Testing for nodal dependence in relational data matrices

TL;DR

This paper introduces an exact likelihood ratio test for detecting row and column dependence in relational data matrices, addressing a gap in formal testing methods for such dependence structures.

Contribution

It develops a likelihood ratio test within the matrix normal model framework to formally assess dependence in relational matrices, including extensions for common data features.

Findings

Provides an exact test for row and column dependence.

Extends the test to handle missing diagonal entries and non-normal data.

Offers a practical tool for analyzing relational data dependence.

Abstract

Relational data are often represented as a square matrix, the entries of which record the relationships between pairs of objects. Many statistical methods for the analysis of such data assume some degree of similarity or dependence between objects in terms of the way they relate to each other. However, formal tests for such dependence have not been developed. We provide a test for such dependence using the framework of the matrix normal model, a type of multivariate normal distribution parameterized in terms of row- and column-specific covariance matrices. We develop a likelihood ratio test (LRT) for row and column dependence based on the observation of a single relational data matrix. We obtain a reference distribution for the LRT statistic, thereby providing an exact test for the presence of row or column correlations in a square relational data matrix. Additionally, we provide extensions of the test to accommodate common features of such data, such as undefined diagonal entries, a non-zero mean, multiple observations, and deviations from normality.