Morphological traits often covary within and among species according to simple power laws referred to as allometry. Such allometric relationships may result from common growth regulation, and this has given rise to the hypothesis that allometric exponents may have low evolvability and constrain trait evolution. We formalize hypotheses for how allometry may constrain morphological trait evolution across taxa, and test these using more than 300 empirical estimates of static (within-species) allometric relations of animal morphological traits. Although we find evidence for evolutionary changes in allometric parameters on million-year, cross-species time scales, there is limited evidence for microevolutionary changes in allometric slopes. Accordingly, we find that static allometries often predict evolutionary allometries on the subspecies level, but less so across species. Although there is a large body of work on allometry in a broad sense that includes all kinds of morphological trait–size relationships, we found relatively little information about the evolution of allometry in the narrow sense of a power relationship. Despite the many claims of microevolutionary changes of static allometries in the literature, hardly any of these apply to narrow-sense allometry, and we argue that the hypothesis of strongly constrained static allometric slopes remains viable.

Most quantitative traits are highly evolvable (Houle 1992; Hansen et al. 2011) and respond rapidly to both artificial (e.g., Hill and Caballero 1992) and natural selection (e.g., Endler 1986; Hendry and Kinnison 1999; Kinnison and Hendry 2001; Hendry et al. 2008). At the same time there is ample evidence for inertia (slow adaptation) from phylogenetic comparative studies (Hansen 2012), and prolonged episodes of stasis are commonly observed in the fossil record (e.g., Simpson 1944; Eldredge and Gould 1972; Gingerich 2001; Gould 2002; Estes and Arnold 2007; Hunt 2007; Uyeda et al. 2011; Bell 2012). These contrasting patterns have puzzled evolutionary biologists for decades and call for a better understanding of the relative contribution of natural selection and evolutionary constraints on the location of species in morphospace (Bradshaw 1991; Williams 1992; Björklund 1996; Schluter 2000; Arnold et al. 2001; Hansen and Houle 2004; Brakefield and Roskam 2006; Polly 2008; Futuyma 2010; Hansen 2012). Evolutionary stasis may be reconciled with high evolvability if individual traits are constrained by genetic correlations (e.g., Walsh and Blows 2009). In particular, many morphological traits are strongly correlated with body size through scaling relations that often follow a power law, Y = AXb, where Y is trait size and X is body size (Huxley 1924, 1932; Huxley and Teissier 1936). Such allometric scaling relationships are also common for physiological and life-history traits (e.g., Schmidt-Nielsen 1984; Charnov 1993; Kozlowski and Weiner 1997; West et al. 1997; Frank 2009). On a log–log scale, a power law yields a linear equation with intercept log(A) and slope b. Ontogenetic, static, and evolutionary allometries are recognized depending on whether the relation is taken over the development of an individual, across individuals at a similar developmental stage within a population, or across separate evolutionary lineages (Cock 1966; Gould 1966a; Cheverud 1982).

Hypotheses to explain evolutionary, cross-species, allometries fall in two broad classes: those based on functional adaptation between traits and those based on developmental constraints on the evolution of the traits. Huxley and others considered within-species allometry as a potential constraint on morphological trait evolution (Huxley 1932; Simpson 1944; Rensch 1959; Gould and Lewontin 1979; Gould 1971, 2002). This presupposes that parameters of the ontogenetic and static allometries represent meaningful biological traits that have limited potential to evolve. A common interpretation of the ontogenetic and static allometric slopes has been that they represent a ratio of proportional growth between the trait and overall size (Huxley 1932; Savageau 1979; Lande 1985; Stevens 2009) with low evolvability (e.g., Huxley 1932, p. 214). The interpretation and evolvability of allometric intercepts have been less clear (but see White and Gould 1965; Egset et al. 2012).

A strict form of the allometric-constraint hypothesis is that evolutionary changes are bound to follow trajectories imposed by ontogenetic or static allometries, so that evolutionary allometries must resemble these. This requires that both the allometric slope and intercept stay constant. Simpson (1944) used this assumption to derive “selective” explanations for seemingly maladaptive extreme trait values such as the antlers of the Irish elk (Megaloceros giganteus), and Rensch (1959, p. 156) went as far as to elevate it into a methodological principle when he claimed that allometry “is an important help to paleontologists, as one can find out about the size of organs in animals of a body size exceeding that of the types so far.” A related argument is that evolutionary allometries would arise from static allometries if selection acts solely on body size (Gould 1975; Lande 1979, 1985). Gould also assumed a constant allometric slope when he used shifts in allometric intercepts to measure heterochronic acceleration and retardation across taxa (White and Gould 1965; Gould 1966a, 1971, 1977). Hence, there are different allometric-constraint hypotheses depending on whether it is the slope or the slope and the intercept that are thought to be evolutionary fixed, and on whether this happens at the ontogenetic or at the static level.

During the last decades, this complex of allometric-constraint hypotheses has been the subject of many studies that have used differences in static allometry as evidence for evolvability and against the constraint hypotheses (e.g., Weber 1990; Emlen and Nijhout 2000; Wilkinson 1993; Frankino et al. 2005, 2007; Shingleton et al. 2007; McGuigan et al. 2010). However, Houle et al. (2011) pointed out that the majority of these studies have conceptualized allometry as any monotonic relation between trait and body size, which is much broader than the classic narrow sense of allometry as a power relation sensu Huxley (1932), Huxley and Teissier (1936), Gould (1966a, 1971, 1977), Cheverud (1982), Lande (1979, 1985), Schmidt-Nielsen (1984), Eberhard (2009), and many others. Houle et al. (2011) argued that it is the derivation of the allometric power law from simple morphological growth regulation (Huxley 1924, 1932; Savageau 1979; Lande 1985; Stevens 2009) or from allocation models (Bonduriansky and Day 2003) that motivates the constraint hypothesis. Although studying the evolvability and evolution of broad-sense allometry, as done by Emlen (1996), Baker and Wilkinson (2001), Frankino et al. (2005, 2007, 2009), Okada and Miyatake (2009) and others, is interesting and fully justified, it is not a test of the classic allometric-constraint hypotheses. The current consensus of labile allometric relationships is therefore largely based on studies that have neither estimated nor directly selected upon narrow-sense allometric parameters as defined by Huxley.

In this study, we first clarify the theoretical relation between the three levels of allometry (ontogenetic, static, and evolutionary) to articulate different versions of the constraint hypothesis and to provide testable predictions to distinguish between them. We then test some of these predictions using morphological data from more than 300 static allometric relationships. Finally, we evaluate the evolutionary potential of the static allometric parameters based on reports in the literature.



Huxley (1924, 1932) showed that if two morphological traits grew in proportion to each other, variation in growth rate would necessarily produce a pattern of covariation between them that follows a power law. Later this was refined by Savageau (1979), who showed that the essential feature for the power law to arise is that the growth dynamics is one dimensional in the sense that the growth of the whole trait complex is regulated by a single underlying variable (see also von Bertalanffy 1969 and Lande 1985). To illustrate this developmental model, let us assume that the specific growth rate of two metric traits, Yt and Xt, at any one point in time, t, is controlled by a common factor, zt, as

display math(1a)
display math(1b)

where ry and rx are constants, and the common factor zt can follow any form of complicated time-dependent dynamics. It follows that trait values at any time fulfill the equations

display math(2a)
display math(2b)

where math formula is a complex biological variable describing growth regulation, and Xo and Yo are initial values of the traits. This model implies

display math(3)

where a = ln(Yo) – b ln(Xo). Note that this allometric relation does not depend on the common growth factor, Z.

