Supplemental Online Appendix A:
Development of Allometric Null Models and Comparisons to Metabolic Theory
We employed allometric relationships linking body mass and r in mammals to develop
allometric-null models against which to judge our PIC-based methods. Allometric models for r,
which require data on body mass but not on other life history traits, have been widely used to
explore the interspecific relationships in r for decades (e.g., Cole 1954, Blueweiss et al. 1978,
Robinson and Redford 1986) and also feature prominently in metabolic scaling theory (e.g.,
Savage et al. 2004).
Historically, allometric regression models did not attempt to account for phylogenetic
non-independence among species in the datasets. However, phylogenetically based statistical
approaches may be used in allometric analyses to incorporate covariation due to shared
evolutionary history among species (Duncan et al. 2007, Fagan et al. 2010). This joint approach
is most appropriate when building allometric models from existing databases which may be
biased in their taxonomic representation (Fagan et al. 2010).
To build our allometric null models, we used phylogenetically corrected least squares
regression to account for correlated errors due to phylogenetic relatedness (Ives et al. 2007).
Note that while our allometric-null models include phylogenetic information for the sample of
species in the original analysis, they do not incorporate the phylogenetic position of species for
which predicted values are sought. In other words, the allometric null regression models
represent static mappings between female body size and r for the suite of species under
consideration whereas the PIC models customize predictions for the target species based
additionally on their shared evolutionary history with the rest of the clade (Garland & Ives 2000).
According to metabolic theory, population growth rates, 𝑟, can be estimated based on
scaling relationships with other life-history parameters, most commonly species mass, 𝑀, and
age of first reproduction, 𝛽. In this appendix, we test how our PIC-r model based on
comparative phylogenetics relates to metabolic theory. First, we consider whether or not our
model supports the metabolic theory scaling relationships: 𝑟 ∝ 𝛽 −1 and 𝑟 ∝ 𝑀−0.25 (Savage et
al. 2004, Duncan et al. 2007). To do this, we consider linear fits of 𝑙𝑜𝑔(𝑟) vs. 𝑙𝑜𝑔⁡(𝛽) and
𝑙𝑜𝑔(𝑟) vs. 𝑙𝑜𝑔⁡(𝑀) where 𝑟 for each species is found from equations (2-3) in the main text.
Figure A.1 shows a scatterplot of 𝑙𝑜𝑔(𝑟) vs. 𝑙𝑜𝑔⁡(𝛽) while Table A.1 defines regression lines for
𝑙𝑜𝑔(𝑟) vs. 𝑙𝑜𝑔⁡(𝛽) assuming four alternative regression models (Fagan et al. 2010): (i) an
ordinary least squares (OLS) regression, (ii) a phylogenetically corrected ordinary least squares
(pgOLS) regression, (iii) a reduced major axis (RMA) regression and (iv) a phylogenetically
corrected reduced major axis (pgRMA) regression.
Table A.1
Allometric scaling exponents for 𝑟 vs. 𝛽
Caniformia
exponent, (95% CI)
Cervidae
exponent, (95% CI)
OLS
−1.15⁡(−1.25, −1.05) −0.42⁡(−0.38, −0.66)
pgOLS
−0.95⁡(−1.16, −0.74) −0.20⁡(−0.50, −0.10)
RMA
−1.22⁡(−1.32, −1.12) −0.58⁡(−0.36, −0.86)
pgRMA
−1.25⁡(−1.31, −1.19) −0.61⁡(−0.99, −0.37)
In keeping with metabolic theory, we see that, for both Caniformia and Cervidae, there is a
negative linear relationship between 𝑙𝑜𝑔(𝑟) and 𝑙𝑜𝑔⁡(𝛽). While the theoretically predicted -1
exponent from metabolic theory only falls within the 95% CI for the pgOLS regression of the
Caniformia dataset, the predicted exponents for Caniformia are within 25% of the theoretical
value. Results for Cervidae are somewhat more ambiguous; however, given the small size of the
dataset (N = 15, smaller than is recommended for this type of allometric analysis [Wharton et al.
