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Misconceptions about logarithmic transformation and the traditional allometric method
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Review
Misconceptions about logarithmic transformation and the traditional allometric method Gary C. Packard ∗ Department of Biology, Colorado State University, Fort Collins, CO 80523, USA
a r t i c l e
i n f o
Article history: Received 16 March 2017 Received in revised form 17 July 2017 Accepted 17 July 2017 Available online xxx Keywords: Allometry Logarithms Nonlinear regression Power laws Scaling
a b s t r a c t Logarithmic transformation is often assumed to be necessary in allometry to accommodate the kind of variation that accompanies multiplicative growth by plants and animals; and the traditional approach to allometric analysis is commonly believed to have important application even when the bivariate distribution of interest is curvilinear on the logarithmic scale. Here I examine four arguments that have been tendered in support of these perceptions. All the arguments are based on misunderstandings about the traditional method for allometric analysis and/or on a lack of familiarity with newer methods of nonlinear regression. Traditional allometry actually has limited utility because it can be used only to fit a two-parameter power equation that assumes lognormal, heteroscedastic error on the original scale. In contrast, nonlinear regression can fit two- and three-parameter power equations with differing assumptions about structure for error directly to untransformed data. Nonlinear regression should be preferred to the traditional method in future allometric analyses. © 2017 Elsevier GmbH. All rights reserved.
Contents 1. 2.
3.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 Misconceptions about bivariate allometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 2.1. Form of the allometric equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 2.2. Proportional change and relative growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 2.3. Multiplicative variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 2.4. Non-loglinear allometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
1. Introduction The early part of the 20th century was marked by widespread interest among biologists in the use of simple power functions of the form y = a ∗ xb to describe pattern in bivariate observations that follow a curvilinear path on the arithmetic (=linear) scale. The predictor variable (x) in the two-parameter equation typically was a measure of body
∗ Corresponding author. Present address: 865 Three Corner Gate Rd, Livermore, CO 80536, USA. E-mail address: [email protected]
size (e.g., body length or body mass), and the response variable (y) was some measurement taken on the structure, organ, or process of special concern. Some investigators at the time apparently fitted the equation directly to scatterplots of untransformed data by a series of trial-and-error approximations (e.g., Nomura, 1926; Kleiber, 1932), or by fitting a curve by eye and then reading from the graph (e.g., Hecht, 1913, 1916; Crozier and Hecht, 1914; Kleiber, 1932). Other workers, however, estimated the slope and intercept of a straight line drawn by hand on a graph displaying logarithmic transformations (or on a graph with logarithmic coordinates) and then took antilogs for the coefficients to obtain parameters in the power equation (e.g., Pearsall, 1927; Huxley, 1927a, b, 1932; Kunkel and Robertson, 1928). And yet a fourth group of investigators fitted straight lines to logarithmic transformations by ordinary least
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squares regression and then back-transformed the resulting equation to the arithmetic scale (e.g., Clark, 1928; Galtsoff, 1931; Brody and Proctor, 1932; Green and Green, 1932). This last approach to fitting the power function continues, for all intents and purposes, to be in general use today (Warton et al., 2006; White et al., 2012; White and Kearney, 2014) and has come to be known as the traditional allometric method (e.g., Packard, 2014). The traditional allometric method has had its critics over the years (e.g., Thompson, 1943; Smith, 1980, 1984; Lovett and Felder, 1989; Bales, 1996), and it recently has come under renewed scrutiny (e.g., Lagergren et al., 2007; Sartori and Ball, 2009; Packard, 2014, 2015, 2016, 2017a,b). Supporters of the protocol are understandably concerned that a large body of published research might be undermined if criticisms of the method were taken seriously, and they consequently have mounted a spirited defense of their research paradigm (e.g., Klingenberg, 1998; Nevill et al., 2005; Kerkhoff and Enquist, 2009; Xiao et al., 2010; White et al., 2012; Ballantyne, 2013; Glazier, 2013; Lai et al., 2013; Mascaro et al., 2014; Niklas and Hammond, 2014; Lema&tre et al., 2015). However, the defense is based in many instances on ill-defined arguments and/or misunderstanding of various statistical methods. Here I examine four of the most common misconceptions. 2. Misconceptions about bivariate allometry 2.1. Form of the allometric equation Philip Gingerich has proposed that “the allometric equation is not a power function of x and y as is so often stated, but rather a linear function of log x and log y” (Gingerich, 2000, p. 220). The suggestion was based on his study of normal vs. lognormal distributions for random error (i.e., residuals) in samples of biological data and not on an explicit analysis of allometric variation. The assertion consequently rests on something of an extrapolation, but it may, nonetheless, describe contemporary research on allometric variation fairly accurately. Logarithmic transformation was used in the 1920s and 1930s to fit simple, two-parameter power functions to observations expressed on the arithmetic scale. As time went on, however, transformation became less and less a means to fit a power function to untransformed data, and the equation of simple allometry simultaneously became more and more a justification for performing the transformation. Thus, contemporary analyses of allometric variation typically begin with a token nod to the equation of simple allometry, proceed with the rote transformation of data to logarithms, and then continue with the fitting of a straight line to the new distribution (Smith, 1980, 1984). Validation of the fitted model usually is limited to a graphical display of the equation in log domain, and R2 may also be reported as a measure of goodness of fit. However, the quality of the model seldom is assessed graphically on the original scale (Packard, 2017b), and R2 for the fit to logarithms has no bearing on strength of the relationship between the untransformed variables. Thus, it is easy to believe that the allometric equation in current practice is, in fact, log(y) = log(a) + b∗ log(x). If the allometric equation actually is a linear function of log(y) and log(x), future reports of allometric variation should be framed differently. For instance, reports of new research need not (and probably should not) begin with a reference to a two-parameter power function, because the power function is neither necessary nor relevant to the analysis; and no attempt needs to be made at interpretation on the original scale, because the original scale also is irrelevant. These recommendations are consistent with the common belief that observations in logarithmic form are at least as meaningful as untransformed values (e.g., Peters, 1983; Kerkhoff
and Enquist, 2009; Glazier, 2013; Lai et al., 2013), and the relationship of putative interest is, after all, between log(y) and log(x). It is unclear how this relationship between log(y) and log(x) is to be interpreted in biologically meaningful terms when the findings cannot be placed in the context of the original measurements (e.g., Reyment, 1971; Finney, 1989; Osborne, 2002; Feng et al., 2013). This problem of interpretation is an unavoidable consequence of accepting at face value the aforementioned premise about the allometric equation. Gingerich had an important point to make about the kind of random error that may occur in biological data, but his characterization of the allometric equation should not be taken literally. The equation of simple allometry is y = a ∗ xb , as so often stated (e.g., Huxley and Teissier, 1936; Huxley, 1950), and the linearized expression log(y) = log(a) + b∗ log(x) merely provides a way to estimate parameters in the allometric equation via an intermediate step involving logarithms (e.g., West and West, 2012). In other words, a straight line fitted to logarithms is a means to an end and not an end in itself. The linearized expression was an essential tool early in the last century, because neither statistical theory nor statistical practice was sufficiently advanced at the time to permit fitting the power function directly to untransformed observations. However, the equation now can be fitted to untransformed data by nonlinear regression, and the issue of normal vs. lognormal error can be addressed simultaneously by the computational algorithm (Packard, 2015, 2016, 2017a). 2.2. Proportional change and relative growth Several investigators − all of whom cite Huxley (1932) for their rationale − have argued that logarithmic transformation (i.e., the traditional allometric method) is necessary in allometric research because allometry is all about proportional change and relative growth (e.g., Kerkhoff and Enquist, 2009; Glazier, 2013). But what, exactly, is meant by “proportional change” and “relative growth,” and is logarithmic transformation really necessary for describing these processes? Julian Huxley’s treatise on “Problems of Relative Growth” (Huxley, 1932) focused on how some part of the body changes in its proportion to the body as a whole as both the part and body increase in size. At the time of Huxley’s writing, proportional (or relative) size of a part was commonly expressed as a percentage of size of the body, and relative growth was reflected in the changing percentage as the animals (or plants) increased in overall size (e.g., Huxley, 1924a). This expression of relative growth could be represented graphically (see Fig. 2 in Huxley, 1932), but the actual relationship between the structure of interest and body size could not be quantified accurately owing to x being a component of both the predictor and response variables (see Pearson, 1897; Atchley et al., 1976; Albrecht et al., 1993; Kronmal, 1993). Huxley’s desire to express the relationship between x and y mathematically is what led to his independent discovery of the formula for simple allometry (Reeve and Huxley, 1945), which provides a simple, yet explicit, mathematical description for the relationship between the variable of interest and body size. The two-parameter power function can be rearranged algebraically to show that the response variable, y, varies as a constant proportion of xb (e.g., Newcombe, 1948; White et al., 2012), or a = y/xb where a and b are fitted constants. Because y maintains a fixed proportional relationship to xb as both y and x increase in size,
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the proportion describes a pattern of relative growth by the bodily part of interest (i.e., the general subject of Huxley’s book). The point here, of course, is that the proportion of y to xb can readily be extracted from an equation fitted to data by nonlinear regression, so transformation is not a necessity. A proportion (or ratio) of even greater interest to Huxley, however, was the one between the instantaneous rate of growth by the part to the instantaneous rate of growth by the body (Huxley, 1924b, 1932). Assuming that the two-parameter power function is a good descriptor for pattern in the bivariate distribution of untransformed data (Huxley, 1932, p. 6), then constancy of the exponent implies that the ratio of the relative growth rates is also a constant, that is (dy/dt∗ 1/y)/(dx/dt∗ 1/x) = b (Huxley, 1924b, 1927a, 1932; Kleiber, 1932). This constant differential growth-ratio raised the possibility in Huxley’s mind that the formula of simple allometry is a general law of relative growth (Huxley, 1932, 1950; Reeve and Huxley, 1945). Of course, Huxley used the slope of a straight line fitted to logarithms to estimate the exponent in the equation of simple allometry, but the exponent also can be estimated by fitting a power function directly to untransformed observations by nonlinear regression (Packard, 2015, 2016, 2017a). Again, transformation is not a necessity. Huxley also noted that a graph of logarithmic transformations can be used to assess multiplicative growth visually (Huxley, 1927b, 1932). This property of the plot may be the basis for claims about the necessity to use logarithms to evaluate proportional change. After all, equal intervals along the fitted line represent equal amounts of multiplication, so the intervals represent identical proportional changes in y relative to x (Huxley, 1927b, 1932). Nevertheless, this graphical representation of proportionality contributes little to understanding the relationship between the variables, and it was a secondary consideration even for Huxley (Huxley, 1932, p. 11). A plot of logarithms has little import (other than to confirm linearity of the distribution) unless it is accompanied by a graph of the equation of simple allometry on the arithmetic scale − a point that Huxley failed to appreciate fully, but a point that was not lost on other investigators at the time (Clark, 1928; Galtsoff, 1931; Green and Green, 1932). Huxley used logarithms as an indirect way to fit the formula for simple allometry to untransformed data, because no direct method was available to him (see Snedecor, 1937). However, the equation of simple allometry (even one making the same assumptions about random error) now can be fitted directly to untransformed observations by nonlinear regression (e.g., Packard, 2015, 2016, 2017a). Thus, two of the three proportionalities identified here can be readily estimated by nonlinear regression. Only the third proportionality requires transformation, but the resulting bivariate plot is of limited value.
