Department of Ecology and Evolutionary Biology, University of Toronto, 25 Willcocks Street, Toronto, Ontario, Canada M5S 3B2
Department of Palaeobiology, Royal Ontario Museum, 100 Queen's Park, Toronto, Ontario, Canada M5S 2C6
Abstract
Background
Body size is intimately related to the physiology and ecology of an organism. Therefore, accurate and consistent body mass estimates are essential for inferring numerous aspects of paleobiology in extinct taxa, and investigating largescale evolutionary and ecological patterns in the history of life. Scaling relationships between skeletal measurements and body mass in birds and mammals are commonly used to predict body mass in extinct members of these crown clades, but the applicability of these models for predicting mass in more distantly related stem taxa, such as nonavian dinosaurs and nonmammalian synapsids, has been criticized on biomechanical grounds. Here we test the major criticisms of scaling methods for estimating body mass using an extensive dataset of mammalian and nonavian reptilian species derived from individual skeletons with live weights.
Results
Significant differences in the limb scaling of mammals and reptiles are noted in comparisons of limb proportions and limb length to body mass. Remarkably, however, the relationship between proximal (stylopodial) limb bone circumference and body mass is highly conserved in extant terrestrial mammals and reptiles, in spite of their disparate limb postures, gaits, and phylogenetic histories. As a result, we are able to conclusively reject the main criticisms of scaling methods that question the applicability of a universal scaling equation for estimating body mass in distantly related taxa.
Conclusions
The conserved nature of the relationship between stylopodial circumference and body mass suggests that the minimum diaphyseal circumference of the major weightbearing bones is only weakly influenced by the varied forces exerted on the limbs (that is, compression or torsion) and most strongly related to the mass of the animal. Our results, therefore, provide a muchneeded, robust, phylogenetically corrected framework for accurate and consistent estimation of body mass in extinct terrestrial quadrupeds, which is important for a wide range of paleobiological studies (including growth rates, metabolism, and energetics) and metaanalyses of body size evolution.
Background
In extant taxa, body size is recognized as one of the most important biological properties because it strongly correlates with numerous physiological and ecological factors, such as metabolic rate
Due to the biological implications of body size, it is not surprising that numerous paleontological studies have used body mass estimates to reconstruct and interpret: patterns of body size evolution
Currently, there are two types of methods used to estimate body mass in extinct animals: volumetric reconstructions and skeletal scaling relationships. The latter method is commonly used to predict body mass in extinct members of relatively recent crown clades (that is, of Mesozoic origin) such as Mammalia and Aves
