A cranial correlate of body mass in proboscideans

A cranial correlate of body mass in proboscideans Abstract Allometric scaling relationships are often used to estimate the body mass (BM) of extinct mammalian taxa. For proboscideans, shoulder height or limb bone dimensions have typically been used to estimate BM. However, these skeletal correlates are only useful when a complete forelimb is available, or when limb elements can be identified to the species level. Several taxa are known or can be identified from cranial remains alone, which poses a problem for BM estimation. Here, we develop allometric equations to predict the total length and the minimum circumference of the humerus or femur of derived proboscideans from the breadth of the occipital condyles. These predicted measurements are then used to estimate body size from existing equations derived from regressions on limb bones. We developed equations using a combined sample of both extinct and extant proboscidean taxa. Equations for specific families were derived when possible. We find that occipital condyle breadth is a robust predictor of limb bone dimensions. Estimated BM values from predicted limb measurements were a good match to actual BM, and estimates from actual limb measurements. Our method will allow researchers to study BM evolution in proboscideans using a greater range of specimens. allometry, body size, occipital condyle breadth INTRODUCTION Body size is recognized as one of the most important traits of an organism. Studies on extant species have shown that body size affects metabolism (Klieber, 1932; Peters, 1983), reproductive fitness (Brown, Marquet & Taper, 1993), life history traits (Kaplan & Salthe, 1979), development (Gillooly et al., 2002), population density (Damuth, 1981, 2007), abundance (Blackburn et al., 1993) and range size (Gaston & Blackburn, 1996). Body size is particularly useful because it can be measured in common units (e.g. mass, biovolume) across all of life. Because of the relationship with an organism’s biology and ecology and the utility of measurement, paleobiologists strive to accurately estimate the body size of extinct taxa (Damuth & Macfadden, 1990). While estimating the body mass (BM) of an extinct species is a challenging task, paleobiologists have made immense progress over the last few decades, especially for fossil mammals (Damuth & Macfadden, 1990). Classically, researchers have studied the allometric relationships between skeletal measurements and BM of extant taxa and used the predictive equations generated to estimate the BM of extinct counterparts. For example, the linear or areal dimensions of the molars correlate well with BM in extant members of the Artiodactyla and Perissodactyla (Damuth, 1990; Janis, 1990), while the length of the skull correlates well with BM in carnivorans (Van Valkenburgh, 1990). Within the Tetrapoda as a whole, the length, diameter and circumference of the humerus and femur also scale with BM (Alexander et al., 1979; Anderson, Hallmartin & Russell, 1985; Gingerich, 1990; Campione & Evans, 2012). For proboscideans, two allometric scaling relationships are typically used to estimate BM – the first one relates BM to shoulder height (Roth, 1990), and another relates BM with various dimensions of the limb bones (Christiansen, 2004). Unfortunately, estimating the BM of extinct proboscideans remains a challenge for researchers. Using the shoulder height requires a complete forelimb, which is often unavailable for many fossil specimens. Moreover, both equations require that limbs be referable to a species. More recently, volumetric methods have been used to estimate the BM of extant and extinct proboscideans (Larramendi, 2016). However, this method relies on accurate physical or digital reconstructions, which may not always be possible from fragmentary fossil remains. Therefore, estimating BM of extinct proboscideans from the linear dimensions of limb bones remains widely used (e.g. see Lister & Stuart, 2010). The challenge is that some extinct species are known or can only be identified from cranial and dental remains, as is the case with a number of extinct proboscidean taxa from South and Southeast Asia. Unless, limb bones, a complete forelimb or complete skeleton of that species are found, it is not possible to estimate its BM. Here, we address this problem by developing a cranial correlate of BM. We use the occipital condyle breadth (OCB) to estimate the BM of proboscideans. This can be especially useful to researchers attempting to estimate the BM of proboscidean species, which are only known or identified from cranial remains. OCB has been used as a predictor of BM in a variety of taxa, from marine mammals such as cetaceans (Marino et al., 2003; Marino, McShea & Uhen, 2004; Bianucci, Lambert & Post, 2010; Pyenson & Sponberg, 2011) and sirenians (Sarko et al., 2010), to small mammals such as lagomorphs, rodents, shrews and weasels (Martin, 1980; Moncunill-Solé et al., 2014, 2015), and even hominids (Martin, 1981). The expectation is that, as the overall BM of proboscideans increases, so should the size of the skulls, resulting in an increase in the breadth of the occipital condyles, the primary locus of contact of the head with the rest of the body. The dearth of extant elephant specimens that had been weighed before death, or shortly after, prevented us from directly investigating the relationship between OCB and BM. Therefore, we investigated the allometric relationship between OCB and the length and circumference of the humerus and femur of extant and extinct proboscideans, both variables which are known to scale with BM. The allometric equations generated are used to predict the limb bone dimensions, which are then used to estimate BM using equations developed by Christiansen (2004). We compare estimates of BM generated from our predicted limb bone variables to those generated from the actual limb bone dimensions and test our approach on two extant elephant specimens with known BM to assess the precision of our methodology. MATERIAL AND METHODS Specimens Forty-one specimens of both extant and extinct proboscideans spanning the four more derived families, the Elephantidae, Stegodontidae, Gomphotheriidae and Mammutidae, were sampled. Measurements were taken either by directly measuring museum specimens or from published descriptions in peer-reviewed literature. A list of specimens and species sampled are given in Supporting Information, Table S1. Only adult specimens were used in this study to reduce any effects of ontogeny on allometric scaling relationships. The age of extant proboscideans was determined using ontogenetic dental scales developed by Laws (1966) and Roth & Shoshani (1988). For extinct species, the eruption of the third molar was taken as evidence of adulthood. Skeletal variables OCB was measured as the maximum linear distance across both condyles (Fig. 1). Christiansen (2004) determined that the antero-posterior diameter, total length and the minimum circumference of the humerus were the best forelimb predictors of BM. However, he did not mention where on the diaphysis the diameter was measured. To avoid error in measurement, we only measured the total length and minimum circumference of the humerus, both being unambiguous measurements. Similarly, for the hindlimb, Christiansen (2004) showed that the length and the antero-posterior diameter of the femur best predicted BM. For the reasons mentioned above, we measured the total length, but not the diameter. We measured the minimum circumference of the femur as it is also a reliable predictor of BM (Roth, 1990). The total length of the humerus (HL) was measured between the most proximal end of lateral tuberosity to most distal end of lateral condyle, and the total length of the femur (FL) was measured between the proximal end of the femoral head and the distal-most end of the medial condyle (Fig. 1). Minimum circumference was measured by passing a loop of measuring tape down the diaphysis of the limb bones until the minimum circumference was reached (Fig. 1). For the femur minimum circumference (FMC), this is roughly at mid-shaft; for the humerus minimum circumference (HMC), it was close to the distal end of the diaphysis. Figure 1. View largeDownload slide Skeletal measurements used in this study. A, occipital condyle breadth (OCB), measured as the linear distance across both condyles. B, total length of the humerus (HL), measured between the most proximal end of lateral tuberosity to most distal end of lateral condyle, and the minimum circumference of the humerus (HMC) which is measured close to the waist of the diaphysis. C, total length of the femur (FL) measured between the proximal end of the femoral head and the distal-most end of the medial condyle, and the minimum circumference of the femur (FMC) measured near the midpoint of the diaphysis. Figure 1. View largeDownload slide Skeletal measurements used in this study. A, occipital condyle breadth (OCB), measured as the linear distance across both condyles. B, total length of the humerus (HL), measured between the most proximal end of lateral tuberosity to most distal end of lateral condyle, and the minimum circumference of the humerus (HMC) which is measured close to the waist of the diaphysis. C, total length of the femur (FL) measured between the proximal end of the femoral head and the distal-most end of the medial condyle, and the minimum circumference of the femur (FMC) measured near the midpoint of the diaphysis. Predictive equations Ordinary least square (OLS) regressions were fitted to log10-transformed data using the statistical software program R (R Core Team, 2014). OLS regressions are preferred in estimation because they minimize the sum of squared errors (Smith, 2009; Sokal & Rohlf, 2012). The four variables (HL, FL, HMC and FMC) could not be measured for all specimens. Therefore, sample sizes varied among the four variables tested. Because of differences in the proportions of the limb bones among proboscidean families (Christiansen, 2007), we expect the allometric scaling relationships between OCB and the limb bone variables to vary among these clades. Hence, we repeated our analyses by family where possible. Since the majority of the specimens came from the Elephantidae, separate predictive equations were generated for this family. For the Gomphotheriidae, humerus length was the only variable measured for at least six individuals; therefore, a separate predictive equation could be generated for this variable. Only one specimen of the Stegodontidae and three specimens of the Mammutidae were included in the overall sample, thus, separate equations could not be generated for these families. The strength of the relationship between OCB and the limb variables was determined using the coefficient of determination (R2). We calculated the per cent standard error of the estimate (%SEE), a measure of the residual variation following Brody (1945) and Smith (1984). The %SEE measures the per cent error within which 68% of the estimated values match the observed values (Smith, 1984). According to Brody (1945), the SEE of log-transformed data can be calculated as follows: if we take a predicted value of the dependent variable, lets say 100, we get a value of 2 on the regression line (log10100 = 2). The SEE, which is calculated using the formula Σ(Y−Ypredicted)2(n−2), is then added to 2. The antilog of the resulting number is taken, and the value greater than 100 is converted to a percentage of 100. This is the positive %SEE. This can be reproduced using any value taken from the regression line. The predictive power of OCB was determined using the mean absolute per cent predictive error (MAPE). MAPE is a measurement of the average per cent deviation between the predicted and actual limb bone measurements and is calculated using untransformed data. It is calculated as follows: raw values of per cent predictive error (%PE) are estimated for each specimen using the formula (Observed−PredictedPredicted)×100 (Smith, 1984), and then, the mean of the absolute values is taken. Family-specific raw %PE and MAPE were calculated where appropriate. The raw %PE values indicate the degree of over or underestimation of a particular measurement. Negative values indicate overestimation while positive values indicate underestimation. The family-specific MAPE shows the average per cent deviation of the predicted values from the actual values of the limb bone variables for a particular family when estimated using the regression equations. The combined proboscidean equations were developed using all four families, and thus, they can be used to predict the limb bone variables of all the constituent clades. The utility of the family-specific equations was tested only on the respective families sampled to generate those equations. Body mass estimation We used the predictive equations for the total length and minimum circumference developed by Christiansen (2004) to estimate BM (Table 1). Since we could not directly assess the relationship between BM and OCB, we estimated BM