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four, four become eight, and so on as the organism and its parts increase in size (Huxley, 1924b; Katz, 1980). If multiplication of tissue proceeds more rapidly in the part than in the whole, the untransformed data commonly follow the path of a power function with an exponent greater than 1 (e.g., Green and Green, 1932). Conversely, if growth by the whole is more rapid than growth by the part, the exponent in the power function typically lies between 0 and 1. And finally, if growth by the part remains in the same proportion to growth by the whole, the data will follow the path of a straight line passing through the origin and the exponent in the power equation will be 1. The point here is that multiplicative growth sensu Huxley is described by a power function, but a power function can be fitted to untransformed data by nonlinear regression. Huxley used logarithms because no other method was available in the 1920s and 1930s to fit a power function to observations expressed on the original arithmetic scale (Snedecor, 1937). Transformation is not necessary today to describe multiplicative growth. The second version of the argument about necessity for logarithmic transformation concerns variance in the statistical model expressed on the arithmetic scale (e.g., Nevill et al., 2005; Kerkhoff and Enquist, 2009; Xiao et al., 2011; White et al., 2012; Ballantyne, 2013; Lai et al., 2013; Niklas and Hammond, 2014; Pélabon et al., 2014; White and Kearney, 2014; Boldina and Beninger, 2016). “Multiplicative growth” refers here to variance that increases in the response variable in concert with increases in both the predictor variable and the mean function for the fitted model. In other words, the statistical model should incorporate heteroscedasticity in the response variable. The traditional allometric method fits such a model. Consider the full model for a straight line fitted to logarithms, log(yi ) = log(a) + b∗ log(xi ) + εi
εi ∼ N(0, 2 ),
(1)
where ε represents random error. Residuals in the model have a normal distribution and sum to zero, and variance equals 2 over the full range for the predictor variable. When the model is backtransformed (i.e., exponentiated) to the arithmetic scale, it becomes yi = (a∗ xi b )∗ exp(εi )
εi ∼ N(0, 2 ).
(2)
Residuals still are normally distributed, and variance equals 2 , because the residuals are expressed in logarithmic form. However, functional variance increases on the arithmetic scale as a function of both the predictor and the predicted value for the response, owing to the way that error enters the model. The model fitted by traditional allometry (Eq. (2)) typically is contrasted with one fitted by “standard” nonlinear regression, in which residuals are assumed to be normally distributed and variance is assumed to equal 2 over the full range for the predictor variable, or εi ∼ N(0, 2 ).
2.3. Multiplicative variation
yi = (a∗ xi b ) + εi
Logarithmic transformation often is thought to be necessary in allometry because growth is multiplicative by its very nature (e.g., Kerkhoff and Enquist, 2009; Glazier, 2013). This argument is based on two different perceptions of multiplicative variation (Packard, 2014), and both perceptions frequently are presented in the same article without making a clear distinction between them. Ambiguity is thereby introduced into the conversation about the statistical model that is needed to describe pattern in the data (Packard, 2014). The first version for the argument is framed in the context of Huxley’s monograph on Problems of Relative Growth, where growth is described as the “multiplication of living substance” (Huxley, 1932, p. 11). In other words, one cell becomes two, two cells become
Variance is constant, and the model is homoscedastic. A complex procedure has been articulated for comparing this homoscedastic model with the aforementioned heteroscedastic one (Xiao et al., 2011; Ballantyne, 2013) in the apparent conviction that model selection in allometry requires only that a choice be made between Eqs. (2) and (3) (e.g., Nevill et al., 2005; Kerkhoff and Enquist, 2009; Xiao et al., 2011; White et al., 2012; Ballantyne, 2013; Lai et al., 2013; Niklas and Hammond, 2014; White and Kearney, 2014; Boldina and Beninger, 2016). Contrary to widespread belief, however, the traditional method is not the only way by which to fit a simple power function with heteroscedastic error. For example, models fitted directly to
(3)
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untransformed data by nonlinear regression can incorporate heteroscedasticity that increases as a power of the predictor yi = (a∗ xi b ) + εi
εi ∼ N(0, ( 2 ∗ (xi )2 )),
(4)
where residuals again are assumed to be normally distributed and sum to zero, or as a power of the mean yi = (a∗ xi b ) + εi
εi ∼ N(0, ( 2 ∗ (a∗ xi b )2 )).
(5)
The computational algorithm incorporates an additional parameter, , into the model (Pinheiro and Bates, 2000; Ritz and Streibig, 2008). It is worth noting that both Eq. (2), which is estimated by traditional allometry, and Eq. (5), which is estimated by nonlinear regression, assume that variance increases in a multiplicative (or proportional) way in concert with the mean function; the only real difference between them concerns normality vs. lognormality for the distribution for residuals (which is something that can be addressed in the statistical modeling). In any event, claims that logarithmic transformation is necessary in allometry because of the multiplicative nature of error are demonstrably wrong. 2.4. Non-loglinear allometry Logarithmic transformation commonly fails to linearize the bivariate distribution of interest, thereby giving rise to a curvilinear pattern of variation on the logarithmic scale. The transformations are said then to conform to a pattern of “nonlinear allometry” (Knell, 2009) or, to be more precise, a pattern of “non-loglinear allometry” (Strauss, 1993). Investigators encountering non-loglinear variation generally distinguish polyphasic loglinear allometry from complex allometry (Strauss, 1993). In cases of polyphasic loglinear allometry, putative break-points are identified in the log-log plot, thereby separating the distribution into two (or more) quasi-linear segments that are typically fitted with straight lines. In other instances, observations in log space follow the path of a smooth curve that usually is fitted with a quadratic equation. In both forms of non-loglinearity the exponent in the two-parameter allometric equation is thought to change during development: abruptly in the case of polyphasic loglinear variation (e.g., Huxley, 1924b; Glazier, 2013) and as a continually varying function of body size in the case of complex variation (e.g., Gould, 1966; Strauss, 1993). However, the distinction probably is artificial (except in cases involving metamorphosis or polymorphism), with polyphasic allometry merely representing a form of complex allometry in which artifacts of sampling (and transformation itself) have created putative gaps in the log distribution (Reeve and Huxley, 1945; Gould, 1966; Strauss, 1993). Despite widespread acceptance of the concept, non-loglinear allometry is, in most instances, a statistical artifact resulting from application of the traditional allometric method to situations for which it is unsuited. For example, the traditional method is based on the implicit assumption that the original data will be well described by a function passing through the origin of a graph on the arithmetic scale, that is, that the intercept for the model is zero (e.g., Bales, 1996; Sartori and Ball, 2009; Packard, 2015, 2016, 2017a). Yet, when differentiation and growth by the part (or process) of interest begins somewhat later than that of the body as a whole, the fitted function may require an explicit, non-zero intercept to describe the arithmetic distribution adequately (Ebert and Russell, 1994). This possibility actually was anticipated by Huxley himself (1932, p. 241), and has been confirmed by more recent investigations in which nonlinear modeling was used to fit functions with explicit, non-zero intercepts (Bales, 1996; Sartori and Ball, 2009; Packard, 2016). Data that follow the path of a three-parameter power function on the arithmetic scale will follow a curvilinear path in log domain (Sartori and Ball, 2009; Packard, 2017a). In other words,
Fig. 1. (A) Measurements for rostrum length (mm) and body length (mm) of paddlefish (Polyodon spathula) were compiled by Thompson (1934) and reanalyzed by Packard (2012). The three-parameter power function fitted to the observations is y ˆ = 2.948 × 0.701 − 19.567. The negative y-intercept points to a positive x-intercept, which indicates in turn that the rostrum does not begin to grow until post-larval animals have attained a body length of approximately 15 mm (see Thompson, 1934; Larimore, 1949). (B) Logarithmic transformations of data for Polyodon follow a curvilinear path. Such a pattern of variation in log domain is commonly taken to mean that the exponent in a two-parameter equation of simple allometry is itself a changing function of body size (Strauss, 1993). (C) Instantaneous slope for the transformed power function varies as a function of ln(x) and ranges from 2.36 in the smallest fishes to 0.82 in the largest. However, the exponent, b, in the power equation is a constant value of 0.701.