An alternative method to reconstructions, and one that can be used to test and constrain scale and computational models (
Dinosaurian body masses are still generally estimated using reconstructions, with the exception of two studies
1. The widely cited Anderson method, especially among nonavian dinosaur researchers, is criticized based on its use of a taxonomically biased sample towards ungulates (for example,
2. Differences in gait and limb posture impart different stress regimes on the limbs
3. Residual outliers (large residual values) and extreme outliers (values at the upper and lower extremes of the dataset) can have a large effect on regression coefficients
All three of these criticisms are tested for the first time, within the context of 200 mammal and 47 nonavian reptile species [See Additional file
Limb measurement and body mass data. Table of measurements of all the extant taxa used in the present study, as well as the limb measurements of the nonavian dinosaurian taxa shown in Table
Click here for file
Results
Raw data results
Results from the standardized major axis (SMA) analyses comparing clades based on the raw nonphylogenetically corrected data are provided in Figures
Limb scaling patterns in mammalian clades
Limb scaling patterns in mammalian clades. Lines are fitted based on the SMA results presented in Table 1. (A) Log femoral length and circumference plotted against log body mass. (B) Log humeral length and circumference against log body mass. (C) Log femoral length plotted against log humeral length. (D) The log of combined humeral and femoral circumference against log body mass. SMA, standardized major axis.
Limb scaling patterns in quadrupedal terrestrial tetrapods
Limb scaling patterns in quadrupedal terrestrial tetrapods. Lines are fitted based on the SMA results presented in Table 1. Lissamphibians are plotted (green) but no line was fitted due to its small sample size and body mass range. (A) Log femoral length and circumference plotted against log body mass. (B) Log humeral length and circumference against log body mass. (C) Log femoral length plotted against log humeral length. (D) The log combined humeral and femoral circumference against log body mass. SMA, standardized major axis.
Stylopodial scaling in mammals and nonavian reptiles.
Analysis
(
Sample
N
95% +CI
R^{2}
Sim.
L_{F }vs. C_{F}
All
234
1.0301
1.0616 to 0.9996
0.6020
0.542 to 0.6619
0.9459
G
Mammalia
188
1.0332
1.0677 to 0.9999
0.6148
0.5469 to 0.6827
0.9484
G
Reptilia
46
1.1751
1.3184 to 1.0474
0.8115
0.5884 to 1.0347
0.8560
> G, < E
Ungulata
32
1.2014
1.3338 to 1.0821
0.9810
0.6676 to 1.2943
0.9211
> G, < E
Carnivora
46
0.9840
1.0888 to 0.8893
0.5409
0.3317 to 0.75
0.8887
G
Marsupialia
14
1.0774
1.1467 to 1.0123
0.7317
0.6057 to 0.8577
0.9902
> G, < E
Euarchonta
14
1.0251
1.2141 to 0.8656
0.7382
0.3835 to 1.0929
0.9269
G
Glires
66
0.9542
1.0334 to 0.8811