from the limb bone variables predicted from OCB, and BM from the actual limb bone measurements of the specimens sampled. BMs were estimated for specimens in every family using the limb bone measurements predicted using the combined proboscidean equations, and when possible, using the family-specific equations. The difference in BM estimation (i.e. difference in BM estimated from predicted limb variables vs. actual limb variables) was measured using the equation %BMdiff=|(BMobserved−BMpredictedBMpredicted)× 100|,  that is, the degree of mismatch between the estimate from actual and predicted limb measurements. The differences were compared among the four variables for each family. The %BMdiff measures the difference in estimated BM caused by the PE of the regression equations we developed. Table 1. Predictive equations for body mass of extant elephants from Christiansen (2004) Variable (X)  α ± 95% CI  β ± 95% CI  MAPE  %SEE  R2  HL  −4.14 ± 1.25  2.64 ± 0.43  6.74  11.52  0.98  HMC  −1.60 ± 0.58  2.06 ± 0.23  5.54  7.78  0.99  FL  −5.57 ± 1.84  3.04 ± 0.61  6.15  14.54  0.97  FMC  −1.61 ± 1.30  2.07 ± 0.52  11.52  18.46  0.95  Variable (X)  α ± 95% CI  β ± 95% CI  MAPE  %SEE  R2  HL  −4.14 ± 1.25  2.64 ± 0.43  6.74  11.52  0.98  HMC  −1.60 ± 0.58  2.06 ± 0.23  5.54  7.78  0.99  FL  −5.57 ± 1.84  3.04 ± 0.61  6.15  14.54  0.97  FMC  −1.61 ± 1.30  2.07 ± 0.52  11.52  18.46  0.95  Equations are in the form log10Body mass = α + β(log10X). The variable X is the independent variable. α, intercept; β slope; CI, confidence interval; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; MAPE, mean absolute per cent prediction error; %SEE, per cent standard error of the estimate. View Large Table 1. Predictive equations for body mass of extant elephants from Christiansen (2004) Variable (X)  α ± 95% CI  β ± 95% CI  MAPE  %SEE  R2  HL  −4.14 ± 1.25  2.64 ± 0.43  6.74  11.52  0.98  HMC  −1.60 ± 0.58  2.06 ± 0.23  5.54  7.78  0.99  FL  −5.57 ± 1.84  3.04 ± 0.61  6.15  14.54  0.97  FMC  −1.61 ± 1.30  2.07 ± 0.52  11.52  18.46  0.95  Variable (X)  α ± 95% CI  β ± 95% CI  MAPE  %SEE  R2  HL  −4.14 ± 1.25  2.64 ± 0.43  6.74  11.52  0.98  HMC  −1.60 ± 0.58  2.06 ± 0.23  5.54  7.78  0.99  FL  −5.57 ± 1.84  3.04 ± 0.61  6.15  14.54  0.97  FMC  −1.61 ± 1.30  2.07 ± 0.52  11.52  18.46  0.95  Equations are in the form log10Body mass = α + β(log10X). The variable X is the independent variable. α, intercept; β slope; CI, confidence interval; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; MAPE, mean absolute per cent prediction error; %SEE, per cent standard error of the estimate. View Large We also tested our equations on two specimens of extant species in our sample for which pre-mortem BM was recorded. This test determines how well our predictive approach estimates actual BM. RESULTS Allometric equations OLS regressions were used to determine the relationship between OCB and the length and minimum circumference of the humeri and femora (Fig. 2). For the combined proboscidean sample, the four limb bone variables had moderately strong positive relationships with OCB; all R2 values ranged between 0.68 and 0.75. The minimum circumference of the humerus and the femur showed the strongest relationship, followed by the length of the femur and the length of the humerus (Table 2). The MAPE for all four variables were very similar, ranging from 7.31 to 8.45%. While the circumference variables showed the strongest correlation, the length variables were most precisely predicted (Table 2). All variables also had low %SEE (Fig. 2, Table 2). Table 2. Predictive equations for limb bone measurements from OCB Taxon  Variable (Z)  α ± 95% CI  β ± 95% CI  n  P value  R2  MAPE  %SEE  All proboscideans  HL  0.66 ± 1.05  0.99 ± 0.45  38  <0.0001***  0.68  8.21  12.09  FL  1.10 ± 0.90  0.83 ± 0.38  34  <0.0001***  0.70  7.31  9.48  HMC  −0.22 ± 1.20  1.19 ± 0.52  30  <0.0001***  0.75  8.44  11.22  FMC  −0.09 ± 1.32  1.12 ± 0.56  26  <0.0001***  0.73  8.45  12.18  Elephantidae  HL  0.61 ± 0.84  1.01 ± 0.36  30  <0.0001***  0.82  6.83  8.78  FL  1.16 ± 0.82  0.81 ± 0.35  25  <0.0001***  0.79  5.56  7.59  HMC  0.15 ± 1.39  1.03 ± 0.60  23  <0.0001***  0.69  8.10  11.28  FMC  0.17 ± 1.49  1.01 ± 0.64  20  <0.0001***  0.69  8.17  12.24  Gomphotheriidae  HL  −1.02 ± 5.75  1.67 ± 2.44  6  0.019*  0.78  7.24  10.39  Taxon  Variable (Z)  α ± 95% CI  β ± 95% CI  n  P value  R2  MAPE  %SEE  All proboscideans  HL  0.66 ± 1.05  0.99 ± 0.45  38  <0.0001***  0.68  8.21  12.09  FL  1.10 ± 0.90  0.83 ± 0.38  34  <0.0001***  0.70  7.31  9.48  HMC  −0.22 ± 1.20  1.19 ± 0.52  30  <0.0001***  0.75  8.44  11.22  FMC  −0.09 ± 1.32  1.12 ± 0.56  26  <0.0001***  0.73  8.45  12.18  Elephantidae  HL  0.61 ± 0.84  1.01 ± 0.36  30  <0.0001***  0.82  6.83  8.78  FL  1.16 ± 0.82  0.81 ± 0.35  25  <0.0001***  0.79  5.56  7.59  HMC  0.15 ± 1.39  1.03 ± 0.60  23  <0.0001***  0.69  8.10  11.28  FMC  0.17 ± 1.49  1.01 ± 0.64  20  <0.0001***  0.69  8.17  12.24  Gomphotheriidae  HL  −1.02 ± 5.75  1.67 ± 2.44  6  0.019*  0.78  7.24  10.39  Equations are in the form log10Z= α + β(log10OCW). The variable Z is the dependent variable. α, intercept; β slope; CI, confidence interval; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; MAPE, mean absolute per cent prediction error; n, sample size; OCB, occipital condyle breadth; %SEE, per cent standard error of the estimate. *Significant at alpha of 0.05. ***Significant at alpha of 0.001. View Large Table 2. Predictive equations for limb bone measurements from OCB Taxon  Variable (Z)  α ± 95% CI  β ± 95% CI  n  P value  R2  MAPE  %SEE  All proboscideans  HL  0.66 ± 1.05  0.99 ± 0.45  38  <0.0001***  0.68  8.21  12.09  FL  1.10 ± 0.90  0.83 ± 0.38  34  <0.0001***  0.70  7.31  9.48  HMC  −0.22 ± 1.20  1.19 ± 0.52  30  <0.0001***  0.75  8.44  11.22  FMC  −0.09 ± 1.32  1.12 ± 0.56  26  <0.0001***  0.73  8.45  12.18  Elephantidae  HL  0.61 ± 0.84  1.01 ± 0.36  30  <0.0001***  0.82  6.83  8.78  FL  1.16 ± 0.82  0.81 ± 0.35  25  <0.0001***  0.79  5.56  7.59  HMC  0.15 ± 1.39  1.03 ± 0.60  23  <0.0001***  0.69  8.10  11.28  FMC  0.17 ± 1.49  1.01 ± 0.64  20  <0.0001***  0.69  8.17  12.24  Gomphotheriidae  HL  −1.02 ± 5.75  1.67 ± 2.44  6  0.019*  0.78  7.24  10.39  Taxon  Variable (Z)  α ± 95% CI  β ± 95% CI  n  P value  R2  MAPE  %SEE  All proboscideans  HL  0.66 ± 1.05  0.99 ± 0.45  38  <0.0001***  0.68  8.21  12.09  FL  1.10 ± 0.90  0.83 ± 0.38  34  <0.0001***  0.70  7.31  9.48  HMC  −0.22 ± 1.20  1.19 ± 0.52  30  <0.0001***  0.75  8.44  11.22  FMC  −0.09 ± 1.32  1.12 ± 0.56  26  <0.0001***  0.73  8.45  12.18  Elephantidae  HL  0.61 ± 0.84  1.01 ± 0.36  30  <0.0001***  0.82  6.83  8.78  FL  1.16 ± 0.82  0.81 ± 0.35  25  <0.0001***  0.79  5.56  7.59  HMC  0.15 ± 1.39  1.03 ± 0.60  23  <0.0001***  0.69  8.10  11.28  FMC  0.17 ± 1.49  1.01 ± 0.64  20  <0.0001***  0.69  8.17  12.24  Gomphotheriidae  HL  −1.02 ± 5.75  1.67 ± 2.44  6  0.019*  0.78  7.24  10.39  Equations are in the form log10Z= α + β(log10OCW). The variable Z is the dependent variable. α, intercept; β slope; CI, confidence interval; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; MAPE, mean absolute per cent prediction error; n, sample size; OCB, occipital condyle breadth; %SEE, per cent standard error of the estimate. *Significant at alpha of 0.05. ***Significant at alpha of 0.001. View Large Figure 2. View largeDownload slide Ordinary least square regressions between log10 occipital condyle breadth (OCB) and log10 limb bone variables for the combined proboscidean sample. A, relationship with total humerus length (HL). B, relationship with total femur length (FL). C, relationship with minimum circumference of the humerus (HMC). D, relationship with minimum circumference of the femur (FMC). Regression equations are shown in the format y = α + βx, where α is the intercept and β is the regression coefficient. The coefficient of determination (R2), per cent standard error of the estimate (%SEE) and mean absolute per cent predictive error (MAPE) are shown along with the equations. Blue circles represent the Elephantidae, orange triangles represent the Gomphotheriidae, green squares represent the Mammutidae and purple diamonds represent the Stegodontidae. Solid black lines are the lines of best fit. Dashed grey lines represent 95% confidence intervals (CIs). Dotted lines represent 95% prediction intervals (PIs). Figure 2. View largeDownload slide Ordinary least square regressions between log10 occipital condyle breadth (OCB) and log10 limb bone variables for the combined proboscidean sample. A, relationship with total humerus length (HL). B, relationship with total femur length (FL). C, relationship with minimum circumference of the humerus (HMC). D, relationship with minimum circumference of the femur (FMC). Regression equations are shown in the format y = α + βx, where α is the intercept and β is the regression coefficient. The coefficient of determination (R2), per cent standard error of the estimate (%SEE) and mean absolute per cent predictive error (MAPE) are shown along with the equations. Blue circles represent the Elephantidae, orange triangles represent the Gomphotheriidae, green squares represent the Mammutidae and purple diamonds represent the Stegodontidae. Solid black lines are the lines of best fit. Dashed grey lines represent 95% confidence intervals (CIs). Dotted lines represent 95% prediction intervals (PIs). The utility of the combined proboscidean equations was tested using MAPE and raw %PE specific for each family (Table 3 and Fig. 4, respectively). For the Elephantidae, total length measurements tended to be underestimated, while predicted minimum circumference variables tended to be overestimated (Fig. 4). On average, OCB most precisely predicted the total lengths of the femur and humerus (Table 3). For the Gomphotheriidae, the total length variables tended to be overestimated, while minimum circumference variables were generally underestimated (Fig. 4). OCB predicted the minimum circumference of the femur and humerus, and total length of the femur with more precision (Table 3). The Mammutidae showed results similar to the Gomphotheriidae, with the minimum circumferences of the femur and humerus being predicted with better precision (Table 3). The small sample of mastodons did not allow us to discern general trends in over or underestimation. Our equations overestimated the total length and the minimum circumference of the humerus of the one specimen of Mammut americanum for which this bone was measured (Fig. 4). For the two M. americanum specimens for which the femur was measured, the minimum circumference of one specimen was overestimated, while that for the other was underestimated; the total lengths of the femur for both specimens were overestimated (Fig. 4). Only the total femur length of Mammut borsoni was underestimated using these equations (Fig. 4). The Stegodontidae were similar to the Elephantidae, with OCB best predicting the total lengths of the humerus and the femur (Table 4). All limb bone variables for this specimen except for total humerus length were underestimated (Fig. 4). Table 3. Family-specific mean %PE and %BM differences Predictive equation  Family  Variable  n  MAPE (±SD)  %BMdiff (mean ± SD)  All proboscideans  Elephantidae  FL  23  6.56 ± 3.87  20.85 ± 13.15  FMC  18  8.33 ± 7.78  17.52 ± 17.58  HL  28  6.73 ± 4.55  18.19 ± 12.81  HMC  21  8.54 ± 6.63  17.82 ± 18.55  Gomphotheriidae  FL  4  6.79 ± 6.94  18.14 ± 17.95  FMC  3  4.61 ± 2.09  9.81 ± 4.56  HL  6  16.21 ± 8.89  36.05 ± 16.86  HMC  5  6.85 ± 4.28  14.72 ± 9.49  Mammutidae  FL  3  14.24 ± 4.19  41.03 ± 11.44  FMC  2  7.31 ± 9.00  16.17 ± 20.10  HL  1  12.58  29.88  HMC  1  2.59  5.27  Stegodontidae  FL  1  7.94  26.17  FMC  1  25.13  59.06  HL  1  0.02  0.06  HMC  1  19.88  45.28  Elephantidae  Elephantidae  FL  23  5.56 ± 4.10  16.64 ± 11.85  FMC  18  8.17 ± 7.22  16.85 ± 15.36  HL  28  6.83 ± 4.79  18.68 ± 13.83  HMC  21  8.10 ± 6.36  16.78 ± 13.50  Gomphotheriidae  Gomphotheriidae  HL  6  7.24 ± 4.22  19.32 ± 12.71  Predictive equation  Family  Variable  n  MAPE (±SD)  %BMdiff (mean ± SD)  All proboscideans  Elephantidae  FL  23  6.56 ± 3.87  20.85 ± 13.15  FMC  18  8.33 ± 7.78  17.52 ± 17.58  HL  28  6.73 ± 4.55  18.19 ± 12.81  HMC  21  8.54 ± 6.63  17.82 ± 18.55  Gomphotheriidae  FL  4  6.79 ± 6.94  18.14 ± 17.95  FMC  3  4.61 ± 2.09  9.81 ± 4.56  HL  6  16.21 ± 8.89  36.05 ± 16.86  HMC  5  6.85 ± 4.28  14.72 ± 9.49  Mammutidae  FL  3  14.24 ± 4.19  41.03 ± 11.44  FMC  2  7.31 ± 9.00  16.17 ± 20.10  HL  1  12.58  29.88  HMC  1  2.59  5.27  Stegodontidae  FL  1  7.94  26.17  FMC  1  25.13  59.06  HL  1  0.02  0.06  HMC  1  19.88  45.28  Elephantidae  Elephantidae  FL  23  5.56 ± 4.10  16.64 ± 11.85  FMC  18  8.17 ± 7.22  16.85 ± 15.36  HL  28  6.83 ± 4.79  18.68 ± 13.83  HMC  21  8.10 ± 6.36  16.78 ± 13.50  Gomphotheriidae  Gomphotheriidae  HL  6  7.24 ± 4.22  19.32 ± 12.71  %BMdiff, absolute per cent difference in body mass estimated from actual limb measurements and predicted limb variables; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; MAPE, mean absolute per