The statistical analysis of fluctuating asymmetry: REML estimation of a mixed regression model

The unbiased estimation of fluctuating asymmetry (FA) requires independent repeated measurements on both sides. The statistical analysis of such data is currently performed by a two-way mixed ANOVA analysis. Although this approach produces unbiased estimates of FA, many studies do not utilize this method. This may be attributed in part to the fact that the complete analysis of FA is very cumbersome and cannot be performed automatically with standard statistical software. Therefore, further elaboration of the statistical tools to analyse FA should focus on the usefulness of the method, in order for the correct statistical approaches to be applied more regularly. In this paper we propose a mixed regression model with restricted maximum likelihood (REML) parameter estimation to model FA. This routine yields exactly the same estimates of FA as the two-way mixed ANOVA . Yet the advantages of this approach are that it allows (a) testing the statistical significance of FA, (b) modelling and testing heterogeneity in both FA and measurement error (ME) among samples, (c) testing for nonzero directional asymmetry and (d) obtaining unbiased estimates of individual FA levels. The switch from a mixed two-way ANOVA to a mixed regression model was made to avoid overparametrization. Two simulation studies are presented. The first shows that a previously proposed method to test the significance of FA is incorrect, contrary to our mixed regression approach. In the second simulation study we show that a traditionally applied measure of individual FA [abs(left – right)] is biased by ME. The proposed mixed regression method, however, produces unbiased estimates of individual FA after modelling heterogeneity in ME. The applicability of this method is illustrated with two analyses.     

Accuracy and power in fluctuating asymmetry studies: effects of sample size and number of within-subject repeats

This paper presents results from simulations investigating the effect of sample size, number of within-subject repeats and relative degree of measurement error on the power and accuracy of test for fluctuating asymmetry (FA). These data confirm that sampling variation of population-level FA-estimates is large and that high sample size is required to obtain reasonably high power when testing for FA or comparing FA levels between populations. The results also clearly show that increasing the number of within-subject repeats can dramatically increase accuracy and power when measurement error is relatively high.     

A simple model of the relationship between asymmetry and developmental stability

The relationship between developmental stability and morphological asymmetry is derived under the standard view that structures on each side of an individual develop independently and are normally distributed. I use developmental variance of sizes of parts, V_D, as the converse of developmental stability, and assume that V_D follows a gamma distribution. Repeatability of asymmetry, a measure of how informative asymmetry is about V_D, is quite insensitive to the variance in V_D, for example only reaching 20% when the coefficient of variation of V_D is 100%. The coefficient of variation of asymmetry, CV_FA, also increases very slowly with increasing population variation in V_D. CV_FA values from empirical data are sometimes over 100%, implying that developmental stability is sometimes more variable than any previously studied type of trait. This result suggests that alternatives to this model may be needed.     

Mixture analysis of asymmetry: modelling directional asymmetry, antisymmetry and heterogeneity in fluctuating asymmetry

The occurrence of different forms of asymmetry complicates the analysis and interpretation of patterns in asymmetry. Furthermore, between-individual heterogeneity in developmental stability (DS) and thus fluctuating asymmetry (FA), is required to find relationships between DS and other factors. Separating directional asymmetry (DA) and antisymmetry (AS) from real FA and understanding between-individual heterogeneity in FA is therefore crucial in the analysis and interpretation of patterns in asymmetry. In this paper we introduce and explore mixture analysis to (i) identify FA, DA and AS from the distribution of the signed asymmetry, and (ii) to model and quantify between-individual heterogeneity in developmental stability and FA. In addition, we expand mixtures to the estimation of the proportion of variation in the unsigned FA that can be attributed to between-individual heterogeneity in the presumed underlying developmental stability (the so-called hypothetical repeatability). Finally, we construct weighted normal probability plots to investigate the assumption of underlying normality of the different components. We specifically show that (i) model selection based on the likelihood ratio test has the potential to yield models that incorporate nearly all heterogeneity in FA; (ii) mixtures appear to be a powerful and sensitive statistical technique to identify the different forms of asymmetry; (iii) restricted measurement accuracy and the occurrence of many zero observations results in an overestimation of the hypothetical repeatability on the basis of the model parameters; and (iv) as judged from the high correlation coefficients of the normal probability plots, the underlying normality assumption appears to hold for the empirical data we analysed. In conclusion, mixtures provide a useful statistical tool to study patterns in asymmetry.     