0.4716
0.3454 to 0.5979
0.8978
G
L_{H }vs. C_{H}
All
234
1.0644
1.0971 to 1.0326
0.6229
0.5626 to 0.6831
0.9452
> G, < E
Mammalia
187
1.0603
1.0967 to 1.0252
0.6199
0.5511 to 0.6887
0.9459
> G, < E
Reptilia
47
1.2190
1.3355 to 1.1126
0.8536
0.6724 to 1.0348
0.9072
> G, < E
Ungulata
32
1.3083
1.4325 to 1.1949
1.1391
0.8529 to 1.4254
0.9407
> G, < E
Carnivora
46
1.0814
1.1777 to 0.9929
0.7101
0.5193 to 0.9009
0.9209
G
Marsupialia
14
1.0472
1.187 to 0.9238
0.6059
0.3774 to 0.8343
0.9601
G
Euarchonta
14
0.9175
1.0816 to 0.7782
0.4785
0.1826 to 0.7744
0.9309
G
Glires
66
0.9296
0.9931 to 0.8702
0.4116
0.3166 to 0.5066
0.9300
G
L_{F }vs. BM
All
234
2.9307
3.0323 to 2.8325
2.1677
1.9744 to 2.3611
0.9306
G
Mammalia
188
2.9930
3.0974 to 2.8922
2.3410
2.1359 to 2.5461
0.9439
G
Reptilia
46
3.2500
3.7486 to 2.8177
2.4800
1.7132 to 3.2468
0.7778
G
Ungulata
32
3.4979
3.8785 to 3.1545
3.4591
2.5578 to 4.3603
0.9230
> G, < E
Carnivora
46
2.7472
3.0791 to 2.451
1.8012
1.1427 to 2.4597
0.8584
G
Marsupialia
14
2.9980
3.4286 to 2.6215
2.4690
1.7121 to 3.2258
0.9542
G
Euarchonta
14
3.0622
3.7695 to 2.4877
2.9486
1.6443 to 4.253
0.8893
G
Glires
66
2.7702
2.9779 to 2.5769
1.9621
1.6297 to 2.2944
0.9160
0
C_{F }vs. BM
All
247
2.8479
2.8997 to 2.7969
0.4587
0.3845 to 0.5328
0.9794
< G, > E
Mammalia
200
2.8977
2.9504 to 2.8459
0.5615
0.4829 to 0.64
0.9834
< G, > E
Reptilia
47
2.7943
2.9801 to 2.6201
0.2653
0.057 to 0.4736
0.9540
E
Ungulata
41
2.9204
3.1192 to 2.7344
0.6173
0.232 to 1.0027
0.9586
G
Carnivora
48
2.7893
2.9182 to 2.6661
0.2895
0.0946 to 0.4844
0.9768
E
Marsupialia
14
2.7827
3.1222 to 2.4801
0.4328
0.0138 to 0.8518
0.9664
G, E, S
Euarchonta
15
2.9728
3.1874 to 2.7727
0.7271
0.4393 to 1.0149
0.9864
G
Glires
66
2.9031
3.084 to 2.7328
0.5929
0.3965 to 0.7893
0.9413
G
L_{H }vs. BM
All
234
2.8653
2.9489 to 2.7841
1.8284
1.6745 to 1.9823
0.9506
0
Mammalia
187
2.8626
2.9522 to 2.7756
1.8476
1.6778 to 2.0175
0.9548
0
Reptilia
47
3.3718
3.704 to 3.0694
2.5472
2.0315 to 3.0629
0.9018
> G, < E
Ungulata
32
3.4092
3.8036 to 3.0558
2.9639
2.0630 to 3.8648
0.9135
> G, < E
Carnivora
46
2.8202
3.0641 to 2.5957
1.8667
1.3831 to 2.3503
0.9253
G
Marsupialia
14
3.1988
3.6508 to 2.8027
2.3972
1.6611 to 3.1333
0.9556
G
Euarchonta
14
2.5359
2.9736 to 2.1627
1.6484
0.8577 to 2.4391
0.9354
0
Glires
66
2.6071
2.766 to 2.4573
1.3946
1.1559 to 1.6332
0.9438
0
C_{H }vs. BM
All
247
2.6861
2.7322 to 2.6406
0.1438
0.0788 to 0.2087
0.9816
E
Mammalia
200
2.6938
2.7445 to 2.6439
0.1655
0.0913 to 0.2398
0.9823
E
Reptilia
47
2.7661
2.9296 to 2.6117
0.1862
0.0049 to 0.3675
0.9634
E
Ungulata
41
2.5273
2.7222 to 2.3464
0.1672
0.544 to 0.2097
0.9473
E
Carnivora
48
2.5959
2.7027 to 2.4933
0.0012
0.1613 to 0.1637
0.9815
E, S
Marsupialia
14
3.0547
3.4219 to 2.7269
0.5465
0.1208 to 0.9721
0.9673
G
Euarchonta
15
2.7558
2.9725 to 2.5548
0.3168
0.0345 to 0.5992
0.9840
E
Glires
66
2.8045
2.9447 to 2.671
0.2403
0.0986 to 0.3819
0.9618
< G, > E
L_{F }vs. L_{H}
All
233
1.0246
1.0469 to 1.0027
0.1206
0.0778 to 0.1634
0.9723

