How many more? Sample size determination in studies of morphological integration and evolvability

1. The variational properties of living organisms are an important component of current evolutionary theory. As a consequence, researchers working on the field of multivariate evolution have increasingly used integration and evolvability statistics as a way of capturing the potentially complex patterns of trait association and their effects over evolutionary trajectories. Little attention has been paid, however, to the cascading effects that inaccurate estimates of trait covariance have on these widely used evolutionary statistics. 2. Here, we analyse the relationship between sampling effort and inaccuracy in evolvability and integration statistics calculated from 10‐trait matrices with varying patterns of covariation and magnitudes of integration. We then extrapolate our initial approach to different numbers of traits and different magnitudes of integration and estimate general equations relating the inaccuracy of the statistics of interest to sampling effort. We validate our equations using a data set of cranial traits and use them to make sample size recommendations. 3. Our results suggest that highly inaccurate estimates of evolvability and integration statistics resulting from small sample sizes are likely common in the literature, given the sampling effort necessary to properly estimate them. We also show that patterns of covariation have no effect on the sampling properties of these statistics, but overall magnitudes of integration interact with sample size and lead to varying degrees of bias, imprecision and inaccuracy. 4. Finally, we provide r functions that can be used to calculate recommended sample sizes or to simply estimate the level of inaccuracy that should be expected in these statistics, given a sampling design

An ontogenetic perspective on the relationship between age and size at maturity

1. Understanding the relationship between age and size at maturity is essential because these traits are pivotal determinants of an organism’s fitness. 2. The relationship between age and size is commonly addressed using optimization and quantitative genetic approaches. Here we argue that the value of such studies is often limited by an insufficient consideration of organismal ontogeny. 3. On the basis of a simple conceptual framework of hierarchical resource allocation, we identify key aspects of ontogeny that prove critical to a fuller understanding of the relationship between age and size, and which, to date, have been insufficiently explored. In particular, these include intrinsic variation in growth rate within and among populations, and the physiological nature of the maturation process that co-ordinates growth and reproductive function in an organism. 4. We also provide some guidance to the empirical investigation of these aspects, anticipating that a wider theoretical, but especially empirical appreciation of ontogenetic detail will greatly increase the explanatory and predictive power of life-history studies