# Using Extreme Value Statistics to Reconceptualize Psychopathology as Extreme Deviations From a Normative Reference Model

## TL;DR

This paper introduces a new statistical framework using extreme value theory to better understand and detect extreme deviations in neuroimaging data, which could improve diagnosis of psychiatric disorders.

## Contribution

The novel contribution is combining normative models with multivariate extreme value statistics to model extreme deviations in neuroimaging data.

## Key findings

- A framework using normative models and extreme value statistics was developed to model extreme deviations in neuroimaging data.
- The approach was demonstrated on the UK Biobank dataset, showing how extreme values can be used to estimate risk and detect atypicality.
- A tail pairwise dependency matrix (TPDM) was introduced to capture multivariate tail dependencies in biological markers.

## Abstract

For many problems in neuroimaging, the most informative features occur in the tail of the distribution. For example, when considering psychiatric disorders as deviations from a ‘norm’, the tails of the distribution are considerably more informative than the bulk of the distribution for understanding risk, stratifying and predicting such disorders, and for anomaly detection. Yet, most statistical methods used in neuroimaging focus on modeling the bulk and fail to adequately capture extreme values occurring in the tails. To address this, we propose a framework that combines normative models with multivariate extreme value statistics to accurately model extreme deviations of a reference cohort for individual participants. Normative models are now widely used in clinical neuroscience and similar to the employment of normative growth charts in pediatric medicine to track a child's weight in relation to their age; normative models can be used with neuroimaging measurements to quantify individual neurophenotypic deviations from a reference cohort. However, formal statistical treatment of how to model the extreme deviations from these models has been lacking until now. In this article, we provide such an approach inspired by applications of extreme value statistics in meteorology. Since the presentation of extreme value statistics is quite technical, we begin with a non‐technical introduction to the fundamental principles of extreme value statistics to accurately map the tails of the normative distribution for biological markers, including mapping multivariate tail dependence across multiple markers. Next, we give a demonstration of this approach to the UK Biobank dataset and demonstrate how extreme values can be used to accurately estimate risk and detect atypicality. This framework provides a valuable tool for the statistical modeling of extreme deviations in neurobiological data, which could provide us with more accurate and effective diagnostic tools for neurological and psychiatric disorders.

Overview of our approach for extreme value statistics. First, we fit a normative model to imaging phenotypes, before employing a ‘peaks‐over‐threshold’ approach widely used in meteorology and finance. This allows us to model tail dependencies and provides a multivariate measure of extremely dependence through a tail pairwise dependency matrix (TPDM).