Data

The primary data for this study are data points on growth rate (mass versus age) for both dinosaurs and extant groups. These data were obtained from the supplemental material of the most recent studies [12,13] or were provided by their authors.

Grady et al. derived this data from their own growth-curve fits to age–mass data, most of which they obtained by digitizing published graphs. The dinosaur age–mass growth data were obtained directly from Grady, and the curve fits reported by Grady et al. [13] were successfully replicated by using the methods disclosed in the paper. In the course of replicating the fits, however, a number of errors in the dinosaur data sets came to light; these data errors are discussed in S1 Text (see also S1 Fig). I prepared an amended version of the data set (S2 Fig, table in S1 Table) that corrects the errors.

Grady et al. used highly unconventional methods to fit growth curves to their data. For most dinosaur taxa and some extant species, they supplemented the specimen-derived data with hypothetical neonate masses. In addition, for most dinosaur taxa and some extant species, they imposed a constraint that fixed the maximum asymptotic mass to a value chosen from the literature rather than deriving that value from the growth-curve fit. Each of these actions raises serious concerns about the degree to which the resulting growth data is influenced by those assumptions.

A related issue arises from the limited nature of the data, which for most dinosaur data sets span only part of the lifespan [14]. Indeed, only one dinosaur taxon (Tyrannosaurus rex) in either the Grady et al. or Werner and Griebeler studies has been shown to have data points that span the age range in which the maximum growth rate occurs. As a result, the maximum growth rate may be an extrapolation far from known data points. Grady et al. supplement the data sets with artificial constraints and hypothetical neonate sizes in part to work around the limitation that the existing data are inadequate to support estimations of maximum growth rates. Because this study focuses on the issue of allometric scaling of growth rate and its interpretation, I nevertheless proceeded with the data derived by Grady et al. using their methodology.

Werner and Griebeler [12] used dinosaur growth data from the literature rather than fitting growth curves themselves. In several cases, they included results that have been shown to be incorrect or irreproducible (see S1 Text), and I thus again produced a corrected data set (S3 Fig, table in S1 Table).

Both the Grady et al. and Werner and Griebeler papers include data for Archaeopteryx, which raises more fundamental concerns. Dinosaur growth series are constructed on the basis of lines of arrested growth (LAG), which appear in thin sections of bone. Correspondence with extant examples calibrates the LAGs to be annual markers and thus a standard “clock” for assessing the timing of growth.

However, Archaeopteryx bones contain no LAGs. For this taxon, the growth series was instead constructed on the basis of four very uncertain assumptions, as described by Erickson et al. [6]. First, the paper assumes that the bone is laid down at a constant rate. Second, it assumes that we can estimate that constant rate accurately. Together, these first two assumptions mean that Archaeopteryx growth is implicitly being modeled as linear growth and that the age of specimens is estimated by dividing the size by that rate. The third assumption is that Archaeopteryx growth is ultimately sigmoidal and that all of the known Archaeopteryx specimens happen to lie within the linear regime of a sigmoidal growth curve. Fourth and finally, the method assumes that the true sigmoidal curve can be recovered from the linear regime data by imposing asymptotic conditions by fiat.

There is little reason to believe that any of these assumptions are valid. We do not know the correct bone growth rate nor whether it was constant. This is particularly true if Archaeopteryx was an ectotherm or mesotherm, as Werner and Griebeler and Grady et al. ultimately conclude, and thus potentially sensitive to environmental variations in temperature. Histological arguments can offer some suggestions, but with very little certainty and very wide bounds. In addition, there is a fundamental contradiction between the assumption that one can model growth as linear for all known specimens and that the growth is ultimately sigmoidal. How does one know that the youngest and oldest specimens observed to date truly lie within the linear regime? We don’t even know that these specimens are “youngest” and “oldest”–all that is known definitively is that they are the smallest or largest described to date.

Moreover, the reasoning that one can use the assumed growth rate to determine the specimen age is highly circular—if the rate is wrong or growth is non-constant, then all of the ages change. Age estimates based on LAG analysis are supported by a tremendous amount of careful research on the bone histology of extant species. It might be possible to calibrate the method used on Archaeopteryx by applying it to extant bird specimens of known ages, but that has not been done.

Indeed, if one does suppose that the assumption is correct that Archaeopteryx specimens are all in the linear phase, then the maximum growth rate is simply the assumed constant rate. There is little point in making a growth curve, which only obscures the fact that the entire exercise rests on the assumed growth rate. In truth, the data point for Archaeopteryx essentially amounts to educated guesses for the value of two parameters: k (or equivalently G_max) and M. The estimates are, unfortunately, highly correlated because any change to k changes the specimen ages and thus the estimate of M, assuming that it is determined by regression.

Without a standard clock (such as LAG) to demonstrate otherwise, the most parsimonious assumption is that Archaeopteryx grew the way most modern birds grow today, achieving full size in a time frame between a few months (i.e., one growing season) to a year or two [35–37]. If so, then its growth rate would have been similar to that of birds of the same mass.

A different argument is that it is taxonomically inappropriate to classify Archaeopteryx with dinosaurs. Most authorities use the term “dinosaur” to mean non-avian dinosaurs, which would exclude Archaeopteryx. It is the only avian dinosaur contained within the group that is called “Mesozoic dinosaurs” by Grady et al. and labelled “Dinosaurs” by Werner and Griebeler. In other taxonomic groups, Grady et al. put extinct taxa in with their modern descendants, a policy that would put Archaeopteryx in with birds or would mean including modern birds with dinosaurs.

The inclusion of Archaeopteryx matters because its body size is much smaller than that of the other members of the group, so the taxa has a disproportionate influence on the regression—and in particular on the value of R² and the width of the confidence interval. Because Archaeopteryx is a very influential but also very uncertain data point, my calculations on the dinosaur data sets were done both with and without Archaeopteryx data.

