## How Big is a Giant? The Importance of Method in Estimating Body Size of Extinct Mammals

## Abstract

Estimating body size of extinct mammals presents problems when size can be estimated only by extrapolation. I examined the influence of phylogenetic, biomechanical, and statistical assumptions on body size estimates for 2 species of fossil castorids, the Pleistocene “giant” beaver *Castoroides* and the fossorial Miocene beaver *Palaeocastor*. Prior descriptions of *Castoroides* as “black-bear sized” were greatly exaggerated; new analyses estimated body masses at 60–100 kg. Estimates for *Palaeocastor* were similar to previously published estimates (0.8–1.2 kg). Biologically realistic size estimates were based on femur length and on interspecific data covering a wide range of body mass; skull length measurements or extrapolations from an ontogenetic single-species data set resulted in excessively large body mass estimates. Body mass must be estimated with due attention to the choice of both morphological trait and reference taxa.

- allometry
- beaver
- body size
*Castor**Castoroides*- fossils
*Palaeocastor*- phylogeny
- regression

Body mass is a trait that influences nearly every aspect of mammalian function, behavior, and ecology. Because body mass per se cannot be preserved in the fossil record, methods of estimating body mass are of considerable importance for reconstructing the natural history of extinct mammals (Damuth and MacFadden 1990).

In general, body size is inferred from skeletal or fossil remains in a series of indirect steps. First, the investigator must select a living species, or species group, which can be considered a reasonable analog for the fossil animals studied. These analogs are chosen with respect to their hypothesized affinity with the extinct species; affinity may be phylogenetic, functional, or a combination of the two (Bryant and Russell 1992). Second, data obtained from the surrogate species are used to establish a quantitative relationship between one or more morphological traits and body mass. Finally, body mass of the extinct species is estimated by back-calculation from the relationship derived from the surrogate species.

This apparently straightforward approach to size estimation has several logistic problems. First, there may be no close living relatives with which to make comparisons. Second, specific morphological adaptations may be correlated with shifts in overall body size over evolutionary time; therefore, even close sister taxa may not show the same scaling relationship between size and a given trait (Janis 1990). Third, it is assumed that the morphological trait considered is a reasonable proxy for body mass. This in turn assumes that gravity is the predominant factor affecting mass–structure relationships, rather than some other selective factor unrelated to load bearing. For example, limb bones are subject to gravitational loading and show similar scaling relationships over a wide size range and across a variety of unrelated species (Alexander 1985). In contrast, skull morphology is determined primarily by food acquisition and processing requirements, design constraints unrelated to weight bearing (Janis 1990). As a result, although skull length is correlated with body size, it may not preserve similarity of size–scaling relationships across species. Finally, the choice of scaling method itself will greatly influence the mass estimates. For example, geometric (or isometric) similarity assumes that mass should scale to the cube of length, and thus all linear measurements increase in the same proportion as the overall size increase (Alexander 1985; Schmidt-Nielsen 1984). However, extrapolated estimates of body size based solely on isometric assumptions may be grossly in error. Extrapolation outside the range of predictor variables is explicitly warned against in many statistics texts (Draper and Smith 1998; Zar 1999) and may lead to nonsensical values for the trait being predicted. This seems to have been the case for previous size estimates of some of the classic extinct mammalian giants such as the Oligocene rhinocerotoid *Indicotherium* (Fortelius and Kappelman 1993).

In this paper I performed a computer-intensive assessment of various biological and regression models for estimating the body mass of 2 species of fossil beaver: the Miocene fossorial beavers *Palaeocastor* and the Pleistocene “giant” beaver *Castoroides ohioensis*. I considered 3 major aspects of the body mass estimation process—choice of morphological trait (skull length versus femur length), choice of phylogenetic reference group, and choice of regression model—and compared the resulting body mass estimates for each fossil species.

## Materials and Methods

#### Current body size estimates of extinct castorids

The only surviving modern representatives of the family Castoridae are the North American beaver *Castor canadensis* and its European counterpart *C. fiber* (Jenkins and Busher 1979). *C. canadensis* is the largest extant rodent in North America and is second in size only to the South American capybara *Hydrochaeris hydrochaeris*. Average head-and-body length of adult *C. canadensis* is approximately 80–90 cm (Jenkins and Busher 1979). Body mass of adults is generally 15–20 kg and may exceed 30 kg.