If observations are made on the same individual at different ages or developmental stages, represented by the parameter t, the resulting different values of Zt give rise to different values of y = ln[Y] and x = ln[X]. The variation in y with x during growth will follow a linear relationship on log-scale according to equation (3), which can be estimated by the ontogenetic allometric model

display math(4)

where bo and ao describe the ontogenetic allometry. We use yo and xo to signify that y and x vary over an ontogeny.

Any trait variation generated by genetic or environmental differences in realization of the growth factor Z among individuals in a population will also follow a power law summarized by the static allometric model. We use

display math(5)

to describe static allometry. Note that ontogenetic and static allometries are the same if there is no variation in bo and ao across individuals. It does not matter if the sources of variation in Z are different (genetic, environmental, or age related), because this variation only places individuals at different points along the trajectory defined by the ontogenetic allometry without affecting the shape of the relation.

Assume that the growth trajectories of x and y of every individual in a population follow the ontogenetic allometric model yo = boxo + ao, and that each individual is now observed at a given developmental stage with size xo = xs and trait value yo = ys, then, if the static allometric slope is estimated as an ordinary least-squares regression coefficient, and we assume a multivariate normal distribution of all three parameters ao, bo, and xs, we obtain

display math(6)

where bbo,x and bao,x are linear regression slopes of the ao and bo on log size xs. Hence, the static allometric slope is a sum of the average ontogenetic allometric slope in the population and two regression slopes of the ontogenetic parameters on log size (see Pélabon et al. 2013 for a related decomposition).

The effect of static allometry on the evolutionary allometric slope can be similarly derived by assuming a multivariate normal distribution of the underlying parameters across species:

display math(7)

where bbs,x and bas,x are the across-species linear regression coefficients of the parameters as and bs on species mean log size, xe. Equation (7) tells us that an evolutionary allometric slope may deviate from the static allometric slope if static slopes or intercepts are correlated with body size across species. For example, a correlation between intercept and body size may arise from selection if there is a functional relation between the trait and body size. Hence, for static allometry to act as an absolute constraint on coadaptation between the trait and body size (or two traits), it is necessary that both the static slope and the intercept lack evolvability. In this case, different species must lie along the static allometric trajectory (Fig. S1A). This is the basis of Lande's (1979, 1985) finding that selection acting exclusively on body size will produce an evolutionary allometry along the static allometry if additive genetic variances and covariances stay constant (i.e., constant genetic static allometric slope).


Assuming a static allometry between trait size ys and body size xs (eq. (5)) with body size centered on its mean and a multivariate normal distribution of the parameters (as, bs, xs), the variance in trait size, Var[ys], can be expressed as a function of variation in the underlying parameters of the static allometry:

display math(8)

Equation (8) shows that the contribution of the different parameters to trait variance is not additive (i.e., not orthogonal), and we will have to consider interactions when evaluating the relative contribution of the different parameters. Both variation in size and allometric intercept can generate some variation in the trait by themselves, whereas variation in slope does not generate trait variation unless size also vary (see Fig. S1 for a graphical representation).

To quantify the constraints that arise, we can ask what happens if some of the allometric parameters or body size are held constant in the sense that their means are not allowed to change. This can be answered by calculating the trait variance conditional on different parameter combinations. If the variation is considered to be additive genetic, this gives a direct measure of genetic constraint as it is equivalent to computing conditional evolvabilities (Hansen et al. 2003; Hansen and Houle 2008). In Table 1, we present the relevant conditional variances (see supplementary Appendix for calculations). Interpreted as conditional evolvabilities these give the trait evolvability that would result if stabilizing selection kept the mean of the conditioning parameter(s) constant (Hansen et al. 2003). For example, the additive genetic variance conditional on the allometric intercept, Var[y|a], gives the evolvability of the trait when the allometric intercept is under stabilizing selection .

Table 1. Constraints on trait variance. The constraints are quantified by calculating the decrease in trait variance that results when conditioning on the constraining parameter(s). “Constraint” refers to which variable(s) are kept fixed in the expression for trait variance (eq. (8)). “Model” gives the expression of the variance that is left in the trait given the specified constraint. “Allometry” refers to conditioning on both the intercept, a, and slope, b, in the allometric relation. “Shape” is the trait variance conditional on size Var[y|x]. The constraint “intercept on Var[y|x]” refers to keeping both the intercept and size constant, whereas “slope on Var[y|x]” refers to keeping both the slope and size constant. All relations are computed under the assumptions of a multivariate normal distribution of a, b, and x and that size, x, is mean centered across the compared taxa


  1. Var[a|x] = Var[a] – Cov[a, x]2/Var[x], Var[x|a] = Var[x] – Cov[a, x]2/Var[a], Var[b|a] = Var[b] – Cov[a, b]2/Var[a], Var[a|b] = Var[a] – Cov[a, b]2/Var[b], and Var[x|b] = Var[x] – Cov[b, x]2/Var[b]. Cov[b, x|a] = Cov[b, x] – Cov[a, b]Cov[a, x]/Var[a].

SizeEx[Var[y|x]] = Var[a|x] + Var[b]Var[x] + Cov[b, x]2
InterceptEa[Var[y|a]] = Var[x|a](E[b]2 + Var[b|a]) + Cov[b, x|a]2
SlopeEb[Var[y|b]] = Var[a|b] + Var[x|b] (E[b]2 + Var[b])
AllometryEb[Var[y|a, b]] = (E[b]2 +Var[b]) (Var[a]Var[b]Var[x] + 2Cov[a, b]Cov[a, x]Cov[b, x] – Cov[a, b]2Var[x] – Cov[a, x]2Var[b] – Cov[b, x]2Var[a])/(Var[a]Var[b] – Cov[a, b]2)
Intercept on Var[y|x]Ex[Var[y|a, x]] = (Var[x]) (Var[a]Var[b]Var[x] + 2Cov[a, b]Cov[a, x]Cov[b, x] – Cov[a, b]2Var[x] – Cov[a, x]2Var[b] – Cov[b, x]2Var[a])/(Var[a]Var[x] – Cov[a, x]2)
Slope on Var[y|x]Var[y|b, x] = Var[a|b, x] = (Var[a]Var[b]Var[x] + 2Cov[a, b]Cov[a, x]Cov[b, x] – Cov[a, b]2Var[x] – Cov[a, x]2Var[b] – Cov[b, x]2Var[a])/(Var[b]Var[x] – Cov[b, x]2)

The trait variance conditional on size, Var[y|x], is simply the residual trait variance around the allometric regression. We can view this as a quantification of Gould's (1977) concept of “dissociation” of trait from size. Hence, the conditional variances Var[y|x, a] and Var[y|x, b] quantify the ability of the trait to “dissociate” from size (i.e., for shape evolution) when, respectively, the intercept and slope are kept constant.


