Polychoric correlations under the assumption of elliptical latent traits

Categorical structural equation models (cat-SEM) typically rely on tetrachoric/polychoric correlations under a latent multivariate normality assumption. This can be generalized to an elliptical latent trait, whose radial symmetry justifies treating a single correlation parameter as the target of estimation. This article makes two contributions. First, it introduces an elliptical sieve estimator that profiles the latent correlation over a non-parametric radial scale mixture. Simulations under a variety of latent elliptical densities show reduced pseudo-likelihood bias and stable performance across realistic threshold schemes (including skewed floor/ceiling patterns), category numbers (3-5) and sample sizes typical of cat-SEM. Second, an ordinal tail asymmetry test is proposed that uses checkerboard-copula tail contrasts and a randomization reference distribution to diagnose violations of latent radial symmetry. For items with 4-5 categories and symmetric thresholds, or thresholds skewed in the same direction, the test maintains near-nominal Type I error under elliptical densities, and achieves high power against non-elliptical copulas. However, with binary items and with three-category items whose thresholds are strongly asymmetric and oriented in opposite directions, Type I error inflates even under ellipticity. In such coarse, highly unbalanced designs, fully parametric latent-copula models remain preferable to semiparametric ordinal diagnostics.