Euclidean distance matrix analysis: Confidence intervals for form and growth differences

Analysis of biological forms using landmark data has received substantial attention recently. Much of the statistical work in this area has concentrated on the estimation of average form, average form difference, and average growth difference. From the statistical, as well as the scientific point of view, it is important that any estimate of a scientifically relevant quantity be accompanied by a statement regarding its accuracy. Such a statement is contained in a confidence interval. The purpose of this paper is to provide a method to obtain confidence intervals for form difference and growth difference estimators. The estimators are based on Euclidean distance matrix analysis. The confidence intervals are calculated using the model independent bootstrap method. We illustrate the method by using three examples: morphological differences between samples of craniofacial patients and normal controls using two dimensional data from head X-rays, sexual dimorphism of craniofacial morphology in Cebus apella, and sexual dimorphism of facial growth in Cebus apella using three-dimensional data.