Star graphs induce tetrad correlations: for Gaussian as well as for binary variables

TL;DR

This paper explores how star graphs can produce tetrad correlations in both Gaussian and binary variables, linking graphical models to latent variable structures and deriving conditions for their existence.

Contribution

It establishes tetrad correlation conditions for latent variables in Gaussian and binary distributions, connecting star graphs to traceable regressions and expanding understanding of dependence structures.

Findings

Derived tetrad correlation conditions for Gaussian variables

Extended tetrad correlation conditions to binary variables

Presented applications with binary items demonstrating the theory

Abstract

Tetrad correlations were obtained historically for Gaussian distributions when tasks are designed to measure an ability or attitude so that a single unobserved variable may generate the observed, linearly increasing dependences among the tasks. We connect such generating processes to a particular type of directed graph, the star graph, and to the notion of traceable regressions. Tetrad correlation conditions for the existence of a single latent variable are derived. These are needed for positive dependences not only in joint Gaussian but also in joint binary distributions. Three applications with binary items are given.