Data sets for extant species were not checked in the same ways that I analyzed the dinosaur data. The two papers used different definitions of extant groups: for example, Grady et al. split fish into two groups (teleosts and sharks), whereas Werner and Griebeler left fish in one group. I used the same group definitions as in the original papers. Two groups within Grady et al. (monotremeta and testudines) consisted of just two data points each and were omitted from the analysis here.

In addition to growth-rate data, Grady et al. also present metabolic data on basal metabolic rate (BMR) for 120 species across both endotherms and ectotherms. These data are potentially quite difficult to compare; they are gathered under differ circumstances, with individual animals that are at different points on their growth–some are juveniles, some are adults. There are important technical differences between BMR and metrics like resting metabolic rate (RMR). The data for ectotherms have also been adjusted to a standard environmental temperature of 27°C, and the adjustment approach could be questioned. As a result of these issues, the data may be ill-suited to draw like-to-like comparisons across the diverse species in the study. For the purposes of the current work, however, the data were used without attempting to address these issues.

Statistical Methods

Commercial statistical software (Mathematica v. 10.1) was used to perform the calculations. The models fit to data are presented in S2 Table. Detailed regression results are given in S4 Table and S5 Table.

Grady et al. and Werner and Griebeler used both ordinary-least-squares (OLS) and phylogenetic regressions. Here, OLS regression alone is used because it is sufficient to illustrate the relevant phenomena discussed in the study. The use of phylogenetically independent contrasts (PIC) or phylogenetic generalized least squares (PGLS) would not qualitatively alter the findings because the issues found are based on more fundamental issues, such as the choice of dependent and independent variables.

Werner and Griebeler used a single data point for each extant species but multiple data points for some dinosaur taxa. OLS regressions here are weighted to give the same total weight per taxon as assigned to taxa having a single data point.

Growth Rate and Regression

A fundamental goal of the Case studies was to determine how the maximum growth rate of vertebrates scales with their maximum body mass. In the Case paper [1], the maximum growth rate was determined both by curve fitting and by direct observation. Case made a rather unfortunate choice of the dependent variable G_max for his regressions, which potentially confounds their interpretation, as shown here.

Many growth models are used in biology (see [14] for a list of the most common 75 models). In this case, we wish to model mass as a function of time. Basic dimensional analysis requires that all biological growth models can be written in the form (1) Where M has units of mass, M_c is a constant mass offset, and the function f(s) is a dimensionless function of a dimensionless variable s = k(t−t_c), where the age t and the constant offset t_c have dimensions of time, and the growth constant k has dimensions of inverse time. For example, for linear growth f(s) = s, whereas exponential growth can be represented by f(s) = e^s. Sigmoidal growth functions can also be put in this form. Logistic growth is (2) and Gompertz and von Bertalanffy growth are given by (3)

In this framework, the maximum growth rate G_max is given by the chain rule as (4)

In the case of sigmoidal growth curves, the dimensionless constant factor C depends on the specific growth-curve model. For the three most common sigmoidal growth curves: (5)

When we fit a nonlinear growth curve to age-mass data points, we determine both M and k. To calculate the maximum growth rate, one then multiplies these two variables by C to obtain G_max.

Thus all biological growth models have the property that G_max contains M as a factor, and all such models have a growth rate (i.e., an inverse time constant) k.

The regression posited by Case used M as the independent variable (see discussion below). Regressing MkC versus M is extremely problematic, however, because the dependent variable contains the independent variable as an explicit factor. The effect of this linkage is to exaggerate the degree of correlation and artificially boost the coefficient of determination R², which can create the illusion of a strong biological association between maximum growth rate and body mass. The strong correlation is actually caused by multiplying kC by M to form the dependent variable.

In mathematical terms, this is equivalent to the hypothesis that (6) which can be reformulated using Eq (4) as (7)

Thus there is no reason to multiply kC by M to form G_max; the same hypothesis (6) can equally be tested with (7), a regression on kC.

Indeed, regression on the growth parameter k versus M was done in studies that predate Case [36,38] and these studies were the source for the bird data used by Case. This approach was subsequently used across many species [39].

The biological interpretation of kC is the mass-specific maximum growth rate, given by (8) where t_M is the time it would take to grow to mass M at a constant growth rate equal to G_max. Equivalently, it is the percentage growth per unit time at the peak growth rate.

Of course a real growth trajectory does not stay at the peak rate G_max, so actual growth time is always longer than t_M. The growth time from birth to 90% of M is typically 1.5× to 2× t_M for many species.

Case used log-log transformed variables, in which case the difference between (7) and (6) is the geometric shear transformation (9)

It is a shear transformation that rotates and compresses the (M, kC) plane along the line y = x. Fig 2 shows the geometric effect of the shear transformation between kC and G_max. This geometric effect acts on the data points and as a result will change any calculation based on the data points, including OLS regression or phylogenetic regressions (PIC or PGLS).

Fig 3 shows the effect of this transformation for two key extant groups in each study. Note that the estimate of the slope b and its 95% confidence interval are not affected by the choice of dependent variable, but the coefficient of determination R² is changed, as is the amount of scatter on the plot.

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Fig 3. Regression using G_max or kC, kD as the dependent variable for several extant groups.

Using G_max instead of kC or kD reduces the scatter on the graph and greatly increases R², while leaving the slope b and its 95% confidence interval (in square brackets) unchanged. [w]: Werner and Griebeler [12], [g]: Grady et al. [13]. Red lines delineate the convex hulls encompassing the data points. Shaded regions bounded by dashed lines delineate 95% confidence bands on the regressions.

https://doi.org/10.1371/journal.pone.0163205.g003

Precocial birds are a key group for both studies. Allometric scaling of growth with mass accounts for only 51% of the variation in kC growth rates for precocial birds in Grady et al., and for just 37.5% of the corresponding variation in the Werner and Griebeler study. Corrections for other groups are shown in S4–S9 Figs. The residuals of these regressions are relatively well behaved (S10 and S12 Figs), although for some groups the presence of multiple maxima in the distribution of residuals suggests that there may be some statistical advantage to splitting them into subgroups. Probability plots show that the residuals do not show large deviations from a normal distribution (S11 and S13 Figs).