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the requirement for an intercept in the power function describing untransformed observations is the cause for non-loglinear variation. This peculiar relationship between data expressed in arithmetic and logarithmic domains was recognized by Cumming Robb in 1929 (Robb, 1929), but his finding was largely overlooked, misunderstood, or ignored by later investigators. Indeed, it was only recently that the connection between non-loglinearity and an intercept in the original power function was made unequivocally in an elegant treatment by Sartori and Ball (2009). Take the three-parameter equation
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of a two-parameter power function when they can be described better and more simply by a three- parameter function is not a productive way to go forward with research on allometric variation (Ebert and Russell, 1994). Three-parameter functions with different forms for random error can readily be fitted by nonlinear regression (e.g., Packard, 2015, 2016, 2017a), so the problem of “non-loglinear allometry” can be easily resolved by applying modern statistical methods. 3. Summary
b
y = y0 + (a∗ x ), where y0 is the intercept. The equation can be transformed to ln(y) = ln(y0 + (a∗ xb )), which then can be rewritten as ln(y) = ln(y0 + (a∗ exp(b∗ ln(x)))). This third step is critical because the curvature for the expression in log domain then can be evaluated by differentiating the equation for ln(x), so that the tangent to the curve − or the instantaneous slope (m) − becomes m = (a∗ b∗ exp(b∗ ln(x)))/(y0 + (a∗ exp(b∗ ln(x)))) (Sartori and Ball, 2009). Steps in the process of differentiation can be confirmed by accessing any one of the online sites for taking derivatives. (All of these sites use natural logarithms, which is why I used them here.) Complexity of the aforementioned derivative makes it difficult to mentally visualize the impact of the intercept in the original three-parameter equation. However, the effect can readily be illustrated by drawing on an earlier investigation of relative growth by the spatulate rostrum of paddlefish, Polyodon spathula (Packard, 2012). Recall that the rostrum, which is miniscule in postlarval fishes (Larimore, 1949), grows to substantial size by the time the animals reach adulthood (Thompson, 1934). The pattern of relative growth by the rostrum of paddlefish is described best by a three-parameter power function that points to rostral growth not beginning until fishes have attained a body length of about 15 mm (Fig. 1A). When the data are transformed to logarithms, however, they follow a distinctly curvilinear path (Fig. 1B) that has been cited as a prime example of relative growth in which the exponent in a two-parameter allometric equation is a continuously changing function of body size (Gould, 1966). This perception is merely an illusion. By taking the derivative (m) for the transformed three-parameter power function and then evaluating it at different levels for ln(x) (Fig. 1C), it becomes apparent that the instantaneous slope varies with ln(x) while the exponent, b, is held constant. The curve in log domain is caused by the need for an explicit, non-zero intercept in arithmetic domain (y0 ), and not by a changing exponent in the power function. To carry the example one step further, consider what happens when the intercept for a model fitted to untransformed observations is zero. In that case, the derivative for the equation expressed in log domain can be simplified to the value b, which indicates that the instantaneous slope of the line describing transformations does not change with ln(x). Thus, the model describing logarithmic transformations will be a straight line, thereby confirming that the equation in arithmetic domain is a two-parameter power function. Non-loglinear allometric variation is, in most instances, a statistical deception that is caused by a reliance on log-log displays of bivariate data (see Robb, 1929). This conclusion has major implications for several contemporary theories that are based on the concept of non-loglinear allometry (e.g., Banavar et al., 2014; Bueno and López-Urrutia, 2014). Forcing data to conform to the form
Julian Huxley’s “discovery” of the two-parameter equation for simple allometry was a notable event (Huxley, 1924b, 1932) − not for novelty of the equation (e.g., Reeve and Huxley, 1945; Gould, 1966; Niklas and Hammond, 2014) but because of the way in which he so effectively promoted a simple procedure for fitting the equation to the original bivariate data. Unfortunately, the approach was seriously flawed from the outset because the procedure is capable only of fitting a two-parameter power function that assumes lognormal, heteroscedastic error on the original scale. A three-parameter power function is more appropriate than a two-parameter function for describing pattern in many datasets, and different assumptions may be needed to model random error appropriately. A range of models, including ones with lognormal, heteroscedastic error, can now be fitted directly to untransformed observations by nonlinear regression. The aforementioned misconceptions about logarithmic transformation and the traditional allometric method should not be allowed to impede wider acceptance of newer, more appropriate methods for analysis. Acknowledgments I thank the two referees for helpful suggestions for revision as well as for their support and encouragement. References Albrecht, G.H., Gelvin, B.R., Hartman, S.E., 1993. Ratios as a size adjustment in morphometrics. Am. J. Phys. Anthropol. 91, 441–468. Atchley, W.R., Gaskins, C.T., Anderson, D., 1976. Statistical properties of ratios. I. Empirical results. Syst. Zool. 25, 137–148. Bales, G.S., 1996. Heterochrony in brontothere horn evolution: allometric interpretations and the effect of life history scaling. Paleobiology 22, 481–495. Ballantyne, F., 2013. Evaluating model fit to determine if logarithmic transformations are necessary in allometry: a comment on the exchange between Packard (2009) and Kerkhoff and Enquist (2009). J. Theor. Biol. 317, 418–421. Banavar, J.R., Cooke, T.J., Rinaldo, A., Maritan, A., 2014. Form, function, and evolution of living organisms. Proc. Natl. Acad. Sci. USA 111, 3332–3337. Boldina, I., Beninger, P.G., 2016. Strengthening statistical usage in marine ecology: linear regression. J. Exp. Mar. Biol. Ecol. 474, 81–91. Brody, S., Proctor, R.C., 1932. Relation between basal metabolism and mature body weight in different species of mammals and birds. Missouri Agric. Exp. Sta. Res. Bull. 166, 89–101. Bueno, J., López-Urrutia, A., 2014. Scaling up the curvature of mammalian metabolism. Front. Ecol. Evol. 2, 61. Clark, F.N., 1928. The weight–length relationship of the California sardine (Sardina cærulea) at San Pedro. Fish Bulletin No. 12. Division of Fish and Game of California, Sacramento. Crozier, W.J., Hecht, S., 1914. Correlations of weight, length, and other body measurements in the weakfish, Cynoscion regalis. Bull. U.S. Bur. Fish. 33, 141–147. Ebert, T.A., Russell, M.P., 1994. Allometry and model II non-linear regression. J. Theor. Biol. 168, 367–372. Feng, C., Wang, H., Lu, N., Tu, X.M., 2013. Log transformation: application and interpretation in biomedical research. Statist. Med. 32, 230–239. Finney, D.J., 1989. Was this in your statistics textbook? V. Transformation of data. Exp. Agric. 25, 165–175. Galtsoff, P.S., 1931. The weight-length relationship of the shells of the Hawaiian pearl oyster, Pinctada sp. Am. Nat. 65, 423–433. Gingerich, P.D., 2000. Arithmetic or geometric normality of biological variation: an empirical test of theory. J. Theor. Biol. 204, 201–221. Glazier, D.S., 2013. Log-transformation is useful for examining proportional relationships in allometric scaling. J. Theor. Biol. 334, 200–203. Gould, S.J., 1966. Allometry and size in ontogeny and phylogeny. Biol. Rev. 41, 587–640.
