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 h eterogeneity in developmental stability (DS) and thus fluctuating asymmetry (FA), is required to find relationships between DS and other factors. Sepa rating directional asymmetry (DA) and antisymmetry (AS) from real FA and un derstanding 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 t he distribution of the signed asymmetry, and (ii) to model and quantify bet ween-individual heterogeneity in developmental stability and FA. In additio n, we expand mixtures to the estimation of the proportion of variation in t he unsigned FA that can be attributed to between-individual heterogeneity i n the presumed underlying developmental stability (the so-called hypothetic al repeatability). Finally, we construct weighted normal probability plots to investigate the assumption of underlying normality of the different comp onents. We specifically show that (i) model selection based on the likeliho od ratio test has the potential to yield models that incorporate nearly all heterogeneity in FA; (ii) mixtures appear to be a powerful and sensitive s tatistical technique to identify the different forms of asymmetry; (iii) re stricted 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 coef ficients of the normal probability plots, the underlying normality assumpti on appears to hold for the empirical data we analysed. In conclusion, mixtu res provide a useful statistical tool to study patterns in asymmetry.