Mammalia
187
1.0450
1.0682 to 1.0223
0.1707
0.1248 to 0.2166
0.9771

Reptilia
46
0.9644
1.0644 to 0.8739
0.0190
0.176 to 0.1379
0.8943

Ungulata
32
1.0260
1.107 to 0.9509
0.1452
0.0491 to 0.3395
0.9584

Carnivora
46
0.9741
1.0283 to 0.9227
0.0232
0.1338 to 0.0874
0.9682

Marsupialia
14
0.9372
1.1182 to 0.7856
0.0224
0.2894 to 0.3344
0.9204

Euarchonta
14
1.2075
1.4238 to 1.0241
0.5127
0.106 to 0.9195
0.9307

Glires
66
1.0625
1.1019 to 1.0246
0.2177
0.1536 to 0.2818
0.9788

C_{H+F }vs. BM
All
247
2.7779
2.8191 to 2.7374
1.1564
1.086 to 1.2267
0.9863

Mammalia
200
2.8071
2.8495 to 2.7654
1.2289
1.1541 to 1.3037
0.9886

Reptilia
47
2.7933
2.9496 to 2.6452
1.0833
0.8636 to 1.3031
0.9671

Ungulata
41
2.7319
2.8959 to 2.5773
1.0660
0.6989 to 1.4331
0.9676

Carnivora
48
2.6921
2.7969 to 2.5911
0.9568
0.7669 to 1.1466
0.9834

Marsupialia
14
2.9125
3.1855 to 2.6628
1.3738
0.9658 to 1.7817
0.9797

Euarchonta
15
2.8692
3.0561 to 2.6937
1.3928
1.0911 to 1.6946
0.9889

Glires
66
2.8850
3.0113 to 2.764
1.3260
1.1561 to 1.4960
0.9705

Standardized Major Axis equation shown in the format
Slope and intercept comparisons of stylopodial scaling patterns in mammalian clades.
Standardized major axis equation shown in the format
Slope and intercept comparisons of stylopodial scaling patterns in mammals and nonavian reptiles.
All Data
Mammals < 168 kg^{a}
L_{F }vs C_{F}
°
*
°
L_{H }vs C_{H}
*
*
*
**
°
L_{F }vs BM
C_{F }vs BM
*
L_{H }vs BM
*
**
*
°
*
**
*
C_{H }vs BM
L_{F }vs L_{H}
°
*
C_{H+F }vs BM
Standardized major axis equation shown in the format
Limb scaling patterns in different mammalian size classes
Limb scaling patterns in different mammalian size classes. Lines are fitted based on the SMA results presented in Table 4. All three comparisons plot the log total stylopodial circumference against log body mass in the mammalian sample of the dataset. Size class comparisons are based on previously studied thresholds discussed in the text
Stylopodial scaling in mammals of different sizes.
Analysis
(
Sample
N
R^{2}
Sim
L_{F }vs. C_{F}
< 20 kg
136
0.8868
0.9335 to 0.8424
0.3733
0.2921 to 0.4545
0.9095
0
> 20 kg
52
1.0000
1.1370 to 0.8795
0.4907
0.1714 to 0.8100
0.7945
G
< 50 kg
150
0.9486
0.9987 to 0.9009
0.4715
0.3819 to 0.5611
0.8993
G
> 50 kg
38
1.1331
1.2731 to 1.0084
0.8317
0.4935 to 1.1699
0.8806
> G, < E
< 100 kg
158
0.9659
1.0123 to 0.9216
