Julian Huxley showed that within-species (static) allometric (power-law) relations can arise from proportional growth regulation with the exponent in the power law equaling the factor of proportionality. Allometric exponents may therefore be hard to change and act as constraints on the independent evolution of traits. In apparent contradiction to this, many empirical studies have concluded that static allometries are evolvable. Many of these studies have been based, however, on a broad definition of allometry that includes any monotonic shape change with size, and do not falsify the hypothesis of constrained narrow-sense allometry. Here, we present the first phylogenetic comparative study of narrow-sense allometric exponents based on a reanalysis of data on eye span and body size in stalk-eyed flies (Diopsidae). Consistent with a role in sexual selection, we found strong evidence that male slopes were tracking “optima” based on sexual dimorphism and relative male trait size. This tracking was slow, however, with estimated times of 2–3 million years for adaptation to exceed ancestral influence on the trait. Our results are therefore consistent with adaptive evolution on million-year time scales, but cannot rule out that static allometry may act as a constraint on eye-span adaptation at shorter time scales.

Morphological traits often scale with a power of each other among individuals at the same developmental stage (Huxley 1932; Gould 1966a). An early interpretation of such static morphological allometries was that they resulted from a constant proportionality of growth between the traits under study (e.g., Huxley 1932). Allometric exponents were therefore viewed as characters with low evolvability and as potential constraints on the independent evolution of traits (e.g., Huxley 1932; Rensch 1959; Gould and Lewontin 1979; Lande 1979, 1985; Cheverud 1982; Gould 1989, 2002). A more recent interpretation is that static allometries can be understood as reaction norms, reflecting resource allocations to traits across a range of possible final body sizes (e.g., Emlen and Nijhout 2000; Bonduriansky and Day 2003; Kodric-Brown et al. 2006). In both interpretations, the allometric exponent reflects a developmental constraint that may or may not ensure a functional relation between traits and overall body size.

Over the last decades, the allometric-constraint hypothesis has been challenged by the argument that allometric exponents should be evolutionary labile like most quantitative traits (e.g., Eberhard and Gutiérrez 1991; Emlen and Nijhout 2000). Indeed, many researchers have argued or assumed that the slopes of static allometries are determined by selection on traits and size (Petrie 1988; Green 1992; Eberhard et al. 1998; Bernstein and Bernstein 2002; Hosken et al. 2005; Kodric-Brown et al. 2006; Eberhard 2009). It has been claimed that allometric slopes vary considerably among populations and closely related taxa, hinting at rapid evolution of allometric relationships (Emlen and Nijhout 2000; Shingleton et al. 2007; Frankino et al. 2009). Artificial selection experiments changing linear scaling relationships between traits on an arithmetic scale have also been used as arguments for high evolvability of allometric parameters (e.g., Weber 1990, 1992; Wilkinson 1993; Frankino et al. 2005, 2007, 2009).

Many recent studies of short- or long-term evolution of morphological allometries have, however, relied on a broader definition of allometry compared to how the concept was originally conceived. Instead of defining allometry as the study of proportional changes between traits (censu Huxley 1932; Huxley and Tessier 1936; Gould 1966a; von Bertalanffy 1969; Cheverud 1982; Lande 1985; Eberhard 2009) much recent literature use it to refer to any linear or nonlinear scaling relationship on nonlog-transformed traits (e.g., Emlen 1996; Emlen and Nijhout 2000; Baker and Wilkinson 2001; Moczek and Nijhout 2003; Frankino et al. 2005, 2007). In effect, this broad-sense allometry can be used to refer to practically any monotonic change in shape with size (see, e.g., review in Frankino et al. 2009). In contrast, Huxley's narrow-sense allometry entails the very specific power law relation Y=aXb, between a trait Y and body size X, which then yields the standard linear allometric equation log(Y) = log(a) +blog(X), where b is the allometric slope. It is the derivation of this power law from simple growth laws (e.g., Huxley 1924, 1932; Savageau 1979; Lande 1985; Stevens 2009) that justifies the allometry-as-constraint hypothesis, just as the interest in power laws in physiology, life-history theory, and other fields is motivated by specific models (e.g., Schmid-Nielsen 1984; Charnov 1993; Kozlowski and Weiner 1997; West et al. 1997; Frank 2009). The hypothesis that allometric relations constitute a constraint on evolutionary change must therefore refer to narrow-sense allometry to have general explanatory power. Scale transformations can completely change the meaning of the fitted model and linear regressions on arithmetic and proportional scales are not testing the same hypothesis. When Weber (1990, 1992), Wilkinson (1993), Emlen and Nijhout (2000), Baker and Wilkinson (2001), Frankino et al. (2005, 2007), Okada and Miyatake (2009), and others are reporting on selection responses or evolutionary changes in (broad-sense) allometry, they are studying shape evolution, but they are not estimating the theoretically relevant parameters for testing the allometric-constraint hypothesis of Huxley and Gould (Houle et al. 2011).

Consequently, little information about the evolution and evolutionary potential of narrow-sense allometry has been assembled. For example, Egset et al. (2012) seems to be the first experimental selection study that has specifically selected on and measured the response in narrow-sense static allometric slopes; they found no response over three generations of selection on a tail-fin allometry in guppy despite a robust response in the allometric intercept. Furthermore, although many studies have documented species differences in static allometric slopes (e.g., Simmon and Tomkins 1996; Tomkins and Simmons 1996; Kawano 2004, 2006; Hosken et al. 2005), there are yet no phylogenetic comparative studies that assess the tempo and mode of evolution in narrow-sense static or developmental allometries.

For this reason, we re-analyze data presented by Baker and Wilkinson (2001) on the evolution of eye span in relation to body size in Diopsid stalk-eyed flies to specifically test hypotheses about the evolution of the narrow-sense allometric exponent. The original study investigated the evolution of the within-species arithmetric slope of eye span on body length in relation to sexual dimorphism. Our analysis differs from Baker and Wilkinson (2001) in two important ways. First, we study the slope of log eye span on log body size, which corresponds to Huxley's allometric model. Hence, we test a hypothesis that was not tested in the original study. Second, to test the hypothesis that allometric slopes evolve adaptively in relation to strength of sexual selection, we use a phylogenetic comparative method specifically designed for adaptive evolution (Hansen 1997; Butler and King 2004; Hansen et al. 2008) in place of the independent-contrasts method used by Baker and Wilkinson (2001).