2006]) this is not surprising. Overall, then, we conclude that our data and our definition of 𝑟 in
terms of life history parameters (equations (2-3) in the main text) give rise to a scaling law that
shows the expected negative linear relationship between ⁡𝑙𝑜𝑔(𝑟) and 𝑙𝑜𝑔⁡(𝛽) with an exponent
approximating the theoretically predicted 𝑟 ∝ 𝛽 −1 result.
Figure A.1 Log transformed population growth rate, 𝑙𝑜𝑔(𝑟), as a function of log
transformed age of first reproduction 𝑙𝑜𝑔(𝛽) for (a) the Caniformia group:
Ailuridae (closed circles), Canidae (open circles), Mephitidae (closed triangles),
Mustelidae (open triangles), Otariidae (closed diamonds), Phocidae (open
diamonds), Procyonidae (closed squares), and Ursidae (open squares) and (b) the
Cervidae group (closed circles). Both panels show linear fits for an ordinary least
squares (OLS) regression (red, dashed), a phylogenetically corrected ordinary
least squares (pgOLS) regression (red, solid), a reduced major axis (RMA)
regression (black, dashed), and a phylogenetically corrected reduced major axis
(pgRMA) regression (black, solid).
Figure A.2 shows a scatterplot of 𝑙𝑜𝑔(𝑟) vs. 𝑙𝑜𝑔⁡(𝑀) while Table A.2 defines regression lines for
𝑙𝑜𝑔(𝑟) vs. 𝑙𝑜𝑔⁡(𝑀) assuming (i) OLS regression, (ii) pgOLS regression, (iii) RMA regression
and (iv) pgRMA regression. Again, from Figure A.2, we see that there is a negative linear
relationship between 𝑙𝑜𝑔(𝑟) and 𝑙𝑜𝑔⁡(𝑀) (although the relationship is weaker than was the case
for 𝑙𝑜𝑔(𝑟) and 𝑙𝑜𝑔⁡(𝛽), as indicated by the lower R2 values for the two OLS fits in Figure A.2.a
as compared to Figure A.1.a).
While Table A.2 suggests wide variation in the predicted scaling exponent, the theoretical value
of -0.25 falls within three of the eight 95% CIs. In addition, our data are consistent with other
empirical estimates of scaling exponents. For example, Duncan et al. (2007) reported 𝑙𝑜𝑔(𝑟) vs.
log⁡(𝑀) scaling exponents of -0.35 for the Carnivora (parent phylogenetic order to the
Caniformia) and -0.2 for the Artiodactyla (parent phylogenetic order to the Cervidae). -0.35 falls
within the 95% CI for the OLS fit for Caniformia, while -0.2 falls within three of the four 95%
CIs for the Cervidae group. Thus we conclude that our dataset is at least qualitatively consistent
with metabolic theory to the extent that there is a negative linear relationship between 𝑙𝑜𝑔(𝑟)
and 𝑙𝑜𝑔⁡(𝑀). Scaling exponents approximate both theoretical predictions and results from other
empirical studies based on metabolic theory. This evidence for allometric relationships between
r and M, coupled with historical emphasis on such relationships (Cole 1954, Blueweiss et al.
1978, Hennemann 1984, Schmitz and Lavigne 1984, Robinson and Redford 1986, Ross 1992),
justifies our use of allometric null models as benchmarks against which to judge the performance
of the models based on phylogenetically independent contrasts (PIC).
Table A.2
Predicted scaling relationships for 𝑟 vs. 𝑀
Caniformia exponent
(95% CI)
Cervidae
exponent (95% CI)
OLS
−0.35⁡(−0.42, −0.28)
−0.10⁡(−0.26, 0.05)
pgOLS
−0.10⁡(−0.24, 0.04)⁡
0.005⁡(−0.14, 0.15)
RMA
−0.46⁡(−0.54, −0.39)
−0.28⁡(−0.47, −0.16)
pgRMA
−0.49⁡(−0.55, −0.44)
−0.27, (−0.95, −0.08)
Next, we ask how predicted 𝑟 values based on scaling laws from metabolic theory compare to
predicted 𝑟 values from our PIC-r model. Figure A.3 shows the magnitude of the absolute
prediction error, |𝑟̂ − 𝑟| as a function of 𝑙𝑜𝑔(𝑟) for the PIC-r model (black circles) and also for 𝑟
estimates based on pgOLS fits of 𝑙𝑜𝑔(𝑟) vs 𝑙𝑜𝑔(𝛽) (grey circles) and 𝑙𝑜𝑔(𝑟) vs 𝑙𝑜𝑔(𝑀) (open
circles). These last two methods correspond to using the (empirically fitted) scaling relationships
from metabolic theory to estimate 𝑟, with the latter being the allometric null model from the
main text.
