Model Specification of Three-way Latent Nonlinear Effects: The Constrained Approach
許多統計模型常在線性假設下開展，但愈來愈多的研究提倡非線性的分析架構，或是發現了非線性現象。在行為科學中，最常被探討的非線性效果可能是二階交互作用，或稱為調節效果。當調節效果再被調節，或是調節效果本身是非線性時，都有可能產生三階非線性效果。目前多數三階非線性效果多以迴歸或是徑路模型分析，此類做法忽略了獨變項中可能的測量誤差，因而可能低估非線性效果。有鑑於此，本研究目的在利用結構方程模型以處理測量誤差，針對三個三階潛在非線性效果的典型模型，即三階交互作用模型、調節平方模型與立方模型， 推導利用限制式方法設定模型時所需的限制式， 並呈現LISREL 實作範例，再以模擬資料驗證LISREL程式的正確性。
While many statistical models are traditionally based on the assumption of linearity between variables, the fact that the social and behavioral science literature abounds in empirical studies involving nonlinear effects makes it imperative that statistical methods capable of dealing with nonlinear relations among variables be developed. One of the most common types of nonlinear effect is two-way interaction, or moderation. Cases in which the moderation effect is either quadratic or moderated by another variable call for probing three-way nonlinear effects, most analyses of which are based on regression or path analysis which ignore measurement errors in the independent variables. An alternative to regression and path analysis in analyzing three-way nonlinear effects is the use of structural equation modeling (SEM) in which measurements are incorporated into models. The SEM-based approach utilized in the current study is the constrained approach; our aim is to derive constraints corresponding to three types of threeway latent nonlinear effect models, including the three-way interaction model, the moderated quadratic model and the cubic model. The study also presents LISREL exemplar syntaxes and confirms their validity through simulated datasets.