Dinosaurs are an even more important group for the current study. Results for the original and corrected dinosaur data sets in both studies are shown in Fig 4. The corrected regressions are quite poor: allometric scaling explains only 51% of the variation in the Werner and Griebeler data and only 38% of the variation in the larger dinosaur data set from Grady et al.

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Fig 4. Dinosaur growth-rate allometry.

[g]: data from Grady et al. [13]; [w]: data from Werner and Griebeler [12]. Plots on the left are the original data. Plots on the right, labeled *, are the corrected data sets with Archaeopteryx removed. Red lines delineate the convex hulls encompassing the data points. Shaded regions bounded by dashed lines delineate 95% confidence bands on the regressions. Note that for the [w] plots, N is not the number of taxa because there are multiple data sets for some taxa; the regressions are weighed to account for this. The original data set (upper right) includes 13 taxa; the corrected data set comprises 12 taxa (Archaeopteryx is excluded).

https://doi.org/10.1371/journal.pone.0163205.g004

It is often the case that problems with statistics occur in the interpretation of results rather than in the actual statistical calculations, in part because standard statistical software makes the calculations straightforward. In this instance, the value of R²–and more generally the difference between the variation in the data and the regressions–is important to the interpretation of the results. Case chose regression variables that showed very tight clustering of data points, with high values of R², which gives the appearance of being due to a fundamental biological phenomenon. This outcome encourages the use the regression results for classification (and thus hypothesis H1). The smaller the range of variation in the data, the closer the regression results are to actual data. As a practical matter, this relationship would tend to reduce the difference between using the regression versus the actual data.

We now know that the apparently tight clusters are due to the shear transformation (9), not biology. With the correct choice of variables, the effect of allometry explains only half or less of the effect for some key groups, so there is a huge amount of scatter in the points. This by itself argues strongly against hypothesis H1. If the allometric regression of maximum growth rate versus body mass has low explanatory power, then why should it be used to predict metabolism?

Grady et al. argue that “self-correlation” between M and G_max is actually a positive aspect of their analysis: “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” [40]. This is incorrect; the mass-specific growth rate kC is a biologically reasonable quantity–it is the growth-rate percentage per unit time at peak growth rate. The peak growth rate in absolute terms (i.e. mass per unit time) has a clear and obvious dependence on mass. It is intuitively clear that is easier for a 1000 kg animal to add 1 kg of new growth in a given time interval than it is for a 1 kg animal to add 1 kg in the same interval. For the former this would represent a 0.1% growth increment, while for the latter it would be 100%, or a doubling of growth. So G_max is clearly dependent on mass, whereas the mass-specific growth rate kC is not.

Choice of Independent Variable

Grady et al. followed Case [1,2] and Erickson et al. [5–7,10,11,41] in using maximum asymptotic mass M as the independent variable. This follows a long tradition of using M in this manner in studies of metabolism [42–45] and growth [8,35,46].

Werner and Griebeler took a different approach, arguing that one cannot compare the maximum growth rate from different growth models because it occurs at different ages and masses [12]:

Today, a proper method for estimating growth rates is to fit non-linear growth functions to growth data. The most commonly used growth models describing individual growth are the Logistic, Gompertz or von Bertalanffy growth functions [12–19]. All three functions are similar in shape (sigmoidal), but the location of the point of inflection differs. The von Bertalanffy function has a point of inflection at approximately 30%, the Gompertz function at around 37% and the Logistic function at 50% of asymptotic body mass. Thus, maximum growth, which is observed at the point of inflection and is expressed in absolute maximum growth rate, is not comparable even between these standard models without an appropriate transformation.

This last assertion is not supported by any arguments, and unfortunately is not correct. The value of the maximum growth rate is comparable across all three models because in each case it is defined as the change in mass per time and is given by the first time derivative of the growth curve. In each model, the maxima do occur at different ages and thus different masses, but it is nevertheless statistically and biologically justified to compare them. The concept of fitting a growth curve on a per-species basis is based on the assumption that the growth trajectory–and thus the curve’s parameters (as an approximation)–is encoded in the genome of that species (often called determinate growth, because the properties are pre-determined). In this view, the maximum asymptotic mass of the species M is already built into the biological mechanisms regulating growth at all ages, including at the point of maximum growth rate. It thus does not matter that the mass at the time of fastest growth differs from model to model. That is the philosophy behind most authors’ use of M as an independent variable for a wide range of natural history parameters.

In effect, Werner and Griebeler present a different biological hypothesis than Case, Erickson, or Grady et al.: viz., that the choice of independent variable matters, and that the choice should be body mass at the age when maximum growth occurs (BMatMG). Werner and Griebeler did not discuss their reasoning in detail, nor did they provide a sensitivity analysis assessing the difference this choice makes.

Mathematically, (10) where d is a constant that depends on the growth model. For example, (11)

The effect of using BMatMG as the independent variable instead of M is to introduce the constant factor d on the independent variable, and thus to rescale the dependent variable by the constant D: (12)

As a result, for the Werner and Griebeler data sets I follow their practice and regress kD versus BMatMG.

The physical interpretation of kD is given by (13)

Where t_BMatMG is the time it would take to grow from birth to mass M at a constant growth rate equal to G_max. As with the interpretation of kC, actual growth time is longer than t_BMatMG. The growth time from birth to mass BMatMG is typically 2× to 3× t_BMatMG for many species.

