Products of Variables in Structural Equation Models

A general method is introduced in which variables that are products of other variables in the context of a structural equation model (SEM) can be decomposed into the sources of variance due to the multiplicands. The result is a new category of SEM which we call a Products of Variables Model (PoV). Some useful and practical features of PoV models include the estimation of interactions between latent variables, latent variable moderators, manifest moderators with missing values, and manifest or latent squared terms. Expected means and covariances are analytically derived for a simple product of two variables and it is shown that the method reproduces previously published results for this special case. It is shown algebraically that using centered multiplicands results in an unidentified model, but if the multiplicands have non-zero means, the result is identified. The method has been implemented in OpenMx and Ωnyx and is applied in five extensive simulations.

本文提出一种通用方法，可将结构方程模型（Structural Equation Model, SEM）框架下由其他变量乘积构成的变量，分解为各乘法因子对应的方差来源。由此衍生出一类全新的结构方程模型，我们将其命名为变量乘积模型（Products of Variables Model, PoV）。变量乘积模型具备多项实用特性，可实现潜变量间交互作用、潜调节变量、带有缺失值的观测调节变量，以及显变量或潜变量平方项的估计。针对两变量的简单乘积场景，本文通过解析推导得到了其期望均值与协方差，并验证该方法可复现该特例下已发表的既有研究结论。代数推导表明，若对乘法因子实施中心化处理，则所得模型将陷入不可识别状态；但若乘法因子具有非零均值，则模型可被识别。该方法已在OpenMx与Ωnyx软件中完成实现，并通过五组大规模仿真实验开展了验证应用。