0.5000
0.4155 to 0.5845
0.9119
G
> 100 kg
30
1.1059
1.2659 to 0.9661
0.7572
0.3679 to 1.1465
0.8774
G
L_{H }vs. C_{H}
< 20 kg
135
0.8778
0.9248 to 0.8331
0.3345
0.2567 to 0.4124
0.9073
0
> 20 kg
52
1.1541
1.2900 to 1.0326
0.7954
0.4848 to 1.1060
0.8459
> G, < E
< 50 kg
149
0.9040
0.9990 to 0.9040
0.4445
0.3613 to 0.5277
0.9060
0
> 50 kg
38
1.1856
1.3524 to 1.0394
0.8774
0.4879 to 1.2668
0.8475
> G, < E
< 100 kg
157
0.9710
1.0166 to 0.9274
0.4764
0.3967 to 0.5560
0.9161
G
> 100 kg
30
1.1229
1.3132 to 0.9602
0.7114
0.2646 to 1.1582
0.8352
G
L_{F }vs. BM
< 20 kg
136
2.6288
2.7825 to 2.4836
1.7421
1.4756 to 2.0086
0.8892
0
> 20 kg
52
2.8571
3.2511 to 2.5108
1.8964
0.9788 to 2.8141
0.7920
G
< 50 kg
150
2.7619
2.9166 to 2.6754
1.9510
1.6751 to 2.2270
0.8873
0
> 50 kg
38
3.1523
3.5377 to 2.8089
2.6399
1.7089 to 3.5709
0.8831
G
< 100 kg
158
2.8104
2.9526 to 2.6750
2.0305
1.7717 to 2.2893
0.9025
0
> 100 kg
30
3.0497
3.5022 to 2.6557
2.3587
1.2593 to 3.4582
0.8715
G
C_{F }vs. BM
< 20 kg
138
2.9559
3.0735 to 2.8429
0.6266
0.4858 to 0.7675
0.9471
G
> 20 kg
62
2.8638
3.0353 to 2.7020
0.5040
0.1723 to 0.8358
0.9492
G
< 50 kg
153
2.9054
3.0013 to 2.8126
0.5716
0.4504 to 0.6928
0.9592
G
> 50 kg
47
2.7816
2.9651 to 2.6094
0.3222
0.0436 to 0.6880
0.9546
E
< 100 kg
164
2.9117
2.9945 to 2.8312
0.5784
0.4698 to 0.6870
0.9674
< G, > E
> 100 kg
36
2.7946
3.0538 to 2.5575
0.3497
0.1751 to 0.8745
0.9351
G, E
L_{H }vs. BM
< 20 kg
135
2.4386
2.5604 to 2.3225
1.1878
0.9858 to 1.3898
0.9192
0
> 20 kg
52
2.9807
3.2978 to 2.6940
2.0078
1.2791 to 2.7365
0.8728
G
< 50 kg
149
2.5866
2.7051 to 2.4734
1.4091
1.2063 to 1.6120
0.9245
0
> 50 kg
38
3.0525
3.4941 to 2.6667
2.1794
1.1501 to 3.2088
0.8392
G
< 100 kg
157
2.6465
2.7582 to 2.5394
1.5013
1.3060 to 1.6966
0.9321
0
> 100 kg
30
2.9405
3.4742 to 2.4888
1.8798
0.6326 to 3.1269
0.8127
G
C_{H }vs. BM
< 20 kg
138
2.7768
2.8898 to 2.6683
0.2550
0.1255 to 0.3845
0.9447
E
> 20 kg
62
2.5793
2.7425 to 2.4258
0.0509
0.3671 to 0.2653
0.9434
E, S
< 50 kg
153
2.7188
2.8130 to 2.6277
0.1941
0.0793 to 0.3088
0.9551
E
> 50 kg
47
2.5612
2.7123 to 2.4184
0.1031
0.4070 to 0.2008
0.9635
E, S
< 100 kg
164
2.7253
2.8067 to 2.6463
0.2005
0.0972 to 0.3038
0.9640
E
> 100 kg
36
2.6488
2.8634 to 2.4504
0.0887
0.3518 to 0.5293
0.9500
E, S
L_{F }vs. L_{H}
< 20 kg
135
1.0776
1.1143 to 1.0422
0.2261
0.1618 to 0.2904
0.9619

