Individual and group differences in multivariate domains: What happens when the number of traits increases?

The major domains of psychological variation are intrinsically multivariate, and can be mapped at various levels of resolution-from broad-band descriptions involving a small number of abstract traits to fine-grained representations based on many narrow traits. As the number of traits increases, the corresponding space becomes increasingly high-dimensional, and intuitions based on low-dimensional representations become inaccurate and misleading. The consequences for individual and group differences are profound, but have gone largely unrecognized in the psychological literature. Moreover, alternative distance metrics show distinctive behaviors with increasing dimensionality. In this paper, I offer a systematic yet accessible treatment of individual and group differences in multivariate domains, with a focus on high-dimensional phenomena and their theoretical implications. I begin by introducing four alternative metrics (the Euclidean, Mahalanobis, city-block, and shape distance) and reviewing their geometric properties. I also examine their potential psychological significance, because different metrics imply different cognitive models of how people process information about similarity and dissimilarity. I then discuss how these metrics behave as the number of traits increases. After considering the effects of measurement error and common methods of error correction, I conclude with an empirical example based on a large dataset of self-reported personality.