The trait variance in equation (8) and the conditional variances in Table 1 can be computed for allometric relations at any level (ontogenetic, static, or evolutionary). We will use the conditional variances at the among-taxa level to assess how much of evolution in trait size and shape can be explained by changes in the different static allometric parameters, as well as by changes in size. For example, to assess the influence of changes in the allometric slope on trait evolution, we compute how much the among-population and among-species trait variances are reduced by conditioning on the slope. The relative decrease in trait variance would then indicate to what extent changes in slope have been important in generating trait diversity across populations and species. This procedure allows us to compare the influence of the different parameters on a common scale set by the trait variance.

Materials and Methods


To evaluate the extent to which allometries have evolved, we calculated the among-taxa (populations and species) variances in allometric parameters in a set of published studies obtained by searching for “ontogenetic allometry” and “static allometry” in ISI Web of Knowledge. We also investigated citations in previous reviews on static allometries by Kodric-Brown et al. (2006), Bonduriansky (2007), and Eberhard (2009). We constrained our search to the animal kingdom and considered only studies that estimated the allometric parameters on log-transformed morphological data analyzed using ordinary least-squares regression, that reported standard errors of their estimates, and that reported comparable results for at least two taxa. We did not use studies in which allometric exponents are estimated from loadings on first principal components (Jolicoeur 1963), because it is then unclear whether differences in the estimated exponents are due to differences in allometry or to differences in the direction of the principal component (i.e., due to different measures of “size”). For completeness, we list such studies in supplementary Table S1. In total, we were only able to use 14 of 75 studies that reported what they defined as static (N = 52) or ontogenetic allometries (N = 23). Excluded studies are listed in Tables S2 and S3. In one case, the study of Kelly et al. (2000) on guppy, Poecilia reticulata, we reanalyzed the data, which were kindly provided by the authors. We corrected for sampling error in the among-taxa slope and intercept variances by subtracting the mean of the squared standard errors of the parameters.


None of the studies we found in the literature reported all the necessary parameters to analyze the relation between static and evolutionary slopes using equation (7). We therefore reanalyzed data from Baker and Wilkinson (2001), Kawano (2002, 2004, 2006), and Swallow et al. (2005) kindly provided by the authors. These data from holometabolic insects enabled us to estimate a total of 304 morphological static allometries that could be organized into 21 groups, each containing at least eight homologous static allometries. Twelve groups consisted of static allometries from populations belonging to the same species, whereas nine consisted of static allometries from species belonging to the same genus. All these are based on length measures of both the trait and body size.

We used the same data to investigate how trait and shape variation across taxa depend on variation in body size and the allometric parameters (eq. (8)). We decided that a minimum of 10 homologous static allometries was necessary to estimate the parameters in the models listed in Table 1. Eleven groups of allometries satisfied this criterion, five within species and six across species within the same genus. All analyses were conducted in R (R Development Core Team 2012).


To assess the short-term evolvability of allometry, we searched the literature for studies that had estimated additive genetic variation in both intercept and slope of morphological narrow-sense allometries by breeding studies or artificial-selection experiments. Studies were obtained by complementing the search described in the above section with a search for “allometry” combined with “artificial selection” in ISI Web of Knowledge. In total, we only found two studies that provided estimates of additive genetic variances in allometric slopes on appropriate scales: Atchley and Rutledge (1980) on ontogenetic allometry and Egset et al. (2012) on static allometry. In addition, we reanalyzed data from two studies: Tobler and Nijhout (2010), who used reduced major-axis regression (data for reanalysis kindly provided by the authors) and Cayetano et al. (2011), who did not report R2 values (data for reanalysis obtained from Dryad).



We were only able to use 10 of 52 studies that reported static allometric parameters. Adding the insect data we reanalyze and data on guppy populations from Kelly et al. (2000), we were able to compare 37 sets of homologous static allometric slopes across taxa (Table 2). All but three of these sets showed positive variance after we controlled for sampling error, indicating that allometric slopes do vary, but for the most part this variation was moderate (Figs. 1-3). There were a few cases of extreme differences in slope, but these involved allometries with very small or unreported R2. Among studies with a reasonable fit to the allometric model (median R2 > 50%), the standard deviation (SD) of the slope varied between 0.01 and 0.63 with a median of 0.19. The highest variation in the slopes of well-fitting allometries was found for mandible lengths within the beetle genus Odontolabis. The mean static slope of the 23 analyzed species in this genus was 2.45, with a SD of 0.63, which means that 95% of the slopes are expected to be between 1.18 and 3.72 if normally distributed.

Table 2. Variation in static slopes and intercepts across populations and species. Means and standard deviations (SD) of static slopes (bs) and intercepts (as) across a set of taxa. A SD of zero means that the observed variance was less than the mean sampling variance of the parameter. Median R2 values and their 25–75% quartiles (qt) are given for each group of homologous static allometries. A question mark after trait or size means that the scale was not reported. All traits and overall size measures are in log mm if not otherwise is indicated
TaxonTraitSizeSexMean bs, SDMean as, SDMedian R2Reference
  1. Body size (x) variable centered on the population/species mean before the allometry was estimated.

  2. *Data reanalyzed in this study. 1: D. simulans (2 pop), D. melanogaster (2 pop), D. pseudoobscura (2 pop). 2: C. japonicus, C. opilio, C. japonicus/opilio. 3: Longest distance across carapace of the lower lateral margin. 4: Tip of snout to insertion of the caudal fin. 5: C. damarensis, C. Hottentotus, G. capensis, B. suillus. 6: C. chloris, C. erythrinus, F. coelebs. 7: Trait explanations for Anderson et al. (2012). ACC = m. accelerator linguae; ENT = entoglossal process; HG = m. hyoglossus; TP = tongue pad.

  3. **Front of head to tip of wings.

  4. ***Front of head to tip of elytron.

  5. #Tip of nose to base of tail.