If BMatMG were to be adopted for the Grady et al. study, it would have no practical effect because all of the data points in that study were derived by using a Gompertz growth model. Using BMatMG would rescale the x and y coordinates, but since the growth-rate regression is done with log-transformed data, the rescaling would amount only to translating the points. The slope of each regression would be unchanged, and the intercept would be adjusted by an additive constant. Conversely, if the Werner and Griebeler study were to be redone using M as the independent variable, some points would move relative to each other, which would change some of the regression results. Unfortunately, the published data set does not specify which model is used for each extant species, so I was unable to test the size of this effect.

Intragroup Variation and the Ecological Fallacy

It is common in statistics to compare multiple groups by estimating a statistic, such as the mean, for each group. The larger the sample of the group, the better the estimate of the statistic in question. With sufficiently large samples, one can get sharp estimates of the mean mass of each group, and these means can be quite distinct even if there is considerable overlap among the groups. As an example, the Centers for Disease Control and Prevention’s National Health and Nutrition Examination Survey, in its 2013–2014 data release [47], measured the mass of male and female humans age 20 or older, with the result that the mean male mass was 83. 1 kg and the mean female mass was 72.0 kg, a difference of 11.1 kg.

One could summarize this by noting that, on average, men are heavier than women, but this does not imply that we can infer that men are heavier on an individual basis–i.e. that all men are heavier than all women. The conflict between conclusions drawn on the group average and at the individual level is known as the ecological fallacy.

The original Case studies arguably did not fall into the ecological fallacy because Case was interested in what could be termed the “forward inference” that different groups of animals had different allometry. Grady et al. and Werner and Griebeler do fall into the ecological fallacy because they seek to make the reverse inference–i.e. to classify dinosaur metabolism by comparing their mean growth-rate allometry to that of extant groups.

Returning to the example of human body mass cited above, we cannot use the mean body mass to classify individuals as male or female by their body mass with any reasonable accuracy. Given a group of N adult humans, all of them the same gender, we could do the classification if N is large enough. Simple bootstrap calculations on the human body mass data set cited above show that one needs N ≥ 67 to do the classification to the 95% confidence level. Note that the classification, in effect, applies the group average to the individual (classification as male or female)–this is valid only because we assume up front that everyone in this group is of the same gender.

That is a very important limitation. If instead one permits three kinds of group–all male, all female, or randomly male and female with equal probability–then the group size for 95% confidence level jumps to N ≥ 303. If we chose more groups with known a priori odds of the male/female mix, N becomes bigger still, and if the a priori odds of group composition by gender are unknown, the classification problem becomes effectively impossible for any N. That is the ecological fallacy in action: with sufficiently strong constraints (such as certain knowledge that all members of a group are the same gender and that N is large enough), we can make a valid (although still statistical) inference. But without those strong constraints, we cannot do so.

The relevance to the Grady et al. and Werner and Griebeler studies is that we explicitly do not know that all dinosaurs had the same metabolism. This is particularly true because their dinosaur data sets span the range from basal dinosaurs to highly derived taxa over an interval of more than 100 million years. It is thus analogous to the case of classification with unknown gender ratio. Even if we suppose that the limited data set available gives us a true estimate of the average dinosaur growth-rate allometry, there is no way to use that to classify all dinosaurs because they could have different metabolisms (see inset graphs in Fig 1).

A second point is that there is a clear and obvious biological meaning to the group average of body mass for male and female humans. We know there is a strong genetic difference between men and women involving the X and Y chromosomes. So the biological meaning of the mean mass is presumably the signal due to that chromosomal difference.

But the group average across the Grady et al. or Werner and Griebeler groups has very uncertain biological meaning, as discussed above. An average across a monophyletic group (supplemented with high-quality, time-resolved phylogenetic data) could provide the allometry of the most recent common ancestor. That could not be used to classify the individual members of the group, however–it would apply only to classifying the ancestor.

Since the fossil record has not provided unambiguous, time-resolved phylogeny for all dinosaurs, it is not possible to say with certainty that the phylogenetic regression methods used in [12,13] actually do recover the common ancestor of all dinosaurs. But even if they do, that would be irrelevant for classifying the derived taxa from that ancestor. Instead we have a mean allometry across non-uniformly sampled taxa spanning almost 200 million years. It is far from clear what this means biologically.

Dinosaurs are not the only problem. The groups used in the Grady et al. and Werner and Griebeler studies for extant animals do not have a consistent theme. For altricial and precocial birds the theme is behavioral at best, as discussed above. Other groups are monophyletic, paraphyletic, or polyphyletic, so the biological interpretation of the mean allometry is unclear. Grady et al. and Werner and Griebeler both implicitly assume that a different mean allometry for the group can be interpreted as being due to metabolic differences, but that is far from clear. Altricial and precocial birds do differ in metabolic ways, but also in mass, as can be seen in the convex hull plots of Fig 5B and S23 Fig. Which effect is more important? One cannot tell because the effects of larger body mass may confound the analysis of growth allometry.

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Fig 5. Confidence and predictions bands versus convex hull.

Plot (A) shows the results of regression on G_max for data sets from Grady et al. [13]. Shaded regions are the 95% confidence bands for the regressions. Plots (B) shows the range of variation within each group as a shaded convex hull polygons containing the data from each of the two studies, respectively. Dinosaurs are shown as a black outline; the dashed black outline represents dinosaurs excluding Archaeopteryx. Plot (C) shows the single prediction bands for regression of kC versus M for each group. Plot (D) shows the convex hull polygons for mass and BMR data from [13].

https://doi.org/10.1371/journal.pone.0163205.g005

Indeed if one took the group of eutherian mammals and arbitrarily divided it into two groups above and below a certain growth rate threshold, the difference between the resulting groups would look much like the difference between altricial and precocial birds. Such a division would highlight different mean allometry, but that does not mean there is a true biological basis for it.

