## ABSTRACT

The relationship between basal metabolism *P* and body mass *M* of 391 mammalian species has been analysed by least-squares regression, robust regression and covariance analyses. This relationship is a power function:

*b*is 0.678±0.007 (mean±S.D.) and the mass coefficient

*a*takes different values. Theory of measurement revealed that the 2/3 mass exponent is due to an underlying dimensional relationship between the primary quantity of mass and the secondary quantity of power.

This paper shows that the 2/3 mass exponent is not the physiological problem of interest. It is not the slope of the metabolic regression line, but its location in the mass/power plane, that must be explained. This location is given by the value of the mass coefficient, the explanation of which is, and remains, the central question in comparative physiology.

## INTRODUCTION

The relationship between body size and energy metabolism in animals has been studied for more than 150 years. In recent years there has been a renewed interest in this problem (Calder, 1984; Economos, 1983; Heusner, 1982a,*b*, 1984, 1985, 1987; Jürgens, 1989; McMahon and Bonner, 1983; Peters, 1983; Schmidt-Nielsen, 1984; Wieser, 1984).

*M*and basal metabolism

*P*are related by a power function: where

*a*is the mass coefficient and

*b*is the mass exponent.

Speculations about mechanisms of heat loss in mammals (Sarrus and Rameaux, 1838-1839; Rubner, 1883) or biological similitude (Economos, 1983; Gunther, 1975; Lambert and Teissier, 1927) predicted a mass exponent of 2/3, while statistical analyses actually revealed significantly larger mass exponents of 0.73–0.75 (Brody, 1945; Kleiber, 1961). In, 1963, the Third Symposium on Energy Metabolism (Kleiber, 1965) adopted the 0.75 mass exponent for standardizing basal metabolism. A few theoretical interpretations of the 0.75 mass exponent have been proposed (McMahon, 1973; Blum, 1977). However, none has proved to be a satisfactory explanation (Calder, 1987; Heusner, 1987; Speakman, 1990).

In 1982, the validity of 0.75 mass exponent was challenged. Heusner (1982*a*,*b*) questioned the meaning of the interspecific regression line between the logarithms of body mass and basal metabolism by showing that the average values of basal metabolism in seven mammalian species do not lie on a single regression line. Bartels (1982) reported a 0.66 mass exponent for mammals whose body mass ranged from 2.5 to 380×10^{6}g. A re-analysis of Bartels’ data by Jürgens (1989) revealed a mass exponent *b* of 0.678±0.017 (*N*=81). These studies also indicated that within different mass ranges *b* took significantly different values. For example, in small mammals *b* was significantly lower than in large mammals (*M*<20000g: *b*=0.602±0.031, *N*=67; *M*>20000 g: 6=0.772±0.033, A=14; *t=*5.15, *P*<0.00l). Bartels and Heusner’s results suggested that the metabolic data for mammals may not lie on a single linear regression line.

Hayssen and Lacy (1985) concluded that a linear regression line did not accurately describe the relationship between the logarithms of body mass and basal metabolism in 293 mammalian species. Their overall exponent 6=0.693 ±0.010 was significantly different from 0.75 (*t*=5.7, d.f.=291, *P*<0.00l). A re-analysis of their data by Jürgens (1989) has corroborated his earlier findings in Bartels’ data, i.e. in small mammals the mass exponent (*b*=0.648±0.012, *N*=275) tended to be lower than in large mammals (*b*=0.862±0.167, *N*=18).

Metabolic data published by Elgar and Harvey (1987) revealed the same trends: the overall mass exponent was significantly different from 0.75 (*b*=0.710±0.011, *t*=3.64, d.f.=263, *P*<0.00l), and in small mammals the mass exponent was lower than in large mammals (*b*=0.648 ±0.014, *N*=234 *versus b*=0.837 ± 0.088, *N*=31).

A study by McNab (1988) confirmed that in 320 mammalian species the overall mass exponent is significantly different from 0.75 (*b*=0.714±0.009; *r*=4.25, d.f. =318, *P*<0.001). But these data did not show a significant difference between the mass exponent in small and large mammals (*b*=0.683 ±0.009, *N*=297 for small mammals; *b*=0.601±0.172, *N*=23 for large mammals).

What general conclusions can be drawn from these four studies? In each sample, small mammals represented 83–94% of the data points and in these animals *b* was not significantly different from 2/3. Except in McNab’s study, *b* tended to be greater in large mammals. The inclusion of a relatively small number of large mammals was sufficient to make the overall value of *b* significantly greater than 2/3. These observations can be statistically explained if we assume that large and small mammals lie on different, but parallel, regression lines with slopes of 2/3, the line for large mammals being above that for small ones. Adding a few large mammals to the small ones would then cause the sample estimate of *b* to be greater than 2/3. Also, depending on the width and location of the mass range within which data points are sampled, different values of *b* would be observed.