Scathophagidae (13 sp.)Testis (?)Hind tibia (?)m0.695, 0.2720.56 (0.74 – 0.50)Hosken et al. (2005)
 Clasper (?)Hind tibia (?)m0.287, 0.2700.41 (0.66 – 0.21)Hosken et al. (2005)
 Mandible (?)Hind tibia (?)m1.002, 0.2250.83 (0.94 – 0.73)Hosken et al. (2005)
Diopsidae (30 sp.)Eye spanLength**m1.645, 0.4550.697, 0.697†0.90 (0.92 – 0.87)Voje and Hansen (2013)
 Eye spanLength**f1.188, 0.1920.638, 0.322†0.89 (0.93 – 0.82)Voje and Hansen (2013)
C. dalmanni (8 pop.)Torax widthLength**m0.971, 0.0110.258, 0.019†0.80 (0.73 – 0.86)Voje and Hansen (2013)
 Eye spanLength**m1.818, 0.2460.896, 0.024†0.94 (0.95 – 0.91)Voje and Hansen (2013)
 Torax widthLength**f1.131, 0.0000.236, 0.012†0.85 (0.83 – 0.89)Voje and Hansen (2013)
 Eye spanLength**f1.188, 0.0660.730, 0.010†0.92 (0.91 – 0.95)Voje and Hansen (2013)
2 pop. of 3 sp. of Drosophila1Sex combWing lengthm0.904, 0.1380.16 (0.22 – 0.13)Sharma et al. (2011)
Dermaptera (42 sp.)ForcepsPronotum widthm1.344, 0.566Simmons and Tomkins (1996)
 ElytraPronotum widthm0.932, 0.397Simmons and Tomkins (1996)
Pycnosiphorus (8 sp.)Mandible lengthLength***m2.019, 0.3410.229, 0.0580.87 (0.84 – 0.90)Kawano (2006)*
Prosopocoilus (41 sp.)Mandible lengthLength***m2.253, 0.4470.941, 0.1490.89 (0.88 – 0.95)Kawano (2006)*
Odontolabis (23 sp.)Mandible lengthLength***m2.449, 0.6330.984, 0.2770.83 (0.77 – 0.91)Kawano (2006)*
Nigidius (8 sp.)Mandible lengthLength***m1.396, 0.0930.464, 0.1140.85 (0.77 – 0.90)Kawano (2006)*
Lucanus (17 sp.)Mandible lengthLength***m1.814, 0.3031.096, 0.1170.89 (0.82 – 0.94)Kawano (2006)*
Dorcus (37 sp.)Mandible lengthLength***m2.366, 0.4890.932, 0.1920.92 (0.90 – 0.96)Kawano (2006)*
Aegus (10 sp.)Mandible lengthLength***m1.758, 0.2700.688, 0.1120.90 (0.87 – 0.95)Kawano (2006)*
P. giraffa (8 pop.)Genitalia lengthLength***m2.122, 0.0481.398, 0.0220.95 (0.94 – 0.96)Kawano (2004)*
P. giraffa (8 pop.)Mandible lengthLength***m0.239, 0.0500.882, 0.0180.82 (0.78 – 0.84)Kawano (2004)*
D. titanus (19 pop.)Mandible lengthLength***m1.533, 0.2561.245, 0.0440.95 (0.94 – 0.98)Kawano (2004)*
D. titanus (19 pop.)Genitalia lengthLength***m0.373, 0.0550.798, 0.0490.79 (0.71 – 0.86)Kawano (2004)*
C. atlas (10 pop.)Genitalia lengthLength***m0.438, 0.1141.111, 0.0280.67 (0.57 – 0.76)Kawano (2002)*
C. causasus (10 pop.)Genitalia lengthLength***m0.317, 0.0661.167, 0.0200.67 (0.61 – 0.76)Kawano (2002)*
D. reichei (9 pop.)Genitalia lengthLength***m0.260, 0.0960.649, 0.0580.69 (0.63 – 0.73)Kawano (2004)*
X. gideon (13 pop.)Genitalia lengthLength***m0.317, 0.0431.102, 0.0330.77 (0.70 – 0.83)Kawano (2004)*
3 sp. of Chionoecetes2Eye orbitSize measure3m1.009, 0.0270.95 (0.96 – 0.95)Oh et al. (2011)
 Rostral hornSize measure3m0.950, 0.0000.92 (0.93 – 0.92)Oh et al. (2011)
 Carapace lengthSize measure3m2.067, 0.1460.89 (0.89 – 0.88)Oh et al. (2011)
 Total weight (log g)Size measure3m2.969, 0.4080.95 (0.96 – 0.92)Oh et al. (2011)
B. episcopi (12 pop.)GonopodiumStandard body lengthm0.893, 0.067-0.383, 0.093Jennions and Kelly (2002)
P. reticulata (8 pop.)GonopodiumStandard body lengthm0.649, 0.1750.24 (0.31 – 0.15)Kelly et al. (2000)*
 Color spot (log mm2)Standard body lengthm2.045, 1.2200.08 (0.13 – 0.02)Kelly et al. (2000)*
P. reticulata (21 pop.)Caudal finSize measure4m0.781, 0.179-0.581, 0.4370.17 (0.28 – 0.11)Egset et al. (2011)
Four species5Reproductive tractLength#m0.980, 0.0000.26 (0.48 – 0.10)Kinahan et al. (2008)
 Length of penisLength#m0.245, 0.9980.35 (0.50 – 0.18)Kinahan et al. (2008)
Variation in ontogenetic slopes and intercepts across populations and species 
Three species6Bill widthMass (log g)Mix0.323, 0.023Björklund (1994)
 Bill depthMass (log g)Mix0.353, 0.039Björklund (1994)
 Bill lengthMass (log g)Mix0.410, 0.000Björklund (1994)
 Tarsus lengthMass (log g)Mix0.593, 0.092Björklund (1994)
 Wing lengthMass (log g)Mix0.778, 0.230Björklund (1994)
2 Gallus gallus domesticus F1 crossesShank length (log cm)Mass (log g)m0.399, 0.012Cock (1963)
 Shank length (log cm)Mass (log g)f0.394, 0.013Cock (1963)
 Shank widthMass (log g)m0.256, 0.025Cock (1963)
 Shank widthMass (log g)f0.223, 0.049Cock (1963)
M. fascicularis, M. nemestrina, M. mulatta, P. cynocephalusSymphysis widthMandibular lengthMix0.955, 0.000Vinyard and Ravosa (1998)
 Mandibular arch widthMandibular lengthMix0.753, 0.000Vinyard and Ravosa (1998)
C. calyptratus, F. pardalis, T. johnstoniJaw length7Snout–vent lengthMix0.870, 0.066Anderson et al. (2012)
 ENT length7Snout–vent lengthMix0.797, 0.058Anderson et al. (2012)
 Mass (log g)7Snout–vent lengthMix2.993, 0.197Anderson et al. (2012)
 HG mass (log g)7Snout–vent lengthMix2.593, 0.333Anderson et al. (2012)
 ACC and TP mass (log g)7Snout–vent lengthMix2.340, 0.122Anderson et al. (2012)
 ACC mass (log g)7Snout–vent lengthMix2.293, 0.171Anderson et al. (2012)
 Jaw length7Mass (log g)Mix0.281, 0.033Anderson et al. (2012)
 ENT length7Mass (log g)Mix0.261, 0.014Anderson et al. (2012)
 HG mass (log g)7Mass (log g)Mix0.860, 0.056Anderson et al. (2012)
 ACC and TP mass (log g)7Mass (log g)Mix0.777, 0.059Anderson et al. (2012)
 ACC mass (log g)7Mass (log g)Mix0.753, 0.086Anderson et al. (2012)
 ENT length7Jaw lengthMix0.793, 0.074Anderson et al. (2012)
 HG mass (log g)7Jaw lengthMix2.933, 0.604Anderson et al. (2012)
 ACC and TP mass (log g)7Jaw lengthMix2.610, 0.185Anderson et al. (2012)
 ACC mass (log g)7Jaw lengthMix2.543, 0.129Anderson et al. (2012)
 HG mass (log g)7ENT lengthMix3.153, 0.323Anderson et al. (2012)
 ACC and TP mass (log g)7ENT lengthMix2.813, 0.045Anderson et al. (2012)
 ACC mass (log g)7ENT lengthMix2.723, 0.150Anderson et al. (2012)
Figure 1.

The relationship between evolutionary (stippled lines) and static (black lines) allometries of nongenital traits across populations within species. The regression of evolutionary slopes on mean static slope across the six groups in this figure is: evolutionary slope = −0.24(±0.31) + 1.14(±0.21) × mean static slope, R2 = 88%. Data for Prosopocoilus giraffa and Dorcus titanus originally published in Kawano (2004) and for Cyrtodiopsis dalmanni in Swallow et al. (2005)

Figure 2.