It has been argued that sexual selection in general favors the evolution of positive allometries (Petrie 1988; Green 1992, 2000; Simmons and Tomkins 1996, see Fairbairn 1997; Bonduriansky 2007; Eberhard 2009 for review and critical discussion of this hypothesis). The effects of selection on growth rates are poorly understood, however, and the theoretical basis of claims to the effect that directional sexual selection should favor steeper allometries has been questioned (Bonduriansky and Day 2003). The lack of a general theory does not invalidate the empirical hypothesis however, and Bonduriansky and Day (2003) also showed that steeper static allometries may evolve if there is a trade-off in resources between the trait and size, and the selection gradient on the trait is increasing faster with size than the selection gradient on size itself. This may fit with condition-dependent sexual selection. Here, we test the hypothesis that intra- and intersexual selection on eye span is selecting indirectly for steep static allometric slopes in species in which eye span is the main character determining male reproductive success. Species in the Diopsidae family are either sexually mono- or dimorphic in eyestalk length (Baker and Wilkinson 2001). In dimorphic species, eye span is used in male–male competition for access to females (e.g., Burkhardt and de la Motte 1983, 1985; Panhuis and Wilkinson 1999). Strong female preferences for males with large eye spans have also been documented for dimorphic species (Burkhardt and de la Motte 1988; Wilkinson et al. 1998; Hingle et al. 2001; Cotton et al. 2006), whereas such preferences have not been found in monomorphic species (Wilkinson et al. 1998).

Given the current consensus of evolvable static allometries and the strong sexual selection pressure favoring males with large eye span in dimorphic Diopsids, one could predict rapid adaptation of eye-span allometries to different levels of sexual selection. We test this in two ways by using sexual dimorphism and relative trait size as proxies for the influence of sexual selection. In addition to the cross-species dataset, we also analyze a dataset of seven highly sexually dimorphic populations of the species Cyrtodiopsis dalmanni from Swallow et al. (2005). We also test Gould's idea (1966a,b) that species with extreme body size would evolve shallow allometries to avoid potentially maladaptive consequences of extrapolating an allometric relation. Further, the Diopsidae is the only group of hypercephalic flies in which females also possess prominent head projections (Baker and Wilkinson 2001). Wilkinson (1993) demonstrated that females exhibit a correlated response to selection on male eye span in C. dalmanni, suggesting that males and females share some genetic mechanisms influencing eyestalk morphology. We therefore test whether female eye-span allometries evolve as a correlated response to changes in male allometries.

Material and Methods


Measurements of eye span and body length in 30 species of stalk-eyed flies used in the comparative analyses come from Baker and Wilkinson (2001). Data on eye span and body length in seven populations of C. dalmanni come from Swallow et al. (2005). All data (morphological and sequence data) were kindly provided by G. S. Wilkinson, R. H. Baker, and J. G. Swallow (pers. comm.). Eye span was measured across the outer edge of each eye and body length was measured from the front of the head to the end of the wing. Baker and Wilkinson (2001) and Swallow et al. (2005) provide more information on how the measurements were performed.


Comparative analyses require phylogenies with branching points reflecting the timing of speciation events. Data from three mitochondrial genes (cytochrome oxidase II, 12S ribosomal RNA, and 16S ribosomal RNA) and three nuclear genes (elongation factor-1a, wingless, and white) are available for all 30 species from which we have morphological data (Baker et al. 2001). Fragments of all six genes were compiled together and we constructed an alignment consisting of 3287 base pars.

Bayesian inference analysis was performed with MrBayes version 3.1.2 (Huelsenbeck and Ronquist 2001). A general time-reversible model with invariable sites and a gamma distribution (GTR + I +Γ) fitted the sequences best according to the information criteria AICc, AIC, and BIC in jModeltest version 0.1.1 (Posada 2008). We set the priors to match this model, but we did not fix any of the parameters. Two independent analyses were run simultaneously, each starting from different random trees. Each search was run with four Markov chains for 10,000,000 generations and trees were sampled every 10,000 generation. We discarded the first 2,750,000 generations as burn-in. Plots showing generation versus the log likelihood were inspected to ensure the two independent analyses had reached stationarity before the end of the burn-in.

The Bayesian inference tree is shown in Figure 1. The clade support (posterior probabilities) is generally high (mean 0.97), but the tree includes one polytomy, which includes the only Chaetodiopsis specimen. Paraphyly is common at the genus level and Sphyracephala and Diopsis are the only monophyletic genera.

Figure 1.

Bayesian inference of the phylogeny of the species included in the study, based on fragments of three mitochondrial genes and three nuclear genes (see text for more information). Priors were set to match a GTR+I+Γ model. All branching points have a posterior probability of 1.00, except for the ones with values indicated and the polytomy. The scale bar on the lower left side represents one change per 10 nucleotide positions. Gray boxes indicate sexual dimorphism in slope. The phylogeny (including branch lengths) has been deposited in Dryad.

Dating phylogenetic trees is challenging. Stalk-eyed diopsid flies occurred in North America and Europe during the Late Eocene and Oligocene (Grimaldi and Engel 2005). The fossil stalk-eyed fly Prosphyracephala succini (Loew), often found embedded in Bitterfeld amber (Kotrba 2004; Rossi et al. 2005), has been placed as a sistergroup to the Diopsinae, which includes all species in our study, except Teloglabrus entabenensis (Kotrba 2004). The age of Bitterfeld amber has been debated, but recent studies have argued that it is independent of the older and better known Baltic amber with a suggested absolute age of 25.3–23.8 million years (e.g., Dunlop and Mitov 2009 and references therein). A rough estimate of the minimal length of our phylogenetic tree is therefore 25 million years, but a much older root cannot be excluded.


Static allometric relationships were estimated using the linear version of the allometric equation log(Y) = log(a) +blog(X) using ordinary least-squares regression. We did not correct for measurement error, because information was not available. We centered all body-size data for each population on the population mean before estimating the allometries to control for the inherent negative correlation between slope and intercept (White and Gould 1965; Gould 1966a, Egset et al. 2011). Centering the dependent variable before the analysis changes only the estimate of the intercept. Confidence intervals for regression parameters were calculated using the MBESS package in R.