Please cite this article in press as: Packard, G.C., Misconceptions about logarithmic transformation and the traditional allometric method. Zoology (2017), http://dx.doi.org/10.1016/j.zool.2017.07.005
G Model ZOOL-25590; No. of Pages 6 6
ARTICLE IN PRESS G.C. Packard / Zoology xxx (2017) xxx–xxx
Green, C.V., Green, S.K., 1932. Shell growth in the periwinkle, Littorina litorea. Am. Nat. 66, 371–376. Hecht, S., 1913. The relation of weight to length in the smooth dog fish, Mustelus canis. Anat. Rec. 7, 39–42. Hecht, S., 1916. Form and growth in fishes. J. Morphol. 27, 379–400. Huxley, J.S., 1924a. The variation in the width of the abdomen in immature fiddler crabs considered in relation to its relative growth-rate. Am. Nat. 58, 468–475. Huxley, J.S., 1924b. Constant differential growth-ratios and their significance. Nature 114, 895–896. Huxley, J.S., 1927a. Further work on heterogonic growth. Biolog. Zentralb. 47, 151–163. Huxley, J.S., 1927b. On the relation between egg-weight and body-weight in birds. J. Linn. Soc. Lond. Zool. 36, 457–466. Huxley, J.S., 1932. Problems of Relative Growth. Methuen; London. Huxley, J.S., 1950. Relative growth and form transformation. Proc. Roy. Soc. Lond. B 137, 465–469. Huxley, J.S., Teissier, G., 1936. Terminology of relative growth. Nature 137, 780–781. Katz, M.J., 1980. Allometry formula: a cellular model. Growth 44, 89–96. Kerkhoff, A.J., Enquist, B.J., 2009. Multiplicative by nature: why logarithmic transformation is necessary in allometry. J. Theor. Biol. 257, 519–521. Kleiber, M., 1932. Body size and metabolism. Hilgardia 6, 315–353. Klingenberg, P.C., 1998. Heterochrony and allometry: the analysis of evolutionary change in ontogeny. Biol. Rev. 73, 79–123. Knell, R.J., 2009. On the analysis of non-linear allometries. Ecol. Entomol. 34, 1–11. Kronmal, R.A., 1993. Spurious correlation and the fallacy of the ratio standard revisited. J. Roy. Stat. Soc. A 156, 379–392. Kunkel, B.W., Robertson, J.A., 1928. Contributions to the study of relative growth in Gammarus cheureuxi. J. Mar. Biol. Assoc. U.K. 15, 655–682. Lagergren, R., Svensson, J.-E., Stenson, J.A.E., 2007. Models of ontogenetic allometry in cladoceran morphology studies. Hydrobiologia 594, 109–116. Lai, J., Yang, B., Lin, D., Kerkhoff, A.J., Ma, K., 2013. The allometry of coarse root biomass: log-transformed linear regression or nonlinear regression? PLoS ONE 8, e77007. Larimore, R.W., 1949. Changes in the cranial nerves of the paddlefish, Polyodon spathula, accompanying development of the rostrum. Copeia 1949, 204–212. Lemaître, J.F., Vanpé, C., Plard, F., Pélabon, C., Gaillard, J.M., 2015. Response to Packard: make sure we do not throw out the biological baby with the statistical bath water when performing allometric analyses. Biol. Lett. 11, 20150144. Lovett, D.L., Felder, D.L., 1989. Application of regression techniques to studies of relative growth in crustaceans. J. Crust. Biol. 9, 529–539. Mascaro, J., Litton, C.M., Hughes, R.F., Uowolo, A., Schnitzer, S.A., 2014. Is logarithmic transformation necessary in allometry? Ten, one-hundred, one-thousand-times yes. Biol. J. Linn. Soc. 111, 230–233. Nevill, A.M., Bate, S., Holder, R.L., 2005. Modeling physiological and anthropometric variables known to vary with body size and other confounding variables. Yearb. Phys. Anthropol. 48, 141–153. Newcombe, C.L., 1948. An application of the allometry equation to the study of growth in Callinectes sapidus Rathbun. Am. Nat. 82, 315–325. Niklas, K.J., Hammond, S.T., 2014. Assessing scaling relationships: uses, abuses, and alternatives. Int. J. Plant Sci. 175, 754–763. Nomura, E., 1926. An application of a = kbx in expressing the growth relation in the freshwater bivalve, Sphaerium heterodon Pils. Sci. Rept. Tohoku Imp. Univ. (Ser. 4) 2, 57–62. Osborne, J.W., 2002. Notes on the use of data transformations. Pract. Assess. Res. Eval. 8, 6. Packard, G.C., 2012. Is non-loglinear allometry a statistical artifact? Biol. J. Linn. Soc. 107, 764–773.
Packard, G.C., 2014. Multiplicative by nature: logarithmic transformation in allometry. J. Exp. Zool. B 322, 202–207. Packard, G.C., 2015. Quantifying the curvilinear metabolic scaling in mammals. J. Exp. Zool. A 323, 540–546. Packard, G.C., 2016. Relative growth by the elongated jaws of gars: a perspective on polyphasic loglinear allometry. J. Exp. Zool. B 326, 168–175. Packard, G.C., 2017a. Is complex allometry in field metabolic rates of mammals a statistical artifact? Comp. Biochem. Physiol. A 203, 322–327. Packard, G.C., 2017b. The essential role for graphs in allometric analysis. Biol. J. Linn. Soc. 120, 468–473. Pearsall, W.H., 1927. Growth studies. VI. On the relative sizes of growing plant organs. Ann. Bot. 41, 549–556. Pearson, K., 1897. Mathematical contributions to the theory of evolution. On a form of spurious correlation which may arise when indices are used in the measurement of organs. Proc. R. Soc Lond. 60, 489–498. Pélabon, C., Firmat, C., Bolstad, G.H., Voje, K.L., Houle, D., Cassara, J., Le Rouzic, A., Hansen, T.F., 2014. Evolution of morphological allometry. Ann. New York Acad. Sci. 1320, 58–75. Peters, R.H., 1983. The Ecological Implications of Body Size. Cambridge University Press, Cambridge. Pinheiro, J.C., Bates, D.M., 2000. Mixed-Effects Models in S and S-PLUS. Springer, New York. Reeve, E.C.R., Huxley, J.S., 1945. Some problems in the study of allometric growth. In: Le Gros Clark, W.E., Medawar, P.B. (Eds.), Essays on Growth and Form Presented to D’Arcy Wentworth Thompson. Oxford University Press, Oxford, pp. 121–156. Reyment, R.A., 1971. Multivariate normality in morphometric analysis. Math. Geol. 3, 357–368. Ritz, C., Streibig, J.C., 2008. Nonlinear Regression with R. Springer, New York. Robb, R.C., 1929. On the nature of hereditary size limitation. II. The growth of parts in relation to the whole. Brit. J. Exp. Biol. 6, 311–324. Sartori, A.F., Ball, A.D., 2009. Morphology and postlarval development of the ligament of Thracia phaseolina (Bivalvia: Thraciidae), with a discussion of model choice in allometric studies. J. Mollusc. Stud. 75, 295–304. Smith, R.J., 1980. Rethinking allometry. J. Theor. Biol. 87, 97–111. Smith, R.J., 1984. Allometric scaling in comparative biology: problems of concept and method. Am. J. Physiol. 246, R152–R160. Snedecor, G.W., 1937. Statistical Methods Applied to Experiments in Agriculture and Biology. Collegiate Press, Ames, IA, USA. Strauss, R.E., Huxley, J.S., 1993. The study of allometry since Huxley. In: Problems of Relative Growth, new edition. Johns Hopkins University Press, Baltimore, pp. xlviii–lxxv. Thompson, D.H., 1934. Relative Growth in Polyodon. Biological Notes, No. 2. Illinois Natural History Survey, Urbana, IL, USA. Thompson, D.W., 1943. On Growth and Form − A New Edition. Macmillan, New York. Warton, D.I., Wright, I.J., Falster, D.S., Westoby, M., 2006. Bivariate line-fitting methods for allometry. Biol. Rev. 81, 259–291. West, D., West, B.J., 2012. On allometry relations. Int. J. Mod. Physics B 26, 18. White, C.R., Kearney, M.R., 2014. Metabolic scaling in animals: methods, empirical results, and theoretical explanations. Compr. Physiol. 4, 231–256. White, E.P., Xiao, X., Isaac, N.J.B., Sibly, R.M., 2012. Methodological tools. In: Sibly, R.M., Brown, J.H., Kodric-Brown, A. (Eds.), Metabolic Ecology: A Scaling Approach. Wiley, Hoboken, pp. 9–20. Xiao, X., White, E.P., Hooten, M.B., Durham, S., 2011. On the use of log-transformation vs. nonlinear regression for analyzing biological power laws. Ecology 92, 1887–1894.