The relationship between evolutionary (stippled lines) and static (black lines) allometries of nongenital traits across species belonging to the same genus. The regression of evolutionary slopes on mean static slope across the nine groups in this figure is: evolutionary slope = –0.98(±1.06) + 3.48(±0.54) × mean static slope, R2 = 32%. Data for Diasemopsis originally published in Baker and Wilkinson (2001) and for Pycnosiphorus, Prosopocoilus, Odontolabis, Nigidius, Lucanus, Dorcus, and Aegus in Kawano (2006).

Figure 3.

The relationship between evolutionary (stippled lines) and static (black lines) allometries of genital traits across populations within species. The regression of evolutionary slopes on mean static slope across the six groups in this figure is: evolutionary slope = –2.18 (±0.96) + 0.33 (±2.89) × mean static slope, R2 = 12%. Data for Chalcosoma caucasus and Chalcosoma atlas originally published in Kawano (2002) and for Prosopocoilus giraffa, Dorcus titanus, Dorcus reichei, and Xylotrupes gideon in Kawano (2004).

There was much less variation in static slopes at the subspecies level (SD = 0.07) than at the among-species level (SD = 0.27). A SD of 0.07 means that 95% of the populations are expected to fall within ± 0.14 slope units around the mean if the slopes are normally distributed. This may not be more than what could be expected due to phenotypic plasticity and sources of error alone. Two exceptions provided clear evidence for slope variation across populations. These involved the mandibles of the Lucanid beetle Dorcus titanus and the eyestalks in the stalk-eyed fly Cyrtodiopsis dalmanni, which had cross-population SDs of 0.26 and 0.25, respectively.

All 23 sets of comparisons of intercepts showed positive variation with median SDs of 0.02 and 0.15 at the subspecies and species levels, respectively (Table 2). The SD of a trait on log scale is approximately equal to the coefficient of variation of the trait on the original scale, and these two values hence mean that differences in intercept generated trait differences with SDs of 2% and 15% of the trait mean. As a benchmark, we may compare these to the median within-population coefficient of variation of 7.5% for quantitative traits from the review of Hansen et al. (2006). Hence, differences in intercept across subspecies generate only a small amount of trait variance, whereas there are substantial differences across species.


Only four of 23 studies that reported homologous sets of ontogenetic allometries satisfied our criteria for inclusion, but these yielded 29 sets for comparison. In general, these showed low levels of variation and 3 of the 29 sets even showed negative variance after we controlled for sampling error (Table 2). The median SD of ontogenetic slopes across species was 0.058 (range 0.000–0.605). A value of 0.058 means that 95% of the slopes are expected to be between 0.88 and 1.12 if normally distributed around a mean slope of 1 (i.e., isometry). We only found four homologous ontogenetic slopes estimated across population and their SDs ranged from 0.012 to 0.049. None of the studies reported R2, so we cannot judge how well these allometric relationships fit the Huxley model, and ontogenetic allometries may often be nonlinear (e.g., Huxley 1932; Deacon 1990; Pélabon et al. 2013). None of the ontogenetic studies reported intercepts.


Under the strict constraint hypothesis, evolutionary allometry should follow the static allometry, and most trait variation should be generated by differences in mean body size across populations or species. For nongenital traits, we found a good match between static and evolutionary allometries across populations within species in all the six species we considered (Fig 1), but the results were much more variable across species (Fig 2). This may reflect static allometries acting as constraint on shorter time scales, but less so on the million-year time scales that usually separate animal species. Genital traits showed a different pattern with evolutionary allometries much steeper than the static allometries in all cases (Fig 3).

Differences between static and evolutionary allometric slopes were due to nonzero regressions of either the allometric intercept or slope on body size across species (eq. (7); Table 3), although these regression coefficients tend to have large standard errors. The regression coefficient of intercept on log body size b(as,x) ranged from −5.11 to 3.69 (median 0.44), whereas the minimum and maximum coefficient of slope on log body size b(bs,x) ranged from −1.98 to 4.31 (median 0.03). The two coefficients were almost always opposite in sign and had compensatory effects on the difference between the static and evolutionary regressions.

Table 3. Comparisons of evolutionary and static allometries. “E. all.” = slope of the evolutionary allometry; “Pred. be” = predicted evolutionary slope using equation (7); “R2 E. all.” = the amount of variance explained by the evolutionary allometry; “s. all.” = static allometry; “SE” = standard error. The four last columns refer to the parameters in equation (7). All traits and overall size measures are in log mm
TaxonTrait typeSize measureN, sexE. all. (±SE)Pred. beR2 e. all.Mean R2 s. all.b(as,x) (±SE)b(bs,x) (±SE)E[bs] (±SE)E[x] (±SE)
  1. *Front of head to tip of elyton.

  2. **Front of head to tip of wing.

P. giraffaMandible lengthLength*8, m1.95 (±0.39)1.9480.61%94.69%1.96 (±4.73)−1.27 (±2.70)2.12 (±0.06)1.69 (±0.01)
D. titanusMandible lengthLength*19, m1.66 (±0.24)1.6773.64%95.20%3.43 (±2.68)−1.98 (±1.50)1.53 (±0.06)1.66 (±0.01)
C. dalmanniEye span lengthLength**8, m1.17 (±0.10)1.1495.58%92.38%−0.80 (±0.98)0.95 (±1.16)1.19 (±0.03)0.78 (±0.01)
C. dalmanniEye span lengthLength**8, f2.08 (±0.29)2.0889.43%93.72%0.41 (±2.86)−0.19 (±3.38)1.82 (±0.09)0.82 (±0.01)
C. dalmanniThorax widthLength**8, m0.87 (±0.15)0.9184.75%85.02%0.87 (±0.58)−1.40 (±0.72)1.13 (±0.03)0.78 (±0.01)
C. dalmanniThorax widthLength**8, f0.84 (±0.27)0.8762.14%80.04%1.37 (±0.62)−1.81 (±0.85)0.97 (±0.03)0.82 (±0.01)
Mean   1.431.4481.03%90.18%1.21−0.951.461.09
Median   1.421.4182.68%93.05%1.12−1.341.360.82
DiasemophsisEye spanLength**13, m1.78 (±0.18)2.0589.49%89.99%−3.29 (±1.13)4.31 (±1.47)1.91 (±0.10)0.80 (±0.02)
DiasemophsisEye spanLength**13, f1.43 (±0.12)1.4893.03%90.09%−0.57 (±0.45)0.91 (±0.60)1.31 (±0.03)0.81 (±0.01)
PycnosiphorusMandible length (log mm)Length*8, m0.97 (±0.49)0.9638.97%87.04%0.03 (±5.42)−1.00 (±5.23)2.02 (±0.15)1.10 (±0.01)
ProsopocoilusMandible length (log mm)Length*41, m1.37 (±0.17)1.4263.74%89.28%−1.93 (±1.10)0.77 (±0.78)2.25 (±0.08)1.42 (±0.02)
OdontolabisMandible length (log mm)Length*23, m0.78 (±0.11)0.8168.04%82.97%−3.63 (±1.49)1.30 (±0.92)2.45 (±0.14)1.53 (±0.03)
NigidiusMandible length (log mm)Length*8, m2.71 (±1.01)2.7654.39%84.52%0.48 (±3.14)0.73 (±2.22)1.40 (±0.08)1.21 (±0.01)
LucanusMandible length (log mm)Length*17, m2.71 (±0.29)2.8254.39%88.90%−1.94 (±1.65)1.96 (±1.12)1.81 (±0.09)1.50 (±0.02)
DorcusMandible length (log mm)Length*37, m1.28 (±0.09)1.2585.13%92.28%0.62 (±0.81)−1.20 (±0.57)2.37 (±0.09)1.45 (±0.02)
AegusMandible length (log mm)Length*10, m1.27 (±0.09)1.2895.94%89.59%0.81 (±0.55)−1.06 (±0.40)1.76 (±0.10)1.22 (±0.07)
Mean   1.591.6574.89%88.30%−1.050.751.921.23
Median   1.371.4268.04%89.28%−0.570.771.911.22
P. giraffaGenitalLength*8, m0.40 (±0.29)0.4124.81%81.54%0.12 (±1.85)0.03 (±0.97)0.24 (±0.02)1.69 (±0.01)
D. titanusGenitalLength*19, m0.77 (±0.26)0.7933.54%78.54%−0.37 (±0.90)0.47 (±0.43)0.37 (±0.02)1.66 (±0.01)
D. reicheiGenitalLength*9, m1.09 (±0.61)1.1731.44%69.20%0.44 (±2.38)0.32 (±1.35)0.26 (±0.04)1.47 (±0.01)
C. atlasGenitalLength*16, m1.25 (±0.26)1.2563.07%67.09%−5.11 (±2.20)3.36 (±1.24)0.43 (±0.03)1.76 (±0.01)
C. caucasusGenitalLength*10, m1.74 (±0.18)1.7491.99%66.82%3.69 (±3.77)−1.23 (±2.06)0.32 (±0.03)1.84 (±0.00)
X. gideonGenitalLength*13, m0.93 (±0.26)0.9854.19%77.22%1.89 (±0.95)−0.77 (±0.67)0.32 (±0.02)1.59 (±0.01)
Mean   1.031.0649.84%73.40%0.110.360.321.67
Median   1.011.0843.87%73.21%