The best regression model for estimating allometric parameters has been a subject of debate, and major axis and especially reduced major axis regression is often used in place of ordinary regression. Unfortunately, neither of these alternative regressions account for true biological variation around the regression, and will typically give wrong results when such variation exists (Kelly and Price 2004; Hansen and Bartoszek 2012). Furthermore, the reduced major axis estimate of the regression slope is simply the ratio between the SDs of the two traits, and does not even contain the covariance between traits. This means that any change of variance in the traits will be interpreted as changes in the slope, regardless of whether the change is due to changes in the slope itself or to changes in biological or other sources of residual variation. For this reason, the finding of changes in reduced major axis slopes in the experimental selection study of Tobler & Nijhout (2010) does not imply a change in static allometry (Egset et al. 2012). Accordingly, we base all our analyses and arguments on standard regression methods.


The adaptation-inertia model (Hansen 1997; Butler and King 2004; Hansen et al. 2008; Labra et al. 2009) was fitted with R version 2.8.1 (R Development Core Team) using the SLOUCH 1.2 program. In SLOUCH, the evolution of the trait is modeled as an Ornstein–Uhlenbeck process, which can be represented by the stochastic differential equation:


where dy is the change in the trait, y, (e.g., male eye-span allometric slope) over a time step dt, α is a parameter measuring the rate of adaptation toward a primary optimum, θ, and σdW is a white-noise process having independent, normally distributed random changes with mean zero and variance σ2. The Ornstein–Uhlenbeck process includes both a deterministic tendency to evolve toward a primary optimal state (sensu Hansen 1997), as well as a stochastic component, which can be interpreted as evolutionary changes due to secondary noise generated by unmeasured selective forces, genetic drift, etc. The primary optimum, θ, is a linear function of a randomly changing predictor variable (sexual size dimorphism in our case). The predictor variable only influences the trait through its influence on the primary optimum. The predictor variable itself is only observed on the tips of the phylogeny, and its evolution through the phylogeny is modeled as a Brownian-motion process (see Hansen et al. 2008 for details). The method uses generalized least squares for estimation of the regression parameters (i.e., the influence of the predictor on the primary optimum) and maximum likelihood for estimation of α and σ2 in an iterative procedure.

The method returns an estimate of the regression of the primary optimum on the predictor variables that can be interpreted as the optimal adaptive relationship free of ancestral influence. Unless adaptation is instantaneous, this “optimal regression” will be steeper than the usual “evolutionary regression”, which also reflect maladaptation due to phylogenetic inertia. We report both of these regressions, as their ratio (the “phylogenetic correction factor,”ρ) can be used to judge the influence of constraint on adaptation. The relative effects of phylogenetic inertia (resistance of adaptation) and adaptation can also be described as a half-life, t1/2= ln(2)/α, defined as the mean predicted time it takes for a species to evolve halfway toward a new optimum (Hansen 1997). If the half-life is short relative to branch lengths on the phylogeny, then adaptation is quick and past history has little influence. The method also assesses the stochastic variance, σ2, in the evolutionary process, but returns it as an estimate of vy2/2α, which is the expected residual variance when adaptation and secondary changes have come to a stochastic equilibrium.

A phylogenetic effect can arise for two reasons. It can be due to phylogenetic inertia (slowness of adaptation) or it can be due to a phylogenetic effect in the environment to which the species are adapting. It is essential that it is only phylogenetic inertia and not general phylogenetic effects that are corrected for in the analysis (Hansen and Orzack 2005; Labra et al. 2009; Revell 2010). In our model, this is done by estimating phylogenetic inertia as the half-life in a model including a predictor variable (the adaptive effect). The overall phylogenetic effects we report are half-lives obtained from models without predictor variables (i.e., with only an intercept). The difference between these half-lives can tell us whether the phylogenetic effects are due to inertia or to the environment (see Labra et al. 2009).

Observation or measurement errors may reduce the accuracy of regression (Fuller 1987; Buonaccorsi 2010). We included estimation variance in the individual species parameters as observation variance in both predictor and response variables in the SLOUCH analysis. As both predictor and response variables in some of our analyses are derived from measures of eye span, we also estimated and included observation covariance between the variables. Because sample sizes differed by more than one order of magnitude between species, we averaged the sample variance across species (assuming the sampling distributions of the different species are the same) and then estimated the estimation variances of the individual species means by dividing with the sample size of each species (Hansen and Bartoszek 2012). We used a bootstrap procedure to estimate the observation variances of the eye-span ratios as well as the observation covariances between the variables. Data on male and female eye span were randomly sampled with replacement to create 10,000 pseudoreplicate datasets for each species from which the variance and covariances were calculated. We also quantified the bias in all regressions due to observation error by calculating a phylogenetically corrected reliability ratio K (Hansen and Bartoszek 2012), but with one exception this bias was very small.


Our test of the hypothesis that static allometries are formed by sexual selection requires a measure of the relative importance of sexual selection in the species. Unfortunately, there are no direct measures of sexual selection available for most species. Here, we used two different proxy measures: One based on sexual dimorphism and one based on relative development of the male trait. The use of sexual dimorphism to measure influence of sexual selection has a long history (e.g., Andersson 1994), and there has been debate about the best approach (Lovich and Gibbon 1992; Smith 1999; Fairbairn 2007). Here, we used the difference between log male and log female eye span. We also tested the arithmetic ratio of the male and female eye span, which gave similar results (as expected, because the two are the same to a first approximation). We chose to present results from the log difference because it explained slightly more variation and had slightly higher AIC value. Our relative-trait-size measure was the difference between log male eye span and log male body length. Again, we also ran the analyses for the arithmetic ratio and got qualitatively similar results with slightly worse fit.

Note that the SLOUCH model assumes that the predictor variables have a longer phylogenetic half-life than the model residuals (Hansen et al. 2008). This is well supported for log relative eye span, but for sexual dimorphism the half-life was similar to the model residuals, and this may explain the somewhat worse fit to this predictor.

