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

Abstract

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 analyze 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 dataset 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.

Measuring the magnitude of morphological integration: The effect of differences in morphometric representations and the inclusion of size

“…Before doing so, we evaluated two possible sources of bias in empirical estimates of integration: sample size (Haber ; Fruciano et al. ; Adams and Collyer ; Grabowski and Porto ) and total amount of variance (Hallgrímsson et al. ; Young et al.…”

Phenotypic integration of the cervical vertebrae in the Hominoidea (Primates)

“…A high value for means traits are able to evolve to a large degree because constraints are low. Although Hansen and Houle () propose additional statistics to quantify integration and evolvability, these require extremely large sample sizes to capture the population mean (Grabowski and Porto ) and are not included here.…”

“…Correlation matrices estimated from small samples typically reflect the true population value starting at sample sizes of around 40 (Ackermann ). However, following Grabowski and Porto (), I estimate that the ī , ē , and statistics, in combination with the correlation values and trait number used here, require sample sizes of between 80 and 140 to capture the true population values. Conditional evolvability, in particular, is estimated to require extremely large sample sizes beyond 140 and to display high levels of inaccuracy.…”

“…Conditional evolvability, in particular, is estimated to require extremely large sample sizes beyond 140 and to display high levels of inaccuracy. Because such large sample sizes are required, and ī , ē , and statistics have high margins of error even at large sample sizes (Grabowski and Porto ), I calculated 99% confidence intervals for each statistic using an SD estimated from the SE: …”

“…I determined statistical significance of differences between species or elements using the confidence intervals for each variable value, where taxa or elements differed significantly if their confidence intervals did not overlap. The sample sizes used here are generally adequate to capture the true value of ī , ē , and within this confidence interval (Grabowski and Porto ). For analyses of the ventral (vertebral body) and dorsal (neural arch) portions of the cervical vertebrae, I excluded the CB.…”