cent prediction error; n, sample size; %PE, per cent predictive error. View Large Table 3. Family-specific mean %PE and %BM differences Predictive equation  Family  Variable  n  MAPE (±SD)  %BMdiff (mean ± SD)  All proboscideans  Elephantidae  FL  23  6.56 ± 3.87  20.85 ± 13.15  FMC  18  8.33 ± 7.78  17.52 ± 17.58  HL  28  6.73 ± 4.55  18.19 ± 12.81  HMC  21  8.54 ± 6.63  17.82 ± 18.55  Gomphotheriidae  FL  4  6.79 ± 6.94  18.14 ± 17.95  FMC  3  4.61 ± 2.09  9.81 ± 4.56  HL  6  16.21 ± 8.89  36.05 ± 16.86  HMC  5  6.85 ± 4.28  14.72 ± 9.49  Mammutidae  FL  3  14.24 ± 4.19  41.03 ± 11.44  FMC  2  7.31 ± 9.00  16.17 ± 20.10  HL  1  12.58  29.88  HMC  1  2.59  5.27  Stegodontidae  FL  1  7.94  26.17  FMC  1  25.13  59.06  HL  1  0.02  0.06  HMC  1  19.88  45.28  Elephantidae  Elephantidae  FL  23  5.56 ± 4.10  16.64 ± 11.85  FMC  18  8.17 ± 7.22  16.85 ± 15.36  HL  28  6.83 ± 4.79  18.68 ± 13.83  HMC  21  8.10 ± 6.36  16.78 ± 13.50  Gomphotheriidae  Gomphotheriidae  HL  6  7.24 ± 4.22  19.32 ± 12.71  Predictive equation  Family  Variable  n  MAPE (±SD)  %BMdiff (mean ± SD)  All proboscideans  Elephantidae  FL  23  6.56 ± 3.87  20.85 ± 13.15  FMC  18  8.33 ± 7.78  17.52 ± 17.58  HL  28  6.73 ± 4.55  18.19 ± 12.81  HMC  21  8.54 ± 6.63  17.82 ± 18.55  Gomphotheriidae  FL  4  6.79 ± 6.94  18.14 ± 17.95  FMC  3  4.61 ± 2.09  9.81 ± 4.56  HL  6  16.21 ± 8.89  36.05 ± 16.86  HMC  5  6.85 ± 4.28  14.72 ± 9.49  Mammutidae  FL  3  14.24 ± 4.19  41.03 ± 11.44  FMC  2  7.31 ± 9.00  16.17 ± 20.10  HL  1  12.58  29.88  HMC  1  2.59  5.27  Stegodontidae  FL  1  7.94  26.17  FMC  1  25.13  59.06  HL  1  0.02  0.06  HMC  1  19.88  45.28  Elephantidae  Elephantidae  FL  23  5.56 ± 4.10  16.64 ± 11.85  FMC  18  8.17 ± 7.22  16.85 ± 15.36  HL  28  6.83 ± 4.79  18.68 ± 13.83  HMC  21  8.10 ± 6.36  16.78 ± 13.50  Gomphotheriidae  Gomphotheriidae  HL  6  7.24 ± 4.22  19.32 ± 12.71  %BMdiff, absolute per cent difference in body mass estimated from actual limb measurements and predicted limb variables; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; MAPE, mean absolute per cent prediction error; n, sample size; %PE, per cent predictive error. View Large Table 4. Raw %PE for the limb variables estimated for the two specimens and a comparison of actual body mass of with estimates from this study Species  Specimen    Actual  HL  HMC  FL  FMC  Equation  Elephas maximus  CN1399  %PE    16.34  19.26  14.43  31.63  All proboscideans  BM (kg)  3534  2486.6  2401.48  2327.17  2200.78  %Diff    42.13  47.17  51.87  60.59  %PE    17.86  15.12  10.37  26.81  Elephantidae  BM (kg)    2402.37  2582.86  2596.94  2377.2  %Diff    47.11  36.84  36.09  48.67  Loxodonta africana  ROMV R6000  %PE    −5.81  −5.16  −2.31  −0.51  All proboscideans  BM (kg)  6435  7531.77  6790.5  6783.8  5881.76  %Diff    14.56  5.14  5.24  9.4  %PE    −5.37  −2.03  −4.97  0.43  Elephantidae  BM (kg)    7441.4  6350.79  7377.51  5768.53  %Diff    13.52  1.33  12.77  11.55  Species  Specimen    Actual  HL  HMC  FL  FMC  Equation  Elephas maximus  CN1399  %PE    16.34  19.26  14.43  31.63  All proboscideans  BM (kg)  3534  2486.6  2401.48  2327.17  2200.78  %Diff    42.13  47.17  51.87  60.59  %PE    17.86  15.12  10.37  26.81  Elephantidae  BM (kg)    2402.37  2582.86  2596.94  2377.2  %Diff    47.11  36.84  36.09  48.67  Loxodonta africana  ROMV R6000  %PE    −5.81  −5.16  −2.31  −0.51  All proboscideans  BM (kg)  6435  7531.77  6790.5  6783.8  5881.76  %Diff    14.56  5.14  5.24  9.4  %PE    −5.37  −2.03  −4.97  0.43  Elephantidae  BM (kg)    7441.4  6350.79  7377.51  5768.53  %Diff    13.52  1.33  12.77  11.55  BM, body mass in kilograms; %Diff, per cent difference between actual mass and estimated mass; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; %PE, raw per cent predictive error. View Large Table 4. Raw %PE for the limb variables estimated for the two specimens and a comparison of actual body mass of with estimates from this study Species  Specimen    Actual  HL  HMC  FL  FMC  Equation  Elephas maximus  CN1399  %PE    16.34  19.26  14.43  31.63  All proboscideans  BM (kg)  3534  2486.6  2401.48  2327.17  2200.78  %Diff    42.13  47.17  51.87  60.59  %PE    17.86  15.12  10.37  26.81  Elephantidae  BM (kg)    2402.37  2582.86  2596.94  2377.2  %Diff    47.11  36.84  36.09  48.67  Loxodonta africana  ROMV R6000  %PE    −5.81  −5.16  −2.31  −0.51  All proboscideans  BM (kg)  6435  7531.77  6790.5  6783.8  5881.76  %Diff    14.56  5.14  5.24  9.4  %PE    −5.37  −2.03  −4.97  0.43  Elephantidae  BM (kg)    7441.4  6350.79  7377.51  5768.53  %Diff    13.52  1.33  12.77  11.55  Species  Specimen    Actual  HL  HMC  FL  FMC  Equation  Elephas maximus  CN1399  %PE    16.34  19.26  14.43  31.63  All proboscideans  BM (kg)  3534  2486.6  2401.48  2327.17  2200.78  %Diff    42.13  47.17  51.87  60.59  %PE    17.86  15.12  10.37  26.81  Elephantidae  BM (kg)    2402.37  2582.86  2596.94  2377.2  %Diff    47.11  36.84  36.09  48.67  Loxodonta africana  ROMV R6000  %PE    −5.81  −5.16  −2.31  −0.51  All proboscideans  BM (kg)  6435  7531.77  6790.5  6783.8  5881.76  %Diff    14.56  5.14  5.24  9.4  %PE    −5.37  −2.03  −4.97  0.43  Elephantidae  BM (kg)    7441.4  6350.79  7377.51  5768.53  %Diff    13.52  1.33  12.77  11.55  BM, body mass in kilograms; %Diff, per cent difference between actual mass and estimated mass; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; %PE, raw per cent predictive error. View Large Figure 3. View largeDownload slide Ordinary least square regressions between log10 occipital condyle breadth (OCB) and log10 limb bone variables for the Elephantidae-only data set (A–D) and the Gomphotheriidae-only data set (E). A, relationship with total humerus length (HL) of the Elephantidae. B, relationship with total femur length (FL) of the Elephantidae. C, relationship with minimum circumference of the humerus (HMC) of the Elephantidae. D, relationship with minimum circumference of the femur (FMC) of the Elephantidae. E, relationship of the with HL of the Gomphotheriidae. Regression equations are shown in the format y = α + βx, where α is the intercept and β is the regression coefficient. The coefficient of determination (R2), per cent standard error of the estimate (%SEE) and mean absolute per cent predictive error (MAPE) are shown along with the equations. Blue circles represent the Elephantidae, orange triangles represent the Gomphotheriidae. Solid black lines are the lines of best fit. Dashed grey lines represent 95% confidence intervals (CIs). Dotted lines represent 95% prediction intervals (PIs). Figure 3. View largeDownload slide Ordinary least square regressions between log10 occipital condyle breadth (OCB) and log10 limb bone variables for the Elephantidae-only data set (A–D) and the Gomphotheriidae-only data set (E). A, relationship with total humerus length (HL) of the Elephantidae. B, relationship with total femur length (FL) of the Elephantidae. C, relationship with minimum circumference of the humerus (HMC) of the Elephantidae. D, relationship with minimum circumference of the femur (FMC) of the Elephantidae. E, relationship of the with HL of the Gomphotheriidae. Regression equations are shown in the format y = α + βx, where α is the intercept and β is the regression coefficient. The coefficient of determination (R2), per cent standard error of the estimate (%SEE) and mean absolute per cent predictive error (MAPE) are shown along with the equations. Blue circles represent the Elephantidae, orange triangles represent the Gomphotheriidae. Solid black lines are the lines of best fit. Dashed grey lines represent 95% confidence intervals (CIs). Dotted lines represent 95% prediction intervals (PIs). Figure 4. View largeDownload slide A comparison of the raw per cent predictive errors (%PE) for the predicted limb bone variables for each family using box plots. The limb bone variables are predicted using the combined proboscidean equations. Values less than 0 are overestimated while values greater than 0 are underestimated. Single horizontal bars indicate a sample size of 1. Black bars in the box plots show the median value. Dotted horizontal line at %PE of 0 shows perfect prediction. Blue circles are outliers for the Elephantidae. Families are shown in alphabetical order in each panel. FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference. Figure 4. View largeDownload slide A comparison of the raw per cent predictive errors (%PE) for the predicted limb bone variables for each family using box plots. The limb bone variables are predicted using the combined proboscidean equations. Values less than 0 are overestimated while values greater than 0 are underestimated. Single horizontal bars indicate a sample size of 1. Black bars in the box plots show the median value. Dotted horizontal line at %PE of 0 shows perfect prediction. Blue circles are outliers for the Elephantidae. Families are shown in alphabetical order in each panel. FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference. When possible, we generated separate allometric equations for each family (Fig. 3). For equations specific to the Elephantidae, the strongest correlations were between OCB and the total lengths of the humerus and femur (Table 2). These two variables were also the most precisely predicted and had the smallest %SEE (Table 2). Overall, the correlations between OCB and the four limb variables were moderate to strong, with R2 ranging from 0.69 to 0.82 and were more precisely predicted than they were when using the combined proboscidean equations (Table 3). All four limb bone variables were roughly equally over and underestimated, with median values close to 0 (Fig. 5). For the Gomphotheriidae-specific equation, the correlation between OCB and humerus length is moderately strong, and the MAPE is smaller than that when HL is predicted using the combined proboscidean equation (Table 3). The MAPEs for humerus and femur minimum circumference, and femur total length for gomphotheres when estimated using the combined proboscidean equations, are very similar to the MAPE of humerus length estimated using the Gomphotheriidae only equation. Out of the six specimens sampled, four HLs were overestimated, while two were underestimated (Fig. 5). Figure 5. View largeDownload slide A comparison of the raw per cent predictive errors (%PE) for the predicted limb bone variables for the Elephantidae and Gomphotheriidae using box plots. The limb bone variables are predicted using the family-specific equations. Values less than 0 are overestimated while values greater than 0 are underestimated. Black bars in the box plots show the median value. Dotted horizontal line at %PE of 0 shows perfect prediction. Blue circles are outliers for the Elephantidae. For humerus length, the first box plot is for the Elephantidae and the second is for the Gomphotheriidae. FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference. Figure 5. View largeDownload slide A comparison of the raw per cent predictive errors (%PE) for the predicted limb bone variables for the Elephantidae and Gomphotheriidae using box plots. The limb bone variables are predicted using the family-specific equations. Values less than 0 are overestimated while values greater than 0 are underestimated. Black bars in the box plots show the median value. Dotted horizontal line at %PE of 0 shows perfect prediction. Blue circles are outliers for the Elephantidae. For humerus length, the first box plot is for the Elephantidae and the second is for the Gomphotheriidae. FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference. Body mass estimation We used our predicted limb variables to estimate BM using equations developed by Christiansen (2004) and compared these estimates with those derived from actual limb bone measurements. Raw values for both estimations are shown in Supporting Information, Table S2. The per cent difference in BM estimated from actual limb bones and limb variables predicted using the combined proboscidean equations varied between 17.52 ± 17.58% and 20.85 ± 13.15% for the Elephantidae; %BMdiff for each limb bone variable were only marginally different from each other (Table 3). For the Gomphotheriidae, %BMdiff varied from 9.81 ± 4.56% to 36.05 ± 16.86%. The minimum circumferences variables yielded the smallest differences between the two estimates (Table 3). The %BMdiff for the Mammutidae ranged from 5.27% to 41.03 ± 11.44%, with the minimum circumference variables having the smallest difference from the actual limb measurements in estimated BM (Table 3). For the one Stegodon sampled, the %BMdiff ranged from 0.06 to 59.06%. The BM values estimated from the predicted total length variables were most similar to those estimated from the actual limb measurements (Table 3). We used the family-specific equations to predict the limb bone variables for the Elephantidae, and Gomphotheriidae, and subsequently estimated BM using these predicted variables. BM estimates for the Elephantidae more precisely matched estimates from the actual limb measurements, and estimates were marginally better than those derived