*Palaeocastor* was much smaller than the modern beaver *C. canadensis*; skeletal dimensions suggest that it was approximately the size of the modern prairie dog *Cynomys* (0.7–1.0 kg). Unlike modern beavers, *Palaeocastor* was highly fossorial and inhabited the upland grasslands; the famous “devil's corkscrews” are fossil remnants of their burrow systems (Martin and Bennett 1977).

Body mass estimation is difficult for the Pleistocene giant beaver *C. ohioensis* because the size of skeletal elements of this species is much larger than that of all extant rodents (Erickson 1962; Moore 1890, 1899; Stirton 1935). *Castoroides* was approximately twice the length of the modern beaver; Hay (1914) used simple geometric scaling relationships (mass ∝ length^{3}) to argue that *Castoroides* must have weighed at least 8 times as much as *Castor*. By assuming an adult mass of 60 lb (24 kg) for *Castor*, Stirton (1965) calculated that *Castoroides* would have weighed in excess of 480 lb (213 kg). Subsequent descriptions of *Castoroides* as “the size of a black bear” or “bear-sized” (Dawson and Krishtalka 1984; Kurtén and Anderson 1980; McLaughlin 1984) almost certainly resulted from these crude size estimates; adult black bears (*Ursus americanus*) weigh approximately 150–300 kg, and size range for all extant bear species is 50–400 kg (Nowak and Paradiso 1983).

#### Morphometrics

I examined 4 *Castoroides* specimens located at Chicago Field Museum of Natural History, the University of Wisconsin–Madison Zoological Museum (Dallman 1969), Indiana State University Geology Museum, and Joseph Moore Museum of Natural History (Earlham College, Richmond, Indiana—Moore 1890, 1893). The Earlham specimen is probably the most complete skeleton of *Castoroides* ever found (Erickson 1962); the remaining specimens were complete crania. I measured condylobasal skull lengths to the nearest 1 mm with a steel rule and carpenter's square; skull length averaged 254 mm. I measured the maximum articular distance of the left femur on the Earlham specimen (182 mm). Condylobasal lengths of 3 *Palaeocastor* crania (Chicago Field Museum of Natural History) were measured to the nearest 0.1 mm with digital calipers and averaged 74 mm.

I used 2 “standard,” or reference, groups of extant rodents for comparison with fossil castorids. The 1st group was an ontogenetic series of 76 beavers (*C. canadensis*) ranging in age from young of the year to adult. Beavers were trapped in Wood County, Wisconsin, during the winters of 1989–1991 as part of a beaver-control program. I weighed each intact carcass with a spring balance to the nearest 0.1 kg; animals ranged in mass from 5 to 35 kg. I measured the condylobasal length of each cleaned skull to the nearest 0.1 mm with digital calipers; skull lengths were 85–145 mm. I measured the right femur of 26 specimens; femur lengths were 77–120 mm.

The 2nd reference group consisted of 19 extant rodent species. Species were chosen because of their structural and presumed phylogenetic affinity to castorid rodents, their relatively large body size, or both; these included sciurognathous rodents, such as tree squirrels and marmots, and hystricognathous rodents, such as porcupine and capybara (McLaughlin 1984; Woods 1984). Muroid rodents were not included because of their extremely small size relative to the fossil taxa considered here. I measured both condylobasal skull length and femur length to the nearest 0.1 mm with digital calipers. Matched body mass data were obtained for each specimen from specimen tags. For each species, 3–10 adults were measured.

#### Statistical analyses

I estimated body mass from skull length and femur length by both ordinary least-squares regression and reduced major-axis (or geometric mean) regression (McArdle 1988). Because bone measurements were expected to have a smaller random and measurement error component than body mass, these were used as predictor (X) variables (Draper and Smith 1998). All data were log_{10}-transformed before analysis.

