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.

Understanding the patterns of genetic variation and constraint for continuous reaction norms, growth trajectories, and other function-valued traits is challenging. We describe and illustrate a recent analytical method, simple basis analysis (SBA), that uses the genetic variance-covariance (G) matrix to identify "simple" directions of genetic variation and genetic constraints that have straightforward biological interpretations. We discuss the parallels between the eigenvectors (principal components) identified by principal components analysis (PCA) and the simple basis (SB) vectors identified by SBA. We apply these methods to estimated G matrices obtained from 10 studies of thermal performance curves and growth curves. Our results suggest that variation in overall size across all ages represented most of the genetic variance in growth curves. In contrast, variation in overall performance across all temperatures represented less than one-third of the genetic variance in thermal performance curves in all cases, and genetic trade-offs between performance at higher versus lower temperatures were often important. The analyses also identify potential genetic constraints on patterns of early and later growth in growth curves. We suggest that SBA can be a useful complement or alternative to PCA for identifying biologically interpretable directions of genetic variation and constraint in function-valued traits.