结构方程模型中调节效应的标准化估计

Appropriate Standardized Estimates for Moderating Effects in Structural Equation Models

回归分析和结构方程分析中，标准化估计对解释模型和比较效应大小有重要作用。对于调节效应模型（或交互效应模型），通常的标准化估计没有意义。虽然显变量的调节效应模型标准化估计问题已经解决，但潜变量的调节效应模型标准化估计问题复杂得多。本文先介绍回归分析中显变量调节效应模型的标准化估计，然后提出了一种通过参数的原始估计和通常标准化估计来计算潜变量调节效应模型的标准化估计的方法，得到的标准化估计是尺度不变的，说明可以用标准化估计来解释和比较主效应和调节效应。

The analyses of interactions or moderating effects are very important in psychological， behavioral and management researches. The exogenous (predictors) and endogenous (outcomes) constructs in these studies can be simple manifest variables or latent factors (made up of several observed indicators). The present paper discusses the standardized estimates of the models with moderating effects.Standardized estimates are routinely used to summarize the results of multiple regression models of manifest variables and structural equation models (SEM) of latent variables because they facilitate interpretation and comparison. Although the typical standardized estimates of moderating effects are not appropriate for multiple regression models with moderating terms， straightforward alternatives are well known. Whereas the analogous problem exists for the estimation of latent moderating effects in SEM， the situation is more complicated and apparently the difficulty has not been resolved. Here we propose appropriate standardized parameter estimates which can be easily formulated from the raw and completely standardized estimates routinely available from existing SEM software packages.It can be derived that for a structural equation with an moderating term ， the appropriate standardized parameters ， ， should be ， ， ， where ， and are the usual completely standardized estimates of ， and ， while and are the raw estimates of the variances of ， and . Both ， ， and ， are available from commercial SEM softwares.Some scale invariant properties of the appropriate standardized estimates are described， including that the main and moderating effects are scale invariant. That is， irrespective of the metric (e.g.， meter， centimeter) we use for the measurement， the appropriate standardized parameter estimates are identical. These desirable properties of the appropriate standardized estimates are illustrated with a simulation data set using the unconstrained approach to estimate the latent moderating effects. The results support the use of the appropriate standardized estimates in interpreting and comparing SEM estimates in latent moderating effect models.Although the difficult mathematical derivation is a bit beyond most applied SEM users， the ultimate procedures to obtain the appropriate standardized estimates are quite simple and straightforward， easily accomplished in a hand-held calculator or Excel spreadsheets. We demonstrate how applied users can obtain the appropriate standardized parameter estimates for models involving latent moderating terms.