Although its use is controversial, reduced major-axis regression has been recommended as an alternative to ordinary least-squares regression for predictive purposes (Draper and Smith 1998; Ricker 1973, 1984). Whereas ordinary least-squares regression assumes that X is measured without error, or that the error is small relative to Y (Draper and Smith 1998), reduced major-axis regression assumes that the error is distributed between both variables (Clarke 1980; McArdle 1988; Sprent and Dolby 1980; Teissier 1948). Because the correlation between body mass and morphological traits is usually high (*r* > 0.9), parameter estimates obtained from both line-fitting techniques will be very close. However, under most circumstances, simulation investigations have shown that estimates derived from reduced major-axis regression are less biased and more stable than those obtained from least-squares regression, especially if the sample sizes are small (Draper and Smith 1998; Riggs et al. 1978). Other benefits include those of scale-invariance, robustness, and symmetry. These properties indicate, respectively, that predictions will be unaffected by changes in measurements units, the model will be effective for a wide variety of distributions, and the slope can be replaced by its reciprocal if Y and X are reversed (Draper and Smith 1998; McArdle 1988; Ricker 1973, 1984; Riggs et al. 1978).

The precision of body mass estimates is determined most effectively by confidence intervals. Confidence intervals are a method of appraising both variability in a sample and the uncertainty associated with the estimation of the mean value (Gardner and Altman 1986). The width of the confidence interval depends on the variance, the sample size, and the degree of reliability (probability) associated with the interval. Confidence intervals for intraspecific data present no particular problem in their calculation. However, because the effects of phylogenetic relationships between species must be accounted for, it is more difficult to obtain valid estimates for interspecific data. Species are a product of evolution, and species relationships differ in both hierarchy and degree of affinity; consequently, species data in interspecific comparisons are not statistically independent (Felsenstein 1985). Conversely, the underlying biological implication of species “independence” is the assumption that all species radiated simultaneously and instantly from a common ancestor. Although parameter estimates may be relatively unaffected by an incorrect assumption of independence, variance estimates will be severely biased (Garland et al. 1993).

Phylogenetic relationships of rodents used in this study (Fig. 1) were obtained from Hafner (1984), Sarich (1985), and Stirton (1935). Approximate times of divergence were estimated from fossil evidence and biochemical and molecular data (Hafner 1984; Savage and Russell 1983; Stirton 1935). The rodent fossil record is too poor to allow rigorous reconstruction of evolutionary relationships (Hartenberger 1985); the phylogeny used here is thus a composite based on available data. I constructed the phylogenetic tree used in subsequent analyses using the PDTREE module in the Phenotypic Diversity Analysis Program (PDAP, version 5.0—Garland et al. 1998).

I used 2 recently developed computational methods that explicitly incorporate phylogenetic information into subsequent statistical analyses. First, I used Monte Carlo simulations to generate empirical distributions of the predicted body mass values. This technique incorporates phylogenetic information by simulating character evolution numerous times along a known (i.e., user-specified) phylogenetic tree (Garland et al. 1993).

Because it cannot be determined with certainty whether evolution of rodents proceeded either gradually or rapidly during early radiation (Hartenberger 1985), correlated evolution of the dependent variables (skull length and femur length) with body mass was simulated in accordance with 2 models of evolutionary change. The gradual model of phenotypic change assumes that change is random with respect to direction (Felsenstein 1985); in contrast, the speciational model assumes that change occurs in conjunction with a speciation event. Branch lengths are a measure of the degree of relationship between species; thus, in the gradual model, branch lengths are proportional to the expected variance of phenotypic change, and in the speciational model all branch lengths are set equal to 1 (Garland et al. 1993; Martins and Garland 1991). Analyzing data under these 2 competing models of evolutionary change allows evaluation of the robustness of variance estimates with respect to specification of branch lengths (Reynolds and Lee 1996).

Evolutionary correlation between traits was set to 0.95, as determined from empirical species data (Garland et al. 1993; Table 1). Simulation values for each trait were defined by biologically realistic limits for mammals. Lower limits were set to 0.005 kg, 10 mm, and 8 mm, representing mass, skull length, and femur length, respectively, for the least shrew *Cryptotis parva*. The upper limit to skull length was set to 600 mm; this is the approximate skull length of the largest known rodent, the extinct heptaxodontid *Eumegamys* (Dawson and Krishtalka 1984). The upper limit to femur length was set to 400 mm and that for mass to 600 kg, as estimated for Pliocene heptaxodontid genera (Hartenberger 1985). Initial values represented the assumed ancestral size of rodents, approximately that of a “chipmunk” (Wood 1962). I estimated the ancestral size from body mass, skull length, and femur length measurements for 10 adult specimens of the eastern chipmunk, *Tamias striatus*; averages were 0.11 kg, 40 mm, and 31 mm, respectively. Simulations were performed 1,000 times on log_{10}-transformed variables, using PDAP-PDSIMUL (Garland et al. 1998).

