The tetrad constraint is widely used to test whether four observed variables are conditionally independent given a latent variable, based on the fact that if four observed variables following a linear model are mutually independent after conditioning on an unobserved variable, then products of covariances of any two different pairs of these four variables are equal. It is an important tool for discovering a latent common cause or distinguishing between alternative linear causal structures. However, the classical tetrad constraint fails in nonlinear models because the covariance of observed variables cannot capture nonlinear association. In this paper, we propose a generalized tetrad constraint, which establishes a testable implication for conditional independence given a latent variable in nonlinear and nonparametric models. In linear models, this constraint implies the classical tetrad constraint; in nonlinear models, it remains a necessary condition for conditional independence but the classical tetrad constraint no longer is. Based on this constraint, we further propose a formal test, which can control type I error and has power approaching unity under certain conditions. We illustrate the proposed approach via simulations and two real data applications on mental ability tests and on moral attitudes towards dishonesty.

翻译：四元组约束被广泛用于检验四个观测变量在给定潜变量条件下是否条件独立，其依据是：若遵循线性模型的四个观测变量在给定未观测变量后相互独立，则这四个变量任意两对不同组合的协方差乘积相等。该方法是发现潜在共同原因或区分替代线性因果结构的重要工具。然而，经典四元组约束在非线性模型中失效，因为观测变量的协方差无法捕捉非线性关联。本文提出一种广义四元组约束，为非参数和非线性模型中给定潜变量条件下的条件独立性建立了可检验的推论。在线性模型中，该约束蕴含经典四元组约束；在非线性模型中，它仍是条件独立性的必要条件，但经典四元组约束不再成立。基于该约束，我们进一步提出一种形式化检验方法，该方法能控制第一类错误，并在特定条件下检验功效趋于一致。通过仿真实验以及关于智力能力测试和不诚实行为道德态度的两个真实数据应用，我们验证了所提方法的有效性。