Directional asymmetry and the measurement of developmental instability

Three widely used methods of estimating fluctuating asymmetry may yield serious overestimates if directional asymmetry is present. When two sides of a bilateral trait grow at different rates, then the asymmetry variance (Var[l-r]) increases with size, even when developmental noise is nil. But the residual variance around a population's mean developmental trajectory is invariant with respect to size. Thus, it can be used as a measure of developmental instability. We introduce a measure of developmental instability, the residual variance (s²δ), obtainable from either a major axis regression, which is equivalent to a principal component analysis on l and r, or a general structural model. This residual variance can be estimated from directionally asymmetric or even antisymmetric traits. We present examples of developmental instability estimated from directionally asymmetric mandibles (house mouse) and leaves (soybean), and antisymmetric claws (fiddler crab).     

Patterns of fluctuating asymmetry in flowers: Implications for sexual selection in plants

Characters in animals used in signalling and subjected to strong directional selection often demonstrate (i) an elevated level of fluctuating asymmetry (small random deviations from bilateral symmetry) and (ii) a negative relationship between the degree of individual fluctuating asymmetry and the size of a given character. We tested these two predictions in plants since flowers are subjected to strong directional selection and are involved in signalling to pollinators, whereas leaves are supposed not to be directly involved in signalling. The overall level of fluctuating asymmetry in a number of plant species with bilaterally or radially symmetric flowers was not generally higher in floral traits than in leaves. The level of fluctuating asymmetry in plants was sometimes significantly consistent within individuals. The absolute degree of individual fluctuating asymmetry in floral traits was generally negatively related to the size of the trait, while there was a positive relationship for leaves. The degree of individual fluctuating asymmetry in floral traits was marginally negatively related to the degree of individual fluctuating asymmetry in leaf traits. These patterns of fluctuating asymmetry in plants suggest that (i) the degree of asymmetry in flowers signals different aspects of quality than does the degree of asymmetry in leaves, and that (ii) fluctuating asymmetry in flowers often reflects the phenotypic quality of individual plants.     

Fluctuating and directional asymmetry in natural bird populations exposed to different levels of habitat disturbance, as revealed by mixture analysis

While bilateral trait asymmetry is widely recognized to estimate developmental instability, much controversy exists over which types of asymmetry (fluctuating, directional, and/or antisymmetry) to use. Recently it has been hypothesized that the three types are strongly interrelated, and that increased developmental instability may be reflected in a transition from fluctuating to directional asymmetry and/or antisymmetry. Alternatively, habitat disturbance might change the genetic expression of directional asymmetry. We present herein the first empirical evidence for stress-mediated shifts in types of asymmetry in natural populations, by using mixture analysis to model tarsus asymmetry in bird populations exposed to different levels of habitat disturbance. Observed asymmetry patterns almost exclusively consisted of true fluctuating asymmetry in the least disturbed populations, but became progressively mixed with directional asymmetry under increasing disturbance. Failing to unravel these mixtures of different forms of asymmetry may have critical implications for the analysis and interpretation of asymmetry data.     

Limited mass‐independent individual variation in resting metabolic rate in a wild population of snow voles (Chionomys nivalis)

Resting metabolic rate (RMR) is a potentially important axis of physiological adaptation to the thermal environment. However, our understanding of the causes and consequences of individual variation in RMR in the wild is hampered by a lack of data, as well as analytical challenges. RMR measurements in the wild are generally characterized by large measurement errors and a strong dependency on mass. The latter is problematic when assessing the ability of RMR to evolve independently of mass. Mixed models provide a powerful and flexible tool to tackle these challenges, but they have rarely been used to estimate repeatability of mass‐independent RMR from field data. We used respirometry to obtain repeated measurements of RMR in a long‐term study population of snow voles (Chionomys nivalis) inhabiting an environment subject to large circadian and seasonal fluctuations in temperature. Using both uni‐ and bivariate mixed models, we quantify individual repeatability in RMR and decompose repeatability into mass‐dependent and mass‐independent components, while accounting for measurement error. RMR varies among individuals, that is, is repeatable (R = .46) and strongly co‐varies with BM. Indeed, much of the repeatability of RMR is attributable to individual variation in BM, and the repeatability of mass‐independent RMR is reduced by 41% to R = .27. These empirical results suggest that the evolutionary potential of RMR independent of mass may be severely constrained. This study illustrates how to leverage bivariate mixed models to model field data for metabolic traits, correct for measurement error and decompose the relative importance of mass‐dependent and mass‐independent physiological variation.     