The human body mass analogy above is scalar, or one-dimensional. Any approach to using maximum growth rate to determine metabolism must confront a basic issue: species have both a mass M and a maximum growth rate, which could be expressed as G_max or kC or kD. Thus a species is fundamentally represented by a point on a two-dimensional mass vs. growth rate plane rather than by a single scalar quantity. As an example, what is the appropriate way to compare the growth rates of two species, one that has a mass of 1 g and the other of 10⁶ g? Similarly, if one measures metabolism directly via BMR, one also has a two-dimensional problem because a 1 g and a 10⁶ g animal will have vastly different total power outputs in watts.

This issue is one reason to apply regression. By understanding how growth rate scales with maximum body mass, we can reason about species across a range of sizes. Regression is entirely analogous to the mean in a scalar example like body mass–it measures the central tendency of the group, but it yields two parameters (in this case) rather than one. The issues with the ecological fallacy and group definition all apply; the only difference is that we must compare two parameters rather than one.

Fig 5A shows regression results for Grady et al. Each paper [1,5,7,9–13,16] includes a plot of this kind (step 5), either with or without 95% confidence bands. The key message that these plots convey is that groups have clearly separate regression lines that for the most part do not overlap. Endotherms and ectotherms show quite different growth-rate allometry in these plots. The visual appeal of these plots due to clearly separated regression lines with narrow 95% confidence bands is a principal reason that the results of prior studies seemed compelling. Qualitatively similar plots can be made for data from the Werner and Griebeler data (S23 Fig).

Unfortunately this picture is very misleading, as can be seen in Fig 5C, which plots the convex hull (smallest enclosing convex polygon) of kC versus M of the data points for each group. The convex hull plot shows that there is considerable overlap in the growth rate allometry of individual species; it is only the group-wide averages that are distinct.

The conceptual error hiding within Fig 5A is that the 95% confidence bands reflect the constraints on the group central tendency under regression. The tight bounds on the eutherian mammal group tell us that the average allometry across 153 extant mammals is well constrained. Biologically, it is unclear what interpretation should be applied to the average eutherian allometry, but in any event it is not the appropriate statistical method for classification of individuals. The convex hull delineates the total range of variations across individuals in each group. The appropriate statistical measure of the individual variation, and thus classification, is given by the single prediction band (also with 95% confidence) which is shown in Fig 5C. The difference between these two is, in effect, the ecological paradox in action. While the group mean allometry might be tightly constrained, the variation within the group is not.

Rather than being cleanly separated, the growth data for each group overlap considerably. Overlap of this kind also exists in a plot of M and G_max, but the shear transformation (9) tends to hide the overlap in a tight cluster of points (Fig 2 and S15 Fig). The original regressions of Case [1, 2] had the compelling property that they seemed to separate disparate groups into distinct bands (i.e. Fig 4A and 4B). This is not true for the individual species-level data, however.

Note that this is not the case for direct measures of metabolism. Fig 5D shows the BMR versus M data of Grady et al., plotted as convex hulls. There is a clear separation between endotherms and ectotherms, even at the individual species level. Note that BMR here is in watts, and it thus depends on the total mass of the animal. One can also look at mass-specific metabolism BMR/M–i.e., watts per gram of body mass, which is analogous to kC. That metric, too, can cleanly separate endotherms from ectotherms (S14 Fig). This poses a major challenge for hypotheses H1 and H2. Maximum growth rate and metabolism are both properties possessed by individual species and presumably encoded in their genomes. If the two parameters are closely tied, as the hypotheses hold, then why is it that groups can be cleanly separated by metabolism, but not by maximum growth rates?

The overlap in the species-level growth-rate and mass data can be demonstrated in other ways as well. Tables 1 and 2 quantify the overlap by showing a count of the data points in each group that fall within the convex hulls. As an example, data on the eutherian mammals from Grady et al. includes members that fall within the convex hull of every other group except altricial birds. In data from Werner and Griebeler, the eutherians include members that overlap the convex hull of every other group.

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Table 1. Data points that lie inside the convex hull polygon of each group.

The data points from Grady et al. [13] arranged vertically and the convex hull associated with each group horizontally. Because there is substantial overlap among the convex hulls, a data point may belong to more than one convex hull.

https://doi.org/10.1371/journal.pone.0163205.t001

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Table 2. Data points that lie inside the convex hull polygon of each group.

The data points from Werner and Griebeler [12] arranged vertically and the convex hull associated with each group horizontally. Because there is substantial overlap among the convex hulls, a data point may belong to more than one convex hull.

https://doi.org/10.1371/journal.pone.0163205.t002

Of the 21 dinosaur taxa analyzed by Grady et al., 20 fall within the convex hull of eutherian or marsupial mammals. Conversely, the dinosaur convex hull encompasses more data points for mammals (23) than for dinosaurs. In the data obtained from Werner and Griebeler of the 12 dinosaur taxa, seven dinosaur data points fall within the mammal convex hull, and again there are more data points for mammals within the dinosaur convex hull than there are points for dinosaurs.

Inclusion within the convex hull is only one of many possible ways to measure the overlap. One could instead use the regression lines to make a parallelogram bounded above by the regression line shifted vertically to capture the highest positive residual and bounded below by the regression line shifted to capture the smallest negative residual (S16 Fig). The parallelograms thus generated contain the convex hulls and thus yield even more overlap.

A third, more restrictive overlap classification criterion makes use of distance from a point to the regression line nearest each point (S3 Table). This produces results that are qualitatively similar to those shown in Tables 1 and 2. In the Grady et al. data sets, some points for eutherian mammals are closer to every other regression line (except that of altricial birds) than they are to their own regression line.