To test Gould's hypothesis of shallower allometries in species with large trait values, we used mean male body size on an arithmetic scale as a predictor variable. To test for a correlated response in female allometries to changes in male allometries, we used female allometric slope as a response variable and male allometric slope as predictor variable. Because the correlated-evolution hypothesis suggests a direct and not a delayed response as in the case of adaptation, we set the phylogenetic correction factor to 1 such that the optimal and evolutionary regression are forced to be the same (see Hansen and Bartoszek 2012 for discussion). Note that this analysis still allows for phylogenetic correlations in the residuals. Finally, we studied the evolutionary allometry by regressing log mean eye span on log mean body size across species. This was also done both with a direct and a delayed response corresponding either correlated or adaptive evolution (both allowing phylogenetic correlations in the residuals).

A concern with many of these analyses is that the response and predictor variables partially derive from the same data. This introduces a circularity that may create spurious relationships. We do not think this a problem with the analyses we present. First, note that there is no direct statistical correlation between the allometric slope and the size of the trait, because there is no correlation between the slope and the intercept of a regression when the intercept is taken at the mean of the predictor variable. A second issue is that correlated estimation error in the slope and the predictor variables may give a spurious correlation in the comparative analysis. We dealt with this by explicitly modeling the estimation variances and covariances as measurement error in the comparative analysis. This problem was also minor, because the estimation covariances were always rather small. Third, it is of course possible that static allometry may coevolve with male and female trait or body sizes for reasons other than direct sexual selection, but this is not an issue that derives from circularity in the data, and we explore some such alternatives by evaluating the influence of trait and body size on the static allometry. We decided, however, to not present analyses of the relation of eye span itself to sexual dimorphism or relative trait development, as such regressions would be highly vulnerable to spurious correlations.



Mean body length, mean eye span, and eye-span allometric slopes and intercepts for both sexes in the 30 species are given in Table 1. Within-species regressions of log eye span on log body size show strong fit for both males and females in all species: r2-values range from 60% to 98% for males (mean = 87%, median = 89%) and 49% to 96% for females (mean = 85%, median = 89%). Pair-wise comparisons of male allometric slopes show that 267 of 435 comparisons are different from each other on a 95% confidence level. Diasemopsis longipedunculata has the steepest slope (2.69) and is different from all the other 29 species (Fig. 2). In contrast, only 94 of 435 female pair-wise slope comparisons are different from each other on a 95% confidence level. Males and females had statistically different allometric slopes in 18 species and different intercepts (mean trait sizes) in 26 species.

Table 1.  Morphometric measurements of 30 species of stalk-eyed flies. Morphometric measurements of body length (BL) and eye span (ES) in millimeters, allometric slope and intercept are the least-squares regression of log eye span as function of log body size (N= sample size).
 Species N Mean BL in mmMean ES in mmSlope±SEIntercept±SE r 2 (%)
 1 Eurydiopsis argentifera 1 69.4810.3736.5300.2400.875±0.1600.815±0.00385.26
 2 Diopsis apicalis 1 31 7.384 0.481 8.184 0.952 1.730±0.136 0.910±0.004 84.27
 3 Chaetodiopsis meigenii 2 407.3590.5567.4211.1712.025±0.1040.865±0.00390.72
 4 Diasemopsis sp. W 2 21 7.286 0.681 7.287 1.132 1.688±0.097 0.857±0.004 93.78
 5 Diopsis fumipennis 1 257.0900.4876.8570.6211.276±0.0820.834±0.00290.91
 6 Diasemopsis longipedunculata 2 75 7.086 0.696 8.171 2.185 2.691±0.076 0.896±0.003 94.54
 7 Diasemopsis hirsuta 2 677.0620.4757.1471.0552.148±0.0940.849±0.00388.66
 8 Cyrtodiopsis quinqueguttata 102 6.974 0.512 4.255 0.379 1.157±0.044 0.631±0.001 89.70
 9 Diasemopsis elongata 2 306.9170.7707.8651.8692.156±0.0640.883±0.00397.53
10 Diasemopsis silvatica 2 82 6.811 0.388 7.182 0.933 2.252±0.082 0.852±0.002 90.31
11 Diasemopsis aethiopica 1 1046.6070.5426.2770.9441.735±0.0620.793±0.00288.29
12 Diasemopsis conjuncta 2 65 6.508 0.565 5.670 0.841 1.639±0.074 0.749±0.003 88.33
13 Diasemopsis dubia 2 1286.3860.3576.8360.7181.763±0.0850.832±0.00277.23
14 Cyrtodiopsis whitei 2 86 6.367 0.981 7.965 2.164 1.844±0.045 0.880±0.003 95.44
15 Diasemopsis obstans 2 1246.3290.4186.3950.9342.216±0.0630.801±0.00290.88
16 Teleopsis breviscopium 2 57 6.322 0.629 10.985 2.182 1.722±0.175 1.033±0.008 63.02
17 Diasemopsis nebulosa 2 666.2490.4576.0110.8631.926±0.0780.774±0.00390.32
18 Cyrtodiopsis dalmanni 2 92 6.226 0.728 7.509 1.752 1.946±0.061 0.864±0.003 91.91
19 Teloglabrus entabenensis 1 185.9280.3771.1670.0901.194±0.0830.066±0.00292.43
20 Diopsis gnu 1 7 5.698 0.317 4.481 0.230 0.894±0.124 0.651±0.003 89.50
21 Diasemopsis fasciata 2 635.5960.4625.7470.8621.829±0.0670.754±0.00292.22
22 Sphyracephala munroi 45 5.293 0.281 2.856 0.222 1.326±0.118 0.454±0.003 74.11
23 Diasemopsis signata 1 655.0030.4144.0000.4651.356±0.0590.599±0.00289.09
24 Teleopsis quadriguttata 29 4.978 0.522 3.329 0.384 1.082±0.053 0.519±0.002 93.77
25 Teleopsis rubicunda 2 884.9200.5474.9661.0351.679±0.0750.686±0.00485.23
26 Sphyracephala brevicornis 1 32 4.868 0.422 1.831 0.141 0.825±0.064 0.362±0.006 84.13
27 Diasemopsis albifacies 634.7680.3333.3980.3521.415±0.0700.529±0.00286.64
28 Trichodiopsis minuta 2 41 4.652 0.362 4.028 0.590 1.813±0.108 0.600±0.004 87.64
29 Sphyracephala beccarii 2 994.5010.1442.0520.1001.201±0.0990.312±0.00160.07
30 Sphyracephala bipunctipennis 2 23 4.012 0.353 2.341 0.419 1.944±0.155 0.261±0.002 87.65
  Median 63.00 6.326 0.469 6.144 0.852 1.726 0.783 89.30
 Mean sexually dimorphic species61.386.3230.5125.9300.9471.7040.72087.53
  Mean sexually monomorphic species 50.17 5.483 0.389 4.405 0.516 1.408 0.605 85.49
 Species N Mean BL in mmMean ES in mmSlope±SEIntercept±SE r 2 (%)
  1. 1Only intercept differs between the sexes at the 95% confidence level (t-test).