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Review
Misconceptions about logarithmic transformation and the traditional allometric method Gary C. Packard ∗ Department of Biology, Colorado State University, Fort Collins, CO 80523, USA
a r t i c l e
i n f o
Article history: Received 16 March 2017 Received in revised form 17 July 2017 Accepted 17 July 2017 Available online xxx Keywords: Allometry Logarithms Nonlinear regression Power laws Scaling
a b s t r a c t Logarithmic transformation is often assumed to be necessary in allometry to accommodate the kind of variation that accompanies multiplicative growth by plants and animals; and the traditional approach to allometric analysis is commonly believed to have important application even when the bivariate distribution of interest is curvilinear on the logarithmic scale. Here I examine four arguments that have been tendered in support of these perceptions. All the arguments are based on misunderstandings about the traditional method for allometric analysis and/or on a lack of familiarity with newer methods of nonlinear regression. Traditional allometry actually has limited utility because it can be used only to fit a two-parameter power equation that assumes lognormal, heteroscedastic error on the original scale. In contrast, nonlinear regression can fit two- and three-parameter power equations with differing assumptions about structure for error directly to untransformed data. Nonlinear regression should be preferred to the traditional method in future allometric analyses. © 2017 Elsevier GmbH. All rights reserved.
Contents 1. 2.
3.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 Misconceptions about bivariate allometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 2.1. Form of the allometric equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 2.2. Proportional change and relative growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 2.3. Multiplicative variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 2.4. Non-loglinear allometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
1. Introduction The early part of the 20th century was marked by widespread interest among biologists in the use of simple power functions of the form y = a ∗ xb to describe pattern in bivariate observations that follow a curvilinear path on the arithmetic (=linear) scale. The predictor variable (x) in the two-parameter equation typically was a measure of body
∗ Corresponding author. Present address: 865 Three Corner Gate Rd, Livermore, CO 80536, USA. E-mail address: [email protected]
size (e.g., body length or body mass), and the response variable (y) was some measurement taken on the structure, organ, or process of special concern. Some investigators at the time apparently fitted the equation directly to scatterplots of untransformed data by a series of trial-and-error approximations (e.g., Nomura, 1926; Kleiber, 1932), or by fitting a curve by eye and then reading from the graph (e.g., Hecht, 1913, 1916; Crozier and Hecht, 1914; Kleiber, 1932). Other workers, however, estimated the slope and intercept of a straight line drawn by hand on a graph displaying logarithmic transformations (or on a graph with logarithmic coordinates) and then took antilogs for the coefficients to obtain parameters in the power equation (e.g., Pearsall, 1927; Huxley, 1927a, b, 1932; Kunkel and Robertson, 1928). And yet a fourth group of investigators fitted straight lines to logarithmic transformations by ordinary least
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squares regression and then back-transformed the resulting equation to the arithmetic scale (e.g., Clark, 1928; Galtsoff, 1931; Brody and Proctor, 1932; Green and Green, 1932). This last approach to fitting the power function continues, for all intents and purposes, to be in general use today (Warton et al., 2006; White et al., 2012; White and Kearney, 2014) and has come to be known as the traditional allometric method (e.g., Packard, 2014). The traditional allometric method has had its critics over the years (e.g., Thompson, 1943; Smith, 1980, 1984; Lovett and Felder, 1989; Bales, 1996), and it recently has come under renewed scrutiny (e.g., Lagergren et al., 2007; Sartori and Ball, 2009; Packard, 2014, 2015, 2016, 2017a,b). Supporters of the protocol are understandably concerned that a large body of published research might be undermined if criticisms of the method were taken seriously, and they consequently have mounted a spirited defense of their research paradigm (e.g., Klingenberg, 1998; Nevill et al., 2005; Kerkhoff and Enquist, 2009; Xiao et al., 2010; White et al., 2012; Ballantyne, 2013; Glazier, 2013; Lai et al., 2013; Mascaro et al., 2014; Niklas and Hammond, 2014; Lema&tre et al., 2015). However, the defense is based in many instances on ill-defined arguments and/or misunderstanding of various statistical methods. Here I examine four of the most common misconceptions. 2. Misconceptions about bivariate allometry 2.1. Form of the allometric equation Philip Gingerich has proposed that “the allometric equation is not a power function of x and y as is so often stated, but rather a linear function of log x and log y” (Gingerich, 2000, p. 220). The suggestion was based on his study of normal vs. lognormal distributions for random error (i.e., residuals) in samples of biological data and not on an explicit analysis of allometric variation. The assertion consequently rests on something of an extrapolation, but it may, nonetheless, describe contemporary research on allometric variation fairly accurately. Logarithmic transformation was used in the 1920s and 1930s to fit simple, two-parameter power functions to observations expressed on the arithmetic scale. As time went on, however, transformation became less and less a means to fit a power function to untransformed data, and the equation of simple allometry simultaneously became more and more a justification for performing the transformation. Thus, contemporary analyses of allometric variation typically begin with a token nod to the equation of simple allometry, proceed with the rote transformation of data to logarithms, and then continue with the fitting of a straight line to the new distribution (Smith, 1980, 1984). Validation of the fitted model usually is limited to a graphical display of the equation in log domain, and R2 may also be reported as a measure of goodness of fit. However, the quality of the model seldom is assessed graphically on the original scale (Packard, 2017b), and R2 for the fit to logarithms has no bearing on strength of the relationship between the untransformed variables. Thus, it is easy to believe that the allometric equation in current practice is, in fact, log(y) = log(a) + b∗ log(x). If the allometric equation actually is a linear function of log(y) and log(x), future reports of allometric variation should be framed differently. For instance, reports of new research need not (and probably should not) begin with a reference to a two-parameter power function, because the power function is neither necessary nor relevant to the analysis; and no attempt needs to be made at interpretation on the original scale, because the original scale also is irrelevant. These recommendations are consistent with the common belief that observations in logarithmic form are at least as meaningful as untransformed values (e.g., Peters, 1983; Kerkhoff
and Enquist, 2009; Glazier, 2013; Lai et al., 2013), and the relationship of putative interest is, after all, between log(y) and log(x). It is unclear how this relationship between log(y) and log(x) is to be interpreted in biologically meaningful terms when the findings cannot be placed in the context of the original measurements (e.g., Reyment, 1971; Finney, 1989; Osborne, 2002; Feng et al., 2013). This problem of interpretation is an unavoidable consequence of accepting at face value the aforementioned premise about the allometric equation. Gingerich had an important point to make about the kind of random error that may occur in biological data, but his characterization of the allometric equation should not be taken literally. The equation of simple allometry is y = a ∗ xb , as so often stated (e.g., Huxley and Teissier, 1936; Huxley, 1950), and the linearized expression log(y) = log(a) + b∗ log(x) merely provides a way to estimate parameters in the allometric equation via an intermediate step involving logarithms (e.g., West and West, 2012). In other words, a straight line fitted to logarithms is a means to an end and not an end in itself. The linearized expression was an essential tool early in the last century, because neither statistical theory nor statistical practice was sufficiently advanced at the time to permit fitting the power function directly to untransformed observations. However, the equation now can be fitted to untransformed data by nonlinear regression, and the issue of normal vs. lognormal error can be addressed simultaneously by the computational algorithm (Packard, 2015, 2016, 2017a). 2.2. Proportional change and relative growth Several investigators − all of whom cite Huxley (1932) for their rationale − have argued that logarithmic transformation (i.e., the traditional allometric method) is necessary in allometric research because allometry is all about proportional change and relative growth (e.g., Kerkhoff and Enquist, 2009; Glazier, 2013). But what, exactly, is meant by “proportional change” and “relative growth,” and is logarithmic transformation really necessary for describing these processes? Julian Huxley’s treatise on “Problems of Relative Growth” (Huxley, 1932) focused on how some part of the body changes in its proportion to the body as a whole as both the part and body increase in size. At the time of Huxley’s writing, proportional (or relative) size of a part was commonly expressed as a percentage of size of the body, and relative growth was reflected in the changing percentage as the animals (or plants) increased in overall size (e.g., Huxley, 1924a). This expression of relative growth could be represented graphically (see Fig. 2 in Huxley, 1932), but the actual relationship between the structure of interest and body size could not be quantified accurately owing to x being a component of both the predictor and response variables (see Pearson, 1897; Atchley et al., 1976; Albrecht et al., 1993; Kronmal, 1993). Huxley’s desire to express the relationship between x and y mathematically is what led to his independent discovery of the formula for simple allometry (Reeve and Huxley, 1945), which provides a simple, yet explicit, mathematical description for the relationship between the variable of interest and body size. The two-parameter power function can be rearranged algebraically to show that the response variable, y, varies as a constant proportion of xb (e.g., Newcombe, 1948; White et al., 2012), or a = y/xb where a and b are fitted constants. Because y maintains a fixed proportional relationship to xb as both y and x increase in size,
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the proportion describes a pattern of relative growth by the bodily part of interest (i.e., the general subject of Huxley’s book). The point here, of course, is that the proportion of y to xb can readily be extracted from an equation fitted to data by nonlinear regression, so transformation is not a necessity. A proportion (or ratio) of even greater interest to Huxley, however, was the one between the instantaneous rate of growth by the part to the instantaneous rate of growth by the body (Huxley, 1924b, 1932). Assuming that the two-parameter power function is a good descriptor for pattern in the bivariate distribution of untransformed data (Huxley, 1932, p. 6), then constancy of the exponent implies that the ratio of the relative growth rates is also a constant, that is (dy/dt∗ 1/y)/(dx/dt∗ 1/x) = b (Huxley, 1924b, 1927a, 1932; Kleiber, 1932). This constant differential growth-ratio raised the possibility in Huxley’s mind that the formula of simple allometry is a general law of relative growth (Huxley, 1932, 1950; Reeve and Huxley, 1945). Of course, Huxley used the slope of a straight line fitted to logarithms to estimate the exponent in the equation of simple allometry, but the exponent also can be estimated by fitting a power function directly to untransformed observations by nonlinear regression (Packard, 2015, 2016, 2017a). Again, transformation is not a necessity. Huxley also noted that a graph of logarithmic transformations can be used to assess multiplicative growth visually (Huxley, 1927b, 1932). This property of the plot may be the basis for claims about the necessity to use logarithms to evaluate proportional change. After all, equal intervals along the fitted line represent equal amounts of multiplication, so the intervals represent identical proportional changes in y relative to x (Huxley, 1927b, 1932). Nevertheless, this graphical representation of proportionality contributes little to understanding the relationship between the variables, and it was a secondary consideration even for Huxley (Huxley, 1932, p. 11). A plot of logarithms has little import (other than to confirm linearity of the distribution) unless it is accompanied by a graph of the equation of simple allometry on the arithmetic scale − a point that Huxley failed to appreciate fully, but a point that was not lost on other investigators at the time (Clark, 1928; Galtsoff, 1931; Green and Green, 1932). Huxley used logarithms as an indirect way to fit the formula for simple allometry to untransformed data, because no direct method was available to him (see Snedecor, 1937). However, the equation of simple allometry (even one making the same assumptions about random error) now can be fitted directly to untransformed observations by nonlinear regression (e.g., Packard, 2015, 2016, 2017a). Thus, two of the three proportionalities identified here can be readily estimated by nonlinear regression. Only the third proportionality requires transformation, but the resulting bivariate plot is of limited value.
