values, which are more statistically tractable and theoretically meaningful. Perhaps more important, they provide a consistent metric of growth rate that facilitates comparative assessment.

In contrast, calculating maximum growth rate by locating and measuring the discontinuous spurts in growth, as D’Emic urges, greatly increases the difficulty in achieving unbiased analysis. Growth rate varies from day to day, season to season, and year to year; it varies between individuals and environments. Maximum rates are scale dependent; measurement of finer time increments, or more individuals, leads to a higher calculated rate (*10*–*12*). Consequently, comparisons between different temporal resolution or sample sizes, common in macro- and paleo-analyses, will be biased and can generate spurious conclusions without statistical adjustments. No such adjustments are suggested by D’Emic.

Indeed, D’Emic’s recommendation for doubling dinosaur growth rates but leaving other taxa unchanged illustrates the pitfalls of this approach. It appears from D’Emic’s plot that dinosaurs grow relatively faster with this correction, but variation in seasonal growth is documented in virtually all taxa, including endotherms (*13*, *14*). If all growth rates are equally doubled (a seemingly arbitrary value), the relative differences between vertebrate taxa and dinosaurs are unchanged and our conclusions unaffected.

Finally, D’Emic also raises several minor points, including the proper standardization of time for growth rates. Although the number of days in a year and hours in a day have varied over the Phanerozoic, the seconds per year have been approximately constant (*15*). As stated in our methods [supplementary materials for (*1*)], all reported values were standardized to current temporal units, where 1 day = 86,400 s, appropriate for comparisons across geological time scales. In addition, D’Emic questioned our use of phylogenetic and ecological groups and asserts that we did not analyze *Archaeopteryx* with other Mesozoic dinosaurs. On the contrary, we grouped *Archaeopteryx* with other Mesozoic dinosaurs precisely because of their similar biology, as they share more traits with nonavian dinosaurs than derived birds (*16*, *17*). While D’Emic states that “[a] fairer comparison would be to compare clades only,” it is unclear how doing so would advance our understanding of extinct vertebrate physiology. Phylogenetic clades—such as Dinosauria, Aves, Mammalia, or Therapsida—regardless of their Linnaean rank, are not equivalent or inherently suited for comparison. Comparative analyses should be designed with respect to the biology of the taxa of interest, the nature of the question, and relevant published work [e.g., (*17**–**19*]. Grouping Mesozoic dinosaurs with crown birds growing over 10 times as fast, as D’Emic recommends, would obscure more than it illuminates.

In the second Comment, Myhrvold takes issue with our statistical analysis of growth rates. In particular, he argues that our choice of regressing maximum growth rate against body size was inappropriate, that data errors and regression choice inflated the reported *r*^{2} values, and that growth rate is not linked to resting metabolic rate. We disagree with his points but demonstrate that, regardless, following his recommendations has virtually no effect on our conclusions.

Myhrvold argues that regressing maximum growth rate (*G*_{max}) against final or asymptotic body mass (*M*) is inappropriate because both variables share the parameter *M*, an apparent case of self-correlation. Although we agree with Myhrvold’s observation, we disagree that this poses a problem. The issue of self-correlation has been discussed previously in ecology, and the notion that it is inherently flawed has been largely dismissed (*20*–*22*). As noted in one review, “The claim that the correlation between variables sharing a common term is spurious is a pervasive and unfortunate misconception” (*21*). Self-correlation is useful when clarifying a relationship and is statistically equivalent to independent variables in most respects when the shared parameter is removed. Further, in our analysis, self-correlation emerges not because growth rate and final mass are logically dependent but rather because of the particular manner in which growth rate was calculated. As we show, this calculation is justified by independent empirical studies and greatly facilitates analytical efforts.

The alternative that Myhrvold proposes is to regress the rate parameter *k*/*e* against *M*, instead of *G*_{max}. However, this approach simply rescales the analysis, leaving the error structure unchanged. While our regression of *G*_{max} versus *M* yields a slope of α, regressing *k/e *versus *M* gives a slope of α – 1, with an identical standard error (Table 1). Indeed, it is the standard error and not, as Myhrvold argues, the correlation coefficient that is fundamental to our analysis, because it allows statistical differences to be determined. Regardless of which pair of variables are assessed, the significant differences between taxa are identical (Table 1 and Fig. 1). However, a notable contrast between our approaches is that maximum growth rate has a clear biological meaning, unlike the model parameter *k* divided by natural base *e* (units: time^{–1}), a mathematical abstraction that cannot be readily interpreted.