  2. 2Slope and intercept are different between the sexes at the 95% confidence level (t-test).

 1 Eurydiopsis argentifera 1 99.9030.3516.7560.2360.874±0.1740.829±0.00375.21
 2 Diopsis apicalis 1 31 7.549 0.523 7.080 0.680 1.297±0.105 0.848±0.003 83.52
 3 Chaetodiopsis meigenii 2 277.5670.5316.4030.5471.110±0.1110.805±0.00379.37
 4 Diasemopsis sp. W2 22 7.391 0.653 6.014 0.656 1.221±0.053 0.776±0.002 96.23
 5 Diopsis fumipennis 1 217.1520.5915.9520.5511.048±0.0930.773±0.00386.20
 6 Diasemopsis longipedunculata 2 50 7.305 0.707 5.986 0.761 1.299±0.045 0.773±0.002 94.50
 7 Diasemopsis hirsuta 2 677.3430.5546.4310.7091.450±0.0610.805±0.00289.54
 8 Cyrtodiopsis quinqueguttata 57 7.149 0.455 4.283 0.375 1.127±0.065 0.636±0.002 84.08
 9 Diasemopsis elongata 2 297.3700.8126.1910.9281.403±0.0540.786±0.00395.99
10 Diasemopsis silvatica 2 112 6.772 0.489 5.277 0.480 1.210±0.037 0.720±0.001 90.69
11 Diasemopsis aethiopica 1 936.6580.5535.5010.7501.557±0.0600.736±0.00287.79
12 Diasemopsis conjuncta 2 65 6.333 0.601 4.714 0.566 1.215±0.052 0.670±0.002 89.61
13 Diasemopsis dubia 2 1076.5320.4005.1450.4221.268±0.0540.710±0.00183.91
14 Cyrtodiopsis whitei 2 86 5.548 0.663 4.671 0.765 1.324±0.030 0.665±0.002 95.90
15 Diasemopsis obstans 2 1076.3970.3694.8620.4271.373±0.0680.685±0.00279.18
16 Teleopsis breviscopium 2 30 6.382 0.416 6.092 0.463 0.992±0.119 0.783±0.003 70.40
17 Diasemopsis nebulosa 2 666.5980.5835.1590.6081.312±0.0450.709±0.00292.89
18 Cyrtodiopsis dalmanni 2 91 5.854 0.611 5.211 0.725 1.286±0.037 0.713±0.002 93.12
19 Teloglabrus entabenensis 1 126.3680.3481.2230.0811.055±0.1950.086±0.00572.09
20 Diopsis gnu 1 6 6.065 0.401 4.712 0.435 1.380±0.130 0.671±0.004 95.74
21 Diasemopsis fasciata 2 635.8640.5534.8850.6371.316±0.0580.685±0.00289.33
22 Sphyracephala munroi 14 5.960 0.262 2.884 0.147 1.054±0.148 0.459±0.003 79.35
23 Diasemopsis signata 1 635.4250.4634.1450.4461.206±0.0540.615±0.00288.82
24 Teleopsis quadriguttata 18 5.211 0.544 3.368 0.354 0.996±0.064 0.525±0.003 93.44
25 Teleopsis rubicunda 2 515.4450.5164.6500.6121.343±0.0520.665±0.00293.11
26 Sphyracephala brevicornis 1 33 5.277 0.410 1.927 0.133 0.834±0.054 0.322±0.006 87.94
27 Diasemopsis albifacies 675.1500.3883.4390.3211.198±0.0470.534±0.00290.76
28 Trichodiopsis minuta 2 34 4.890 0.415 3.643 0.398 1.242±0.066 0.559±0.003 91.36
29 Sphyracephala beccarii 2 1104.9710.1592.0970.0750.813±0.0720.321±0.00153.66
30 Sphyracephala bipunctipennis 2 22 4.516 0.318 2.106 0.170 0.836±0.182 0.284±0.002 49.00
  Median 50.50 6.375 0.502 4.874 0.471 1.218 0.685 89.08
 Mean sexually dimorphic species53.006.5050.5034.8290.5041.2010.65085.43
  Mean sexually monomorphic species 48.50 5.805 0.429 4.151 0.395 1.136 0.590 83.72
Figure 2.

Male (gray broken lines) and female (black lines) static allometric slopes of log eye span on log body size. Error bars indicate 95% confidence interval. Species are plotted as a decreasing function of mean body size. Number refers to species as they are listed in Table 1. Eighteen of 30 species are sexually dimorphic in slopes given a 95% confidence level.

Data for seven populations of C. dalmanni are given in Table 2. Also here, regressions of eye span on body size show strong fit with r2-values ranging from 92% to 96% for males (mean = median = 94%) and 86% to 97% for females (mean = 92%, median = 94%). All populations are sexual dimorphic in eye span after controlling for body size. Neither males nor females differ in slopes among populations on a 95% confidence level with the exception of the Brastagi population (Fig. 3). Brastagi males are unique in having a steeper slope than males in all the other populations. Also the allometric slope in Brastagi females is significantly steeper than the female slopes in four other populations.

Table 2.  Morphometric measurements of seven populations of Cyrtodiopsis dalmanni. Morphometric measurements of body length (BL) and eye span (ES) in millimeters, allometric slope and intercept are the least-squares regression of log eye span as function of log body size (N= sample size).
  Cyrtodiopsis dalmanni Sex N Mean BL in mmMean ES in mmSlope±SEIntercept±SE r 2 (%)
  1. 1Slope and intercept are different between the sexes at the 95% confidence level (t-test).