> 20 kg
52
0.9586
1.0632 to 0.8642
0.0374
0.2839 to 0.2092
0.8666

< 50 kg
149
1.0672
1.1041 to 1.0315
0.2079
0.1414 to 0.2744
0.9564

> 50 kg
38
1.0327
1.1337 to 0.9407
0.1509
0.0956 to 0.3973
0.9236

< 100 kg
157
1.0613
1.0945 to 1.0291
0.1983
0.1374 to 0.2592
0.9624

> 100 kg
30
1.0371
1.1743 to 0.9160
0.1629
0.1727 to 0.4984
0.8965

C_{H+F }vs. BM
< 20 kg
138
2.9032
2.9989 to 2.8105
1.3628
1.2223 to 1.5032
0.9634

> 20 kg
62
2.7519
2.8828 to 2.6269
1.1186
0.8251 to 1.4120
0.9674

< 50 kg
153
2.8383
2.9165 to 2.7622
1.2743
1.1542 to 1.3945
0.9714

> 50 kg
47
2.6819
2.8173 to 2.5530
0.9409
0.6286 to 1.2531
0.9731

< 100 kg
164
2.8409
2.9084 to 2.7750
1.2778
1.1709 to 1.3846
0.9771

> 100 kg
36
2.7442
2.9343 to 2.5663
1.0954
0.6491 to 1.5416
0.9630

Standardized Major Axis equation shown in the format
Slope and intercept comparisons of stylopodial scaling patterns in different mammalian size classes.
20 kg
50 kg
100 kg
L_{F }vs C_{F}
*
*
L_{H }vs C_{H}
*
**
*
*
*
**
L_{F }vs BM
C_{F }vs BM
L_{H }vs BM
*
**
*
C_{H }vs BM
L_{F }vs L_{H}
°
C_{H+F }vs BM
Standardized major axis equation shown in the format
L_{F}, femoral length; L_{H}, humeral length.
In total, 80 pairwise comparisons are made between mammalian clades (Tables
Regardless of the comparison method used, the most significant variation is noted in the scaling of stylopodial proportions (length to circumference) of the humerus and femur, as well as in the scaling of humeral and femoral lengths with body mass (Figure
In total, ten and 13 significant differences were noted in comparisons between intercepts using confidence intervals and a ttest, respectively, including a significant difference in the intercept of Carnivora and Glires using 95% confidence intervals in the comparison of total stylopodial (humerus + femur) circumference and body mass. However, visual inspection reveals major overlap between the data points at the minimum values along the xaxis (Figure
Mammalian and reptilian scaling patterns show similar scaling coefficients, overall. Of the eight comparisons, two scaling coefficients showed significant differences using both the 95% confidence intervals and the likelihood ratio test. More specifically, the humeral proportions and humeral length to body mass in reptiles scale above that observed for mammals (Figure
Finally, in order to assure that the results obtained for mammals and reptiles are not influenced by differences in body size range in the two samples, we reran the analyses using a subset of the mammalian dataset (N = 174), which corresponds to all mammals equal to, or below, the mass of the
Table S1. Raw and PIC stylopodial scaling in a subset of the mammalian dataset and nonavian reptiles. Mammalian subset corresponds to all taxa < 168 kg in order to better approximate body mass range in the sample of nonavian reptiles. Standardized Major Axis equation shown in the format
Click here for file
Size class comparisons, based on the mammalian dataset (N = 200), at three different thresholds reveal greater variation in scaling patterns between subsamples at lower body size thresholds (Tables
Independent contrast results
Overall, phylogenetically corrected scaling relationships reveal lower coefficients of determination than the raw data. The mean R^{2 }(0.9126 ± 0.0105) for the corrected data is significantly lower than that obtained from the raw data (two tailed ttest: t = 4.4721;
Table S2. Phylogenetically corrected stylopodial scaling in mammals and nonavian reptiles. Scaling equation shown in the format
Click here for file
Most importantly, however, based on the confidence intervals, comparisons between scaling coefficients obtained from the raw data (Table
Discussion
Skeletal limb morphology in vertebrates is considered to reflect a tradeoff between the energetic requirements imposed by movement and the functional requirements imposed by loadings on the bone from behavioral qualities and/or body size
Interspecific limb scaling patterns are often used to test theoretical biomechanical models, such as geometric, elastic, and static similarity, which predict scaling patterns based on biomechanical observations and/or assumptions
The results obtained here suggest that limb scaling in mammalian and reptilian clades exhibits a great deal of variation with respect to elastic and geometric similarity, and as suggested by Christiansen
The results presented here reveal that general scaling patterns of limb circumference in numerous different terrestrial vertebrates, though not always strictly elastic (as defined by McMahon), follow consistent allometric trajectories. Such allometric relationships indicate that, interspecifically, as animals get larger their limbs increase in robusticity at a higher rate compared to body mass. These changes in the architecture of the limb in relation to size support the dynamic similarity hypothesis proposed by Rubin and Lanyon
Stylopodial scaling as a predictor of body mass
As body mass is correlated with numerous physiological and ecological properties, (for example,
Ungulate uniqueness and bias
