The Extremes of Normative Modelling

Abstract

AbstractIn this paper, we propose a new framework for understanding and modelling neurobiological extreme atypicalities for individual participants. We combine the strength of normative models, to make predictions for individual patients, with multivariate extreme value statistics, which allows us to model the outer centiles accurately, enabling accurate estimation of risk and detection of atypicality. In general, most statistical methods focus principally on estimating the mean of the distribution and are not appropriate to capture the tails of the distribution. We believe that the tails carry most of the information for predicting psychopathy in neurological and psychiatric disorders and should therefore be modelled carefully.In the univariate case, we demonstrate how to fit a generalized Pareto distribution with the peaks over threshold method. For the multivariate case, we present a novel framework that allows us to capture the multivariate tails of the distribution fully. The model relies on the so-called tail pairwise dependence matrix (TPDM). The TPDM is similar to a covariance matrix, but specified for the tails. From the TPDM we can create a reduced basis of vectors that explain most of the structure underlying the extreme values. The method is similar to covariance principal component analysis, but uniquely made for the extreme values. To evaluate this framework, we use a range of neuroimaging-derived measures from the UK Biobank dataset, including structural, functional and diffusion image-derived phenotypes. We find that the first extreme principal component mostly consists of whole-brain volume and area measures and the second extreme principal component is mostly compromised of white matter tracts. We further demonstrate the link between extreme brain deviations in the individual from a reference cohort to behaviour.