from the combined proboscidean equations (Table 3). The Gomphotheriidae-specific equation more precisely predicted humerus length, and correspondingly, the %BMdiff was much lower than when the humerus length was estimated using the combined proboscidean equation (Table 3). We compared the BM estimates calculated using our method with two specimens that had pre-mortem BM recorded as a test of our approach. Our estimates closely matched the actual BM (<15% difference) of a specimen of Loxodonta africana, but differed between 36.09 and 60.59% from the actual mass of a specimen of Elephas maximus (Table 4). DISCUSSION We have developed a method to indirectly estimate BM for extinct proboscideans using cranial remains. Allometric relationships between the breadth of the occipital condyles and the total length and minimum circumferences of the humerus and femur are used to predict these limb bone variables. In turn, these predicted variables are used to estimate BM using existing predictive equations developed by Christiansen (2004). Our results only apply to proboscidean families with columnar limbs. Older taxa such as Moeritherium, or the Numidotheriidae, have relatively shorter limbs, and studies have shown that they had a different gait as well (Court, 1994), which would suggest different allometric scaling relationships. The more derived columnar appendicular anatomy, which is well adapted to support the weight of the animal (Shoshani, 1996), is found in proboscideans like the Deinotheriidae, and more derived families such as the Elephantidae, Mammutidae, Gomphotheriidae and Stegodontidae (Tassy, 1996), and results in a unique gait (Hutchinson et al., 2003; Ren & Hutchinson, 2008). Given this similarity in appendicular anatomy, and its adaptation for weight bearing, it is probably that these families share similar allometric scaling relationships with body size. Although we did not include the Deinotheriidae because of a lack of available material, or published measurements, our equations should be applicable because of the similarity in anatomy. While the above-mentioned taxa generally show similar body proportions, family-specific differences do exist (Christiansen, 2007). Gomphothere and mastodon species tend to have shorter humeri and femora when compared to elephantids for similar values of minimum bone circumference, that is, for animals of similar size, elephantids tend to be taller than gomphotheres and mastodons (Christiansen, 2007; Haynes, 1991). Our data show similar patterns for the Elephantidae and Gomphotheriidae. For a given OCB (our proxy for skull size), both elephantids and gomphotheres tend to have similar values of humerus and femur minimum circumference (Fig. 2C, D), but several gomphothere specimens have relatively short humeri for the same OCB as the elephantids in the sample (Fig. 2A, B). These differences increase scatter for these variables, which results in the relatively low R2 values (Fig. 2). Interestingly, the total length variables were predicted with more precision than the minimum circumference variables. This discrepancy between the strength of correlation and precision of prediction is caused by differences in sample sizes of the different families sampled. The majority of the data points in the combined sample belong to the Elephantidae, which have their humerus and femur lengths more strongly correlated with and more precisely predicted by OCB (Table 3). Since there were more total length measurements overall, and since most of them belonged to the Elephantidae, the MAPE for the total length variables were lower on average than those for the minimum circumference variables. The differences in body proportions and sample size among families were the impetus behind repeating the analyses using family-specific subsets of the data set. Overall, the family-specific equations better predicted the limb bone variables than the combined proboscidean equations. Because we had insufficient sample sizes to generate equations specific to FL, FMC and HMC for the Gomphotheriidae, and for all variables for the Mammutidae and Stegodontidae, we assessed how well the combined proboscidean equations predicted the different limb bone variables for these taxa. For the Gomphotheriidae, the most precisely predicted variables were the minimum circumference of the humerus and femur and the total length of the femur. As mentioned above, the combined proboscidean equations were (1) generated using a data set that was biased towards the Elephantidae, and (2) the Elephantidae have proportionally longer limb bones than the Gomphotheriidae for a given minimum circumference, a pattern more evident in humeri than in femora (Christiansen, 2007). Therefore, the equations poorly predicted the total length of the humerus, but the minimum circumferences of the gomphotheres fell close to the regression line, which tended to fit the large number of Elephantidae data points. The results for the Mammutidae and Stegodontidae are hard to generalize because of the very small sample size. Only three mastodons are included in the sample, and one Stegodon (Supporting Information, Table S1). The specimen of M. americanum, for which total humerus length and minimum circumference measurements were available, scaled like the elephantids. Two specimens for which femur minimum circumference was measured also scaled close to the elephantids, and the actual measurements did not deviate much from the regression line. A femur length measurement was available for one specimen of the large species M. borsoni, and for the two other M. americanum specimens. This variable deviated much more from the regression line. It appears that mastodon FL has a different scaling relationship (Fig. 2B), but more data are required to confirm this. Therefore, from the available data and the MAPE for the Mammutidae, we suggest that the combined proboscidean equations that predict the minimum circumferences of the humerus and femur be used for this family. The one Stegodon specimen sampled tended to scale like the elephantids. To our knowledge, there is no comparative study that has explored the allometric relationships between the total length and minimum circumference of stegodontids and elephantids, but the combined proboscidean equations from our study produced low MAPEs for the total humerus and femur length for this specimen. The goal of this study was to estimate BM of proboscideans from cranial characters. Because our method of estimating BM relies on a predicted variable, the accuracy of the estimates will always be limited and will depend on the precisions of the limb variable estimation equations and the precision of the mass estimation equations. The mass estimation equations developed by Christiansen (2004) included a sub-adult specimen in the sample. This is potentially problematic because of changes in allometry during ontogeny (Miller et al., 2008). Christiansen (2004) also used Reduced Major Axis regressions instead of OLS to assess the relationship between BM and limb bone variables, an approach that is not appropriate for predictive models (Smith, 2009). To determine whether the analytical methodology or the addition of a sub-adult specimen has any effect on the relationship between BM and limb variables, we re-analysed Christiansen’s (2004) data with and without the sub-adult, using OLS regressions (Supporting Information, Tables S3, S4). These analyses showed no appreciable differences in the slope, intercept, %SEE or MAPE (Table 1, Supporting Information, Tables S3, S4). The strength of the correlation between the femoral variables and BM did decrease upon the removal of the sub-adult specimen (Supporting Information, Tables S3, S4). Since the measures of residual variation (%SEE and MAPE), which are more informative in determining predictive capacity (Smith, 1984), remain comparable, the published equations are likely robust estimators of BM. The true test of our two-step predictive approach to estimating BM is comparing our estimates to known values of BM. We were able to sample two specimens of known mass and compared our estimates to these values. We more precisely estimated the limb bone variables of the L. africana specimen (ROMV R6000) than the E. maximus specimen (CN 1399), which resulted in close matches to the known mass of the L. africana specimen, but only moderate matches to the known mass of the E. maximus specimen (Table 4). The moderate match can be accounted for by the fact that the limb variables were not as precisely predicted by our equations. The specimen in question has the smallest OCB in our sample and might represent an individual with a disproportionally small skull, which would result in the underestimation of the limb bone variables, and correspondingly smaller estimated BM. Alternatively, this mismatch might be a result of sexual dimorphism in Asian elephants. Female Asian elephants lack tusks and have more gracile skulls with smaller parieto-occipital bosses (Todd, 2010). This would result in a lighter skull with proportionally smaller occipital condyles for the size of the limb bones, since limb bone size is more likely to be determined by overall BM. The effect of sexual dimorphism on these allometric relationships is beyond the scope of this study, but can be tested in the future with a larger sample of elephants with known BM. Despite the degree of mismatch for this one specimen, these estimates are reasonable. The differences are minimized when the mass estimates are log10 transformed, the scale at which most studies on BM are conducted (Maurer, Brown & Rusler, 1992; Smith et al., 2010; Lyons & Smith, 2013). For the remaining sample, we could only compare mass estimates from actual limb measurements to estimates from predicted limb variables. Overall, the precision by which the limb variables were predicted dictated how well the estimated BM matched the BM estimated from actual limb measurements. Therefore, researchers using our approach are recommended to use equations that most precisely predict the limb variables. For the Elephantidae, the family-specific equations best predicted limb variables. For gomphotheres, the combined proboscidean equations more precisely predicted the minimum circumferences of the humerus and femur, and the total length of the femur, while the family-specific equation better predicted the length of the humerus. Until further specific equations are developed for the Mammutidae, we suggest that minimum circumferences be estimated to estimate BM. Lastly for the Stegodontidae, we recommend predicting total length variables to estimate BM. When available, actual limb bones are preferably to estimate BM for all taxa, but our method allows for reasonable values of BM to be estimated when only cranial remains are available for study. SUPPORTING INFORMATION Additional Supporting Information may be found in the online version of this article at the publisher's web-site: Table S1. List of specimens and measurement references. Table S2. Body mass estimated using actual limb bone measurements and predicted limb bone variables. Table S3. Ordinary least square regression equations for body mass of extant elephants modified from Christiansen (2004). Equations are in the form log10Body mass = α + β(log10X). The variable X is the independent variable. Sample size is 7. Table S4.Ordinary least square regression equations for body mass of extant elephants modified from Christiansen (2004). Equations are in the form log10Body mass = α + β(log10X). The variable X is the independent variable. The sub-adult specimen was removed to create these equations; therefore, the sample size was reduced to 6. ACKNOWLEDGEMENTS We would like to thank A. Millhouse and E. Langan for providing access to extinct and extant proboscidean specimens at the Smithsonian’s National Museum of Natural History, J. Galkin and E. Westwig for providing access to extinct and extant proboscidean specimens at the American Museum of Natural History, N. Gilmore providing access to extant elephant specimens at the Academy of Natural Sciences of Drexel University, and S. Gaikwad for providing access to a specimen of E. maximus at the Bombay Veterinary College. We also would like to thank D. K. Johansson, J. Miller, K. Seymour and A. Larramendi for providing additional proboscidean skeletal measurements, and M. Chen for translating a paper on Chinese stegodontids to English. The authors would also like to thank A. Webb and D. Fraser for assisting with R. This study was supported by the George Mason University Provost Fellowship, awarded to AMJ. Lastly, we thank three anonymous reviewers for useful suggestions that helped improve this manuscript. 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New York: Cambridge University Press, 181– 205. © 2018 The Linnean Society of London, Zoological Journal of the Linnean Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Zoological Journal of the Linnean Society Oxford University Press

A cranial correlate of body mass in proboscideans

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The Linnean Society of London