Ungulates, and specifically artiodactyls or bovids, are considered to exhibit scaling patterns distinct from those seen in other mammals. In particular, their limbs are considered to follow an elastic trend
Table S3. Raw and PIC stylopodial scaling in Artiodactyla and Bovidae. Standardized Major Axis equation shown in the format
Click here for file
The different patterns of limb scaling observed in ungulates compared to mammals
Raw OLS regression for body mass estimation and percent prediction error of body mass proxies
Raw OLS regression for body mass estimation and percent prediction error of body mass proxies. (A) The leastsquares regression of the raw data between the log total stylopodial circumference and log body mass in a sample of 245 (talpids removed) mammals and nonavian reptiles. Regression equation shown in the format
Limb scaling patterns at different gaits and limb postures
Extant terrestrial vertebrates have a variety of gaits and limb postures
Outliers
The final criticism made towards the use of skeletal scaling methods, such as the Anderson method, to estimate body mass is related to the effect outliers have on the final predictive equation, especially at large body size where the sample size is low
A recent study by Packard
Extreme outliers, those at the upper and lower extremes of the dataset, also have the potential to significantly affect regression results. In the current dataset, there are no extreme outliers when the data is log transformed. However, as is generally the case with extant size data, there are several positive extreme outliers in the nonlog dataset. Thirtythree extreme outliers are observed in the body mass and combined humeral and femoral circumference data. When these taxa are removed and the loglog analysis is rerun (
The empirical data presented here falsifies the main criticisms forwarded against skeletalbody mass regression models for predicting body mass in extinct taxa, and given the highly conserved nature of the relationship between stylopodial circumference and body mass in extant terrestrial mammals and reptiles, suggests that circumference measurements represent robust proxies of body mass that can be applied to extinct, phylogenetically and morphologically disparate quadrupedal terrestrial amniotes. The examination of eight terrestrial lissamphibian species (one caudatan and seven anurans [Additional file
Implications for body mass estimation
In extinct taxa, skeletal measurement proxies of body size are often preferred to actual body mass estimates. Of the limb measurements taken here, results suggest that the regression between the total circumference of the humerus and femur to body mass exhibits the strongest relationship, with the highest R^{2 }values, and the lowest PPE, standard error of the estimate (SEE), and Akaike Information Criterion (AIC) values of all bivariate regression models (Figure
Table S4. Predictive power of various body mass estimation equations. Bivariate and multiple regression statistics for various body mass proxies discussed here (that is, circumference and length of the humerus and femur). Statistics include the Percent Prediction Error (PPE), along with its upper and lower 95% PPE Confidence Intervals (PPE CI), the Standard Error of the Estimate (SEE), the Coefficient of Determination (R^{2}), and the Akaike Information Criterion Score (AIC).
Click here for file
Based on our results, we propose the following scaling equation as a robust predictor of body mass in quadrupedal tetrapods:
where C_{H+F }is the sum of humeral and femoral circumferences needed to estimate body mass. This regression exhibits a very high coefficient of determination (R^{2 }= 0.988), and a mean PPE of 25.6%. When adjusted for phylogenetic correlation/covariance between observations (that is, species) using a phylogenetic generalized least squares model, the equation is:
which has an almost identical mean PPE (25%) as equation 1 (Figure
In addition to examining bivariate estimates of body mass, we tested the predictive power of a variety of estimations based on multiple regressions by comparing their PPE, SEE, and AIC with those obtained from the bivariate regression of total circumference with body mass. Analyses including all proximal limb bone measurements also reveal low statistical values for both the raw data:
and the phylogenetically corrected data:
Equally low regression statistics were obtained for the multiple regression including only the circumference measurements, raw data:
phylogenetically corrected data:
None of the equations presented above are significantly better at predicting body mass than the combined humeral and femoral circumference (Equations 1 and 2); therefore, any of these equations are likely to provide robust estimates of body mass (Figure
Not surprisingly, the masses estimated for several commonly cited nonavian dinosaurs provided by Equation 2 are more consistent with estimates generated from Anderson
Body mass estimates of some commonly cited nonavian quadrupedal dinosaurs.
Taxon
Sp #
C1962
A1985§
P1997
H1999
S2001
This study
IRSNB R51
4510
7204
3200
3790
3776
8680
651010850
ROM 845
3820
3030
2800