I calculated regression statistics for each simulated data set using Turbo-Pascal 6.0 (Reynolds and Lee 1996). Body mass for each extinct species was estimated from the respective skull or femur length. A reference distribution of body mass was obtained by repeating these calculations over the entire simulated data set. Point estimates of predicted body mass were obtained from the median values of each reference distribution (Maritz 1981). The upper and lower values of the 95% confidence intervals for mass estimates were obtained from the 2.5th and the 97.5th percentile of each reference distribution (Garland et al. 1993; Maritz 1981; Noreen 1989).

An alternative method of generating prediction intervals has been proposed recently by Garland and Ives (2000). This method involves the calculation of independent contrasts for the values of a given trait; independent contrasts are calculated from the differences between sister taxa throughout the phylogenetic tree and result in statistically independent trait values. The method has been described in detail elsewhere (Felsenstein 1985; Garland et al. 1993). Prediction intervals for the unknown body mass were estimated by reformulation of the least-squares regression of standardized contrasts for body mass as a function of the standardized contrasts for skull length and femur length, respectively. Body mass was the dependent variable and was expressed as a function of the change between the current node and the ancestral node for each independent variable. The error term was calculated as the product of random error and the variance assumed under the model of Brownian motion evolution. Next, phenotypic values for the unknown species were estimated by “re-rooting” the phylogenetic tree, so that the body masses of species to be predicted (here, *Castoroides* and *Palaeocastor*) were adjacent to the basal node. This procedure resulted in a phylogenetically weighted mean estimate for the predicted value and concomitant prediction intervals (Garland and Ives 2000). Calculations were performed in PDAP version 5.0 (Garland et al. 1998).

#### Isometric scaling

Isometric similarity arguments suggest that mass should scale as a function of the cube of all linear measurements (Schmidt-Nielsen 1984); that is, the regression coefficient expected under the hypothesis of isometry should equal 3.0 (Alexander 1985; Schmidt-Nielsen 1984). For intraspecific data, I evaluated the observed coefficient with respect to the expected value by 1-sample *t*-tests (Neter et al. 1985). For interspecific data, significance of the deviation of the slope from isometry was evaluated by examination of confidence intervals. Because the expected value of the slope is 3.0 under the null hypothesis, a significant difference between observed and expected values was indicated if the confidence interval did not include 3.0 (Gardner and Altman 1986). For simulated data sets, confidence intervals were obtained from the 2.5th and the 97.5th percentile of the empirical null distribution of simulated values (Garland et al. 1993; Noreen 1989).

I evaluated the amount of bias associated with assuming isometry by the formula
D = 100[10^{λ·δ} − 1]
(1) (Prothero 1986), where D is the maximum proportional deviation of the isometric regression coefficient b_{i} from the observed values b_{o} (%), δ = b_{i} − b_{o}, λ = log(maximum X/minimum X), and X is the predictor variable (skull length and femur length).

## Results

Table 1 presents estimates of regression coefficients based on data for an ontogenetic series of modern beaver *Castor* (Fig. 2) and an interspecific set of 19 extant rodent species (Fig. 3). For the assumption of isometric scaling to hold, all estimates of the regression slope b should be 3.0. For the beaver data, scaling exponents were not significantly different from the expected value (*P* > 0.05), regardless of the trait (skull length or femur length) and regression method (ordinary least-squares or reduced major-axis regression) used. For interspecific data, exponents were consistently higher than the expected value of 3.0, regardless of the model of evolutionary change assumed. However, the isometry hypothesis was rejected only if phylogeny was not considered; exponents did not differ significantly from 3.0 when phylogeny was incorporated, regardless of the model of evolutionary change assumed (Table 1). Although this result suggests that isometric approximations might be useful for subsequent size estimations when a data-based regression equation is not available, the associated bias (estimated by D) indicated that considerable error in estimation could occur with increasing extrapolation. The ratio between maximum and minimum metrics was 1.1 for both skull length and femur length; thus, proportional deviation D could be attributed primarily to the difference between the observed and the expected exponents. D varied from 1% to 8% overall and between 2% and 4% for most interspecific comparisons (Table 1).