Figure A.2 Log transformed population growth rate, 𝑙𝑜𝑔(𝑟), as a function of
log transformed species mass 𝑙𝑜𝑔(𝑀) for (a) the Caniformia group: Ailuridae
(closed circles), Canidae (open circles), Mephitidae (closed triangles), Mustelidae
(open triangles), Otariidae (closed diamonds), Phocidae (open diamonds),
Procyonidae (closed squares), and Ursidae (open squares). (b) the Cervidae group
(closed circles). Both panels show linear fits for an ordinary least squares (OLS)
regression (grey, dashed), a phylogenetically corrected ordinary least squares
(pgOLS) regression (grey, solid), a reduced major axis (RMA) regression (black,
dashed), and a phylogenetically corrected reduced major axis (pgRMA)
regression (black, solid).
Figure A.3 Magnitude of the absolute prediction error |𝑟̂ − 𝑟|, as a function of
log transformed population growth rate 𝑙𝑜𝑔(𝑟) for (a) the Caniformia group and
(b) the Cervidae group. Errors for the PIC-r mass model are shown in black,
errors based on a pgOLS of 𝑙𝑜𝑔(𝑟) vs. 𝑙𝑜𝑔(𝛽) are shown in grey and errors based
on a pgOLS of 𝑙𝑜𝑔(𝑟) vs. 𝑙𝑜𝑔(𝑀), also the allometric null model in the main text,
are shown in red.
From Figure A.3 it is clear that our PIC-r model performs better than the allometric null model
based on pgOLS of 𝑙𝑜𝑔(𝑟) vs. 𝑙𝑜𝑔(𝑀). PIC-r offers an improvement over the model based on
pgOLS of 𝑙𝑜𝑔(𝑟) vs. 𝑙𝑜𝑔(𝛽) for ~50% of species.
Supplemental Online Appendix B: PIC-r Prediction Errors
To understand the limitations of and biases in the PIC-r approach, we consider trends in the
absolute PIC-r prediction error, (𝑟̂ − 𝑟), as a function of life-history traits. Table B.1 shows the
slopes and corresponding p-values obtained from pGLS regressions of (𝑟̂ − 𝑟) against: species
mass (𝑀), average interval between litters (∆), maximum number of female offspring per litter
(𝑚), minimum age of first reproduction (𝛽), average mortality rate (𝜇), and maximum population
growth rate (𝑟). Slopes shown in bold are significantly non-zero at the 95% confidence level.
Table B.1 pGLS regressions of absolute prediction errors (𝑟̂ − 𝑟) vs. life-history parameters
LifeHistory
Parameter
Caniformia
Cervidae
slope
p-value
slope
p-value
𝑀
0.0003
0.88
5.0×10-5
0.85
∆
0.29
0.088
0.24
0.046
𝑚
-0.42
4.1×10-8
-0.23
0.037
𝛽
0.13
0.048
0.032
0.52
𝜇
-1.71
0.030
-0.37
0.78
𝑟
-1.02
0
-1.09
4.24×10-8
Based on results in Table B.1, it is clear that absolute prediction error correlates with a number
of life-history parameters. For example, large litters are associated with underestimation of 𝑟
while small litters are associated with overestimation of 𝑟. Similar trends can be seen for
mortality rate and age of first reproduction in Caniformia, and the interval between litters in
Cervidae. It is tempting to assume that these dependencies arise as a result of correlations
between life-history traits and key aspects of evolution (e.g. rate), and thus reflect inaccuracies in
estimation of the underlying phylogenetic tree (e.g. branch length). However, there is a much
simpler explanation: 𝑟 scales with each of the life-history parameters shown in Table A.1.