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© 2018 The Linnean Society of London, Zoological Journal of the Linnean Society
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0024-4082
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10.1093/zoolinnean/zlx108
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Abstract

Abstract Allometric scaling relationships are often used to estimate the body mass (BM) of extinct mammalian taxa. For proboscideans, shoulder height or limb bone dimensions have typically been used to estimate BM. However, these skeletal correlates are only useful when a complete forelimb is available, or when limb elements can be identified to the species level. Several taxa are known or can be identified from cranial remains alone, which poses a problem for BM estimation. Here, we develop allometric equations to predict the total length and the minimum circumference of the humerus or femur of derived proboscideans from the breadth of the occipital condyles. These predicted measurements are then used to estimate body size from existing equations derived from regressions on limb bones. We developed equations using a combined sample of both extinct and extant proboscidean taxa. Equations for specific families were derived when possible. We find that occipital condyle breadth is a robust predictor of limb bone dimensions. Estimated BM values from predicted limb measurements were a good match to actual BM, and estimates from actual limb measurements. Our method will allow researchers to study BM evolution in proboscideans using a greater range of specimens. allometry, body size, occipital condyle breadth INTRODUCTION Body size is recognized as one of the most important traits of an organism. Studies on extant species have shown that body size affects metabolism (Klieber, 1932; Peters, 1983), reproductive fitness (Brown, Marquet & Taper, 1993), life history traits (Kaplan & Salthe, 1979), development (Gillooly et al., 2002), population density (Damuth, 1981, 2007), abundance (Blackburn et al., 1993) and range size (Gaston & Blackburn, 1996). Body size is particularly useful because it can be measured in common units (e.g. mass, biovolume) across all of life. Because of the relationship with an organism’s biology and ecology and the utility of measurement, paleobiologists strive to accurately estimate the body size of extinct taxa (Damuth & Macfadden, 1990). While estimating the body mass (BM) of an extinct species is a challenging task, paleobiologists have made immense progress over the last few decades, especially for fossil mammals (Damuth & Macfadden, 1990). Classically, researchers have studied the allometric relationships between skeletal measurements and BM of extant taxa and used the predictive equations generated to estimate the BM of extinct counterparts. For example, the linear or areal dimensions of the molars correlate well with BM in extant members of the Artiodactyla and Perissodactyla (Damuth, 1990; Janis, 1990), while the length of the skull correlates well with BM in carnivorans (Van Valkenburgh, 1990). Within the Tetrapoda as a whole, the length, diameter and circumference of the humerus and femur also scale with BM (Alexander et al., 1979; Anderson, Hallmartin & Russell, 1985; Gingerich, 1990; Campione & Evans, 2012). For proboscideans, two allometric scaling relationships are typically used to estimate BM – the first one relates BM to shoulder height (Roth, 1990), and another relates BM with various dimensions of the limb bones (Christiansen, 2004). Unfortunately, estimating the BM of extinct proboscideans remains a challenge for researchers. Using the shoulder height requires a complete forelimb, which is often unavailable for many fossil specimens. Moreover, both equations require that limbs be referable to a species. More recently, volumetric methods have been used to estimate the BM of extant and extinct proboscideans (Larramendi, 2016). However, this method relies on accurate physical or digital reconstructions, which may not always be possible from fragmentary fossil remains. Therefore, estimating BM of extinct proboscideans from the linear dimensions of limb bones remains widely used (e.g. see Lister & Stuart, 2010). The challenge is that some extinct species are known or can only be identified from cranial and dental remains, as is the case with a number of extinct proboscidean taxa from South and Southeast Asia. Unless, limb bones, a complete forelimb or complete skeleton of that species are found, it is not possible to estimate its BM. Here, we address this problem by developing a cranial correlate of BM. We use the occipital condyle breadth (OCB) to estimate the BM of proboscideans. This can be especially useful to researchers attempting to estimate the BM of proboscidean species, which are only known or identified from cranial remains. OCB has been used as a predictor of BM in a variety of taxa, from marine mammals such as cetaceans (Marino et al., 2003; Marino, McShea & Uhen, 2004; Bianucci, Lambert & Post, 2010; Pyenson & Sponberg, 2011) and sirenians (Sarko et al., 2010), to small mammals such as lagomorphs, rodents, shrews and weasels (Martin, 1980; Moncunill-Solé et al., 2014, 2015), and even hominids (Martin, 1981). The expectation is that, as the overall BM of proboscideans increases, so should the size of the skulls, resulting in an increase in the breadth of the occipital condyles, the primary locus of contact of the head with the rest of the body. The dearth of extant elephant specimens that had been weighed before death, or shortly after, prevented us from directly investigating the relationship between OCB and BM. Therefore, we investigated the allometric relationship between OCB and the length and circumference of the humerus and femur of extant and extinct proboscideans, both variables which are known to scale with BM. The allometric equations generated are used to predict the limb bone dimensions, which are then used to estimate BM using equations developed by Christiansen (2004). We compare estimates of BM generated from our predicted limb bone variables to those generated from the actual limb bone dimensions and test our approach on two extant elephant specimens with known BM to assess the precision of our methodology. MATERIAL AND METHODS Specimens Forty-one specimens of both extant and extinct proboscideans spanning the four more derived families, the Elephantidae, Stegodontidae, Gomphotheriidae and Mammutidae, were sampled. Measurements were taken either by directly measuring museum specimens or from published descriptions in peer-reviewed literature. A list of specimens and species sampled are given in Supporting Information, Table S1. Only adult specimens were used in this study to reduce any effects of ontogeny on allometric scaling relationships. The age of extant proboscideans was determined using ontogenetic dental scales developed by Laws (1966) and Roth & Shoshani (1988). For extinct species, the eruption of the third molar was taken as evidence of adulthood. Skeletal variables OCB was measured as the maximum linear distance across both condyles (Fig. 1). Christiansen (2004) determined that the antero-posterior diameter, total length and the minimum circumference of the humerus were the best forelimb predictors of BM. However, he did not mention where on the diaphysis the diameter was measured. To avoid error in measurement, we only measured the total length and minimum circumference of the humerus, both being unambiguous measurements. Similarly, for the hindlimb, Christiansen (2004) showed that the length and the antero-posterior diameter of the femur best predicted BM. For the reasons mentioned above, we measured the total length, but not the diameter. We measured the minimum circumference of the femur as it is also a reliable predictor of BM (Roth, 1990). The total length of the humerus (HL) was measured between the most proximal end of lateral tuberosity to most distal end of lateral condyle, and the total length of the femur (FL) was measured between the proximal end of the femoral head and the distal-most end of the medial condyle (Fig. 1). Minimum circumference was measured by passing a loop of measuring tape down the diaphysis of the limb bones until the minimum circumference was reached (Fig. 1). For the femur minimum circumference (FMC), this is roughly at mid-shaft; for the humerus minimum circumference (HMC), it was close to the distal end of the diaphysis. Figure 1. View largeDownload slide Skeletal measurements used in this study. A, occipital condyle breadth (OCB), measured as the linear distance across both condyles. B, total length of the humerus (HL), measured between the most proximal end of lateral tuberosity to most distal end of lateral condyle, and the minimum circumference of the humerus (HMC) which is measured close to the waist of the diaphysis. C, total length of the femur (FL) measured between the proximal end of the femoral head and the distal-most end of the medial condyle, and the minimum circumference of the femur (FMC) measured near the midpoint of the diaphysis. Figure 1. View largeDownload slide Skeletal measurements used in this study. A, occipital condyle breadth (OCB), measured as the linear distance across both condyles. B, total length of the humerus (HL), measured between the most proximal end of lateral tuberosity to most distal end of lateral condyle, and the minimum circumference of the humerus (HMC) which is measured close to the waist of the diaphysis. C, total length of the femur (FL) measured between the proximal end of the femoral head and the distal-most end of the medial condyle, and the minimum circumference of the femur (FMC) measured near the midpoint of the diaphysis. Predictive equations Ordinary least square (OLS) regressions were fitted to log10-transformed data using the statistical software program R (R Core Team, 2014). OLS regressions are preferred in estimation because they minimize the sum of squared errors (Smith, 2009; Sokal & Rohlf, 2012). The four variables (HL, FL, HMC and FMC) could not be measured for all specimens. Therefore, sample sizes varied among the four variables tested. Because of differences in the proportions of the limb bones among proboscidean families (Christiansen, 2007), we expect the allometric scaling relationships between OCB and the limb bone variables to vary among these clades. Hence, we repeated our analyses by family where possible. Since the majority of the specimens came from the Elephantidae, separate predictive equations were generated for this family. For the Gomphotheriidae, humerus length was the only variable measured for at least six individuals; therefore, a separate predictive equation could be generated for this variable. Only one specimen of the Stegodontidae and three specimens of the Mammutidae were included in the overall sample, thus, separate equations could not be generated for these families. The strength of the relationship between OCB and the limb variables was determined using the coefficient of determination (R2). We calculated the per cent standard error of the estimate (%SEE), a measure of the residual variation following Brody (1945) and Smith (1984). The %SEE measures the per cent error within which 68% of the estimated values match the observed values (Smith, 1984). According to Brody (1945), the SEE of log-transformed data can be calculated as follows: if we take a predicted value of the dependent variable, lets say 100, we get a value of 2 on the regression line (log10100 = 2). The SEE, which is calculated using the formula Σ(Y−Ypredicted)2(n−2), is then added to 2. The antilog of the resulting number is taken, and the value greater than 100 is converted to a percentage of 100. This is the positive %SEE. This can be reproduced using any value taken from the regression line. The predictive power of OCB was determined using the mean absolute per cent predictive error (MAPE). MAPE is a measurement of the average per cent deviation between the predicted and actual limb bone measurements and is calculated using untransformed data. It is calculated as follows: raw values of per cent predictive error (%PE) are estimated for each specimen using the formula (Observed−PredictedPredicted)×100 (Smith, 1984), and then, the mean of the absolute values is taken. Family-specific raw %PE and MAPE were calculated where appropriate. The raw %PE values indicate the degree of over or underestimation of a particular measurement. Negative values indicate overestimation while positive values indicate underestimation. The family-specific MAPE shows the average per cent deviation of the predicted values from the actual values of the limb bone variables for a particular family when estimated using the regression equations. The combined proboscidean equations were developed using all four families, and thus, they can be used to predict the limb bone variables of all the constituent clades. The utility of the family-specific equations was tested only on the respective families sampled to