Morphological integration of fluctuating asymmetry in the mouse mandible

Morphological integration of fluctuating asymmetry (FA) was assessed for nine mandibular characters in random-bred house mice to test the hypothesis that there should be a significant integration or concordance of FAs at the population level, but not at the individual level. FA estimates for each of the nine characters were made for each of 16 subpopulations (two replicates each for mice varying in sex and age) and their correlations indicated a moderate level of integration (index of integration = 0.42). A matrix permutation test of the differences of the correlations within two (‘incisor’ and ‘muscle’) developmentally different character groups versus correlations between groups was significant, indicating the presence of morphological integration. Kendall's coefficient of concordance also indicated a significant population asymmetry parameter for FAs within the two character groups. Correlations among individual FA estimates were generally lower than those calculated from subpopulations, the index of integration being 0.21. The matrix permutation test failed to show significant morphological integration among individual FAs, but Kendall's coefficient of concordance was significant, indicating the presence of an ‘individual asymmetry parameter’ among the nine characters. Principal components analysis and canonical correlation analysis confirmed the overall higher level of integration of FAs among the subpopulations and within the two character groupings.     

How to compare fluctuating asymmetry of different traits

Comparing fluctuating asymmetry (FA) between different traits can be difficult because traits vary at different scales. FA is generally quantified either as the variance of the difference between left and right (σ²_L?R) or the mean of the absolute value of this difference (μ_|R?L|). Corrections for scale differences are obtained by dividing by trait size mean. We show that a third index, one minus the correlation coefficient between left and right (1 ? r_L,R), is equivalent to σ²_L?R standardized by trait size variance. The indices are compared with Monte‐Carlo simulations. All achieve the expected correction for scale differences. High type I error rates (false indication of differences) occur only for σ²_L?R and μ_|R?L| if trait sizes close to or below 0 occur. 1 ? r_L,R with a bootstrap test has always low error rates. Recommendation of the index to be used should be based on whether standardization of FA by trait size mean or trait size variance is preferred. A survey of 36 traits in the Speckled Wood Butterfly (Pararge aegeria) indicated that σ²_L?R is slightly higher correlated to trait size variance than to trait size mean. Thus 1 ? r_L,R seems to be the superior index and should be reported when FA of different traits is compared.     

The allometry of fluctuating asymmetry in southern African plants: flowers and leaves

Several studies of fluctuating asymmetry (FA) in animals show that secondary sexual characters used in signalling have a negative relationship between size and asymmetry. Larger sexual traits are presumably more costly to produce, which should lead to greater developmental stress and corresponding increases in asymmetry. In the absence of among individual variation in the ability to handle these costs, the relationship between size and asymmetry should thus be positive. A negative relationship therefore suggests that expression of these traits is condition-dependent. In plants, flowers act as signals for pollinators and may show similar trends to animal signals. Leaves which are uninvolved in signalling should not. Moller & Eriksson (1994) found that 89% of species ( n = 16 of 18) with insect-pollinated flowers showed a negative relationship between petal size and asymmetry, while 79% of species ( n = 15 of 19) showed a positive relationship between leaf size and asymmetry. I carried out a similar study of 18 plant species. The average relationship between petal size and asymmetry did not differ significantly from zero in those species showing measurable FA in flowers ( n = 12). The relationship was significantly negative in one species, and significandy positive in another. On average, leaves in species with FA did not show a significant positive relationship between size and asymmetry ( n = 7). There was no significant difference in the slopes of the relationship between size and asymmetry for leaves and flowers. Levels of floral asymmetry for species with FA were significandy repeatable on individual plants in 33% ( n = 4 of 12) of species, but leaf asymmetry was not significantly repeatable in any species. It is argued that condition-dependence of traits need not result in a negative relationship between size and asymmetry.     

Mandible asymmetry and genetic diversity in island populations of the common shrew, Sorex araneus

Mandibles from 13 island and six mainland populations of common shrews from the west coast of Scotland were subjected to geometric morphometric analysis in order to investigate the relationship between genetic diversity and fluctuating asymmetry. Although population mean shape fluctuating asymmetry (FA) and size FA were significantly inversely correlated with population genetic diversity this result was substantially due to one island. Sanda, the smallest island with by far the lowest genetic diversity, also had the highest FA. When Sanda was removed from the analysis, the relationship was not significant. There was no relationship between genetic diversity and FA at the individual level, whether measured as mean locus heterozygosity or d(2). In general, if genetic variation affects FA at all, the effect is weak and may only be of biological interest in very small populations.     

