[Paper Review] Qualitative Evaluation of Associations by the Transitivity of the Association Signs

Zhichao Jiang, Peng Ding+1 more|arXiv (Cornell University)|May 16, 2014

Advanced Causal Inference Techniques21 references9 citations

TL;DR

This paper introduces four association measures—density, distribution, expectation, and correlation—assessing their transitivity in sign under conditional independence. It establishes conditions under which positive associations between X and Y, and between Y and Z, imply a positive association between X and Z, especially under exponential family distributions, enabling qualitative causal inference from separate studies without joint data.

ABSTRACT

We say that the signs of association measures among three variables {X, Y, Z} are transitive if a positive association measure between the variable X and the intermediate variable Y and further a positive association measure between Y and the endpoint variable Z imply a positive association measure between X and Z. We introduce four association measures with different stringencies, and discuss conditions for the transitivity of the signs of these association measures. When the variables follow exponential family distributions, the conditions become simpler and more interpretable. Applying our results to two data sets from an observational study and a randomized experiment, we demonstrate that the results can help us to draw conclusions about the signs of the association measures between X and Z based only on two separate studies about {X, Y} and {Y, Z}.

Motivation & Objective

• To formalize the transitivity of association signs among three variables X, Y, and Z, where X→Y and Y→Z imply X→Z.
• To identify conditions under which the sign of association between X and Z can be inferred from associations between X–Y and Y–Z without joint data.
• To extend transitivity results to both conditional independence and non-conditional independence settings.
• To apply theoretical findings to real data from an observational study and a randomized experiment, demonstrating practical utility.
• To provide interpretable conditions for transitivity when variables follow exponential family distributions.

Proposed method

• Introduces four association measures: density (log-likelihood second derivative), distribution (conditional CDF derivative), expectation (conditional mean derivative), and correlation coefficient.
• Defines transitivity as the implication: non-negative X–Y and Y–Z associations imply non-negative X–Z association.
• Derives sufficient conditions for transitivity using stochastic monotonicity and conditional independence assumptions.
• Applies results to exponential family distributions, simplifying conditions into interpretable forms.
• Uses Lemma 1 to show that compositions of monotonic functions preserve monotonicity in conditional densities.
• Employs proof by contradiction and covariance bounds to validate transitivity under linear models and correlation constraints.

Experimental results

Research questions

• RQ1 Under what conditions is the sign of association between X and Z transitive through an intermediate variable Y?
• RQ2 How do different association measures (density, distribution, expectation, correlation) compare in their stringency for transitivity?
• RQ3 What conditions ensure transitivity when X and Z are conditionally independent given Y?
• RQ4 How can transitivity be assessed when conditional independence does not hold?
• RQ5 Can theoretical transitivity conditions be applied to real-world data from separate studies?

Key findings

• Transitivity of association signs holds when the joint density satisfies certain monotonicity conditions on partial derivatives, particularly when ∂²lnf(x,z|y)/∂x∂z ≥ 0.
• For exponential family distributions, the transitivity conditions become more interpretable and easier to verify using conditional densities.
• When X and Z are conditionally independent given Y, transitivity is guaranteed if the conditional density f(z|x,y) is non-decreasing in x and y, and f(x|y) is non-decreasing in x.
• In linear models, transitivity of the expectation association holds if the regression coefficients β₁, β₂, and β₄ are non-negative, with a contradiction-based proof showing impossibility of violating transitivity under correlation bounds.
• The density and expectation associations are equivalent when X or Z is binary, simplifying transitivity assessment.
• All conditions for transitivity can be evaluated from the conditional density f(x,z|y), enabling inference from marginal or conditional data alone.