#### Body mass estimates for Castoroides

Both skull length (Fig. 2a) and femur length (Fig. 2b) of *Castoroides* were considerably larger than the largest skull and femur measured in the *Castor* series, although within range of the largest extant rodent in the interspecific data set, the capybara *H. hydrochaeris* (Figs. 3a and 3b). When size estimates were extrapolated from *Castor*, the predicted mean body mass for *Castoroides* was approximately 200 kg, similar to Stirton's (1965) original estimate of 213 kg. However, estimates were reduced by almost half (98–150 kg) when derived from interspecific data (Table 2). Body masses estimated from skull length were almost twice as great as those obtained from femur length; body mass estimates based on skull length as a predictor were 100–200 kg but only 45–80 kg when based on femur length. Reduced major-axis regression resulted in body mass estimates that were approximately 20 kg higher than those obtained from least-squares regression.

Monte Carlo simulations on interspecific data gave almost identical results for body mass estimates, regardless of the model of evolutionary change assumed. Point estimates for body size were identical for phylogenetic and nonphylogenetic models. Confidence intervals derived from phylogenetically corrected models overlapped and were at least an order of magnitude wider than those obtained from standard regression methods.

#### Body mass estimates for Palaeocastor

Body mass estimates for *Palaeocastor* were approximately 4 kg when extrapolated from the mass–skull regression for *Castor* (Fig. 2a; Table 2) and 1.0–1.2 kg when obtained from regression on interspecific data (Table 2; Fig. 3). Note that the skull length of *Palaeocastor* (74 mm) was close to the median skull length (83 mm) of all rodents measured in this study (Fig. 3a). There were no differences in the magnitude of body mass estimates between regression models. Confidence intervals ranged between 0.5 and 2.0 kg, well within the size range of extant ground squirrel species (Table 2).

As mentioned previously, analysis of interspecific data by both gradual and speciational models of evolutionary change gave mass estimates for *Palaeocastor* that were very close to those obtained from nonphylogenetic regression. Point estimates for body size were identical for phylogenetic and nonphylogenetic models. Confidence intervals on body mass estimates derived from phylogenetically corrected methods overlapped those obtained from standard regression methods but did not differ substantially from each other, with the exception of those derived from the method of independent contrasts (Table 2).

## Discussion

It is clear that earlier size estimates of the Pleistocene beaver *Castoroides* were considerably exaggerated, as has been the case with certain other Pleistocene “giant” mammals (Fortelius and Kappelman 1993). Revised size estimates based on femur length suggest an average mass of 50–100 kg rather than the 200+ kg usually cited. Body mass estimates for *Palaeocastor* are less problematical because this species is comparable in size with extant rodent species; however, size estimates were also hugely inflated when extrapolated downward from the larger extant beaver *Castor*. This study emphasizes the dangers of extrapolation when extinct species differ considerably in size from extant species. Further, it illustrates that the range of mass estimates can be considerable, depending on the bias associated with the choice of trait or functional complex, the species reference group, models of phenotypic change over evolutionary time, the statistical model employed, and the degree of extrapolation required. A second problem highlighted by this study relates to the precision of such estimates; a range of estimates must be wide enough to incorporate a variety of sources of error but narrow enough to allow for useful inference. This study indicates the problems inherent in model selection when bias arising from choice of reference group is controlled, and models are statistically significant, but the resulting predictions are too imprecise to be meaningful.

Choosing the best body size estimate for an extinct species ultimately requires subjective judgement. For comparison, I performed a crude check on the reliability of mass estimates for *Castoroides* from regression equations of mass on head–body length of extant terrestrial mammals (Economos 1981; data from Eisenberg 1981:appendix 2). Assuming a head–body length for *Castoroides* of approximately 1,500 mm (Hay 1914; Moore 1890), I obtained body mass estimates of 52–60 kg. These estimates are very similar to estimates obtained from both intraspecific and interspecific regressions of mass on femur length. These results suggest that the particular trait or skeletal element chosen will have the greatest influence on subsequent size estimates.