Testing different 3D techniques using geometric morphometrics: Implications for cranial fluctuating asymmetry in humans

This study aimed to test the performance of 3D digitizer, CT scanner, and surface scanner in detecting cranial fluctuating asymmetry. Sets of 32 landmarks (6 in the midline and 13 bilateral) were acquired from 14 archeological crania using a 3D digitizer, and from 3D models generated from a CT scanner and surface scanner using Viewbox 4. Levels of shape variation were analyzed in MorphoJ using Procrustes analysis of variance and Principal component analysis. Intra-observer error accounted for 1.7%, 1.8%, and 4.5% of total shape variation for 3D digitizer, CT scanner, and surface scanner respectively. Fluctuating asymmetry accounted for 15%–16% of total shape variation. Variation between techniques accounted for 18% of total shape variation. We found a higher level of missing landmarks in our surface scan data than for both 3D digitizer and CT scanner data, and both 3D model-based techniques sometimes obscured taphonomic damage. All three 3D techniques are appropriate for measuring cranial fluctuating asymmetry. We advise against combining data collected with different techniques.     

Genetics of wing size asymmetry in Drosophila buzzatii

Contemporary approaches that use fluctuating asymmetry (FA) as a possible target for natural and sexual selection are based on the premise that FA is a quantifiable expression of developmental instability (DI) that is inherited. Previous work with Drosophila buzzatii found that male mating success was correlated positively to body size (wing length) and negatively to FA, but these relationships seem to be environmentally induced. Heritability of FA was low and not significantly different from zero, but statistical power was also estimated to be very low and, hence, no conclusive evidence could be obtained. A large half‐sib mating design is used here to examine the relationships of different aspects of development for wing size. Consistently with previous findings, I found high heritabilities for wing length (WL) and wing width (WW), and positive correlations between both traits. Heritabilities of FA (FA_WL, FA_WW) were low (0.037) but significantly different from zero, and the genetic correlation between FA_WL and FA_WW was estimated as ?1 because the absolute value for the genetic covariance was similar in magnitude or even larger than the estimated genetic variances of both traits. This suggests that these two traits should be considered to be the same character. The between‐trait phenotypic correlation in FA, which reduces to the repeatability in this situation, was positive and statistically significant thus rendering an estimate of heritability for DI in D. buzzatii of . Nevertheless, the fact that left/right wing sizes were found to be determined by the same set of genes is difficult to reconcile with the presence of special genetic mechanisms that stabilize left/right development in this species. A qualitatively different pattern for asymmetry was observed when the nonlinear composite character wing area (WA ≈ WL × WW) was used, and . Although the results could be made compatible with the existence of a diallelic locus with antagonistic pleiotropic effects on FA_WL and FA_WW that combine multiplicatively to produce overdominance for FA_WA, the available evidence is extremely weak at best. Finally, a test to the null hypothesis of a nongenetic basis of FA, particularly relevant to those situations when directional asymmetry may be heritable, is suggested.     

Morph‐dependent form of asymmetry in mandibles of the stag beetle Prosopocoilus inclinatus (Coleoptera: Lucanidae)

Abstract 1. The form of asymmetry in bilateral organs usually follows the same pattern within single populations. However, some exceptions may occur when a population consists of different phenotypes that are from different ontogenic backgrounds and under different selective pressures. We investigated the asymmetric patterns of mandibles of larvae, females, and males in the stag beetle Prosopocoilus inclinatus. 2. Larval mandibles exhibited directional asymmetry both in length and cross direction, whereas female mandibles showed directional asymmetry in cross direction. These asymmetric structures might be more effective in cutting wood fibres. 3. For the relation of male mandible length to body size, a model with a switch point showed a better fit to the data than a convex curve model. This shows that the males are dimorphic with two distinct morphs. 4. The form of asymmetry in male mandible length differed between the morphs. The smaller males exhibited left‐biased directional asymmetry in common with larvae, whereas the larger males exhibited fluctuating asymmetry. 5. This is a novel finding of a morph‐dependent asymmetry. The morph‐dependent asymmetry in males may be as a result of different selection on each morph or a developmental constraint from larval mandibles to adult ones.     

空间诱变条件下蜜蜂后代的波动性不对称

利用航天搭载的雄蜂精液对处女王进行人工授精,产卵后大量培育后代蜂王;以SP1、SP2及SP3代蜂王后代工蜂为研究对象,考查波动性不对称(fluctuating asymmetry,FA)在不同代次种群中各对称性状间的表现情况。结果表明:SP1、SP2和SP3代工蜂的前翅长及肘脉a均表现出FA,SP1、SP2代的对照组中均未出现,而SP3代的对照组蜂群则全部表现出FA;与对照蜂群相比,空间诱变后代的前翅长及翅脉a的FA值均较高。此外,讨论FA在蜜蜂种群对生态环境适应性上应用以及利用蜜蜂的FA来监测环境污染的可能性及发展前景。