The aim of this paper is to show that, indeed, small and large mammals he on parallel regression lines with slopes of 2/3. The physical implications of this statistical model will be discussed.

## THE DATA SAMPLE

Metabolic data for 391 mammalian species from 17 orders (Artiodactyla, Carnivora, Chiroptera, Edentata, Hyracoidea, Insectivora, Lagomorpha, Mac-roscelidea, Marsupiala, Monotremata, Perissodactyla, Pholidota, Primates, Pro-boscidea, Rodentia, Scandentia and Sirenia) have been compiled from their original sources. I verified that body mass and energy metabolism were measured on the same animals and that the data were valid estimates of basal metabolism. The data have been standardized with respect to the units. Basal metabolism is expressed in Watts (W) and body mass in g (see Appendix). On the graphs, body mass is expressed in kg, to reduce the number of digits on the mass scale.

The coordinates of each data point were the averages of measurements from the same laboratory and made on individuals of the same mammalian species. When data for the same species, but from different sources, were available, the data for which the experimental conditions most closely approximated those of basal metabolism were selected. Each mammalian species was used only once in the statistical analysis, irrespective of the actual number of individuals from which the data point was derived.

Fig. 1 shows the bilogarithmic scatter diagram of the 391 data points. Each data point represents one species.

The dashed lines intersecting the x-axis represent the lower hinge, the median and the upper hinge of the box plot of the logarithm of body mass (Chambers *et al*. 1983). These lines divide the data sample into four equally sized groups or quartiles: 25% of the mammals in this sample are smaller than 41.4 g, 50% are smaUer than 157.8 g, 75% are smaller than 1581 g. Within the mass range from 2.5 to 3672000g there is an overwhelming predominance of small mammals: only 29 mammals or 7.4% are larger than 20000 g.

## LEAST-SQUARES REGRESSION ANALYSIS

Least-squares regression between the logarithms of body mass and basal metabolism yielded the following statistics for the 391 mammals:

The regression was very significant: Snedecor’s ratio of variances was *F*=7808, d.f. = 1, 389, *P*<0.001; the coefficient of determination was *r*^{2}=0.952. The mass exponent *b* was significantly different from 0.75 (*r*=5.05, d.f. =389, *P*<0.001) and from 2/3 (*t*=5.37, d.f. =389, *P*<0.001).

In 362 small mammals (*M*<20000g) the mass exponent (*b*=0.663±0.009) was significantly lower than in 29 large mammals (*M*>20000 g: *b*=0.794± 0.094; *t*=2.05, *P*<0.05). Clearly, a single rectilinear regression line does not accurately describe these data.

## ROBUST REGRESSION ANALYSIS

Subdividing the range of body mass according to arbitrary mass limits is too subjective for analyzing the relationship between body mass and basal metabolism. An objective approach is to determine how many data points within the total mass range can be accurately described by one regression line. Statistically, this amounts to the identification and regrouping of data points (outliers) whosa probability of falling about the same regression line is very low (*P*⩽0.006). The determination of outliers was performed by robust regression analysis (Rousseeuw and Leroy, 1987).

This technique, based on the least median of squares line (LMS line), partitioned the data sample into two groups. Group 1 consisted of 363 mammals (92.8%) within a mass range from 2.5 to 407 000g. Their data points were randomly distributed about the LMS line with a slope of *b*=0.665 (unfortunately there was no method for estimating the error on the slope). The coefficient of determination was 0.959 and the scale estimate (or standard error of estimate) was 0.144. Group 2 consisted of 28 outliers (7.2% of the sample), 25 positive outliers (positive standardized residuals) and 3 negative outliers (negative standardized residuals). Since the leverage (Kleinbaum *et al*. 1988) of these negative outliers was small, they have been added to group 1. The positive outliers belonged to the following mammalian orders: Artiodactyla, 10 species (66%); Carnivora, 7 species (22%); Insectivora, 2 species (8%); Proboscidea, 1 species (100%) and Rodentia, 5 species (3%). The negative outliers belonged to the following orders: Chiroptera, 1 species (3%) and Insectivora, 2 species (8%).

Fig. 2 shows the data and regression lines for both groups. Least-squares regression analysis of group 1 yielded the following results: mass exponent *b*=0.677± 0.008, coefficient of determination *r*^{2}=0.952, standard error of estimate *s*_{y|x}=0.146, *F*=7203, d.f. = l, 364. Least-squares regression analysis of group 2 yielded the following results: *b*=0.679± 0.009, *r*^{2}=0.996, *s*_{y|x}=0.077, *F*=5948, d.f. = 1, 23, *P*<0.00l. In both groups, the regression was very significant; the mass exponents were not different and did not differ from 2/3.