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four, four become eight, and so on as the organism and its parts increase in size (Huxley, 1924b; Katz, 1980). If multiplication of tissue proceeds more rapidly in the part than in the whole, the untransformed data commonly follow the path of a power function with an exponent greater than 1 (e.g., Green and Green, 1932). Conversely, if growth by the whole is more rapid than growth by the part, the exponent in the power function typically lies between 0 and 1. And finally, if growth by the part remains in the same proportion to growth by the whole, the data will follow the path of a straight line passing through the origin and the exponent in the power equation will be 1. The point here is that multiplicative growth sensu Huxley is described by a power function, but a power function can be fitted to untransformed data by nonlinear regression. Huxley used logarithms because no other method was available in the 1920s and 1930s to fit a power function to observations expressed on the original arithmetic scale (Snedecor, 1937). Transformation is not necessary today to describe multiplicative growth. The second version of the argument about necessity for logarithmic transformation concerns variance in the statistical model expressed on the arithmetic scale (e.g., Nevill et al., 2005; Kerkhoff and Enquist, 2009; Xiao et al., 2011; White et al., 2012; Ballantyne, 2013; Lai et al., 2013; Niklas and Hammond, 2014; Pélabon et al., 2014; White and Kearney, 2014; Boldina and Beninger, 2016). “Multiplicative growth” refers here to variance that increases in the response variable in concert with increases in both the predictor variable and the mean function for the fitted model. In other words, the statistical model should incorporate heteroscedasticity in the response variable. The traditional allometric method fits such a model. Consider the full model for a straight line fitted to logarithms, log(yi ) = log(a) + b∗ log(xi ) + εi
εi ∼ N(0, 2 ),
(1)
where ε represents random error. Residuals in the model have a normal distribution and sum to zero, and variance equals 2 over the full range for the predictor variable. When the model is backtransformed (i.e., exponentiated) to the arithmetic scale, it becomes yi = (a∗ xi b )∗ exp(εi )
εi ∼ N(0, 2 ).
(2)
Residuals still are normally distributed, and variance equals 2 , because the residuals are expressed in logarithmic form. However, functional variance increases on the arithmetic scale as a function of both the predictor and the predicted value for the response, owing to the way that error enters the model. The model fitted by traditional allometry (Eq. (2)) typically is contrasted with one fitted by “standard” nonlinear regression, in which residuals are assumed to be normally distributed and variance is assumed to equal 2 over the full range for the predictor variable, or εi ∼ N(0, 2 ).
2.3. Multiplicative variation
yi = (a∗ xi b ) + εi
Logarithmic transformation often is thought to be necessary in allometry because growth is multiplicative by its very nature (e.g., Kerkhoff and Enquist, 2009; Glazier, 2013). This argument is based on two different perceptions of multiplicative variation (Packard, 2014), and both perceptions frequently are presented in the same article without making a clear distinction between them. Ambiguity is thereby introduced into the conversation about the statistical model that is needed to describe pattern in the data (Packard, 2014). The first version for the argument is framed in the context of Huxley’s monograph on Problems of Relative Growth, where growth is described as the “multiplication of living substance” (Huxley, 1932, p. 11). In other words, one cell becomes two, two cells become
Variance is constant, and the model is homoscedastic. A complex procedure has been articulated for comparing this homoscedastic model with the aforementioned heteroscedastic one (Xiao et al., 2011; Ballantyne, 2013) in the apparent conviction that model selection in allometry requires only that a choice be made between Eqs. (2) and (3) (e.g., Nevill et al., 2005; Kerkhoff and Enquist, 2009; Xiao et al., 2011; White et al., 2012; Ballantyne, 2013; Lai et al., 2013; Niklas and Hammond, 2014; White and Kearney, 2014; Boldina and Beninger, 2016). Contrary to widespread belief, however, the traditional method is not the only way by which to fit a simple power function with heteroscedastic error. For example, models fitted directly to
(3)
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untransformed data by nonlinear regression can incorporate heteroscedasticity that increases as a power of the predictor yi = (a∗ xi b ) + εi
εi ∼ N(0, ( 2 ∗ (xi )2 )),
(4)
where residuals again are assumed to be normally distributed and sum to zero, or as a power of the mean yi = (a∗ xi b ) + εi
εi ∼ N(0, ( 2 ∗ (a∗ xi b )2 )).
(5)
The computational algorithm incorporates an additional parameter, , into the model (Pinheiro and Bates, 2000; Ritz and Streibig, 2008). It is worth noting that both Eq. (2), which is estimated by traditional allometry, and Eq. (5), which is estimated by nonlinear regression, assume that variance increases in a multiplicative (or proportional) way in concert with the mean function; the only real difference between them concerns normality vs. lognormality for the distribution for residuals (which is something that can be addressed in the statistical modeling). In any event, claims that logarithmic transformation is necessary in allometry because of the multiplicative nature of error are demonstrably wrong. 2.4. Non-loglinear allometry Logarithmic transformation commonly fails to linearize the bivariate distribution of interest, thereby giving rise to a curvilinear pattern of variation on the logarithmic scale. The transformations are said then to conform to a pattern of “nonlinear allometry” (Knell, 2009) or, to be more precise, a pattern of “non-loglinear allometry” (Strauss, 1993). Investigators encountering non-loglinear variation generally distinguish polyphasic loglinear allometry from complex allometry (Strauss, 1993). In cases of polyphasic loglinear allometry, putative break-points are identified in the log-log plot, thereby separating the distribution into two (or more) quasi-linear segments that are typically fitted with straight lines. In other instances, observations in log space follow the path of a smooth curve that usually is fitted with a quadratic equation. In both forms of non-loglinearity the exponent in the two-parameter allometric equation is thought to change during development: abruptly in the case of polyphasic loglinear variation (e.g., Huxley, 1924b; Glazier, 2013) and as a continually varying function of body size in the case of complex variation (e.g., Gould, 1966; Strauss, 1993). However, the distinction probably is artificial (except in cases involving metamorphosis or polymorphism), with polyphasic allometry merely representing a form of complex allometry in which artifacts of sampling (and transformation itself) have created putative gaps in the log distribution (Reeve and Huxley, 1945; Gould, 1966; Strauss, 1993). Despite widespread acceptance of the concept, non-loglinear allometry is, in most instances, a statistical artifact resulting from application of the traditional allometric method to situations for which it is unsuited. For example, the traditional method is based on the implicit assumption that the original data will be well described by a function passing through the origin of a graph on the arithmetic scale, that is, that the intercept for the model is zero (e.g., Bales, 1996; Sartori and Ball, 2009; Packard, 2015, 2016, 2017a). Yet, when differentiation and growth by the part (or process) of interest begins somewhat later than that of the body as a whole, the fitted function may require an explicit, non-zero intercept to describe the arithmetic distribution adequately (Ebert and Russell, 1994). This possibility actually was anticipated by Huxley himself (1932, p. 241), and has been confirmed by more recent investigations in which nonlinear modeling was used to fit functions with explicit, non-zero intercepts (Bales, 1996; Sartori and Ball, 2009; Packard, 2016). Data that follow the path of a three-parameter power function on the arithmetic scale will follow a curvilinear path in log domain (Sartori and Ball, 2009; Packard, 2017a). In other words,
Fig. 1. (A) Measurements for rostrum length (mm) and body length (mm) of paddlefish (Polyodon spathula) were compiled by Thompson (1934) and reanalyzed by Packard (2012). The three-parameter power function fitted to the observations is y ˆ = 2.948 × 0.701 − 19.567. The negative y-intercept points to a positive x-intercept, which indicates in turn that the rostrum does not begin to grow until post-larval animals have attained a body length of approximately 15 mm (see Thompson, 1934; Larimore, 1949). (B) Logarithmic transformations of data for Polyodon follow a curvilinear path. Such a pattern of variation in log domain is commonly taken to mean that the exponent in a two-parameter equation of simple allometry is itself a changing function of body size (Strauss, 1993). (C) Instantaneous slope for the transformed power function varies as a function of ln(x) and ranges from 2.36 in the smallest fishes to 0.82 in the largest. However, the exponent, b, in the power equation is a constant value of 0.701.