1Brastagi, Sumatra1F386.7600.7306.0500.9101.400±0.0401.002±0.00696.57
   M 27 7.290 1.020 10.590 3.390 2.310±0.100 0.777±0.002 95.14
2Belalong, Borneo1F145.9500.7205.1900.7101.110±0.1100.880±0.00490.22
   M 19 6.700 0.590 7.670 1.160 1.670±0.110 0.711±0.006 93.25
   Malaysia1 M 149 6.680 0.740 8.740 1.580 1.940±0.030 0.713±0.002 95.65
4Bogor, Java1F396.3900.6105.6200.6301.160±0.0500.867±0.00294.52
   M 74 6.500 0.740 7.530 1.540 1.780±0.040 0.747±0.002 95.69
5Bt. Lawang, Sumatra1F496.0300.7405.2300.7501.150±0.0400.849±0.00394.97
   M 73 6.370 0.680 7.220 1.540 1.970±0.060 0.714±0.002 94.44
6Soraya, Sumatra1F435.6900.6005.0700.6901.230±0.0800.004±0.08785.65
   M 24 6.110 0.710 7.080 1.590 1.690±0.110 0.701±0.004 91.61
7Gombak, Malaysia1F895.7400.6405.0700.6701.150±0.0400.834±0.00291.31
   M 90 6.090 0.630 6.950 1.290 1.780±0.050 0.701±0.002 93.92
  Median F 43.00 5.950 0.720 5.220 0.710 1.160 0.854 93.67
  Median M 73.00 6.500 0.710 7.530 1.540 1.780 0.713 94.44
Figure 3.

Male (gray broken lines) and female (black lines) static allometric slopes of log eye span on log body size from seven populations of Cyrtodiopsis dalmanni. Error bars indicate 95% confidence interval. Species are plotted as a decreasing function of mean body size. Number refers to species as they are listed in Table 2. All populations are sexually dimorphic in slopes on a 95% confidence level.


Both male and female static allometric slopes showed indications of phylogenetic effects (Fig. 4; Table 3a), meaning that more related species have more similar static allometries. The maximum-likelihood estimate of the half-life for male allometric slope was t1/2= 22% of tree height (two-unit support region = 0–199%), whereas for female allometric slope it was t1/2= 61% of tree height (two-unit support region = 16%–∞). Assuming a total tree height of 25 million years, the best estimates correspond to 5.5 and 15.25 million years to lose half the ancestral influence for males and females, respectively. The wide support regions show that these estimates are highly uncertain, and the evidence for a stronger effect in females than males is only a tentative indication.

Figure 4.

Support surface for phylogenetic half-life, t1/2, in units of tree length (total tree length = 1) and stochastic equilibrium variance, vy, in a model including only an intercept and no predictor variables for allometric slopes of males (A) and females (B). The maximum-likelihood estimates of the half-lives are 0.22 and 0.61 for male and female allometric slope, respectively. The flat parts of the diagrams represent parameter combinations that are more than two support (log-likelihood) units worse than the best estimate. For all estimated parameter values for these two models, see Table 3a.

Table 3a.  Phylogenetic comparative analyses of the evolution of static allometry. For each model, we show the maximum-likelihood estimates of phylogenetic half-life, t1/2, in units of tree length (total tree length = 1) with its two-unit support region, the stationary variance, vy, and GLS estimates of the intercept and slope of the optimal regression with standard errors (SE). We also show estimated slopes corrected for bias due to measurement error (BCS). Model fit is described with a phylogenetically corrected r2, and the log likelihood (logL) and AICc of the model. Lower AICc scores indicate a better model. When a “–” is entered for the predictor variable, only an intercept is included in the model. In this cases, the phylogenetic half-life is a measure of the overall phylogenetic effect on the response variable.
Response variablePredictor variable t 1/2 (supp. region) vy Intercept±SESlope±SEBCS r 2 (%) logL AICc
Male slope0.22 (0–1.99)0.241.52±0.14−17.1 41.2
log(male eye span/female eye span)   0.11 (0–0.57) 0.01 0.06±0.02      36.7 −66.4
Male slopelog(male eye span/female eye span)0.13 (0–0.72)0.081.33±0.11 5.13±1.075.1442.81 −8.8 27.2
log(male eye span/male body length)   5.3 (0.28–∞) 4.67 −1.87±0.36     −25.1  57.1
Male slopelog(male eye span/male body length)0.10 (0–0.33)0.052.52±0.13 0.68±0.090.6863.66 −2.1 13.7
Male slope Male log body size 0.24 (0–1.99) 0.23 0.27±0.52  1.66±1.10 1.66 7.05 −16.0  41.7
Male slopeMale log eye span0.30 (0–1.26)0.170.38±0.23 2.17±0.612.1729.48−11.9 33.5
Multiple regressions          
Male slope0.17 (0–0.90)0.091.03±0.24  43.52 −8.8 30.0
  log(male eye span/female eye span)      4.32±1.61 4.31    
 log male eye span    0.59±0.590.59   
Male slope   0.06 (0–0.27) 0.04 2.88±0.35    65.81  −1.4  15.3
 log(male eye span/male body length)    0.73±0.130.73   
  log male eye span     −0.52±0.38 −0.52    

These phylogenetic effects in static allometries are weaker than the phylogenetic effects in the traits themselves. Analyses of both log eyestalk length and log body size gave half-life values many times the total tree length, indicating that the traits themselves evolve as if by Brownian motion. This probably reflects more bounded evolution of the allometric slopes, which tends to remove the relationship with phylogeny.


We found clear evidence for adaptation of male static allometry to our two proxies for sexual selection (Table 3a, Fig. 5). Sexual dimorphism in eye span explained more than 40% of the variation in static slopes, and relative eye span explained more than 60%. In both cases, the relationship was positive with the slope of the optimal regression being 5.13 ± 1.07 on log (male eye span/female eye span) and 0.68 ± 0.09 on log (male eye span/male body size). The former number means that a 10% increase in male eye span relative to female eye span would lead to about a 0.5 increase in the (optimal) male static allometric slope. The second number means that a 10% increase of male eye span relative to male body length would give about a 0.07 increase in the (optimal) static allometric slope. These changes would probably not be instantaneous, however, as the best estimates of the phylogenetic half-lives for these two models were 13% and 10% of tree height, respectively, corresponding to 3.25 and 2.50 million years (Table 3a). This is less than the phylogenetic effect of 5.5 million years, which indicates that some, but not all, of the phylogenetic effect in static allometry is due to phylogenetic inertia with the rest an indirect effect of a phylogenetic signal in the predictor variables. We caution, however, that the half-life estimates are uncertain and even instantaneous adaptation (t1/2= 0) is included in the support region.