As Myhrvold notes, regressing *k/e* rather than *G*_{max} against *M* has the effect of lowering the *r*^{2}. Although this does not alter our ability to statistically distinguish between taxa, he nonetheless argues that it undercuts our conclusions. We disagree and emphasize that our reported *r*^{2} values are, in fact, appropriate. Maximum growth rate and asymptotic mass are logically independent entities; one variable can be measured without any knowledge of the other. In such cases, similarly high *r*^{2} values are reported (*23*). Growth over ontogeny has a well-documented mathematical form (*4*, *5*, *24*), and equations are useful for reconstructing organismal growth curves and standardizing comparisons. Due to its empirical validation, calculations of *G*_{max} from model equations are effectively independent of *M*. Indeed, our results are consistent, irrespective of what model is used to calculate *G*_{max}, as we demonstrated in our supplementary methods [figure S10 in (*1*)]. Finally, the use of growth models avoids a number of statistical issues associated with measuring growth at different scales, as discussed above. For its conceptual clarity, analytical utility, and empirical justification, we prefer analysis of *G*_{max} rather than *k/e*. In either case, though, the conclusions are the same.

The peril of misapplying statistical techniques is perhaps best illustrated by Myhrvold’s choice of range polygons to present the growth data [figure 1, in (*3*)]. Range polygons enclose the entire space occupied by a data set and will generally increase with sample size. Although large samples with large ranges may suggest increased uncertainty, in fact, the converse is true. Enhanced sampling reduces the standard error and increases the statistical power for detecting real differences between taxa. Myhrvold’s use of range polygons is misleading in that it implies considerable overlap between taxa that are statistically quite distinct. In contrast, we computed 95% confidence bands that provide a quantitative illustration of the statistical differences. As the confidence bands reveal, regardless of whether *k/e* or *G*_{max} versus *M* is plotted, dinosaurs grew significantly faster than ectotherms throughout most of their body range, intersecting with endotherms at the largest sizes (Fig. 1). They are statistically indistinguishable from mesothermic tuna.

Subsequent analyses by Myhrvold to reassess the relationship between basal metabolic rate and growth are conceptually flawed. Although any two variables may be regressed against each other, the results are not necessarily illuminating. Basal or standard metabolic rate (*B*) can be meaningfully regressed against maximum growth rate if body size is approximately the same or if effects of body size are removed from both variables (e.g., mass-normalized growth rate versus mass-normalized metabolic rate). In contrast, Myhrvold regresses mass-specific rate *k* against total *B.* For many large and unwieldy ectotherms, such as crocodilians and sharks, *B* was measured on small juveniles. For small ectotherms and endotherms, however, *B* was typically measured on adults. Unsurprisingly, there is little correlation between growth and metabolic rate when body size is not standardized. We note that this issue was discussed at some length in our original paper (*1*) and supplementary materials (e.g., see figure 2 and supplementary figures S3 to S5).

Myhrvold states that he discovered 11 errors in our data set and critiques our handling of *Archaeopteryx.* We disagree that including *Archaeopteryx* with other Mesozoic dinosaurs was inappropriate, as we discussed above. It is unfortunate, but not surprising, that some errors may be found among our ~29,000 growth values. A larger data set, however, makes it unlikely that any particular error will influence our results, and that is the case here. The errors Myhrvold mentions are mostly quite minor, generally on the order of a few kilograms for animals weighing multiple tons. In one case, a transcription error of a species’ final mass led to a moderately higher, but still biologically realistic, value for *Gorgosaurus*. Correcting these errors leads to an almost imperceptible change in the fitted regression: The phylogenetically informed slope of *G*_{max} versus *M* changes from 0.76 to 0.77 and the *r*^{2} from 0.92 to 0.91. The nonphylogenetically informed equation for dinosaur maximum growth rate shifts from *G*_{max} = 0.0029*M*^{0.82} to *G*_{max} = 0.0026*M*^{0.83}. The corrected *r*^{2} is slightly higher at 0.97 instead of 0.96.

This discussion highlights the important role that growth rate plays in reconstructing dinosaur biology. Characterization of dinosaur growth offers a rare metric for assessing paleoenergetics because growth is a continuous, quantitative trait that can be directly linked to metabolism and body temperature. Our analyses suggest that dinosaurs were energetically intermediate and that mesothermy was likely widespread (*1*). Despite our disagreements, we concur with both D’Emic and Myhrvold that dinosaur growth studies have room for improvement. In particular, we suggest that the field will be advanced by (i) focusing on greater representation of dinosaur size over ontogeny, particularly small juveniles and adults near asymptotic mass; (ii) validation of mass estimation equations through further study of bone and mass scaling during development; and (iii) elucidating factors affecting growth rate other than size and metabolism, such as vulnerability to predation.