Approximate lower bound sampling errors of maximum likelihood estimates of covariance components and their linear functions can be obtained from the inverse of the information matrix. For non-linear functions, sampling variances are commonly determined as the variance of their first order Taylor series expansions. This is used to obtain sampling errors for estimates of heritabilities and correlations, and these quantities can be computed with most software performing such analyses. In other instances, however, more complicated functions are of interest or the linear approximation is difficult or inadequate. A pragmatic alternative then is to evaluate sampling characteristics by repeated sampling of parameters from their asymptotic, multivariate normal distribution, calculating the function(s) of interest for each sample and inspecting the distribution across replicates. This paper demonstrates the use of this approach and examines the quality of approximation obtained.

Many central questions in ecology and evolutionary biology require characterizing phenotypes that change with time and environmental conditions. Such traits are inherently functions, and new 'function-valued' methods use the order, spacing, and functional nature of the data typically ignored by traditional univariate and multivariate analyses. These rapidly developing methods account for the continuous change in traits of interest in response to other variables, and are superior to traditional summary-based analyses for growth trajectories, morphological shapes, and environmentally sensitive phenotypes. Here, we explain how function-valued methods make flexible use of data and lead to new biological insights. These approaches frequently offer enhanced statistical power, a natural basis of interpretation, and are applicable to many existing data sets. We also illustrate applications of function-valued methods to address ecological, evolutionary, and behavioral hypotheses, and highlight future directions.