Covariance analysis of these two groups showed that the intragroup slope *b* was 0.678±0.007 and the intragroup correlation coefficient *r*_{1} was 0.980. The difference in elevation of the two regression lines was very significant: log(*a*_{2}) = −1.623±0.020, *N*=366 and log(*a*_{2}) = *#x2212;1.207±0.037, *N=*25, *F*=182, d.f. = l, 388, *P*<0.00l.

In summary, the relationship between basal metabolism and body mass of 391 mammalian species can be accurately described by two regression lines with a common slope of 2/3 and different intercepts.

## THEORETICAL INTERPRETATION OF THE 2/3 METABOLIC MASS EXPONENT

The foregoing statistical analysis has shown that a power function (equation 1) with a mass exponent of 2/3 is a valid statistical model for describing the interspecific relationship between body mass and basal metabolism in mammals. This analysis has also shown that, contrary to common belief, the mass coefficient is not constant. Is this power function also a valid physical model – i.e. can a physical or physiological meaning be given to its parameters?

The power function or the so-called allometric equation has commonly been interpreted as revealing some kind of similitude among mammals (Gunther, 1975). In particular, the 2/3 mass exponent has been considered to be a criterion of biological similitude (Lambert and Teissier, 1927) and the 3/4 mass exponent a criterion of elastic similarity (McMahon, 1973). But comparative morphology clearly shows that small and large mammals are not geometrically similar. Furthermore, mammals and birds may fall on the same regression line with a slope of 2/3. These observations call for a different explanation of the 2/3 mass exponent.

The interspecific metabolic mass exponent of 2/3, instead of revealing biological similitude among mammals, may very well reflect an underlying dimensional relationship between the physical quantities of mass and power. The following example from geometry illustrates this idea.

*L*, surface area.S’, and volume

*V*. These three quantities are dimensionally related: therefore, The dimensional relationship (equation 4) is numerically implemented in any given solid by relating the measures of surface area 5 and volume v by means of the volume coefficient

*c*. When 5 and

*v*are expressed in units derived from the same unit of length (cm→cm

^{2}, cm→cm

^{3}), then

*c*is determined by the form of the solid (

*c*=6 in cubes and 4.84 in spheres). However, except for spheres,

*c*is not form-specific, i.e. different forms may have the same value of

*c*. Note that, depending on the values of

*c*, the dimensional relationship (equation 4) can be numerically implemented in many different ways.

Equation 5 also expresses the relationship between volume and surface area in geometrically similar solids of different size, where changes in *s* and *v* are due only to changes in size. Equation 5 is not valid in solids that are not geometrically similar, where changes in s are also due to changes in form (c). The relationship between *v* and *S* is still a power function, but the underlying dimensional 2/3 exponent is masked by the relationship between size and form. Now, the values of the volume exponent and volume coefficient are determined by the distribution of form within the volume range. In general, the volume exponent is different from 2/3, with one exception – when form is randomly distributed within the volume range. In this case, the 2/3 volume exponent does not reveal geometric similitude of the solids but the underlying dimensional relationship between the quantities of surface area and volume. The corollary of this is that the 2/3 volume exponent is a necessary, but not sufficient, condition for geometric similitude.

This example shows that we must distinguish the dimensional relationship between quantities (equation 4) from the numerical relationship between the measures of these quantities (equation 5). We shall now apply these same principles to find the underlying dimensional relationship between the primary quantity of mass and the secondary quantity of power in the metabolic power function.

Primary quantities, such as mass *M*, length *L* and time *T*, are directly measurable. They have well-defined standards of measurement from which their units are derived. A secondary quantity, such as velocity (*L T*^{−1}), is a combination of the arbitrarily chosen primary quantities and has no standard.

*M*, length

*L*and time

*T*as primary quantities, the dimensional equation of any secondary quantity

*Q*is given by: where the exponents

*α*

_{1,}

*β*

_{1}and

*γ*

_{1}are the dimensions of mass, length and time, respectively (Bridgeman, 1931; Ellis, 1968). Since

*α*

_{1}=l,

*β*

_{1}

*=2*and

*γ*

_{1}= 3 for energy metabolism, its dimensional equation is: To establish the dimensional relationship between body mass

*M*and basal metabolism P, we define a standard mammal that embodies the standards of volume, mass and time. In real, complex forms such as mammals, length and surface areas are fractals, i.e. their measure depends on the chosen units (Mandelbrot, 1982). To avoid this complication we choose volume

*V*, which is directly measurable, as one of the primary quantities. The period of a periodic process (cardiac or respiratory cycle) in this standard mammal is taken as the standard of time. With mass

*M*, volume V and time

*T*as primary quantities, the secondary quantities are then given by: where

*α*

_{2},

*β*

_{2}and

*γ*

_{2}are the dimensions of mass, volume and time, respectively.