Eight of the 21 dinosaur data points are closest to the regression lines of endothermic groups (eutherians, marsupials and precocial birds). The result is much the same when this method is applied to data from the study by Werner and Griebeler: eutherians have at least some members that are closest to every other regression line. Only three of the 19 dinosaur data points lie closest to an endothermic regression line, but the dinosaur regression line is the nearest to at least some members of eutherians, marsupials, reptiles, and fish.

These figures and tables all illustrate an important problem: a plot of maximum growth rate versus mass does not allow clean separation of groups. The seeming simplicity of the regression lines (Fig 5A) arises only because they are averages that show the central tendency of each group. The 95% confidence bands summarize the amount of variation in the central tendency, but do not account for the variation in the underlying data, most of which lie outside the 95% confidence bands.

The regression lines in Fig 5A are distinct, but that is irrelevant from a biological standpoint. The biological question is whether we can determine metabolism for individual species by analyzing the group properties (i.e., the regression parameters a and b) or the maximum growth rate (kC or kD or G_max) and mass M data points for the individual species.

The various measures of overlap (convex hull, parallelogram, distance to regression lines) tell us that using individual species data to classify metabolism yields different answers than the group-wide averages do.

As an example, suppose that we did not know the metabolic state of eutherian mammals. The regression lines in Fig 5A would classify them with other endotherms (marsupial mammals, birds). The individual species data for eutherians show overlap (Fig 5B and 5C) with both endothermic and ectothermic groups, rendering classification impossible. This would be true regardless of the classification approach used. If instead we look at metabolism directly (Fig 5D and S14 Fig), we see that eutherians have entirely endothermic growth rates that are very easy to distinguish from ectotherms, but entirely overlap other endotherms.

Another issue with averages is their high sensitivity to the sample used. For example, if we choose the subset of eutherian mammals having the lowest growth rate, the average would be classified as an ectotherm. It is only when this subset is averaged together with faster-growing cousins that the regression results predict endothermy. In the case of extant groups, we know the implications of group composition, but it is unclear whether we can have the same confidence with extinct taxa. In particular, we know that there is a fast growing, endothermic subset of dinosaurs–birds. Is it valid to include them? Phylogenetic analysis (i.e. Fig 1) would suggest that it is, but doing so would completely change the group-wide averages.

We could pool all endotherms and all ectotherms together into just two groups, in the hope that this might give the average allometry some valid biological meaning. Perhaps metabolism-specific groups could average out differences due to other factors, leaving metabolism as the only salient difference. Unfortunately, considerable overlap between the two groups remains in both the convex hull and the single prediction bands (S24 Fig).

When group-wide averages give a different answer than individual data, we have precisely the conditions required for the ecological fallacy [17–19]. As discussed in the introduction, there is a strong biological reason to rely on individual data. There is no proposed biological mechanism by which group-wide averages of maximum growth-rate allometry could determine the metabolism of an individual species and override what its own maximum growth rate would dictate.

Grady et al. have argued [40] that convex hulls are an inappropriate tool for evaluation because they “enclose the entire space occupied by a data set” rather than summarizing its statistical properties. They further argued that “Myhrvold’s use of range polygons is misleading in that it implies considerable overlap between taxa that are statistically quite distinct.”

The crux of the issue is indeed determining where data at the species level show considerable overlap and where they are quite distinct (Fig 5B). The groups are not statistically distinct at the individual level–otherwise the single prediction bands would be separated (Fig 5C). Finally, there is no overlap between endotherms and ectotherms in the species-level data for metabolism itself (Fig 5D and S14 Fig). The fact that an average across a group can yield a tight bounds, if the group is of sufficient size, does not imply that this average is very informative about individual group members.

One Dimensional Comparisons

A different way to approach the two-dimensional comparison problem for growth-rate and mass data is to reduce it to one dimension. Both Grady et al. and Werner and Griebeler used this approach.

Werner and Griebeler performed regressions on their data using an alternate method that fixed the slope of the regression at b = 0.75 and used a alone to fit the data. Note that regression with one parameter is still an average and thus has the same ecological fallacy issues discussed above for regression with two parameters. The value of 0.75 for b has a long history from empirical studies of metabolic scaling [8,42–44] and is predicted theoretically by MTE, as discussed in the next section. Grady et al. perform an equivalent calculation for both b = 0.75 and b = 0.66, which they refer to as the “mass-independent growth rate”–a misnomer because the parameter does depend explicitly on mass.

The correct interpretation of this approach is that it adjusts the growth rate of an individual species to the growth rate it would have if it had a body mass of 1 g (i.e., log M = 0), assuming that the mass dependence is M^b. Because the slope b is fixed, adjusting for any other value of M effectively adds a constant to the distribution. This general method has been explored by others [48].

Fig 6 shows the results where b = 0.75 for both the Grady et al. (Fig 6A) and Werner and Griebeler (Fig 6B) data sets (S25 Fig shows corresponding results where b = 0.66). If a clear separation among groups were possible, then one could draw a vertical line dividing the endotherms on the left from the ectotherms on the right. In actuality, the same mass-adjusted growth rate is associated with both endotherms and ectotherms within a wide band (shaded areas of Fig 6).

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Fig 6. Fixed-slope regression residuals.

Each plot shows the histogram of growth rates adjusted to a mass of 1 g, assuming a fixed slope of b = 0.75 for data sets from (A) Grady et al. [13] and (B) Werner and Griebeler [12]. Histograms for endothermic groups are shaded light red; those for ectotherms are shaded light blue. The shaded band shows the growth rates that overlap among extant endotherms, ectotherms and dinosaurs (yellow). Plots (C) and (D) show the equivalent fixed-slope regressions for BMR data from [13].

https://doi.org/10.1371/journal.pone.0163205.g006

Fig 6 plots these results in essentially the same way as Werner and Griebeler do in their Fig 2 and Grady et al. do in their S6 and S11 Figs, except using distribution histograms rather than box-and-whisker plots. The overlap pictured in Fig 6 is also evident in their figures.

