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 nonzero mean, multiple observations, and deviations from normality. Supplementary materials for this article are available online.
Highlights
Networks or relational data among m actors, nodes or objects are frequently presented in the form of an m × m matrix Y = {yij : 1 ≤ i, j ≤ m}, where the entry yij corresponds to a measure of the directed relationship from object i to object j
We show that T (Ỹ) is invariant under diagonal transformations and we can approximate the null distribution of the test statistic based on for data drawn from a matrix normal distribution where the diagonal entries are replaced with zeros
Unlike the previous testing literature for matrix normal models that required multiple observations, and concentrated on testing a null of separable covariances versus an unstructured alternative, we proposed testing a null of no row or column correlations versus an alternative of full row and column correlations using a single observation of a network
Summary
Networks or relational data among m actors, nodes or objects are frequently presented in the form of an m × m matrix Y = {yij : 1 ≤ i, j ≤ m}, where the entry yij corresponds to a measure of the directed relationship from object i to object j. Such data are of interest in a variety of scientific disciplines: Sociologists and epidemiologists gather friendship network data to study social development and health outcomes among children (Fletcher et al, 2011; Potter et al, 2012), economists study markets by analyzing networks of business interactions among companies or countries (Westveld and Hoff, 2011a; Lazzarini et al, 2001), and biologists study gene-gene interaction networks to better understand biological pathways (Bergmann et al, 2003; Stuart et al, 2003).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