3079
3620
27204530
MPCD 100/504
177
68
164

23.7
79
59  98
AMNH 5372
3690
3649
1800


4370
32805460
NSM PV 20379
8480
5310
6400
3938
4964
7400
55509250
SMA 0018‡
1780
4131
2200
2530
2611
4950
37206190
USNM 10865*
10560
9061
11400
13421
19655
10940
820013670
HMN SII†
78260
29336
31500
25789∫
28655
35780
2684044730
Body masses estimated in this study are based on the phylogenetically corrected total stylopodial circumference equation (Equation 2) and the error range is based on the 25% mean prediction error obtained from the equation. References: A1985, Anderson
Conclusions
Body size is an important biological descriptor, and as a result, is critical to understanding the paleobiology of extinct organisms and ecosystems. This study presents an extensive dataset of extant quadrupedal terrestrial amniotes, which allows testing of the main criticisms that have been put forth against the use of scaling relationships to estimate body mass in extinct taxa. Our results demonstrate a highly conserved relationship between body mass and stylopodial circumference with minimal variation between clades and groups of different gait and size, compared over a large phylogenetic scope. This general relationship allows the estimation of body mass in extinct quadrupedal groups, and is particularly important for a wide range of paleobiological studies, including growth rates
Methods
Database construction
In order to test the hypotheses outlined in the introduction, we amassed an extensive dataset of limb bone measurements of 200 mammal and 47 nonavian reptile species from individuals that were weighed on a scale either prior to death or skeletonization; no extant body masses were estimated. For the most part, the dataset was built with newly measured specimens; however, it was augmented with published measurements from Christiansen and Harris
Taxon sampling
Taxa were chosen based on three criteria: 1) The dataset must include a large range in body mass, so that sizerelated postural differences can be assessed
Avian taxa were not included in the current study because they are bipedal. The forces exerted by body mass in a biped are transmitted through two limbs compared to four in a quadruped, and therefore direct comparisons of limb to body mass scaling between birds and quadrupedal tetrapods are difficult to interpret. A small sample of lissamphibians (one caudatan and seven anurans) for which live body mass is known was examined in this study. Unfortunately, the current sample size does not provide enough power to make meaningful slope and intercept comparisons, and lissamphibians are not included in the main comparisons presented in the results section.
Statistical analyses
The distribution of the variables used in this study are all positively skewed and, therefore, highly different from a normal distribution; as such all variables were logarithmically transformed (at base 10) to approximate a lognormal distribution. In addition to normality, log transforming reduces the level of heteroscedasticity in the data set, minimizes the effect of extreme outliers, and allows for the visualization of data in a linear fashion, which simplifies the visual comparisons of slopes
Interspecific limb scaling
All measurements were incorporated into a variety of bivariate plots and analyzed using the SMA linefitting method (also known as Reduced Major Axis)
To address the criticisms raised against the Anderson method, subgroups within the data were compared. These include comparisons between mammalian clades for which a sample size greater than ten could be obtained, such as Ungulata, Carnivora, Marsupialia, Euarchonta, and Glires. In addition, comparisons were made between different size classes. Size class comparisons were based on three body mass thresholds: 20 kg, which was previously used by Economos
Fitted lines of different subsamples were compared based on the 95% confidence intervals of the slope and intercept, and differences were considered to be significant when intervals did not overlap. However, given that statistical significance can still be obtained even though confidence intervals overlap
where
In addition to comparing limb scaling patterns between different groups, scaling coefficients were used to test theoretical scaling models, such as geometric (GS), elastic (ES), and static (SS) similarity
Phylogenetic independent contrasts
In addition to plotting the raw data, as was done by Anderson
Figure S1. Phylogenetic tree of mammalian and reptilian taxa included in this study. Topology is based on multiple published analyses mentioned in the text. Numbers indicate the branch lengths used in this study, measured in millions of years. Terminal branch lengths are most often given next to the species name.
Click here for file
Both theoretical and empirical studies of PIC state that in order for contrasts to receive equal weighting and thereby conform to the assumptions stipulated by parametric analyses and statistics, branch lengths must be adjusted so that contrasts are standardized, and therefore have a nonsignificant relationship with their standard deviation
Body mass estimation
In order to provide the best estimation parameter for body mass, a Model I (OLS) regression analysis is preferred. It is the most appropriate model for estimating a value of y based on x, as it accounts for the complete error of the y variable that can be explained by the x variable
In addition to the OLS bivariate regression outlined above, we included all limb measurements into a suite of multiple regression analyses and, given that this technique is highly recommended
Abbreviations
AIC: Akaike Information Criterion; ES: elastic similarity; FDR: false discovery rate; GS: geometric similarity; OLS: ordinary least squares; PIC: phylogenetic independent contrasts; PPE: percent prediction error; SEE: standard error of the estimate; SMA: standardized major axis; SS: static similarity.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
NEC and DCE conceived and designed the study, collected the data, and drafted the manuscript. NEC carried out all the data analyses.
Acknowledgements
We are greatly indebted to the efforts of K. Seymour and S. Woodward, collection managers in the departments of Paleobiology and Mammalogy, respectively, at the Royal Ontario Museum, for their efforts in building, organizing, and providing access to a very useful and important collection. We thank M. Andersen for access to the mammalogy collections at the Zoologisk Museum in Copenhagen. Special thanks to R. Benson, D. Brooks, M. Carrano, P. Christiansen, T. Dececchi, G. De Iuliis, J. Head, D. Henderson, G. Hurlburt, H. Larsson, S. Maidment, R. Reisz, and K. Seymour for discussions on limb scaling and body mass estimation methods. Thanks also to D. Warton for his useful insights on linefitting techniques. We also thank K. Brink, C. Brown, D. Larson, C. VanBuren, and M. Vavrek for proofreading various versions of the manuscript. Finally, we appreciate the comments made by M. Carrano, M. Laurin, and an anonymous reviewer, which greatly improved the quality of this study. Funding for this project was provided by a National Sciences and Engineering Research Council PostGraduate Scholarship and a Queen Elizabeth II Graduate Scholarship in Science and Technology (to NEC), as well as a National Sciences and Engineering Research Council Discovery Grant (to DCE). The funding bodies had no role in the study design, collection of data, analyses, or the decision to publish.