Instead of using fixed b for all groups, one can instead perform the adjustment for each group using a value of b determined from the regression for the group. As that value varies group to group, adjusting to a specific value of M can yield small differences among groups, but the qualitative picture remains the same: there is no way to separate endotherms and ectotherms at the species level.

One can perform the same analysis, with fixed b = 0.75, on basal metabolic rate (BMR) versus mass data from Grady et al. (Fig 6C and 6D). In this case, however, a clear separation is possible with little or no overlap between endotherms and ectotherms. Growth rate is very different from BMR in that respect, just as it is in Fig 5.

Fig 6 also shows that the groups having the fewest data points (dinosaurs and marsupials) also have the narrowest ranges of residuals, likely due to small sample sizes. As dinosaur data accumulates and the sample size increases, the range for that group will likely overlap even more groups. There is no reason to believe that the small number of dinosaur taxa sampled (20 in Grady et al., 12 in Werner and Griebeler) spans the entire range of maximum growth-rate variation for the clade Dinosauria.

Theoretical Arguments

Both Grady et al. [13] and Werner and Griebeler [12] offer theoretical explanations based on MST, MTE, and the body of work that has risen around the allometric scaling of BMR with mass first discussed by Kleiber in 1932 [42,43]. Kleiber found that BMR ∝ M^0.75, a result that set in motion a tremendous amount of empirical and theoretical work in the intervening decades [49–58]. This topic is far too vast to adequately summarize here, but some brief observations will suffice to put it in context.

MTE seeks to explain Kleiber’s exponent 0.75 as a result of the fractal scaling of metabolic networks [58]. This theory generates two fundamental predictions: that the exponents b = ±0.75 and in some cases b = ±0.25 should be found universally in biology. In particular, the theory holds that G_max ∝ M^0.75, a relationship of direct relevance here.

Although MTE sparked great theoretical interest, recent work has shown that it is both controversial from a theoretical perspective and apparently refuted by empirical studies [51,58]. The allometry BMR ∝ M^b has been reappraised by many studies [49–58]. Some hold that the correct universal exponent is b = 0.67, whereas others support a universal b = 0.75. Most recent studies indicate, however, that there is no single universal value of b and that it instead varies by taxonomic group and by data set. These results cast grave doubt on the applicability of MTE for growth-rate allometry.

Moreover, BMR ∝ M^b appears to be only an approximation. Rather than this simple power law, which is a straight line on a log-log plot, the actual relationship has curvature [57–60]. The curvature potentially explains why different studies find different exponents: the exponent varies among data sets that measure different parts of the range.

One can also test for curvature in the regression between maximum growth rate and body mass. Table 3 shows the differences in the corrected Akaike information criterion (ΔAICc) resulting from fitting linear, quadratic, and cubic curves (models in S2 Table) to the kC and kD growth data. Following conventional practice [61], the best fit is obtained when ΔAIC_c = 0. Models are considered to have strong statistical support when ΔAIC_c < 2. The regressions are plotted in S18 and S19 Figs.

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Table 3. Model selection results.

The difference between the corrected Akaike information criterion (ΔAIC_c) was calculated for models fit to log-log transformed kC versus M data (from Grady et al.) or kD versus BMatMG data (from Werner and Griebeler), on a group-by-group basis. The models having ΔAIC_c = 0 (unshaded cells) are the best fit. Models having ΔAIC_c < 2 (bold numbers) have strong statistical support. The results show that substantial curvature effects exist for many groups. Models are defined in S2 Table. Best fits are plotted in S18 and S19 Figs.

https://doi.org/10.1371/journal.pone.0163205.t003

These results show curvature in growth-rate allometry in many, but not all, groups. In the Grady et al. data sets, the strongest rejection of the linear model (i.e., where the linear model has the highest ΔAIC_c) is found for eutherian mammals, marsupials, and dinosaurs. For the Werner and Griebeler data sets, in contrast, it is among the altricial and precocial birds where the linear model has the highest ΔAIC_c. In the case of BMR, it is known that there can be a phylogenetic signal and that there is inconsistency among taxonomic groups [62]. That may also be the case for growth rate, in which case phylogenetic regression methods may be more suitable than the OLS method used here.

In studies of BMR, quadratic models have been the most commonly used nonlinear models [59,62–64]. For growth-rate allometry, however, cubic models yield the best fits for all of the cases in which the linear model is strongly rejected.

Curvature rules out the simple allometric power laws predicted by MTE. It implies that the effective exponent is a function of mass [59,62–64], a relationship that may be important for dinosaurs, since that group spans an enormous range of body sizes. Indeed, this curvature may be due to factors that would impose an upper bound on body size [59].

Finally, it should be noted that the one-dimensional comparison discussed above and shown in Fig 6 assumes, in effect, that MTE is correct by performing scaling with b = 0.75. This still does not help with the fundamental problem, however, which is that the growth rates in Fig 6 overlap.

Regression of Maximum Growth Rate versus Metabolism

Both Werner and Griebeler [12] and Grady et al. [13] make the argument that scaling properties of BMR and maximum growth rate imply that the two are related. Grady et al., wrote that “Empirical evidence (13) indicates that G_max scales similarly to B, where G_max = G₀M^α. This suggests that BMR ∝ G_max and thus that metabolic rate may be inferred from growth” [13]. (Their reference 13 is the Case paper [1], and B is their notation for BMR.) Werner and Griebeler similarly noted that “Additionally, similar slopes are observed in regression models of growth rate against body mass or metabolic rate against body mass [5,9,10,12,14,20,37,49,59–61]. Thus a link between growth rate and metabolic rate seems very likely” [12].