Using the equations in Table 1, we can evaluate how much trait variation across taxa is linked to variation in the allometric parameters. For nongenital traits, only a median of 26% (range: 7–59%) of log-trait variance across species remained after conditioning on body size, whereas a median of 60% (range: 20–92%) remained after conditioning on slope and intercept combined (Table 4). The variance left when conditioning only on intercept (median 71%, range: 21–96%) was slightly lower than when conditioning only on slope (median 88%, range: 51–96%). The only nongenital trait we analyzed on the among-population level showed qualitatively the same result as the across-species analyses (Table 4).

Table 4. Groups of homologous allometries compared across populations. The numbers are percent of the predicted total trait variance (eq. (8)) that remains when conditioning on different parameters as listed in Table 1. Intrasp. comp. = intraspecific comparisons; Intersp. comp. = interspecific comparisons; m = male; f = female; N = number of taxa. All traits are linear measurements with units log mm
        Intercept onSlope on
    Ex[Var[y|x]]/Ea[Var[y|a]]/Eb[Var[y|b]]/Eb[Var[y|a, b]]/Ex[Var[y|a, x]]/Var[y|b, x]/
Intrasp. comp.         
C. atlasm16Genital38.31%4.84%78.24%3.69%0.41%37.90%
C. caucasusm10Genital8.43%0.38%96.43%0.38%0.19%8.23%
D. titanusm19Genital65.73%6.93%87.49%4.96%0.17%38.11%
X. gideonm13Genital43.65%3.94%99.04%2.54%0.19%30.35%
D. titanusm19Mandible28.93%59.97%63.76%55.23%0.91%14.26%
Intersp. comp.         
Diasemopsisf13Eye span6.93%61.55%75.21%57.18%0.35%5.40%
Diasemopsism13Eye span11.41%74.70%51.14%43.24%1.47%9.04%

For genital traits, the situation is different. Here, the allometric intercept was by far the most important parameter (Table 4). The median variance left in (log) genitals after conditioning on the intercept was 4.4%, whereas 41.0% and 92.0% was left when conditioning on body size and slope, respectively.


Conditioning traits on body size generally removes the majority of the variance across species and populations and the remaining (residual) variance can be considered as “shape” variance. How much of this residual variance is due to variation in allometric slopes and intercepts indicates their relative importance for “shape” evolution. The results in Table 4 show that this is overwhelmingly due to changes in the intercept. For nongenital traits, the intercept removes a median of 96.3% of the “shape” variance across species (Table 4). For genitals, a median of 99.8% of the variance is removed across populations by conditioning on the intercept (Table 4).

When “shape” is conditioned on the slope, a median of 80.9% of the shape variance in nongenital traits is removed at the species levels. For genital traits a median of 65.9% is removed. Given the large effect of the intercept, most of the shape variance accounted for by the slope must be due to interactions with the intercept.


Almost all studies that claim to estimate the evolutionary potential of allometric relations have either not analyzed the data on a log scale or used line-fitting methods that are not appropriate for estimation of parameters in the narrow-sense definition of allometry (see Discussion; excluded studies are reported in Table S4). Eventually, we were left with only four studies. Egset et al. (2012) found no indication of evolvability of the static slope of caudal fin area on body area in an artificial-selection experiment in the guppy. In contrast, our reanalysis of Tobler and Nijhout (2010) shows that females lines of tobacco hornworms selected for smaller body mass have steeper static slopes for wing mass than lines selected for larger body size (Fig. 4). The data of Cayetano et al. (2011) on genital traits in Callosobruchus showed a very poor fit to the allometric model (Fig. 5), and we argue in the discussion that it is not informative about the evolvability of narrow-sense allometry. Both Egset et al. (2012) and the reanalyzed results from Tobler and Nijhout (2010) show strong evidence of evolvability of the intercept whereas the results based on Cayetano et al. (2011) do not. For ontogenetic slopes, the only study is Atchley and Rutledge (1980), who found positive heritability for the slope of chest circumference (h2 = 0.25 ± 0.07) and tail length (h2 = 0.39 ± 0.08) on body weight within six laboratory strains of rats selected for more than 20 generations for bigger and smaller body weight. They did not report the variance components, however, and the genetic variances may thus have been very small and may also have contained components of dominance and prenatal maternal variance. Consistent with this possibility, there were no changes in the allometric slopes over the 20 generations of selection (range across lines: 0.35–0.37 for chest circumference and 0.45–0.47 for tail length). We regard this study as inconclusive.

Figure 4.

The only known example of a change in a narrow-sense static allometric slope under artificial selection. Based on reanalysis of an experiment reported in Tobler and Nijhout (2010). Static allometric relationships of log total wing mass, y, against log body mass, x, in females of the moth Manduca sexta are shown for strains after 10 generations of artificial selection for larger body mass (crosses, stippled line), smaller body mass (circles, dotted line), or not selected (triangles, unbroken line). The three regression lines are: up-selected: y = – 2.460 (±0.009) + 0.415 (±0.038)x, R2 = 52%; down-selected: y = – 2.608 (±0.030) + 0.639 (±0.056)x, R2 = 61%; not selected: y = – 2.409 (±0.007) + 0.581 (±0.061)x, R2 = 43%. The results for males are very similar (Fig. S2).

Figure 5.