generate those equations. Body mass estimation We used the predictive equations for the total length and minimum circumference developed by Christiansen (2004) to estimate BM (Table 1). Since we could not directly assess the relationship between BM and OCB, we estimated BM from the limb bone variables predicted from OCB, and BM from the actual limb bone measurements of the specimens sampled. BMs were estimated for specimens in every family using the limb bone measurements predicted using the combined proboscidean equations, and when possible, using the family-specific equations. The difference in BM estimation (i.e. difference in BM estimated from predicted limb variables vs. actual limb variables) was measured using the equation %BMdiff=|(BMobserved−BMpredictedBMpredicted)× 100|,  that is, the degree of mismatch between the estimate from actual and predicted limb measurements. The differences were compared among the four variables for each family. The %BMdiff measures the difference in estimated BM caused by the PE of the regression equations we developed. Table 1. Predictive equations for body mass of extant elephants from Christiansen (2004) Variable (X)  α ± 95% CI  β ± 95% CI  MAPE  %SEE  R2  HL  −4.14 ± 1.25  2.64 ± 0.43  6.74  11.52  0.98  HMC  −1.60 ± 0.58  2.06 ± 0.23  5.54  7.78  0.99  FL  −5.57 ± 1.84  3.04 ± 0.61  6.15  14.54  0.97  FMC  −1.61 ± 1.30  2.07 ± 0.52  11.52  18.46  0.95  Variable (X)  α ± 95% CI  β ± 95% CI  MAPE  %SEE  R2  HL  −4.14 ± 1.25  2.64 ± 0.43  6.74  11.52  0.98  HMC  −1.60 ± 0.58  2.06 ± 0.23  5.54  7.78  0.99  FL  −5.57 ± 1.84  3.04 ± 0.61  6.15  14.54  0.97  FMC  −1.61 ± 1.30  2.07 ± 0.52  11.52  18.46  0.95  Equations are in the form log10Body mass = α + β(log10X). The variable X is the independent variable. α, intercept; β slope; CI, confidence interval; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; MAPE, mean absolute per cent prediction error; %SEE, per cent standard error of the estimate. View Large Table 1. Predictive equations for body mass of extant elephants from Christiansen (2004) Variable (X)  α ± 95% CI  β ± 95% CI  MAPE  %SEE  R2  HL  −4.14 ± 1.25  2.64 ± 0.43  6.74  11.52  0.98  HMC  −1.60 ± 0.58  2.06 ± 0.23  5.54  7.78  0.99  FL  −5.57 ± 1.84  3.04 ± 0.61  6.15  14.54  0.97  FMC  −1.61 ± 1.30  2.07 ± 0.52  11.52  18.46  0.95  Variable (X)  α ± 95% CI  β ± 95% CI  MAPE  %SEE  R2  HL  −4.14 ± 1.25  2.64 ± 0.43  6.74  11.52  0.98  HMC  −1.60 ± 0.58  2.06 ± 0.23  5.54  7.78  0.99  FL  −5.57 ± 1.84  3.04 ± 0.61  6.15  14.54  0.97  FMC  −1.61 ± 1.30  2.07 ± 0.52  11.52  18.46  0.95  Equations are in the form log10Body mass = α + β(log10X). The variable X is the independent variable. α, intercept; β slope; CI, confidence interval; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; MAPE, mean absolute per cent prediction error; %SEE, per cent standard error of the estimate. View Large We also tested our equations on two specimens of extant species in our sample for which pre-mortem BM was recorded. This test determines how well our predictive approach estimates actual BM. RESULTS Allometric equations OLS regressions were used to determine the relationship between OCB and the length and minimum circumference of the humeri and femora (Fig. 2). For the combined proboscidean sample, the four limb bone variables had moderately strong positive relationships with OCB; all R2 values ranged between 0.68 and 0.75. The minimum circumference of the humerus and the femur showed the strongest relationship, followed by the length of the femur and the length of the humerus (Table 2). The MAPE for all four variables were very similar, ranging from 7.31 to 8.45%. While the circumference variables showed the strongest correlation, the length variables were most precisely predicted (Table 2). All variables also had low %SEE (Fig. 2, Table 2). Table 2. Predictive equations for limb bone measurements from OCB Taxon  Variable (Z)  α ± 95% CI  β ± 95% CI  n  P value  R2  MAPE  %SEE  All proboscideans  HL  0.66 ± 1.05  0.99 ± 0.45  38  <0.0001***  0.68  8.21  12.09  FL  1.10 ± 0.90  0.83 ± 0.38  34  <0.0001***  0.70  7.31  9.48  HMC  −0.22 ± 1.20  1.19 ± 0.52  30  <0.0001***  0.75  8.44  11.22  FMC  −0.09 ± 1.32  1.12 ± 0.56  26  <0.0001***  0.73  8.45  12.18  Elephantidae  HL  0.61 ± 0.84  1.01 ± 0.36  30  <0.0001***  0.82  6.83  8.78  FL  1.16 ± 0.82  0.81 ± 0.35  25  <0.0001***  0.79  5.56  7.59  HMC  0.15 ± 1.39  1.03 ± 0.60  23  <0.0001***  0.69  8.10  11.28  FMC  0.17 ± 1.49  1.01 ± 0.64  20  <0.0001***  0.69  8.17  12.24  Gomphotheriidae  HL  −1.02 ± 5.75  1.67 ± 2.44  6  0.019*  0.78  7.24  10.39  Taxon  Variable (Z)  α ± 95% CI  β ± 95% CI  n  P value  R2  MAPE  %SEE  All proboscideans  HL  0.66 ± 1.05  0.99 ± 0.45  38  <0.0001***  0.68  8.21  12.09  FL  1.10 ± 0.90  0.83 ± 0.38  34  <0.0001***  0.70  7.31  9.48  HMC  −0.22 ± 1.20  1.19 ± 0.52  30  <0.0001***  0.75  8.44  11.22  FMC  −0.09 ± 1.32  1.12 ± 0.56  26  <0.0001***  0.73  8.45  12.18  Elephantidae  HL  0.61 ± 0.84  1.01 ± 0.36  30  <0.0001***  0.82  6.83  8.78  FL  1.16 ± 0.82  0.81 ± 0.35  25  <0.0001***  0.79  5.56  7.59  HMC  0.15 ± 1.39  1.03 ± 0.60  23  <0.0001***  0.69  8.10  11.28  FMC  0.17 ± 1.49  1.01 ± 0.64  20  <0.0001***  0.69  8.17  12.24  Gomphotheriidae  HL  −1.02 ± 5.75  1.67 ± 2.44  6  0.019*  0.78  7.24  10.39  Equations are in the form log10Z= α + β(log10OCW). The variable Z is the dependent variable. α, intercept; β slope; CI, confidence interval; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; MAPE, mean absolute per cent prediction error; n, sample size; OCB, occipital condyle breadth; %SEE, per cent standard error of the estimate. *Significant at alpha of 0.05. ***Significant at alpha of 0.001. View Large Table 2. Predictive equations for limb bone measurements from OCB Taxon  Variable (Z)  α ± 95% CI  β ± 95% CI  n  P value  R2  MAPE  %SEE  All proboscideans  HL  0.66 ± 1.05  0.99 ± 0.45  38  <0.0001***  0.68  8.21  12.09  FL  1.10 ± 0.90  0.83 ± 0.38  34  <0.0001***  0.70  7.31  9.48  HMC  −0.22 ± 1.20  1.19 ± 0.52  30  <0.0001***  0.75  8.44  11.22  FMC  −0.09 ± 1.32  1.12 ± 0.56  26  <0.0001***  0.73  8.45  12.18  Elephantidae  HL  0.61 ± 0.84  1.01 ± 0.36  30  <0.0001***  0.82  6.83  8.78  FL  1.16 ± 0.82  0.81 ± 0.35  25  <0.0001***  0.79  5.56  7.59  HMC  0.15 ± 1.39  1.03 ± 0.60  23  <0.0001***  0.69  8.10  11.28  FMC  0.17 ± 1.49  1.01 ± 0.64  20  <0.0001***  0.69  8.17  12.24  Gomphotheriidae  HL  −1.02 ± 5.75  1.67 ± 2.44  6  0.019*  0.78  7.24  10.39  Taxon  Variable (Z)  α ± 95% CI  β ± 95% CI  n  P value  R2  MAPE  %SEE  All proboscideans  HL  0.66 ± 1.05  0.99 ± 0.45  38  <0.0001***  0.68  8.21  12.09  FL  1.10 ± 0.90  0.83 ± 0.38  34  <0.0001***  0.70  7.31  9.48  HMC  −0.22 ± 1.20  1.19 ± 0.52  30  <0.0001***  0.75  8.44  11.22  FMC  −0.09 ± 1.32  1.12 ± 0.56  26  <0.0001***  0.73  8.45  12.18  Elephantidae  HL  0.61 ± 0.84  1.01 ± 0.36  30  <0.0001***  0.82  6.83  8.78  FL  1.16 ± 0.82  0.81 ± 0.35  25  <0.0001***  0.79  5.56  7.59  HMC  0.15 ± 1.39  1.03 ± 0.60  23  <0.0001***  0.69  8.10  11.28  FMC  0.17 ± 1.49  1.01 ± 0.64  20  <0.0001***  0.69  8.17  12.24  Gomphotheriidae  HL  −1.02 ± 5.75  1.67 ± 2.44  6  0.019*  0.78  7.24  10.39  Equations are in the form log10Z= α + β(log10OCW). The variable Z is the dependent variable. α, intercept; β slope; CI, confidence interval; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; MAPE, mean absolute per cent prediction error; n, sample size; OCB, occipital condyle breadth; %SEE, per cent standard error of the estimate. *Significant at alpha of 0.05. ***Significant at alpha of 0.001. View Large Figure 2. View largeDownload slide Ordinary least square regressions between log10 occipital condyle breadth (OCB) and log10 limb bone variables for the combined proboscidean sample. A, relationship with total humerus length (HL). B, relationship with total femur length (FL). C, relationship with minimum circumference of the humerus (HMC). D, relationship with minimum circumference of the femur (FMC). Regression equations are shown in the format y = α + βx, where α is the intercept and β is the regression coefficient. The coefficient of determination (R2), per cent standard error of the estimate (%SEE) and mean absolute per cent predictive error (MAPE) are shown along with the equations. Blue circles represent the Elephantidae, orange triangles represent the Gomphotheriidae, green squares represent the Mammutidae and purple diamonds represent the Stegodontidae. Solid black lines are the lines of best fit. Dashed grey lines represent 95% confidence intervals (CIs). Dotted lines represent 95% prediction intervals (PIs). Figure 2. View largeDownload slide Ordinary least square regressions between log10 occipital condyle breadth (OCB) and log10 limb bone variables for the combined proboscidean sample. A, relationship with total humerus length (HL). B, relationship with total femur length (FL). C, relationship with minimum circumference of the humerus (HMC). D, relationship with minimum circumference of the femur (FMC). Regression equations are shown in the format y = α + βx, where α is the intercept and β is the regression coefficient. The coefficient of determination (R2), per cent standard error of the estimate (%SEE) and mean absolute per cent predictive error (MAPE) are shown along with the equations. Blue circles represent the Elephantidae, orange triangles represent the Gomphotheriidae, green squares represent the Mammutidae and purple diamonds represent the Stegodontidae. Solid black lines are the lines of best fit. Dashed grey lines represent 95% confidence intervals (CIs). Dotted lines represent 95% prediction intervals (PIs). The utility of the combined proboscidean equations was tested using MAPE and raw %PE specific for each family (Table 3 and Fig. 4, respectively). For the Elephantidae, total length measurements tended to be underestimated, while predicted minimum circumference variables tended to be overestimated (Fig. 4). On average, OCB most precisely predicted the total lengths of the femur and humerus (Table 3). For the Gomphotheriidae, the total length variables tended to be overestimated, while minimum circumference variables were generally underestimated (Fig. 4). OCB predicted the minimum circumference of the femur and humerus, and total length of the femur with more precision (Table 3). The Mammutidae showed results similar to the Gomphotheriidae, with the minimum circumferences of the femur and humerus being predicted with better precision (Table 3). The small sample of mastodons did not allow us to discern general trends in over or underestimation. Our equations overestimated the total length and the minimum circumference of the humerus of the one specimen of Mammut americanum for which this bone was measured (Fig. 4). For the two M. americanum specimens for which the femur was measured, the minimum circumference of one specimen was overestimated, while that for the other was underestimated; the total lengths of the femur for both specimens were overestimated (Fig. 4). Only the total femur length of Mammut borsoni was underestimated using these equations (Fig. 4). The Stegodontidae were similar to the Elephantidae, with OCB best predicting the total lengths of the humerus and the femur (Table 4). All limb bone variables for this specimen except for total humerus length were underestimated (Fig. 4). Table 3. Family-specific mean %PE and %BM differences Predictive equation  Family  Variable  n  MAPE (±SD)  %BMdiff (mean ± SD)  All proboscideans  Elephantidae  FL  23  6.56 ± 3.87  20.85 ± 13.15  FMC  18  8.33 ± 7.78  17.52 ± 17.58  HL  28  6.73 ± 4.55  18.19 ± 12.81  HMC  21  8.54 ± 6.63  17.82 ± 18.55  Gomphotheriidae  FL  4  6.79 ± 6.94  18.14 ± 17.95  FMC  3  4.61 ± 2.09  9.81 ± 4.56  HL  6  16.21 ± 8.89  36.05 ± 16.86  HMC  5  6.85 ± 4.28  14.72 ± 9.49  Mammutidae  FL  3  14.24 ± 4.19  41.03 ± 11.44  FMC  2  7.31 ± 9.00  16.17 ± 20.10  HL  1  12.58  29.88  HMC  1  2.59  5.27  Stegodontidae  FL  1  7.94  26.17  FMC  1  25.13  59.06  HL  1  0.02  0.06  HMC  1  19.88  45.28  Elephantidae  Elephantidae  FL  23  5.56 ± 4.10  16.64 ± 11.85  FMC  18  8.17 ± 7.22  16.85 ± 15.36  HL  28  6.83 ± 4.79  18.68 ± 13.83  HMC  21  8.10 ± 6.36  16.78 ± 13.50  Gomphotheriidae  Gomphotheriidae  HL  6  7.24 ± 4.22  19.32 ± 12.71  Predictive equation  Family  Variable  n  MAPE (±SD)  %BMdiff (mean ± SD)  All proboscideans  Elephantidae  FL  23  6.56 ± 3.87  20.85 ± 13.15  FMC  18  8.33 ± 7.78  17.52 ± 17.58  HL  28  6.73 ± 4.55  18.19 ± 12.81  HMC  21  8.54 ± 6.63  17.82 ± 18.55  Gomphotheriidae  FL  4  6.79 ± 6.94  18.14 ± 17.95  FMC  3  4.61 ± 2.09  9.81 ± 4.56  HL  6  16.21 ± 8.89  36.05 ± 16.86  HMC  5  6.85 ± 4.28  14.72 ± 9.49  Mammutidae  FL  3  14.24 ± 4.19  41.03 ± 11.44  FMC  2  7.31 ± 9.00  16.17 ± 20.10  HL  1  12.58  29.88  HMC  1  2.59  5.27  Stegodontidae  FL  1  7.94  26.17  FMC  1  25.13  59.06  HL  1  0.02  0.06  HMC  1  19.88  45.28  Elephantidae  Elephantidae  FL  23  5.56 ± 4.10  16.64 ± 11.85  FMC  18  8.17 ± 7.22  16.85 ± 15.36  HL  28  6.83 ± 4.79  18.68 ± 13.83  HMC  21  8.10 ± 6.36  16.78 ± 