On Substantive Research Hypotheses, Conditional Independence Graphs and Graphical Chain Models

TLDR

Graphs of points and lines or arrows are used to formulate hypotheses about variable relationships, where points denote variables, connections denote associations, and absent connections imply conditional independence, thereby characterizing conditional independence structures. Examples show how conditional independences manifest in model‑derived summary statistics and how the graphs reveal analogies and equivalences among models. Graphical chain models unify numerous previously useful statistical techniques and offer tools for novel analyses.

Abstract

SUMMARY Graphs consisting of points, and lines or arrows as connections between selected pairs of points, are used to formulate hypotheses about relations between variables. Points stand for variables, connections represent associations. When a missing connection is interpreted as a conditional independence, the graph characterizes a conditional independence structure as well. Statistical models, called graphical chain models, correspond to special types of graphs which are interpreted in this fashion. Examples are used to illustrate how conditional independences are reflected in summary statistics derived from the models and how the graphs help to identify analogies and equivalences between different models. Graphical chain models are shown to provide a unifying concept for many statistical techniques that in the past have proven to be useful in analyses of data. They also provide tools for new types of analysis.