Moreover, because PIC-r predicts a focal species’ 𝑟 based on the weighted average of 𝑟 values
for other species in the tree, absolute prediction errors naturally exhibit a negative dependence on
𝑟 itself (i.e. predictions for species with large 𝑟 tend to be underestimates while predictions for
species with small 𝑟 tend to be overestimates). Absolute prediction errors thus scale with lifehistory parameters by virtue of 𝑟 scaling with life-history parameters and absolute prediction
errors scaling with 𝑟.
To see the argument above, consider a null model wherein estimates of a species’ 𝑟 are based on
non-phylogenetically corrected averages of the 𝑟 values of all other species:
𝑟̂𝑖𝑛𝑢𝑙𝑙 =
∑𝑛≠𝑖 𝑟𝑛
𝑁−1
(B.1)
In equation (B.1), 𝑟̂𝑖𝑛𝑢𝑙𝑙 is the estimate of 𝑟 for species 𝑖, 𝑟𝑛 is the known value of 𝑟 for species 𝑛,
and there are 𝑁 species in the dataset (including the focal species 𝑖). For large 𝑁, the right hand
side of equation (B.1) will be very nearly constant for all 𝑖, thus the absolute prediction error,
(𝑟̂𝑖𝑛𝑢𝑙𝑙 − 𝑟), will be close to a straight line with a slope of -1. This simple example illustrates why
averaging tends to give high estimates for species with low 𝑟 and low estimates for species with
high 𝑟.
In Figure B.1, we show absolute prediction error as a function of maximum population growth
rate for the PIC-r model of the Caniformia. We also show an OLS regression of absolute
prediction error against maximum population growth rate for the null model in equation (B.1)
(black dashed line). To the extent that PIC-r prediction errors fall between the black dashed line
and its reflection about the x-axis (red dashed line), phylogenetic correction improves estimates
as compared to the null model. However, despite this improvement, the negative trend in
absolute prediction error with maximum population growth rate is retained, explaining error
dependencies on other life-history traits as well.
Figure B.1
Absolute prediction error (𝑟̂ − 𝑟) as a function of maximum
population growth rate 𝑟. Prediction errors from the PIC-r model of the
Caniformia group are shown as: Ailuridae (closed circles), Canidae (open circles),
Mephitidae (closed triangles), Mustelidae (open triangles), Otariidae (closed
diamonds), Phocidae (open diamonds), Procyonidae (closed squares), and Ursidae
(open squares). Prediction errors from the null model in equation (B.1) are shown
as an OLS regression (black dashed line). The grey dashed line is the reflection
of the black dashed line through the x-axis. PIC-r prediction errors falling
between the red and black dashed lines indicate an improvement in PIC-r as
compared to the basic null model in equation (B.1).
Interestingly, the only life-history parameter that does not seem to have a strong effect on PIC-r
prediction error is species mass, 𝑀. Indeed, for both Caniformia and Cervidae, slope estimates
from pGLS regressions of prediction error against mass are very nearly zero and not significantly
different from zero. Figure B.2 shows scatterplots of PIC-r prediction error as a function of
mass. Again, the apparent lack of correlation is obvious to the eye: over- and under- estimates of
𝑟 appear equally likely regardless of species size. We suggest that this apparent lack of
correlation between prediction error and mass is at the root of the improved performance of the
simple PIC-r model relative to the more complex PIC-r-mass model. An alternative possibility,
wherein noise in the 𝑙𝑜𝑔(𝑟) vs 𝑙𝑜𝑔(𝑀) relationship is responsible for the lack of improvement
when using the PIC-r-mass model, is not supported by the data. Outlier species in the allometric
regression are neither more nor less likely to be better estimated via the PIC-r-mass model than
by the PIC-r model (Fig. B3).
Figure B.2
Absolute prediction error (𝑟̂ − 𝑟) as a function of species mass 𝑀.
(a) Prediction errors from the PIC-r model of the Caniformia group are shown as:
Ailuridae (closed circles), Canidae (open circles), Mephitidae (closed triangles),
Mustelidae (open triangles), Otariidae (closed diamonds), Phocidae (open
diamonds), Procyonidae (closed squares), and Ursidae (open squares). (b)
Prediction errors from the PIC-r model of the Cervidae group are shown as closed
circles.
Figure B.3 Log(r) vs. log(M) for Caniformia and Cervidae. Species for which
PIC-r-mass performs better are denoted by green circles. Species for which PIC-r
performs better are denoted by red circles. The black dotted lines indicate
phylogenetically corrected RMA fits.