*m*

_{1;}

*m*

_{2}) volume (

*v*

_{1;}

*v*

_{2}) and time (

*t*

_{1;}

*t*

_{2}) hold: where

*μ, v*and

*τ*are similitude constants.

*μ, v*and

*τ*of the measures of

*M, V*and

*T*are themselves related by the following relationships. If the two standard mammals differ in size only, their density is the same. This is expressed by: Assuming that the measure of the time standard is proportional to length, then: Under these conditions, the relationship between the measures

*q*

_{1}and

*q*

_{2}of any secondary quantity is given by: Equation 18 defines a new system of units in which the dimensions of all the secondary quantities can be expressed in terms of only one primary quantity, mass. Dimensions have no intrinsic significance (Staicu, 1982). They are the result of arbitrarily chosen primary Quantities, and. therefore, deoend on the number and nature of these primary quantities. Different quantities may have the same dimensional equation in the same system of units. For example, frequency and angular velocity have the same dimension

*T*

^{−1}in the

*M, L, T*system. Flow, force and power are dimensionally equivalent ([

*M*

^{2/3}]) in the system based on mass alone.

*p*

_{1;}

*p*

_{2}) in two standard mammals is then: which can be rewritten as: where

*a*, the mass coefficient, is: Equation 20 is also the statistical model derived from regression analysis. The 2/3 mass exponent is the dimension of the secondary quantity of power in terms of the primary quantity of mass. The 2/3 mass exponent reveals the dimensional relationship between the units of mass and power, and not biological similitude in mammals. Indeed, with the large number of mammals, the values of the mass coefficient

*a*tend to be randomized within the mass range. This randomization reveals the underlying, dimensional 2/3 mass exponent.

Physiologically, *a* or the ratio *P/M*^{2/3} is the energy spent per unit mass and per unit of the defined time scale: *a* is the mass-specific physiological power (Heusner, 1982*b*). The ratio *P/M*^{2/3} is a mass-independent measure of basal metabolism (MIM, Heusner, 1985). Changes in this ratio measure the effect of metabolic factors other than mass. Animals with the same MIM lie on the same regression line. At the same body mass, these animals would have the same basal metabolism. Conversely, animals in which the MIM is different cannot have the same basal metabolism at the same body mass.

The system of units based solely on mass shows that the so-called allometric relationships are necessary consequences of the underlying dimensional relationships between mass and the various quantities that are measured in an organism and do not reveal special properties such as geometric, biological or elastic similarity in these organisms.

## CONCLUSION

Theory of measurement reveals a necessary underlying dimensional relationship between mass and power, and statistical analysis confirms the existence of this relationship in mammals. Statistics and the theory of measurement are both concerned with describing relationships accurately, not with providing physical explanations. An accurate description is a first and necessary step towards a physical explanation of a relationship. This paper shows that the mass exponent is not the physiological problem of interest. It is not the slope of the metabolic regression line, but its location in the mass/power plane that must be explained. This location is given by the value of the mass coefficient, the explanation of which is, and remains, the central question in comparative physiology.

I gratefully acknowledge the criticisms and suggestions offered by Dr M. L. Heusner.

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*(Saimin sciureus)*

*J. appl. Physiol*

*Physiol. Zool*

*Am. J. Physiol*

*Am. J. Physiol*

*Comp. Biochem. Physiol*

*Am. J. Physiol*

*Acta theriol*

*Anns physiol, physicochim. Biol*

*Perognathus californiens*

*J. cell. comp. Physiol*

*Thomomys bottae*

*Physiol. Zool*

*Du Bois-Reymond Arch. Anat. Physiol*

*Perognathus hispidus*

*Comp. Biochem. Physiol*

*Tamias striatus*

*Comp. Biochem. Physiol*

*(Lepus arcticus)*

*Can. J. Zool*

*Acta theriol*

*J. comp. Physiol*

*Lasiorhinus latifrons*(Owen)

*Aust. J. Zool*

*Comp. Biochem. Physiol*

*(Ptilocercus lowit)*

*J. Mammal*

*Echinosorex gymnurus*

*J. Mammal*

*(Tagulus javanicus;)*

*Comp. Biochem. Physiol*

*Respir. Physiol*

*Eutamias minimum*. I. Metabolic rate

*Comp. Biochem. Physiol*

*S. Afr. J. Sci*

*J. Mammal*

*Eutamias merriami*

*Comp. Biochem. Physiol*

*Equus asinus*

*J. appl. Physiol*

*Physiol. Zool*

*(Tayassu tajacu)*

*Comp. Biochem. Physiol*