Figure 5.

(A) Support surface for phylogenetic half-life (t1/2) and stochastic equilibrium variance (vy) in the analysis of phylogenetic inertia in male allometric slope using the log of the ratio of male eye span on male body length as a predictor variable (best estimate: t1/2= 0.10, vy= 0.05). (B) Phylogenetic regression of male allometric slope on the log of the ratio of male eye span to male body length. Both the evolutionary (dotted line, slope = 0.53) and the optimal (solid line, slope = 0.68) regressions are shown. The regressions explain 64% of the variation in allometric slope.

Using log eye span and log body size as predictor variables explained much less of the variation in slopes than our proxy variables above (Table 3a). This supports sexual dimorphism in eye span and relative eye span as the best predictors of species differences in static allometry, which is what the sexual selection hypothesis predicts. Furthermore, sexual dimorphism and relative eye span retain a strong effect when the logs of trait and body size are included as covariates in a multiple regression (Table 3a).


We found no support for Gould's hypothesis that static allometries should be more shallow in large-bodied species to avoid extreme trait expression. In fact, the estimated optimal relation between static allometry and mean body size was not negative, but positive with a slope of 0.17 ± 0.10 (0.38 ± 0.22 when corrected for bias due to measurement error). This model, explaining 8.5% of the variance, is marginally better than a model with no effect of body size according to AICc (Table 3b). The corresponding optimal regression in the females explained less than 1% of the variance, gave a slope of about 0, and was not supported by AICc.

Table  b.  .  Relationship between body size and allometric slope.
Response variablePredictor variable t 1/2 (supp. region) vy Intercept±SESlope±SEBCS r 2 (%) logL AICc
Male slope0.22 (0—1.99)0.241.52±0.14−17.1 41.2
Male slope Mean male body size 0.27 (0–∞) 0.21 0.52±0.37 0.17±0.10 0.38 8.45 −15.6  40.7
Female slope0.61 (0.16–∞)0.061.09±0.08 12.1−17.2
Female slope Mean absolute body size 0.48 (0.11–∞) 0.05 0.90±0.16 0.03±0.06 0.08 0.97  12.2 −14.9


There was little evidence for strong correlated evolution between male and female allometric slopes. A direct (no-delay) regression of log female slope on log male slope returned a coefficient of 0.14 ± 0.25, and explained only 14% of the variation (Table 3c). The phylogenetic half-life was 40% of tree height (support region = 0–∞), slightly weaker than the phylogenetic effect.

Table  c.  .  Evolution of female allometric slope.
Response variablePredictor variable t 1/2 (supp. region) vy Intercept±SESlope±SEBCS r 2 (%) logL AICc
Female slope0.61 (0.16–∞)0.061.09±0.0812.1−17.2
Female slope (Direct effect of) male slope 0.40 (0–∞) 0.04 0.90±0.44 0.14±0.25   13.92 14.2 −18.9


The evolutionary allometry is the log–log regression of mean eye span on mean body length across species. We did this based on both an adaptation model with a delay in the approach to an optimal relation, and on a direct-effect model with an assumed immediate response of eye span to changes in body size. For males, both models explained about 50% of the variance in log eye span, and the direct-effect model showed slightly better fit (Table 3d). For females, the direct-effect model was much better than the adaptation model. The estimated allometric exponents in the direct effect models were 1.68 ± 0.52 for the males and 1.29 ± 0.30 for females, which are both almost exactly equal to the average static allometries (Table 1). This result is consistent with the evolutionary allometry being a result of constrained evolution along static allometries. The alternative adaptation models had long half-lives and must be interpreted as consistent with trends toward progressively steeper evolutionary allometries. Also the direct effect models had long half-lives indicating Brownian-motion-like evolution of the residual variation. Also the traits themselves evolve in a Brownian-motion-like pattern (Table 3d).

Table  d.  .  Evolutionary allometries of eye span.
Response variablePredictor variable t 1/2 (supp. region) vy Intercept±SESlope±SEBCS r 2 (%) logL AICc
log male eye span 7.19 (0.35–∞)0.55  0.46±0.1111.3−15.6
log female eye span   10.0 + (0.97–∞) 0.26   0.45±0.07     23.7 −40.5
log male eye spanlog male body length 3.84 (2.41–5.93)0.00−12.72±0.2817.23±4.9517.2545.4622.2−34.7
log male eye span (Direct effect of) log male body length  5.00 (0.97–∞) 0.18  −0.78±0.41  1.68±0.52   53.28 22.7 −35.8
log female eye spanlog female body length 1.52 (0.28–5.61)0.02 −2.49±0.13 3.98±1.11 3.9829.8529.3−49.0
log female eye span (Direct effect of) log female body length 10.0 + (1.15–∞) 0.12  −0.55±0.24  1.29±0.30   51.47 33.5 −57.4


Despite the interest in the evolvability of static allometries over the last decades, little empirical information about the evolution of narrow-sense allometric parameters has accumulated for morphological traits. This stems from a shift in the notion of allometry from its original narrow-sense meaning as a power relationship (Huxley 1932) toward a broad-sense meaning as any change in shape related to size (Mosimann 1970; Frankino et al. 2009). There is nothing wrong with studying broad-sense allometry and shape evolution, but narrow-sense allometry was associated with theoretical ideas and models that do not transfer to broad-sense allometry (Houle et al. 2011). As theoretical context was forgotten, students of allometry chose scales and models for statistical rather than biological reasons. As a consequence, some of the specific hypotheses that relates to narrow-sense allometry remain substantially untested. In particular, the idea that narrow-sense allometric slopes are evolutionary constrained remains almost untested for morphological traits and our study is the first that investigates the evolution of narrow-sense static allometric slopes using phylogenetic comparative methods.

Our main findings were that static allometric slopes indeed varied considerably across the 30 species of stalk-eyed flies and they did so in a way that is consistent with adaptation to sexual selection with our measures of sexual dimorphism and relative trait size explaining, respectively, 43% and 64% of the among-species variance in male eye-span static allometric slope. This adds to studies cited in the introduction showing that static allometries are evolvable on million-year time scales. Related species still have similar static allometries, however. If we assume that the base of our phylogeny is approximately 25 million years back in time, the best estimate is that it takes 5.5 million years to lose half the ancestral effect in males. We caution that this parameter estimate has a wide support region and only unconstrained Brownian motion can be definitely excluded (Fig. 4A).