13.50  Gomphotheriidae  Gomphotheriidae  HL  6  7.24 ± 4.22  19.32 ± 12.71  %BMdiff, absolute per cent difference in body mass estimated from actual limb measurements and predicted limb variables; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; MAPE, mean absolute per cent prediction error; n, sample size; %PE, per cent predictive error. View Large Table 3. Family-specific mean %PE and %BM differences Predictive equation  Family  Variable  n  MAPE (±SD)  %BMdiff (mean ± SD)  All proboscideans  Elephantidae  FL  23  6.56 ± 3.87  20.85 ± 13.15  FMC  18  8.33 ± 7.78  17.52 ± 17.58  HL  28  6.73 ± 4.55  18.19 ± 12.81  HMC  21  8.54 ± 6.63  17.82 ± 18.55  Gomphotheriidae  FL  4  6.79 ± 6.94  18.14 ± 17.95  FMC  3  4.61 ± 2.09  9.81 ± 4.56  HL  6  16.21 ± 8.89  36.05 ± 16.86  HMC  5  6.85 ± 4.28  14.72 ± 9.49  Mammutidae  FL  3  14.24 ± 4.19  41.03 ± 11.44  FMC  2  7.31 ± 9.00  16.17 ± 20.10  HL  1  12.58  29.88  HMC  1  2.59  5.27  Stegodontidae  FL  1  7.94  26.17  FMC  1  25.13  59.06  HL  1  0.02  0.06  HMC  1  19.88  45.28  Elephantidae  Elephantidae  FL  23  5.56 ± 4.10  16.64 ± 11.85  FMC  18  8.17 ± 7.22  16.85 ± 15.36  HL  28  6.83 ± 4.79  18.68 ± 13.83  HMC  21  8.10 ± 6.36  16.78 ± 13.50  Gomphotheriidae  Gomphotheriidae  HL  6  7.24 ± 4.22  19.32 ± 12.71  Predictive equation  Family  Variable  n  MAPE (±SD)  %BMdiff (mean ± SD)  All proboscideans  Elephantidae  FL  23  6.56 ± 3.87  20.85 ± 13.15  FMC  18  8.33 ± 7.78  17.52 ± 17.58  HL  28  6.73 ± 4.55  18.19 ± 12.81  HMC  21  8.54 ± 6.63  17.82 ± 18.55  Gomphotheriidae  FL  4  6.79 ± 6.94  18.14 ± 17.95  FMC  3  4.61 ± 2.09  9.81 ± 4.56  HL  6  16.21 ± 8.89  36.05 ± 16.86  HMC  5  6.85 ± 4.28  14.72 ± 9.49  Mammutidae  FL  3  14.24 ± 4.19  41.03 ± 11.44  FMC  2  7.31 ± 9.00  16.17 ± 20.10  HL  1  12.58  29.88  HMC  1  2.59  5.27  Stegodontidae  FL  1  7.94  26.17  FMC  1  25.13  59.06  HL  1  0.02  0.06  HMC  1  19.88  45.28  Elephantidae  Elephantidae  FL  23  5.56 ± 4.10  16.64 ± 11.85  FMC  18  8.17 ± 7.22  16.85 ± 15.36  HL  28  6.83 ± 4.79  18.68 ± 13.83  HMC  21  8.10 ± 6.36  16.78 ± 13.50  Gomphotheriidae  Gomphotheriidae  HL  6  7.24 ± 4.22  19.32 ± 12.71  %BMdiff, absolute per cent difference in body mass estimated from actual limb measurements and predicted limb variables; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; MAPE, mean absolute per cent prediction error; n, sample size; %PE, per cent predictive error. View Large Table 4. Raw %PE for the limb variables estimated for the two specimens and a comparison of actual body mass of with estimates from this study Species  Specimen    Actual  HL  HMC  FL  FMC  Equation  Elephas maximus  CN1399  %PE    16.34  19.26  14.43  31.63  All proboscideans  BM (kg)  3534  2486.6  2401.48  2327.17  2200.78  %Diff    42.13  47.17  51.87  60.59  %PE    17.86  15.12  10.37  26.81  Elephantidae  BM (kg)    2402.37  2582.86  2596.94  2377.2  %Diff    47.11  36.84  36.09  48.67  Loxodonta africana  ROMV R6000  %PE    −5.81  −5.16  −2.31  −0.51  All proboscideans  BM (kg)  6435  7531.77  6790.5  6783.8  5881.76  %Diff    14.56  5.14  5.24  9.4  %PE    −5.37  −2.03  −4.97  0.43  Elephantidae  BM (kg)    7441.4  6350.79  7377.51  5768.53  %Diff    13.52  1.33  12.77  11.55  Species  Specimen    Actual  HL  HMC  FL  FMC  Equation  Elephas maximus  CN1399  %PE    16.34  19.26  14.43  31.63  All proboscideans  BM (kg)  3534  2486.6  2401.48  2327.17  2200.78  %Diff    42.13  47.17  51.87  60.59  %PE    17.86  15.12  10.37  26.81  Elephantidae  BM (kg)    2402.37  2582.86  2596.94  2377.2  %Diff    47.11  36.84  36.09  48.67  Loxodonta africana  ROMV R6000  %PE    −5.81  −5.16  −2.31  −0.51  All proboscideans  BM (kg)  6435  7531.77  6790.5  6783.8  5881.76  %Diff    14.56  5.14  5.24  9.4  %PE    −5.37  −2.03  −4.97  0.43  Elephantidae  BM (kg)    7441.4  6350.79  7377.51  5768.53  %Diff    13.52  1.33  12.77  11.55  BM, body mass in kilograms; %Diff, per cent difference between actual mass and estimated mass; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; %PE, raw per cent predictive error. View Large Table 4. Raw %PE for the limb variables estimated for the two specimens and a comparison of actual body mass of with estimates from this study Species  Specimen    Actual  HL  HMC  FL  FMC  Equation  Elephas maximus  CN1399  %PE    16.34  19.26  14.43  31.63  All proboscideans  BM (kg)  3534  2486.6  2401.48  2327.17  2200.78  %Diff    42.13  47.17  51.87  60.59  %PE    17.86  15.12  10.37  26.81  Elephantidae  BM (kg)    2402.37  2582.86  2596.94  2377.2  %Diff    47.11  36.84  36.09  48.67  Loxodonta africana  ROMV R6000  %PE    −5.81  −5.16  −2.31  −0.51  All proboscideans  BM (kg)  6435  7531.77  6790.5  6783.8  5881.76  %Diff    14.56  5.14  5.24  9.4  %PE    −5.37  −2.03  −4.97  0.43  Elephantidae  BM (kg)    7441.4  6350.79  7377.51  5768.53  %Diff    13.52  1.33  12.77  11.55  Species  Specimen    Actual  HL  HMC  FL  FMC  Equation  Elephas maximus  CN1399  %PE    16.34  19.26  14.43  31.63  All proboscideans  BM (kg)  3534  2486.6  2401.48  2327.17  2200.78  %Diff    42.13  47.17  51.87  60.59  %PE    17.86  15.12  10.37  26.81  Elephantidae  BM (kg)    2402.37  2582.86  2596.94  2377.2  %Diff    47.11  36.84  36.09  48.67  Loxodonta africana  ROMV R6000  %PE    −5.81  −5.16  −2.31  −0.51  All proboscideans  BM (kg)  6435  7531.77  6790.5  6783.8  5881.76  %Diff    14.56  5.14  5.24  9.4  %PE    −5.37  −2.03  −4.97  0.43  Elephantidae  BM (kg)    7441.4  6350.79  7377.51  5768.53  %Diff    13.52  1.33  12.77  11.55  BM, body mass in kilograms; %Diff, per cent difference between actual mass and estimated mass; FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference; %PE, raw per cent predictive error. View Large Figure 3. View largeDownload slide Ordinary least square regressions between log10 occipital condyle breadth (OCB) and log10 limb bone variables for the Elephantidae-only data set (A–D) and the Gomphotheriidae-only data set (E). A, relationship with total humerus length (HL) of the Elephantidae. B, relationship with total femur length (FL) of the Elephantidae. C, relationship with minimum circumference of the humerus (HMC) of the Elephantidae. D, relationship with minimum circumference of the femur (FMC) of the Elephantidae. E, relationship of the with HL of the Gomphotheriidae. Regression equations are shown in the format y = α + βx, where α is the intercept and β is the regression coefficient. The coefficient of determination (R2), per cent standard error of the estimate (%SEE) and mean absolute per cent predictive error (MAPE) are shown along with the equations. Blue circles represent the Elephantidae, orange triangles represent the Gomphotheriidae. Solid black lines are the lines of best fit. Dashed grey lines represent 95% confidence intervals (CIs). Dotted lines represent 95% prediction intervals (PIs). Figure 3. View largeDownload slide Ordinary least square regressions between log10 occipital condyle breadth (OCB) and log10 limb bone variables for the Elephantidae-only data set (A–D) and the Gomphotheriidae-only data set (E). A, relationship with total humerus length (HL) of the Elephantidae. B, relationship with total femur length (FL) of the Elephantidae. C, relationship with minimum circumference of the humerus (HMC) of the Elephantidae. D, relationship with minimum circumference of the femur (FMC) of the Elephantidae. E, relationship of the with HL of the Gomphotheriidae. Regression equations are shown in the format y = α + βx, where α is the intercept and β is the regression coefficient. The coefficient of determination (R2), per cent standard error of the estimate (%SEE) and mean absolute per cent predictive error (MAPE) are shown along with the equations. Blue circles represent the Elephantidae, orange triangles represent the Gomphotheriidae. Solid black lines are the lines of best fit. Dashed grey lines represent 95% confidence intervals (CIs). Dotted lines represent 95% prediction intervals (PIs). Figure 4. View largeDownload slide A comparison of the raw per cent predictive errors (%PE) for the predicted limb bone variables for each family using box plots. The limb bone variables are predicted using the combined proboscidean equations. Values less than 0 are overestimated while values greater than 0 are underestimated. Single horizontal bars indicate a sample size of 1. Black bars in the box plots show the median value. Dotted horizontal line at %PE of 0 shows perfect prediction. Blue circles are outliers for the Elephantidae. Families are shown in alphabetical order in each panel. FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference. Figure 4. View largeDownload slide A comparison of the raw per cent predictive errors (%PE) for the predicted limb bone variables for each family using box plots. The limb bone variables are predicted using the combined proboscidean equations. Values less than 0 are overestimated while values greater than 0 are underestimated. Single horizontal bars indicate a sample size of 1. Black bars in the box plots show the median value. Dotted horizontal line at %PE of 0 shows perfect prediction. Blue circles are outliers for the Elephantidae. Families are shown in alphabetical order in each panel. FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference. When possible, we generated separate allometric equations for each family (Fig. 3). For equations specific to the Elephantidae, the strongest correlations were between OCB and the total lengths of the humerus and femur (Table 2). These two variables were also the most precisely predicted and had the smallest %SEE (Table 2). Overall, the correlations between OCB and the four limb variables were moderate to strong, with R2 ranging from 0.69 to 0.82 and were more precisely predicted than they were when using the combined proboscidean equations (Table 3). All four limb bone variables were roughly equally over and underestimated, with median values close to 0 (Fig. 5). For the Gomphotheriidae-specific equation, the correlation between OCB and humerus length is moderately strong, and the MAPE is smaller than that when HL is predicted using the combined proboscidean equation (Table 3). The MAPEs for humerus and femur minimum circumference, and femur total length for gomphotheres when estimated using the combined proboscidean equations, are very similar to the MAPE of humerus length estimated using the Gomphotheriidae only equation. Out of the six specimens sampled, four HLs were overestimated, while two were underestimated (Fig. 5). Figure 5. View largeDownload slide A comparison of the raw per cent predictive errors (%PE) for the predicted limb bone variables for the Elephantidae and Gomphotheriidae using box plots. The limb bone variables are predicted using the family-specific equations. Values less than 0 are overestimated while values greater than 0 are underestimated. Black bars in the box plots show the median value. Dotted horizontal line at %PE of 0 shows perfect prediction. Blue circles are outliers for the Elephantidae. For humerus length, the first box plot is for the Elephantidae and the second is for the Gomphotheriidae. FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference. Figure 5. View largeDownload slide A comparison of the raw per cent predictive errors (%PE) for the predicted limb bone variables for the Elephantidae and Gomphotheriidae using box plots. The limb bone variables are predicted using the family-specific equations. Values less than 0 are overestimated while values greater than 0 are underestimated. Black bars in the box plots show the median value. Dotted horizontal line at %PE of 0 shows perfect prediction. Blue circles are outliers for the Elephantidae. For humerus length, the first box plot is for the Elephantidae and the second is for the Gomphotheriidae. FL, total femur length; FMC, femur minimum circumference; HL, total humerus length; HMC, humerus minimum circumference. Body mass estimation We used our predicted limb variables to estimate BM using equations developed by Christiansen (2004) and compared these estimates with those derived from actual limb bone measurements. Raw values for both estimations are shown in Supporting Information, Table S2. The per cent difference in BM estimated from actual limb bones and limb variables predicted using the combined proboscidean equations varied between 17.52 ± 17.58% and 20.85 ± 13.15% for the Elephantidae; %BMdiff for each limb bone variable were only marginally different from each other (Table 3). For the Gomphotheriidae, %BMdiff varied from 9.81 ± 4.56% to 36.05 ± 16.86%. The minimum circumferences variables yielded the smallest differences between the two estimates (Table 3). The %BMdiff for the Mammutidae ranged from 5.27% to 41.03 ± 11.44%, with the minimum circumference variables having the smallest difference from the actual limb measurements in estimated BM (Table 3). For the one Stegodon sampled, the %BMdiff ranged from 0.06 to 59.06%. The BM values estimated from the predicted total length variables were most similar to those estimated from the actual limb measurements (Table 3). We