Body mass estimates of *Castoroides* extrapolated from skull length data were double of those derived from femur length data. The discrepancy between estimates can probably be attributed to the influence of biomechanical considerations on the evolution of each character and the consequent relationship of these characters with mass. For rodents, the usual choice of metric is skull length, because of the importance of cranial characteristics in taxonomic identification and the abundance of cranial material in museum collections compared with postcranial material. Rodents are extremely generalized in overall body shape. However, because the skull is not subject to gravitational loading, the relationship of skull length to mass will not necessarily exhibit the same allometric relationships observed for the axial and appendicular skeleton. The issue is further complicated because considerable functional adaptation is associated with feeding. Although skull-based size comparisons of rodents tacitly assume the same basic skull structure, rodent groups differ considerably in the regional anatomy of the masseteric–zygomatic insertion (Eisenberg 1981; Wood 1959). More important, rodent incisors grow in a logarithmic spiral (Thompson 1942) so that the facial portion of the skull will display disproportionate elongation, and thus strong positive allometry, with increasing size. Hence, skull length is probably not the most appropriate metric for estimating the mass of rodents.

Selection of an appropriate species reference group may require the inclusion of several taxa rather than merely the most closely related species. Choice of group need not be determined only by strict relationship criteria; in fact, this may be impossible when considering taxa, like the family Castoridae, that consist of 1 or a few species only distantly related to the nearest modern relatives. Nevertheless, it is important to realize that the attributes of the species selected may determine the magnitude of subsequent size estimates. For example, mean skull length of *Palaeocastor* was significantly smaller than that of *Castor* but was well within the range of that observed for extant sciurid taxa. Femur length of *Castoroides* (Earlham specimen) was significantly larger than that of the largest adult *Castor* but not much larger than the average dimensions for capybara femora measured in this study. As a result, predicted body masses for both species were more biologically realistic when derived from interspecific data but were inappropriately large when extrapolated from data for a single (albeit more closely related) species. There is a greater scope for inference if a wide variety of taxa are sampled; a broader comparative study has the potential of capturing most of the essential features of interest to the investigator.

On statistical grounds, point estimates for body size based on interspecific data should be less biased if phylogenetic relationships were specifically incorporated into the analyses (Garland et al. 1993). However, the absurdly wide confidence intervals generated from Monte Carlo simulations compared with those from nonphylogenetic models do not suggest that these methods are any better at quantifying the uncertainty in the predictions of individual body masses. Although obtaining “reasonable” size estimates is usually the priority, quantifying the range of likely estimates is a recurrent problem in the paleontological literature, and there are few satisfactory methods for doing so (Damuth and MacFadden 1990; Fortelius and Kappelman 1993); further work on this problem is clearly indicated.

Scaling of length and diameters of limb bones may conform to isometric similarity when data are available for a number of species over a wide size range (Alexander 1985). Isometric scaling considerations suggest that body mass should scale as a function of the cube of all linear measurements; i.e., the regression exponent expected under the hypothesis of isometry should equal 3.0. Monte Carlo simulations indicated no statistically significant differences between the expected and the observed values of the slope b. These data suggest that 3.0 might be a reasonable estimate for b in the absence of empirically based estimates. However, even a relatively small bias, of the order of 3%, can lead to grossly misleading predictions if the body size to be estimated must be extrapolated much beyond the range of the database from which the regression equation is derived. In this study the bias associated with this substitution was affected primarily by the deviation of the observed exponents from the expected value. Body mass estimates for other large Pleistocene mammals have also been shown to be considerably in error when extensive extrapolation is involved and when allometric coefficients differ from the values expected from isometry (Fortelius and Kappelman 1993).

These results suggest that the body mass of extinct mammals must be estimated with close attention to the choice of both morphological trait and species reference group. These 2 factors must be a reasonable surrogate for both body mass in general and the extinct species for which mass is to be estimated. These factors become especially critical when body size estimates must be extrapolated from the existing data. Statistical considerations, such as choice of regression model, may be less important if all that is required is a point estimate of body mass. In general, phylogeny must be accounted for if confidence intervals are required. Nevertheless, such estimates may lack precision to the extent that only careful consideration of the underlying biology will result in feasible size range estimates.

## Acknowledgments

I thank J. Dallman (University of Wisconsin), W. Simpson, B. Patterson (Chicago Field Museum of Natural History), J. Cope (Earlham College), and R. C. Howe (Indiana State University) for allowing me to examine the specimens in their care. I am indebted to Wildlife Manager M. Zechmeister, Sandhill Wildlife Demonstration Area, for allowing me to study the beavers on the property and to all the trappers who participated in the study. I am especially grateful to T. Garland, Jr. for supplying PDA programs and patiently answering many questions on their use. I also thank 2 anonymous referees for helpful comments and suggestions on the manuscript.

- American Society of Mammalogists