Reanalysis of the change in static allometry during the experimental-evolution study in the seed beetle Callosobruchus maculatus from Cayetano et al. (2011). A and B show the differences in static allometries for two male genital traits after 18–21 generations of monogamic (black dots and black line) and polygamic mating (gray crosses and gray dashed line). The size measure, log elytra length, was centered on its mean before we estimated the allometric parameters. This means that the intercept gives the predicted trait size for a mean-sized individual.


Although it was developed into its modern form by Julian Huxley, the baptizer of the modern synthesis, allometry has rested uneasily within the neodarwinian paradigm (Gayon 2000). On a background of increasingly dominant functionalism, allometry was one of few concepts associated with structuralist ideas (Gould 2002; Amundson 2005). Indeed, allometry played a central role in Gould's structuralist challenge to the modern synthesis (e.g., Gould 1977, 2002; Gould and Lewontin 1979). For example, the idea of evolution by heterochrony presupposes the existence of ontogenetic constraints along which shape can evolve by shifts in timing or rate of development. Allometry is the simplest and most obvious example of such a constraint, and Gould (1974, 1977) used it to support the hypothesis that the giant antlers of the Irish elk had evolved by heterochrony extrapolating an ancestral allometry. This was not a new idea; architects of the modern synthesis such as Huxley (1932), Simpson (1944, 1953), and Rensch (1959) used extrapolations along ontogenetic allometries as alternative “Darwinian” explanations for apparent patterns of orthogenesis in which traits like antlers evolved beyond what seemed optimal for the trait in isolation. In an extensive review of all known comparisons of ontogenetic allometries across species, populations and lines of breeding, Cock (1966) concluded that “well-established differences in [the allometric slope] are relatively rare.” Accordingly, it became popular to test for matches between ontogenetic and evolutionary allometries, as in the well-debated case of horse skull dimensions (see modern treatment in Radinsky 1984).

More recent research has shifted from allometry as constraints toward a more dynamical view of allometry as an evolving and adapting entity (e.g., Kodric-Brown et al. 2006; Bonduriansky 2007; Eberhard 2009; Frankino et al. 2009). At first glance, the evidence against allometry as a constraint seems overwhelming, but exactly how and at what time scales allometry may act as a constraint is rarely precisely articulated. Moreover, allometry has turned in into a hypernym with different meanings coexisting in the literature, and the majority of evidence for evolving allometry is evidence for various forms of shape evolution and not evidence for changes in static allometric slopes in the original narrow sense of the term (Houle et al. 2011). Although not denying the abundant evidence for evolvability of shape that has been produced both with and without the label “allometry” attached, we suggest that it is too early to conclude that the “decades-old view of allometries as constraints to evolution is inaccurate and misleading” (Emlen and Nijhout 2000, p. 663), because we cannot rule out that the subset of traits that do show a close fit to the narrow-sense allometric model may indeed be constrained by common growth regulation. Our survey of the literature has revealed that narrow-sense allometries are quite invariant and do tend to predict evolutionary divergence below the species level, but less so on longer time scales.


Even after controlling for sampling error, we found variation in static allometric slopes and intercepts across species. This confirms that all aspects of static allometry are evolvable on the million-year time scales that usually separate animal species, but we have much less evidence of changes in ontogenetic allometries, which may simply reflect a lack of data. Some element of the variation in static allometric parameters may be due to phenotypic plasticity (Shingleton et al. 2009), but this is unlikely to account for all the findings, and this explanation is also inconsistent with invariable allometric parameters on short time scales as discussed later.


As expected, ontogenetic and static allometric slopes vary less across populations than across species. Many interpopulation studies of allometry involve traits with poor fit to the allometric model, and we could only identify two cases with both a good fit and substantial variation in slope. One case concerns a single population of the stalk-eyed fly C. dalmanni, which differs from six other populations in the static slope of eye-span (see Voje and Hansen 2013, for detailed analysis). This is truly an exception that confirms the rule, because these populations have been separated for several million years (Swallow et al. 2005). The other case involves populations of the stag beetle D. titanus. Here we have no data on separation times, but it can not be ruled out that these also have a long history of independent evolution. Thus, although we found many cases of phenotypic differences in static slopes across species, unambiguous examples of substantial differences in static slopes between recently diverged taxa are absent. Furthermore, despite many general claims for evolvable allometry in the literature, evidence for a response to selection of well-fitting narrow-sense static slopes is currently limited to the single case of a small change in wing allometry of Manduca sexta (Tobler and Nijhout 2010 and Fig. 4). Hence, the hypothesis that ontogenetic and static allometric slopes have low evolvability and are constrained on microevolutionary time scales is not falsified.


Genital traits differed from nongenital traits by showing a consistent difference between static and evolutionary slopes also on the subspecies level. Static slopes of male genitalia are still near constant, but consistently shallower than the evolutionary slopes. Hence, differences in genitalia size must have arisen through changes in the static intercept. This pattern extends the well-known finding of a fascinatingly consistent tendency for male genitalia to show very shallow allometries within species of arthropods (Eberhard et al. 1998; Eberhard 2009). One of the suggested explanations for this pattern is the “one-size-fits-all” hypothesis (Eberhard et al. 1998), which postulates that the shallow slopes of male genitalia is due to the fitness advantage of having a genital size that is appropriately adjusted to the most common size of females in the population (Eberhard et al. 1998). Our results suggest that differences in genital size across populations and species are mainly due to selection operating on the genitals themselves. Interestingly, secondary sexual traits like eyestalks in stalk-eyed flies do not behave like genital traits, as most of their variation is explained by body size. Extreme secondary sexual characters may thus be reached along the highway of increasing body size.


Changes in the allometric slope seem negligible as a source of morphological evolution. For most traits, changes in body size are the most important source of evolutionary change, but changes in intercept are also important, and for genital traits changes in intercept explain more evolutionary variation than size.

These differences could reflect differences in selection pressures or in evolvability. Although selection pressures are difficult to assess with our data, the numerous indications of low evolvability in the static slope opens the possibility that the lack of influence of the slope may result from its low evolvability. The relative influences of body size and intercept are more likely related to the frequency of directional selection acting on them. Although one can argue that body size is more evolvable than the typical trait intercept, because it is a complex many-dimensional trait (Houle 1998), we have no indications that trait intercepts are constrained on million-year time scales. Moreover, most of the among-population variance in genitalia was generated by variation in the allometric intercept and not by body size. This is not surprising given the generally shallow static allometries of genitalia (Eberhard et al. 1998; Eberhard 2009).


The minor effect of the static slope on trait size and shape variation across taxa does not in itself confirm a low evolvability of this parameter. An alternative hypothesis is that the static allometric slope is under stabilizing selection to secure functional size relationships between traits across different body sizes. One way to differentiate between these two hypotheses is to compare the evolvability of the static slope with the evolvability of the static intercept.

