## The Annals of Applied Statistics

- Ann. Appl. Stat.
- Volume 11, Number 2 (2017), 725-750.

### Hypothesis testing for network data in functional neuroimaging

Cedric E. Ginestet, Jun Li, Prakash Balachandran, Steven Rosenberg, and Eric D. Kolaczyk

#### Abstract

In recent years, it has become common practice in neuroscience to use networks to summarize relational information in a set of measurements, typically assumed to be reflective of either functional or structural relationships between regions of interest in the brain. One of the most basic tasks of interest in the analysis of such data is the testing of hypotheses, in answer to questions such as “Is there a difference between the networks of these two groups of subjects?” In the classical setting, where the unit of interest is a scalar or a vector, such questions are answered through the use of familiar two-sample testing strategies. Networks, however, are not Euclidean objects, and hence classical methods do not directly apply. We address this challenge by drawing on concepts and techniques from geometry and high-dimensional statistical inference. Our work is based on a precise geometric characterization of the space of graph Laplacian matrices and a nonparametric notion of averaging due to Fréchet. We motivate and illustrate our resulting methodologies for testing in the context of networks derived from functional neuroimaging data on human subjects from the 1000 Functional Connectomes Project. In particular, we show that this global test is more statistically powerful than a mass-univariate approach. In addition, we have also provided a method for visualizing the individual contribution of each edge to the overall test statistic.

#### Article information

**Source**

Ann. Appl. Stat., Volume 11, Number 2 (2017), 725-750.

**Dates**

Received: August 2014

Revised: November 2016

First available in Project Euclid: 20 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoas/1500537721

**Digital Object Identifier**

doi:10.1214/16-AOAS1015

**Mathematical Reviews number (MathSciNet)**

MR3693544

**Zentralblatt MATH identifier**

06775890

**Keywords**

Fréchet mean fMRI graph Laplacian hypothesis testing matrix manifold network data object data

#### Citation

Ginestet, Cedric E.; Li, Jun; Balachandran, Prakash; Rosenberg, Steven; Kolaczyk, Eric D. Hypothesis testing for network data in functional neuroimaging. Ann. Appl. Stat. 11 (2017), no. 2, 725--750. doi:10.1214/16-AOAS1015. https://projecteuclid.org/euclid.aoas/1500537721

#### Supplemental materials

- Proofs of theorems. Therein we here provide detailed proofs of the main results in this paper.Digital Object Identifier: doi:10.1214/16-AOAS1015SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.