used the family-specific equations to predict the limb bone variables for the Elephantidae, and Gomphotheriidae, and subsequently estimated BM using these predicted variables. BM estimates for the Elephantidae more precisely matched estimates from the actual limb measurements, and estimates were marginally better than those derived from the combined proboscidean equations (Table 3). The Gomphotheriidae-specific equation more precisely predicted humerus length, and correspondingly, the %BMdiff was much lower than when the humerus length was estimated using the combined proboscidean equation (Table 3). We compared the BM estimates calculated using our method with two specimens that had pre-mortem BM recorded as a test of our approach. Our estimates closely matched the actual BM (<15% difference) of a specimen of Loxodonta africana, but differed between 36.09 and 60.59% from the actual mass of a specimen of Elephas maximus (Table 4). DISCUSSION We have developed a method to indirectly estimate BM for extinct proboscideans using cranial remains. Allometric relationships between the breadth of the occipital condyles and the total length and minimum circumferences of the humerus and femur are used to predict these limb bone variables. In turn, these predicted variables are used to estimate BM using existing predictive equations developed by Christiansen (2004). Our results only apply to proboscidean families with columnar limbs. Older taxa such as Moeritherium, or the Numidotheriidae, have relatively shorter limbs, and studies have shown that they had a different gait as well (Court, 1994), which would suggest different allometric scaling relationships. The more derived columnar appendicular anatomy, which is well adapted to support the weight of the animal (Shoshani, 1996), is found in proboscideans like the Deinotheriidae, and more derived families such as the Elephantidae, Mammutidae, Gomphotheriidae and Stegodontidae (Tassy, 1996), and results in a unique gait (Hutchinson et al., 2003; Ren & Hutchinson, 2008). Given this similarity in appendicular anatomy, and its adaptation for weight bearing, it is probably that these families share similar allometric scaling relationships with body size. Although we did not include the Deinotheriidae because of a lack of available material, or published measurements, our equations should be applicable because of the similarity in anatomy. While the above-mentioned taxa generally show similar body proportions, family-specific differences do exist (Christiansen, 2007). Gomphothere and mastodon species tend to have shorter humeri and femora when compared to elephantids for similar values of minimum bone circumference, that is, for animals of similar size, elephantids tend to be taller than gomphotheres and mastodons (Christiansen, 2007; Haynes, 1991). Our data show similar patterns for the Elephantidae and Gomphotheriidae. For a given OCB (our proxy for skull size), both elephantids and gomphotheres tend to have similar values of humerus and femur minimum circumference (Fig. 2C, D), but several gomphothere specimens have relatively short humeri for the same OCB as the elephantids in the sample (Fig. 2A, B). These differences increase scatter for these variables, which results in the relatively low R2 values (Fig. 2). Interestingly, the total length variables were predicted with more precision than the minimum circumference variables. This discrepancy between the strength of correlation and precision of prediction is caused by differences in sample sizes of the different families sampled. The majority of the data points in the combined sample belong to the Elephantidae, which have their humerus and femur lengths more strongly correlated with and more precisely predicted by OCB (Table 3). Since there were more total length measurements overall, and since most of them belonged to the Elephantidae, the MAPE for the total length variables were lower on average than those for the minimum circumference variables. The differences in body proportions and sample size among families were the impetus behind repeating the analyses using family-specific subsets of the data set. Overall, the family-specific equations better predicted the limb bone variables than the combined proboscidean equations. Because we had insufficient sample sizes to generate equations specific to FL, FMC and HMC for the Gomphotheriidae, and for all variables for the Mammutidae and Stegodontidae, we assessed how well the combined proboscidean equations predicted the different limb bone variables for these taxa. For the Gomphotheriidae, the most precisely predicted variables were the minimum circumference of the humerus and femur and the total length of the femur. As mentioned above, the combined proboscidean equations were (1) generated using a data set that was biased towards the Elephantidae, and (2) the Elephantidae have proportionally longer limb bones than the Gomphotheriidae for a given minimum circumference, a pattern more evident in humeri than in femora (Christiansen, 2007). Therefore, the equations poorly predicted the total length of the humerus, but the minimum circumferences of the gomphotheres fell close to the regression line, which tended to fit the large number of Elephantidae data points. The results for the Mammutidae and Stegodontidae are hard to generalize because of the very small sample size. Only three mastodons are included in the sample, and one Stegodon (Supporting Information, Table S1). The specimen of M. americanum, for which total humerus length and minimum circumference measurements were available, scaled like the elephantids. Two specimens for which femur minimum circumference was measured also scaled close to the elephantids, and the actual measurements did not deviate much from the regression line. A femur length measurement was available for one specimen of the large species M. borsoni, and for the two other M. americanum specimens. This variable deviated much more from the regression line. It appears that mastodon FL has a different scaling relationship (Fig. 2B), but more data are required to confirm this. Therefore, from the available data and the MAPE for the Mammutidae, we suggest that the combined proboscidean equations that predict the minimum circumferences of the humerus and femur be used for this family. The one Stegodon specimen sampled tended to scale like the elephantids. To our knowledge, there is no comparative study that has explored the allometric relationships between the total length and minimum circumference of stegodontids and elephantids, but the combined proboscidean equations from our study produced low MAPEs for the total humerus and femur length for this specimen. The goal of this study was to estimate BM of proboscideans from cranial characters. Because our method of estimating BM relies on a predicted variable, the accuracy of the estimates will always be limited and will depend on the precisions of the limb variable estimation equations and the precision of the mass estimation equations. The mass estimation equations developed by Christiansen (2004) included a sub-adult specimen in the sample. This is potentially problematic because of changes in allometry during ontogeny (Miller et al., 2008). Christiansen (2004) also used Reduced Major Axis regressions instead of OLS to assess the relationship between BM and limb bone variables, an approach that is not appropriate for predictive models (Smith, 2009). To determine whether the analytical methodology or the addition of a sub-adult specimen has any effect on the relationship between BM and limb variables, we re-analysed Christiansen’s (2004) data with and without the sub-adult, using OLS regressions (Supporting Information, Tables S3, S4). These analyses showed no appreciable differences in the slope, intercept, %SEE or MAPE (Table 1, Supporting Information, Tables S3, S4). The strength of the correlation between the femoral variables and BM did decrease upon the removal of the sub-adult specimen (Supporting Information, Tables S3, S4). Since the measures of residual variation (%SEE and MAPE), which are more informative in determining predictive capacity (Smith, 1984), remain comparable, the published equations are likely robust estimators of BM. The true test of our two-step predictive approach to estimating BM is comparing our estimates to known values of BM. We were able to sample two specimens of known mass and compared our estimates to these values. We more precisely estimated the limb bone variables of the L. africana specimen (ROMV R6000) than the E. maximus specimen (CN 1399), which resulted in close matches to the known mass of the L. africana specimen, but only moderate matches to the known mass of the E. maximus specimen (Table 4). The moderate match can be accounted for by the fact that the limb variables were not as precisely predicted by our equations. The specimen in question has the smallest OCB in our sample and might represent an individual with a disproportionally small skull, which would result in the underestimation of the limb bone variables, and correspondingly smaller estimated BM. Alternatively, this mismatch might be a result of sexual dimorphism in Asian elephants. Female Asian elephants lack tusks and have more gracile skulls with smaller parieto-occipital bosses (Todd, 2010). This would result in a lighter skull with proportionally smaller occipital condyles for the size of the limb bones, since limb bone size is more likely to be determined by overall BM. The effect of sexual dimorphism on these allometric relationships is beyond the scope of this study, but can be tested in the future with a larger sample of elephants with known BM. Despite the degree of mismatch for this one specimen, these estimates are reasonable. The differences are minimized when the mass estimates are log10 transformed, the scale at which most studies on BM are conducted (Maurer, Brown & Rusler, 1992; Smith et al., 2010; Lyons & Smith, 2013). For the remaining sample, we could only compare mass estimates from actual limb measurements to estimates from predicted limb variables. Overall, the precision by which the limb variables were predicted dictated how well the estimated BM matched the BM estimated from actual limb measurements. Therefore, researchers using our approach are recommended to use equations that most precisely predict the limb variables. For the Elephantidae, the family-specific equations best predicted limb variables. For gomphotheres, the combined proboscidean equations more precisely predicted the minimum circumferences of the humerus and femur, and the total length of the femur, while the family-specific equation better predicted the length of the humerus. Until further specific equations are developed for the Mammutidae, we suggest that minimum circumferences be estimated to estimate BM. Lastly for the Stegodontidae, we recommend predicting total length variables to estimate BM. When available, actual limb bones are preferably to estimate BM for all taxa, but our method allows for reasonable values of BM to be estimated when only cranial remains are available for study. SUPPORTING INFORMATION Additional Supporting Information may be found in the online version of this article at the publisher's web-site: Table S1. List of specimens and measurement references. Table S2. Body mass estimated using actual limb bone measurements and predicted limb bone variables. Table S3. Ordinary least square regression equations for body mass of extant elephants modified from Christiansen (2004). Equations are in the form log10Body mass = α + β(log10X). The variable X is the independent variable. Sample size is 7. Table S4.Ordinary least square regression equations for body mass of extant elephants modified from Christiansen (2004). Equations are in the form log10Body mass = α + β(log10X). The variable X is the independent variable. The sub-adult specimen was removed to create these equations; therefore, the sample size was reduced to 6. ACKNOWLEDGEMENTS We would like to thank A. Millhouse and E. Langan for providing access to extinct and extant proboscidean specimens at the Smithsonian’s National Museum of Natural History, J. Galkin and E. Westwig for providing access to extinct and extant proboscidean specimens at the American Museum of Natural History, N. Gilmore providing access to extant elephant specimens at the Academy of Natural Sciences of Drexel University, and S. Gaikwad for providing access to a specimen of E. maximus at the Bombay Veterinary College. We also would like to thank D. K. Johansson, J. Miller, K. Seymour and A. Larramendi for providing additional proboscidean skeletal measurements, and M. Chen for translating a paper on Chinese stegodontids to English. The authors would also like to thank A. Webb and D. Fraser for assisting with R. This study was supported by the George Mason University Provost Fellowship, awarded to AMJ. Lastly, we thank three anonymous reviewers for useful suggestions that helped improve this manuscript. 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