This argument is expressed mathematically as follows. Kleiber’s law and MTE focus on the allometric scaling of BMR with body mass (14) and maximum growth rate G_max also has an allometric scaling law (i.e., Eq 6 above) (15)

Grady et al. use (14) and (15) to relate the two via either of the following direct relationships: (16)

Or (17)

Grady et al. argue that (in the notation used here) b₁ = b₂ = 0.75 which makes (16) and (17) become. (18) which combines Eqs (1) and (4) of [13]. Eqs (16) and (18) are the statement of hypothesis H2 presented by Grady et al.

This reasoning behind Eqs (14) through (18) is an example of a well-known fallacy in statistical reasoning called the “fallacy of averages” [65–68]. If (14) and (15) were true equalities then this would be correct, but in general they are statistical correlations, which renders (16) and (17) to be incorrect as general statements [65–68].

By using the Grady et al. data set of 120 extant species with both BMR and G_max, we can test (18) directly. Unfortunately it fails by large margins, as shown in S22A Fig. Both BMR and G_max are increasing functions of M, so we should expect some correlation. However 0.6 G_max gives an estimate of BMR that ranges from a factor of 13.6 too high to a factor of 161.5 too low.

The BMR data from Grady et al. was measured from specimens at body mass M_met, which is in general less than adult body mass M. This could potentially distort the relationship between BMR and G_max. To test this, we can make an adjusted BMR: (19)

Note that (19) implies that Kleiber’s law (14) holds during ontogeny. This is related to the assumption by Werner and Griebeler that one must use BMatMG rather than M–i.e., that the scaling applies at each moment of ontogenetic growth, rather than just the end point. Most studies of Kleiber’s law involve multiple species rather than following the growth of individuals of a single species, so it is far from clear that this assumption is valid. Indeed, intraspecific studies that study ontogenetic scaling have found a different value of the exponent b [69–71] than interspecific studies have.

Using BMR_adj rather than BMR does not yield much improvement, as can be seen in S22B Fig; the estimate now ranges from a factor of 17.7 too high to a factor of 7.28 too low. This is a smaller range of variation than without the adjustment, but it still varies overall by a factor of 129. Eq (18) is a rather poor estimator.

Instead of directly testing (18) by using BMR and G_max, Grady et al. pursued a regression of what they termed “mass-independent growth rate” as the dependent variable, with “mass-independent metabolic rate” as the independent variable (Fig 2 in ref. [13]). These terms are misnomers: the most natural definition of “mass-independent growth rate” is kC, whereas their definition still contains a factor of M^0.25 and thus cannot be “mass-independent.” Their “mass-independent” metabolic rate similarly contains a factor of M^−0.75. It appears to be intended to perform a function analogous to (19).

It is unclear what a regression using these variables has to do with a test of hypothesis H2 or Eqs (16) and (18). In Section III of the supplemental information of [13], Grady et al. present arguments for treating endotherms and ectotherms differently in creating regressions to test the relationship between BMR (or BMR_adj) from G_max. They rely on the hypothesis that Eqs (14) and (15) are true at every point in time during ontogeny (i.e., as with (19)), but appear to transform G_max rather than BMR as in (19).

The weak correlation between BMR and G_max shown in S22 Fig is likely due to the fact that they both are increasing functions of M. To test this, I regressed BMR versus kC, which is a genuinely mass-independent variable, as well as versus M. The results of these regressions, as well as of kC versus M, are shown in Fig 7 and detailed in S20 Fig.

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Fig 7. Correlations among BMR, kC, and M are not transitive.

(A) shows pairwise correlations between BMR and kC, BMR and M, and kC and M (see also S20 Fig). (B) through (D) show how three-dimensional (BMR,kC,M) points map to each of these pairwise correlations. Even though both BMR and kC have correlations to M, they are essentially not correlated to each other (R² = 0.03).

https://doi.org/10.1371/journal.pone.0163205.g007

I find that there is a moderately strong relationship kC ∝ M^1−b, R² = 0.453 (S20 Fig), and BMR ∝ M^b, R² = 0.622 (S20 Fig). But there is essentially no correlation between BMR and kC, where R² = 0.034 (S20 Fig). If we use BMR_adj instead of BMR, the correlation is slightly stronger but still extremely weak, with R² = 0.127.

The fact that we can find regressions between kC and M, and between BMR and M–but not between BMR and kC–may seem surprising. It is a direct refutation of the reasoning in Werner and Griebeler and Grady et al. discussed above in Eqs (14) through (18). This is the fallacy of averages at work [65–68]. The same phenomenon gives rise to the fact that statistical correlation is not transitive [72]. Even though there is statistical support that kC ∝ M^1−b and that BMR ∝ M^b, this does not imply that BMR ∝ kC^c for any value of c.

S21 Fig plots BMR versus kC and unpacks the shear transformation, thus separating the convex hulls of the extant groups. This plot can be compared to Fig 2A of Grady et al., in which the choice of variables causes shearing that compresses the range of variation.

The results here cast grave doubt on any relationship between BMR and maximum growth rate. A direct test of (18) fails to predict BMR (or BMR_adj) from G_max by a large margin. If instead we use mass specific maximum growth rate kC to remove spurious correlation by shared factors of M, we find that there is no relationship between kC and BMR (or BMR_adj), despite the fact that each one is correlated with M. Instead of offering proof of hypothesis H2, I find that the data set gathered by Grady et al., fails to show such a relationship, and could even be offered as proof to the contrary. Lovegrove, who used a much larger data set and more sophisticated phylogenetic regression methods than used here, also concluded that growth rate is not correlated with BMR for mammals [73].

In response to a previous criticism to this effect [74], Grady et al. argued [40] that it is unsurprising that there is no relationship between BMR versus kC, but did not explain how that can be consistent with their hypothesis.