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the requirement for an intercept in the power function describing untransformed observations is the cause for non-loglinear variation. This peculiar relationship between data expressed in arithmetic and logarithmic domains was recognized by Cumming Robb in 1929 (Robb, 1929), but his finding was largely overlooked, misunderstood, or ignored by later investigators. Indeed, it was only recently that the connection between non-loglinearity and an intercept in the original power function was made unequivocally in an elegant treatment by Sartori and Ball (2009). Take the three-parameter equation
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of a two-parameter power function when they can be described better and more simply by a three- parameter function is not a productive way to go forward with research on allometric variation (Ebert and Russell, 1994). Three-parameter functions with different forms for random error can readily be fitted by nonlinear regression (e.g., Packard, 2015, 2016, 2017a), so the problem of “non-loglinear allometry” can be easily resolved by applying modern statistical methods. 3. Summary
b
y = y0 + (a∗ x ), where y0 is the intercept. The equation can be transformed to ln(y) = ln(y0 + (a∗ xb )), which then can be rewritten as ln(y) = ln(y0 + (a∗ exp(b∗ ln(x)))). This third step is critical because the curvature for the expression in log domain then can be evaluated by differentiating the equation for ln(x), so that the tangent to the curve − or the instantaneous slope (m) − becomes m = (a∗ b∗ exp(b∗ ln(x)))/(y0 + (a∗ exp(b∗ ln(x)))) (Sartori and Ball, 2009). Steps in the process of differentiation can be confirmed by accessing any one of the online sites for taking derivatives. (All of these sites use natural logarithms, which is why I used them here.) Complexity of the aforementioned derivative makes it difficult to mentally visualize the impact of the intercept in the original three-parameter equation. However, the effect can readily be illustrated by drawing on an earlier investigation of relative growth by the spatulate rostrum of paddlefish, Polyodon spathula (Packard, 2012). Recall that the rostrum, which is miniscule in postlarval fishes (Larimore, 1949), grows to substantial size by the time the animals reach adulthood (Thompson, 1934). The pattern of relative growth by the rostrum of paddlefish is described best by a three-parameter power function that points to rostral growth not beginning until fishes have attained a body length of about 15 mm (Fig. 1A). When the data are transformed to logarithms, however, they follow a distinctly curvilinear path (Fig. 1B) that has been cited as a prime example of relative growth in which the exponent in a two-parameter allometric equation is a continuously changing function of body size (Gould, 1966). This perception is merely an illusion. By taking the derivative (m) for the transformed three-parameter power function and then evaluating it at different levels for ln(x) (Fig. 1C), it becomes apparent that the instantaneous slope varies with ln(x) while the exponent, b, is held constant. The curve in log domain is caused by the need for an explicit, non-zero intercept in arithmetic domain (y0 ), and not by a changing exponent in the power function. To carry the example one step further, consider what happens when the intercept for a model fitted to untransformed observations is zero. In that case, the derivative for the equation expressed in log domain can be simplified to the value b, which indicates that the instantaneous slope of the line describing transformations does not change with ln(x). Thus, the model describing logarithmic transformations will be a straight line, thereby confirming that the equation in arithmetic domain is a two-parameter power function. Non-loglinear allometric variation is, in most instances, a statistical deception that is caused by a reliance on log-log displays of bivariate data (see Robb, 1929). This conclusion has major implications for several contemporary theories that are based on the concept of non-loglinear allometry (e.g., Banavar et al., 2014; Bueno and López-Urrutia, 2014). Forcing data to conform to the form
Julian Huxley’s “discovery” of the two-parameter equation for simple allometry was a notable event (Huxley, 1924b, 1932) − not for novelty of the equation (e.g., Reeve and Huxley, 1945; Gould, 1966; Niklas and Hammond, 2014) but because of the way in which he so effectively promoted a simple procedure for fitting the equation to the original bivariate data. Unfortunately, the approach was seriously flawed from the outset because the procedure is capable only of fitting a two-parameter power function that assumes lognormal, heteroscedastic error on the original scale. A three-parameter power function is more appropriate than a two-parameter function for describing pattern in many datasets, and different assumptions may be needed to model random error appropriately. A range of models, including ones with lognormal, heteroscedastic error, can now be fitted directly to untransformed observations by nonlinear regression. The aforementioned misconceptions about logarithmic transformation and the traditional allometric method should not be allowed to impede wider acceptance of newer, more appropriate methods for analysis. Acknowledgments I thank the two referees for helpful suggestions for revision as well as for their support and encouragement. References Albrecht, G.H., Gelvin, B.R., Hartman, S.E., 1993. Ratios as a size adjustment in morphometrics. Am. J. Phys. Anthropol. 91, 441–468. Atchley, W.R., Gaskins, C.T., Anderson, D., 1976. Statistical properties of ratios. I. Empirical results. Syst. Zool. 25, 137–148. Bales, G.S., 1996. Heterochrony in brontothere horn evolution: allometric interpretations and the effect of life history scaling. Paleobiology 22, 481–495. Ballantyne, F., 2013. Evaluating model fit to determine if logarithmic transformations are necessary in allometry: a comment on the exchange between Packard (2009) and Kerkhoff and Enquist (2009). J. Theor. Biol. 317, 418–421. Banavar, J.R., Cooke, T.J., Rinaldo, A., Maritan, A., 2014. Form, function, and evolution of living organisms. Proc. Natl. Acad. Sci. USA 111, 3332–3337. Boldina, I., Beninger, P.G., 2016. Strengthening statistical usage in marine ecology: linear regression. J. Exp. Mar. Biol. Ecol. 474, 81–91. Brody, S., Proctor, R.C., 1932. Relation between basal metabolism and mature body weight in different species of mammals and birds. Missouri Agric. Exp. Sta. Res. Bull. 166, 89–101. Bueno, J., López-Urrutia, A., 2014. Scaling up the curvature of mammalian metabolism. Front. Ecol. Evol. 2, 61. Clark, F.N., 1928. The weight–length relationship of the California sardine (Sardina cærulea) at San Pedro. Fish Bulletin No. 12. Division of Fish and Game of California, Sacramento. Crozier, W.J., Hecht, S., 1914. Correlations of weight, length, and other body measurements in the weakfish, Cynoscion regalis. Bull. U.S. Bur. Fish. 33, 141–147. Ebert, T.A., Russell, M.P., 1994. Allometry and model II non-linear regression. J. Theor. Biol. 168, 367–372. Feng, C., Wang, H., Lu, N., Tu, X.M., 2013. Log transformation: application and interpretation in biomedical research. Statist. Med. 32, 230–239. Finney, D.J., 1989. Was this in your statistics textbook? V. Transformation of data. Exp. Agric. 25, 165–175. Galtsoff, P.S., 1931. The weight-length relationship of the shells of the Hawaiian pearl oyster, Pinctada sp. Am. Nat. 65, 423–433. Gingerich, P.D., 2000. Arithmetic or geometric normality of biological variation: an empirical test of theory. J. Theor. Biol. 204, 201–221. Glazier, D.S., 2013. Log-transformation is useful for examining proportional relationships in allometric scaling. J. Theor. Biol. 334, 200–203. Gould, S.J., 1966. Allometry and size in ontogeny and phylogeny. Biol. Rev. 41, 587–640.