Of the eight studies we found that had used artificial selection to test the evolvability of the static slope, only three had estimated the relevant narrow-sense allometric parameters. In an artificial-selection study with guppy, Egset et al. (2012) did not find a response in the allometric slope of caudal fin area in relation to body area despite a robust response in the intercept. Similarly, Atchley and Rutledge (1980) detected no response in ontogenetic slopes during selection on body size in lab rats. They did find positive heritabilities within the lines at the end of the experiment, however, but in the absence of variance estimates these are not informative of evolvability (see Hansen et al. 2011). Cayetano et al. (2011) found changes in the allometric slopes of two of 11 genital traits in the seed beetle Callosobruchus maculates exposed to different levels of sexual selection. It is not clear that this result is interpretable within the Huxley framework, however, as the traits were almost unrelated to body size in the first place. Size (log elytra length) explained 0% and 31%, respectively, of the variance in log spine length and log aedagus flap length in monogamous lines and 19% and 4% in polygamous lines. The claim by Cayetano et al. (2011) that their results provide the first experimental demonstration of evolution of static allometric slopes as a function of sexual selection may therefore be technically correct in the sense that the slopes are statistically significantly different, but has little relevance in the framework of the narrow-sense allometric model. An alternative interpretation is that the traits do not fit the allometric model and that the observed changes reflect changes in nonallometric residual variation. We do not suggest any cut-off value for how strongly two traits should correlate to be meaningful in testing the constraint hypothesis, but we argue that it is important to have these issues in mind when evaluating the constraint hypothesis.

Currently, the only indication of evolvable narrow-sense allometric slopes in animal morphology is therefore the Tobler and Nijhout (2010) artificial-selection study on M. sexta (Fig. 4). This contrasts with several studies finding the allometric intercept to be evolvable on microevolutionary time scales (e.g., Owen and Harder 1995; Tobler and Nijhout 2010; Egset et al. 2012). To make progress, we need more quantitative-genetics studies specifically focused on investigating the evolutionary potential of narrow-sense static or ontogenetic allometric slopes.


Gould (1966a,1966b) proposed that larger-sized animals would evolve a shallower static slope to avoid nonadaptive extreme trait sizes in larger individuals. Assuming that the body size range in our data is large enough to detect such an effect, we found little support for Gould's hypothesis; the regressions of static slope on body size were just as likely to be positive as negative, and only two positive and two negative slopes of our 21 regressions were statistically significantly different from zero (see also Kawano 2006 and Voje and Hansen 2013).

The results from Tobler and Nijhout's (2010) artificial-selection study show that the evolution of static allometries may be linked to size evolution. A better understanding of this relationship will help the evaluation of the constraint hypothesis (Pélabon et al. 2013) and may also be crucial for understanding the evolution of traits with other scaling relationships than the power model, for example, threshold traits (e.g., Emlen 1996; Moczek 1998, 2003; Emlen and Nijhout 2000). Following the reasoning from Gould (1966a,1966b) and Pélabon et al. (2013), it may be expected that larger morphs in polyphenic populations have a shallower slope compared to smaller morphs. We are not aware of any tests of this hypothesis.


How to best estimate allometric relations has been a subject of much debate. Although some statisticians have viewed the fitting of “allometric” slopes merely as a descriptive exercise, it is essential to understand that testing Huxley's hypothesis requires the estimation of specific parameters from a mathematical model, most saliently the narrow-sense allometric exponent. Reduced major-axis regression and other nonparametric line-fitting methods are commonly used in studies of allometry, but these methods simply do not estimate the narrow-sense allometric exponent, and will often give wrong results if used for this purpose (Kelly and Price 2004; Hansen and Bartoszek 2012). Reduced major axis computes the slope as the ratio between the SDs of the variables. Hence, changes in a reduced major-axis slope can result from any change in the variation of either trait or body size, and therefore can not be taken as evidence for change in the narrow-sense allometric exponent.

The errors that can result when interpreting reduced major-axis slopes as estimates of narrow-sense allometric slopes are illustrated by the study of Kelly et al. (2000), who investigated (among other things) whether differences in predator regimes could affect the allometry between gonopodium length (transformed anal fin) and body length in eight populations of the Trinidadian guppy. Eight reduced major-axis regressions of gonopodium length resulted in slopes ranging from 0.99 to 2.01, and five of these were statistically significantly larger than 1 (Table S5). Negative allometry (slope < 1) is generally expected for primary sexual traits (Eberhard et al. 1998, 2009; Bernstein and Bernstein 2002) and these five significant regressions are, according to the authors, the first examples of positive allometric relationships between male genitalia and body size in a vertebrate species. Our reanalysis of the data with ordinary least-squares regression reveals that all eight regressions show negative allometry and three are statistically significantly less than unity (Table S5). Hence, these results are consistent with the expected negative allometric scaling of genital traits. Lüpold et al. (2004) provide a similar example in which a reduced major-axis slope of 2.09 is interpreted as falsifying negative allometric scaling of penis length in the bat Nyctalus noctula, even though the ordinary least-squares slope of 0.81 is consistent with the expected negative allometric scaling.

Choosing a proper scale is essential when estimating allometric parameters as it determines what hypotheses are being tested. Allometry sensu Huxley is about proportional differences in traits, which is why fitting a linear regression on an arithmetic scale is not estimating the relevant parameters in the allometric model. However, whether one should fit the power model by nonlinear regression on arithmetic scale or the linear model on log scale is less obvious. Conceptually, these two approaches estimate the same parameters, but the results can be very different (Zar 1968; Smith 1980, 1984; Kerkhoff and Enquist 2009; Packard 2009, 2012; Packard et al. 2011). Fitting the model on an arithmetic scale is justified if the residual deviations act additively on the arithmetic scale while fitting a model on log-transformed data is justified if the residual deviations act multiplicatively on the arithmetic scale. It is not expected that parameters estimated on a multiplicative (e.g., logarithmic) scale should minimize arithmetic deviations. The choice of scale should relate to a hypothesis of whether multiplicative or additive processes generate the deviations from the model. The challenge is that observational errors (the difference between the measured and the true value of the entity) may cause additive deviations, whereas biological “errors” (the true biological deviations from the model; Riska 1991; Hansen and Bartoszek 2012) are often multiplicative because biological phenomena like growth and metabolism are inherently multiplicative (e.g., Gingerich 2000). Xiao et al. (2011) advocate averaging between the two methods. In our opinion, this is hiding and not solving the problem, and a solution should be based on modeling and a principled argument as to what type of residual deviation is expected. In most cases, linear regression on a logarithmic scale will be preferable as a first approximation, because biological deviations are likely to be the dominant source of residual variation. Observational errors may also be multiplicative in some cases, and if they are additive then their effects on the log scale can be well approximated by use of mean-scaled arithmetic variances, which can be incorporated into the regression analysis as described in, for example, Hansen and Bartoszek (2012).


Eighty years have gone since Huxley (1932) suggested that common growth regulation was the cause of the commonly observed pattern of a linear scaling relationship between morphological traits and overall size on a log–log scale. Although early students of allometry interpreted this as a potential constraint on phenotypic diversity, recent work has more or less dismissed the constraint idea. In contrast, we have shown that the allometric exponent may be difficult to change on short time scales for traits that fits the model well, and may have biased the course of trait adaptation. There is, however, good evidence that allometries evolve on macroevolutionary time scales, and here they may be less important as constraints.


The authors thank R. H. Baker, K. Kawano, C. D. Kelly, H. F. Nijhout, J. G. Swallow, A. Tobler, and G. S. Wilkinson for their helpfulness and willingness to share their data with us. D. Houle and G. P. Wagner were helpful discussion partners at an early stage of this study and M. Björklund, C. J. P. Firmat, Ø. Holen, J. Merilä, T. O. Svennungsen, and two reviewers gave valuable comments on the manuscript. The work was supported by grant 196434⁄V40 from the Norwegian Research Council to CP at NTNU.


The doi for our data is 10.5061/dryad.v460h.