Derive and calibrate the likelihood-ratio test for the latent-variable SEM

Derive and numerically validate the null distribution and operating characteristics of the likelihood-ratio test for the six-indicator latent-variable structural equation model distinguishing a shared obstruction from no shared obstruction.

Background

The paper proposes a full six-indicator SEM as an extension of the primary residualized-correlation analysis. The relevant comparison tests whether the shared-obstruction loading is zero, a boundary hypothesis for which the usual chi-square reference distribution is inappropriate.

Although the manuscript motivates a mixed reference distribution, it explicitly states that the mixture has not been derived for this model, has not been implemented, and that the SEM calibration remains pending. Until this is resolved, the authors restrict the primary confirmatory analysis to the residualized correlation and method-of-moments screen.

References

Calibration for either is not yet established: the operating characteristics rest on twenty cohorts per cell, and while the boundary geometry of $L_\theta=0$ is what motivates a $\tfrac12\chi2_0+\tfrac12\chi2_1$ reference distribution rather than $\chi2_1$, that mixture is asserted here and neither derived for this model nor implemented --- the code compares its likelihood-ratio statistic to a plain $\chi2_1$.

— Binding-Motivated Contextuality: A Cross-Domain Cyclic Test in Perception and Judgment  (2609.23977 - Shavit, 21 Sep 2026) in Section 3, subsection “The single-latent claim, as a structural test”