Please cite this article in press as: Packard, G.C., Misconceptions about logarithmic transformation and the traditional allometric method. Zoology (2017), http://dx.doi.org/10.1016/j.zool.2017.07.005
G Model ZOOL-25590; No. of Pages 6 6
ARTICLE IN PRESS G.C. Packard / Zoology xxx (2017) xxx–xxx
Green, C.V., Green, S.K., 1932. Shell growth in the periwinkle, Littorina litorea. Am. Nat. 66, 371–376. Hecht, S., 1913. The relation of weight to length in the smooth dog fish, Mustelus canis. Anat. Rec. 7, 39–42. Hecht, S., 1916. Form and growth in fishes. J. Morphol. 27, 379–400. Huxley, J.S., 1924a. The variation in the width of the abdomen in immature fiddler crabs considered in relation to its relative growth-rate. Am. Nat. 58, 468–475. Huxley, J.S., 1924b. Constant differential growth-ratios and their significance. Nature 114, 895–896. Huxley, J.S., 1927a. Further work on heterogonic growth. Biolog. Zentralb. 47, 151–163. Huxley, J.S., 1927b. On the relation between egg-weight and body-weight in birds. J. Linn. Soc. Lond. Zool. 36, 457–466. Huxley, J.S., 1932. Problems of Relative Growth. Methuen; London. Huxley, J.S., 1950. Relative growth and form transformation. Proc. Roy. Soc. Lond. B 137, 465–469. Huxley, J.S., Teissier, G., 1936. Terminology of relative growth. Nature 137, 780–781. Katz, M.J., 1980. Allometry formula: a cellular model. Growth 44, 89–96. Kerkhoff, A.J., Enquist, B.J., 2009. Multiplicative by nature: why logarithmic transformation is necessary in allometry. J. Theor. Biol. 257, 519–521. Kleiber, M., 1932. Body size and metabolism. Hilgardia 6, 315–353. Klingenberg, P.C., 1998. Heterochrony and allometry: the analysis of evolutionary change in ontogeny. Biol. Rev. 73, 79–123. Knell, R.J., 2009. On the analysis of non-linear allometries. Ecol. Entomol. 34, 1–11. Kronmal, R.A., 1993. Spurious correlation and the fallacy of the ratio standard revisited. J. Roy. Stat. Soc. A 156, 379–392. Kunkel, B.W., Robertson, J.A., 1928. Contributions to the study of relative growth in Gammarus cheureuxi. J. Mar. Biol. Assoc. U.K. 15, 655–682. Lagergren, R., Svensson, J.-E., Stenson, J.A.E., 2007. Models of ontogenetic allometry in cladoceran morphology studies. Hydrobiologia 594, 109–116. Lai, J., Yang, B., Lin, D., Kerkhoff, A.J., Ma, K., 2013. The allometry of coarse root biomass: log-transformed linear regression or nonlinear regression? PLoS ONE 8, e77007. Larimore, R.W., 1949. Changes in the cranial nerves of the paddlefish, Polyodon spathula, accompanying development of the rostrum. Copeia 1949, 204–212. Lemaître, J.F., Vanpé, C., Plard, F., Pélabon, C., Gaillard, J.M., 2015. Response to Packard: make sure we do not throw out the biological baby with the statistical bath water when performing allometric analyses. Biol. Lett. 11, 20150144. Lovett, D.L., Felder, D.L., 1989. Application of regression techniques to studies of relative growth in crustaceans. J. Crust. Biol. 9, 529–539. Mascaro, J., Litton, C.M., Hughes, R.F., Uowolo, A., Schnitzer, S.A., 2014. Is logarithmic transformation necessary in allometry? Ten, one-hundred, one-thousand-times yes. Biol. J. Linn. Soc. 111, 230–233. Nevill, A.M., Bate, S., Holder, R.L., 2005. Modeling physiological and anthropometric variables known to vary with body size and other confounding variables. Yearb. Phys. Anthropol. 48, 141–153. Newcombe, C.L., 1948. An application of the allometry equation to the study of growth in Callinectes sapidus Rathbun. Am. Nat. 82, 315–325. Niklas, K.J., Hammond, S.T., 2014. Assessing scaling relationships: uses, abuses, and alternatives. Int. J. Plant Sci. 175, 754–763. Nomura, E., 1926. An application of a = kbx in expressing the growth relation in the freshwater bivalve, Sphaerium heterodon Pils. Sci. Rept. Tohoku Imp. Univ. (Ser. 4) 2, 57–62. Osborne, J.W., 2002. Notes on the use of data transformations. Pract. Assess. Res. Eval. 8, 6. Packard, G.C., 2012. Is non-loglinear allometry a statistical artifact? Biol. J. Linn. Soc. 107, 764–773.
Packard, G.C., 2014. Multiplicative by nature: logarithmic transformation in allometry. J. Exp. Zool. B 322, 202–207. Packard, G.C., 2015. Quantifying the curvilinear metabolic scaling in mammals. J. Exp. Zool. A 323, 540–546. Packard, G.C., 2016. Relative growth by the elongated jaws of gars: a perspective on polyphasic loglinear allometry. J. Exp. Zool. B 326, 168–175. Packard, G.C., 2017a. Is complex allometry in field metabolic rates of mammals a statistical artifact? Comp. Biochem. Physiol. A 203, 322–327. Packard, G.C., 2017b. The essential role for graphs in allometric analysis. Biol. J. Linn. Soc. 120, 468–473. Pearsall, W.H., 1927. Growth studies. VI. On the relative sizes of growing plant organs. Ann. Bot. 41, 549–556. Pearson, K., 1897. Mathematical contributions to the theory of evolution. On a form of spurious correlation which may arise when indices are used in the measurement of organs. Proc. R. Soc Lond. 60, 489–498. Pélabon, C., Firmat, C., Bolstad, G.H., Voje, K.L., Houle, D., Cassara, J., Le Rouzic, A., Hansen, T.F., 2014. Evolution of morphological allometry. Ann. New York Acad. Sci. 1320, 58–75. Peters, R.H., 1983. The Ecological Implications of Body Size. Cambridge University Press, Cambridge. Pinheiro, J.C., Bates, D.M., 2000. Mixed-Effects Models in S and S-PLUS. Springer, New York. Reeve, E.C.R., Huxley, J.S., 1945. Some problems in the study of allometric growth. In: Le Gros Clark, W.E., Medawar, P.B. (Eds.), Essays on Growth and Form Presented to D’Arcy Wentworth Thompson. Oxford University Press, Oxford, pp. 121–156. Reyment, R.A., 1971. Multivariate normality in morphometric analysis. Math. Geol. 3, 357–368. Ritz, C., Streibig, J.C., 2008. Nonlinear Regression with R. Springer, New York. Robb, R.C., 1929. On the nature of hereditary size limitation. II. The growth of parts in relation to the whole. Brit. J. Exp. Biol. 6, 311–324. Sartori, A.F., Ball, A.D., 2009. Morphology and postlarval development of the ligament of Thracia phaseolina (Bivalvia: Thraciidae), with a discussion of model choice in allometric studies. J. Mollusc. Stud. 75, 295–304. Smith, R.J., 1980. Rethinking allometry. J. Theor. Biol. 87, 97–111. Smith, R.J., 1984. Allometric scaling in comparative biology: problems of concept and method. Am. J. Physiol. 246, R152–R160. Snedecor, G.W., 1937. Statistical Methods Applied to Experiments in Agriculture and Biology. Collegiate Press, Ames, IA, USA. Strauss, R.E., Huxley, J.S., 1993. The study of allometry since Huxley. In: Problems of Relative Growth, new edition. Johns Hopkins University Press, Baltimore, pp. xlviii–lxxv. Thompson, D.H., 1934. Relative Growth in Polyodon. Biological Notes, No. 2. Illinois Natural History Survey, Urbana, IL, USA. Thompson, D.W., 1943. On Growth and Form − A New Edition. Macmillan, New York. Warton, D.I., Wright, I.J., Falster, D.S., Westoby, M., 2006. Bivariate line-fitting methods for allometry. Biol. Rev. 81, 259–291. West, D., West, B.J., 2012. On allometry relations. Int. J. Mod. Physics B 26, 18. White, C.R., Kearney, M.R., 2014. Metabolic scaling in animals: methods, empirical results, and theoretical explanations. Compr. Physiol. 4, 231–256. White, E.P., Xiao, X., Isaac, N.J.B., Sibly, R.M., 2012. Methodological tools. In: Sibly, R.M., Brown, J.H., Kodric-Brown, A. (Eds.), Metabolic Ecology: A Scaling Approach. Wiley, Hoboken, pp. 9–20. Xiao, X., White, E.P., Hooten, M.B., Durham, S., 2011. On the use of log-transformation vs. nonlinear regression for analyzing biological power laws. Ecology 92, 1887–1894.
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