Extremal Depth for Functional Data and Applications

Statistical modeling of diabetic neuropathy: Exploring the dynamics of nerve mortality

Diabetic neuropathy is a disorder characterized by impaired nerve function and reduction of the number of epidermal nerve fibers per epidermal surface. Additionally, as neuropathy related nerve fiber loss and regrowth progresses over time, the two-dimensional spatial arrangement of the nerves becomes more clustered. These observations suggest that with development of neuropathy, the spatial pattern of diminished skin innervation is defined by a thinning process which remains incompletely characterized. We regard samples obtained from healthy controls and subjects suffering from diabetic neuropathy as realisations of planar point processes consisting of nerve entry points and nerve endings, and propose point process models based on spatial thinning to describe the change as neuropathy advances. Initially, the hypothesis that the nerve removal occurs completely at random is tested using independent random thinning of healthy patterns. Then, a dependent parametric thinning model that favors the removal of isolated nerve trees is proposed. Approximate Bayesian computation is used to infer the distribution of the model parameters, and the goodness-of-fit of the models is evaluated using both non-spatial and spatial summary statistics. Our findings suggest that the nerve mortality process changes as neuropathy advances.

New methods for multiple testing in permutation inference for the general linear model

Permutation methods are commonly used to test the significance of regressors of interest in general linear models (GLMs) for functional (image) data sets, in particular for neuroimaging applications as they rely on mild assumptions. Permutation inference for GLMs typically consists of three parts: choosing a relevant test statistic, computing pointwise permutation tests, and applying a multiple testing correction. We propose new multiple testing methods as an alternative to the commonly used maximum value of test statistics across the image. The new methods improve power and robustness against inhomogeneity of the test statistic across its domain. The methods rely on sorting the permuted functional test statistics based on pointwise rank measures; still, they can be implemented even for large data. The performance of the methods is demonstrated through a designed simulation experiment and an example of brain imaging data. We developed the R package GET, which can be used for the computation of the proposed procedures.

A depth-based global envelope test for comparing two groups of functions with applications to biomedical data

Functional data are commonly observed in many emerging biomedical fields and their analysis is an exciting developing area in statistics. Numerous statistical methods, such as principal components, analysis of variance, and linear regression, have been extended to functional data. The statistical analysis of functions can be significantly improved using nonparametric and robust estimators. New ideas of depth for functional data have been proposed in recent years and can be extended to image data. They provide a way of ordering curves or images from center-outward, and of defining robust order statistics in a functional context. In this paper we develop depth-based global envelope tests for comparing two groups of functions or images. In addition to providing global P-values, the proposed envelope test can be displayed graphically and indicates the specific portion(s) of the functional data (eg, in pixels or in time) that may have led to rejection of the null hypothesis. We show in a simulation study the performance of the envelope test in terms of empirical power and size in different scenarios. The proposed depth-based global approach has good power even for small differences and is robust to outliers. The methodology introduced is applied to test whether children with normal and low birth weight have similar growth pattern. We also analyzed a brain image dataset consisting of positron emission tomography scans of severe depressed patients and healthy controls. The global envelope test was used to find and visualize differences between the two groups.

Functional principal component analysis for identifying multivariate patterns and archetypes of growth, and their association with long-term cognitive development

For longitudinal studies with multivariate observations, we propose statistical methods to identify clusters of archetypal subjects by using techniques from functional data analysis and to relate longitudinal patterns to outcomes. We demonstrate how this approach can be applied to examine associations between multiple time-varying exposures and subsequent health outcomes, where the former are recorded sparsely and irregularly in time, with emphasis on the utility of multiple longitudinal observations in the framework of dimension reduction techniques. In applications to children’s growth data, we investigate archetypes of infant growth patterns and identify subgroups that are related to cognitive development in childhood. Specifically, “Stunting” and “Faltering” time-dynamic patterns of head circumference, body length and weight in the first 12 months are associated with lower levels of long-term cognitive development in comparison to “Generally Large” and “Catch-up” growth. Our findings provide evidence for the statistical association between multivariate growth patterns in infancy and long-term cognitive development.