Such phylogenetic effects can arise for two reasons (Hansen and Orzack 2005; Labra et al. 2009). If closely related species have similar niches, then adaptation to these will cause a nonrandom distribution of trait values on the phylogeny. If phylogenetic effects are generated in this way, they do not imply any constraints on the rate of adaptation. The other possible cause of phylogenetic effects is a low rate of adaptation (phylogenetic inertia). Specifically estimating phylogenetic inertia in the adaptive response of static allometric slopes to sexual dimorphism and relative trait size, we found half-life values around 3 million years. This is less than the phylogenetic effect of 5.5 million years and although the difference may well be due to estimation error, it is consistent with the phylogenetic effect being a result of both adaptation and inertia, and that static allometries may therefore constrain the independent evolution of eye span on time scales below a few million years.

Million-year phylogenetic half-lives can be interpreted as slow adaptation. It is becoming clear, however, that adaptation and cumulative evolutionary change often take place on million-year time scales (Uyeda et al. 2011; Hansen 2012), and our static allometries may be constrained, but not unusually constrained. It would have been valuable with a direct comparison to rates of adaptation in the trait intercept, but this would run into the circularity problems discussed above.

Note that our adaptation models do not account for about half the among-species variance in static allometric slopes. This may be partially due to our measures not fully capturing the relevant sexual selection on the trait, but it likely also reflects other unmeasured factors of selection acting on allometry. Such factors may involve species differences in predicted costs of large eye span (Swallow et al. 2000; Husak et al. 2011). It may also reflect alternative ways of achieving a large eye span. Worthington et al. (2012) have recently found a negative relationship between the length of the eyestalks and the angle that they attach to the head within C. dalmanni. This shows that eye span is more complex than the mere length of the eyestalks, and may also evolve by shape changes that have a different relations to body size than does stalk length.

If it takes several million years to get rid the majority of the ancestral effect, we do not expect stalk-eyed flies with different allometries to have diverged very recently. Our re-estimation of the allometric relationships in seven populations of C. dalmanni from Swallow et al. (2005) shows that the Brastagi population is the only population in which the male allometric slope differs from all the other monophyletic populations on a 95% confidence level. According to the phylogeny published by Swallow et al. (2005), the Brastagi population is the sister group to the rest of the six monophyletic populations in the study. The time of genetic divergence between Brastagi and the other populations reported in Swallow et al. (2005) is on average larger than the genetic divergence times between the other populations. For example, based on the nuclear gene wingless, the mean genetic divergence between Brastagi and the other populations is 3.8 million years (calculated from Table 3 in Swallow et al. 2005), and the authors conclude that their findings are consistent with old divergence times and very little gene flow since these populations started to diverge. The difference in allometry between Brastagi and the other C. dalmanni populations may therefore have had sufficient time to evolve without contradicting the results from our cross-species analysis.


Gould (1966a,b) proposed that natural selection could change the allometric slope so that large-sized animals have a smaller static allometric slope compared to smaller-sized animals to avoid nonadaptive and absurd trait sizes in species with the larger individuals. We found no support for Gould's hypothesis. In fact there was a weak positive, and not negative, relationship between male allometric slope and body size in the among-species dataset, whereas female slope and body size did not correlate at all. Our results therefore indicate that the observed differences in slopes between species are not a solution to an allometrically size-imposed problem in eyestalk length between species of different size.

A relationship between the allometric slope and overall size could be a mechanism for the evolution of altered allometric slopes as a side effect of body size evolution. This can be ruled out as a likely explanation of our results, because we got qualitatively the same results using different proxies for sexual selection, both with and without controlling for the effect of body size on eye-span length. Another reasonable hypothesis is that the allometries could change as a side effect of direct selection on eye span. Although we found that eye-span evolution is somewhat constrained by its allometric relation to body size, eye span is after all the focal trait for sexual selection and is known to be under strong selection in many of the species (e.g., Wilkinson et al. 1998; Panhuis and Wilkinson 1999; Hingle et al. 2001; Cotton et al. 2006). Indeed, male static allometric slopes were positively affected by mean log eye span across species. However, the multiple regressions showed that it was our measures of sexual dimorphism and relative trait size rather than the individual traits that best explained the evolution of static allometries, and although other selection pressures can not be ruled out, our results are very consistent with a strong effect of sexual selection on static allometry in eye span.


Our analysis provides strong evidence that static allometric exponents are evolvable on macroevolutionary time scales and that species likely to have experienced stronger sexual selection have evolved steeper static allometries. On one hand, this may indicate that static allometries are not important as macroevolutionary constraints, but on the other, we have indications that the “adaptive” changes take place on a million-year time scale, and we cannot exclude the possibility that static allometries constitute a significant constraint for trait evolution on microevolutionary time scales. Although the lack of trait-independent measures of sexual selection has precluded an analysis of sexual adaptation in the eyestalk itself, we did find that the estimated evolutionary allometries are almost exactly equal to the median static allometries (1.68 vs. 1.73 for male slopes and 1.29 vs. 1.22 for female slopes). Combined with the fact that the direct-effect “constraint” model outperformed the adaptation model for eye-span evolution, this indicates that evolutionary changes in eye span and body size have tended to broadly follow along static allometries, as expected if constraints were important. In contrast, we did not find support for the hypothesis that female eye span has evolved as a correlated response to male eye span, as less than 14% of the among-species variation in female eye span was explained by male eye span. We think this implies independent selection to maintain the stalks in males and females, because it is unlikely that these will not be genetically correlated, and Wilkinson (1993) indeed demonstrated a correlated response in female eye span to artificial selection on male eye span in C. dalmanni. Together, these results show that selection and adaptation of eye span must be complex and go beyond intrasexual selection on males.

Associate Editor: P. Lindenfors


We would like to thank G. S. Wilkinson, R. H. Baker, and J. G. Swallow for their kindness and willingness to share their data on stalk-eyed flies. We thank J. Pienaar for help with and development of the SLOUCH program. C. Pélabon and two anonymous reviewers provided helpful and insightful comments on the manuscript and we thank members of the Hansen and Pélabon Labs for discussions. The work was supported by grant 196434/V40 from the Norwegian Research Council to Christophe Pélabon at NTNU.