Supplemental Online Appendix C: Raw Data for Caniformia and Cervidae
Supplemental Table C.1
Caniformia life-history parameters
Species
r
M (kg)
 (yr)
 (yr)
µ (yr-1)
m (#)
tip length
Ailuropoda melanoleuca
Ailurus fulgens
0.139544
0.387016
120
4.33
1.43
1.00
6.44
1.63
0.05
0.15
0.77
1.01
0.453258
0.875254
Amblonyx cinereus
0.574116
3
0.50
1.08
0.18
0.71
0.192695
Arctocephalus australis
0.195119
45
1.00
3.00
0.07
0.50
0.012764
Arctocephalus forsteri
0.141511
55
1.00
5.00
0.06
0.50
0.02031
Arctocephalus galapagoensis
0.086798
27
2.00
3.50
0.08
0.50
0.012764
Arctocephalus gazella
0.200314
40.5
1.00
2.83
0.07
0.50
0.006099
Arctocephalus pusillus
0.187227
77.67
1.00
3.54
0.05
0.50
0.099173
Arctocephalus tropicalis
0.16608
50
1.00
4.00
0.06
0.50
0.006099
Bassariscus astutus
0.644558
0.976
1.00
0.89
0.27
1.35
0.202692
Callorhinus ursinus
0.148494
42.4
1.12
4.08
0.06
0.50
0.208686
Canis adustus
1.81319
8.25
0.50
0.64
0.12
2.15
0.124854
Canis latrans
1.097966
11.8
1.00
1.14
0.10
2.75
0.092248
Canis lupus
0.710635
27
1.22
1.82
0.07
2.57
0.092248
Canis mesomelas
1.0542
9.75
1.00
0.90
0.11
1.98
0.26084
Cerdocyon thous
1.51811
6.5
0.53
0.76
0.13
2.05
0.06524
Cystophora cristata
0.223528
212
1.00
2.99
0.04
0.50
0.166204
Eira barbara
1.4694
4.5
0.50
0.50
0.15
1.25
0.262033
Enhydra lutris
0.165412
21.8
1.15
2.69
0.09
0.50
0.28868
Gulo gulo
0.302902
16.3
2.22
2.23
0.09
1.42
0.331999
Halichoerus grypus
0.174065
182
1.08
4.21
0.04
0.50
0.046263
Ictonyx striatus
0.549191
0.765
1.00
0.77
0.29
1.09
0.165972
Lobodon carcinophagus
0.190796
238
1.00
4.01
0.04
0.50
0.229426
Lontra canadensis
0.351403
4.6
1.06
2.50
0.14
1.36
0.336471
Lutra lutra
0.576544
6.75
1.00
1.00
0.13
1.03
0.192695
Martes americana
0.367861
0.607
1.00
1.47
0.32
1.37
0.300865
Martes pennanti
0.536204
2.6
1.00
1.44
0.19
1.46
0.262033
Meles meles
0.687938
13
1.33
1.10
0.10
1.56
0.498783
Mephitis macroura
1.12746
1.2
1.00
0.80
0.25
2.25
0.08313
Mephitis mephitis
1.19265
2
0.83
0.93
0.21
2.55
0.08313
Monachus schauinslandi
0.127559
173
1.44
5.00
0.04
0.50
0.324987
Mustela erminea
4.10122
0.11
1.00
0.26
0.60
3.39
0.153847
Mustela eversmannii
1.69721
1.35
1.00
0.88
0.24
4.71
0.014018
Mustela frenata
1.813366
0.151
1.00
0.53
0.48
3.05
0.145605
Mustela nigripes
0.673729
0.703
1.00
1.00
0.30
1.65
0.028501
Mustela nivalis
2.76656
0.0498
0.60
0.31
0.81
2.69
0.123539
Mustela putorius
1.599601
0.65
0.79
0.83
0.31
3.84
0.014018
Mustela vison
1.26146
0.9
1.00
0.72
0.28
2.38
0.145605
Nasua narica
0.49902
4.6
1.00
2.00
0.15
1.75
0.586311
Neophoca cinerea
0.161991
79.6
1.45
3.00
0.05
0.50
