On the distribution of developmental errors: Comparing the normal, gamma, and log-normal distribution

Abstract

The amount of between-individual variation in the unobservable developmental instability (DI) has been the subject of intense recent debates. The unexpectedly high estimates of between-individual variation in DI based on distributional characteristics of observable asymmetry values (of on average bilaterally symmetric traits) rely on statistical models that assume an underlying normal distribution of developmental errors. This prompted doubts on the assumption of the Gaussian nature of developmental errors. However, when applying other candidate distributions [log-normal and gamma (γ)], recent analyses of empirical datasets have indicated that estimates remain generally high. Yet, all estimates were based on bilaterally symmetric traits, which did not allow for a formal comparison of the alternative distributions. In the present study, we extend a recent statistical model to allow statistical comparison of the different distributions based on traits that developed repeatedly under the same conditions, such as flower traits and regrown feathers. We analyse simulated and empirical data and show that: (1) it is statistically difficult to differentiate among the three alternatives when variances are small relative to the mean, as is often the case with DI; (2) the normal distribution fits the log-normal or γ relatively well under those circumstances; (3) the deviance information criterion (DIC) is able to pick up differences in model fit among the three alternative distributions, yet more strongly so when levels of DI were high; (4) empirical datasets show a better fit of the normal over the log-normal and γ-distributions as judged by the DIC; and (5) estimates of between-individual variation in DI in the three empirical datasets were relatively high (> 50%) under each distributional assumption. In conclusion, and based on our three datasets, the normal approximation appears to be a reasonable choice for statistical models of DI and the remarkably high estimates of variation in DI cannot be attributed to non-normal developmental noise. Nevertheless, our method should be applied to a broad range of traits and organisms to evaluate the generality of this result. We argue that there is an urgent need for studies that reveal the underlying mechanisms of developmental noise and stability, as well as the role of developmental selection, in order to be able to determine the biological importance of the highly skewed distributions of developmental instability often observed. © 2007 The Linnean Society of London.