0.081416
Nyctereutes procyonoides
1.55555
4.23
1.00
1.02
0.16
4.73
0.199755
Ommatophoca rossii
0.202926
188
1.00
3.46
0.04
0.50
0.229426
Otaria byronia
0.192369
140
1.00
3.69
0.04
0.50
0.106778
Otocyon megalotis
1.18362
4.15
0.50
1.00
0.16
1.87
0.304675
Phoca fasciata
0.233474
82.5
1.00
2.45
0.05
0.50
0.124444
Phoca groenlandica
0.177911
128.8
1.00
4.15
0.04
0.50
0.124444
Phoca hispida
0.111449
69.7
1.25
6.16
0.05
0.50
0.036628
Phoca sibirica
0.155588
81.7
1.00
4.82
0.05
0.51
0.036628
Phoca vitulina
0.181563
101
1.00
3.88
0.05
0.50
0.063008
Phocarctos hookeri
0.201567
183
1.00
3.50
0.04
0.50
0.087359
Poecilogale albinucha
0.30648
0.2575
0.80
1.20
0.44
1.10
0.165972
Potos flavus
0.120924
3
1.00
2.27
0.17
0.50
0.803102
Procyon lotor
0.886438
4.4
1.00
0.95
0.14
1.72
0.202692
Pseudalopex culpaeus
0.99461
13
1.00
1.00
0.10
2.00
0.06524
Pteronura brasiliensis
0.390658
24
1.00
2.00
0.08
0.97
0.38378
Speothos venaticus
1.197471
6
0.57
0.96
0.14
1.93
0.163721
Spilogale putorius
1.679876
0.512
0.82
0.51
0.36
2.31
0.273302
Taxidea taxus
0.976827
6.05
1.35
0.61
0.14
1.54
0.550177
Urocyon cinereoargenteus
1.096611
3.3
1.01
0.94
0.17
2.39
0.504537
Ursus americanus
0.197258
87.2
2.41
3.68
0.05
1.10
0.185153
Ursus arctos
0.148576
204
2.89
5.17
0.04
1.09
0.185153
Vulpes macrotis
0.89761
1.9
1.00
1.00
0.20
2.00
0.101789
Vulpes vulpes
1.15437
3.65
1.10
0.81
0.14
2.17
0.101789
Vulpes zerda
0.642439
1.2
1.00
0.82
0.25
1.23
0.258465
Zalophus californianus
0.137426
84.7
1.00
5.67
0.05
0.50
0.123212
Supplemental Figure C.1
Caniformia Phylogenetic Tree from Agnarsson et al. 2010
Supplemental Table C.2
Cervidae life-history parameters
Species
r
M (kg)
 (yr)
 (yr)
µ (yr-1)
m (#)
tip length
Alces alces
Axis axis
Cervus albirostris
Cervus elaphus
Cervus eldii
Cervus nippon
Cervus timorensis
Cervus unicolor
Dama dama
Elaphurus davidianus
Mazama americana
Muntiacus reevesi
Odocoileus hemionus
Ozotoceros bezoarticus
Rangifer tarandus
0.361242
0.34528
0.271815
0.303467
0.296612
0.26731
0.204153
0.273818
0.282863
0.277142
0.318766
0.552882
0.43003
0.33528
0.399106
408
50
125
109
73
96.5
53
171
54.5
149
23
12
56.3
35
93.6
1.13
1.00
1.00
1.00
1.00
1.00
1.50
1.00
1.00
1.00
1.00
0.67
1.03
1.00
0.95
2.22
1.00
2.00
1.52
1.57
1.84
1.74
1.97
1.79
2.25
1.05
0.50
1.43
1.00
1.00
0.03
0.06
0.04
0.05
0.05
0.06
0.06
0.04
0.06
0.04
0.08
0.11
0.06
0.07
0.05
0.86
0.50
0.50
0.50
0.51
0.51
0.50
0.50
0.54
0.56
0.50
0.50
0.96
0.50
0.55
16.6
12.5
7.5
7.4
7.5
7.5
7.6
7.4
12.2
7.6
4
17
4.1
4
11.5
Supplemental Figure C.2
Cervidae Phylogenetic Tree from Bininda-Emonds et al. 2007