Baldasaro Ruth E, Bauer Daniel J
a The University of North Carolina at Chapel Hill.
Multivariate Behav Res. 2011 Nov 30;46(6):1007-8. doi: 10.1080/00273171.2011.636691.
Many approaches have been proposed to estimate interactions among latent variables. These methods often assume a specific functional form for the interaction, such as a bilinear interaction. Theory is seldom specific enough to provide a functional form for an interaction, however, so a more exploratory, diagnostic approach may often be required. Bauer (2005) proposed a semiparametric approach that allows for the estimation of interaction effects of unknown functional form among latent variables. A structural equation mixture model (SEMM) is first fit to the data. Then an approximation of the interaction is obtained by aggregating over the mixing components. A simulation study is used to examine the performance of this semiparametric approach to two parametric approaches: the latent moderated structures approach (Klein & Moosbrugger, 2000) and the unconstrained product-indicator approach (Marsh, Wen, & Hau, 2004). Data were generated from four functional forms: main effects only, quadratic trend, bilinear interaction, and exponential interaction. Estimates of bias and root mean squared error of approximation were calculated by comparing the surface used to generate the data and the model-implied surface constructed from each approach. As expected, the parametric approaches were more efficient than the SEMM. For the main effects model, bias was similar for both the SEMM and parametric approaches. For the bilinear interaction, the parametric approaches provided nearly identical results, although the SEMM approach was slightly more biased. When the parametric approaches assumed a bilinear interaction and the data were generated from a quadratic trend or an exponential interaction, the parametric approaches generated biased estimates of the true surface. The SEMM approach approximated the true data generation surface with a similarly low level of bias for all the nonlinear surfaces. For example, Figure 1 shows the true surface for the bilinear interaction along with the SEMM estimated average surface. The results suggest that the SEMM approach can provide a relatively unbiased approximation to variety of nonlinear relationships among latent variables.
人们已经提出了许多方法来估计潜在变量之间的相互作用。这些方法通常假设相互作用具有特定的函数形式，比如双线性相互作用。然而，理论很少具体到能够为相互作用提供一种函数形式，所以通常可能需要一种更具探索性、诊断性的方法。鲍尔（2005年）提出了一种半参数方法，该方法能够估计潜在变量之间未知函数形式的相互作用效应。首先将一个结构方程混合模型（SEMM）拟合到数据上。然后通过对混合成分进行汇总来获得相互作用的近似值。一项模拟研究被用于检验这种半参数方法与两种参数方法的性能：潜在调节结构方法（克莱因和穆斯布鲁格，2000年）以及无约束乘积指标方法（马什、温、豪，2004年）。数据是从四种函数形式生成的：仅主效应、二次趋势、双线性相互作用以及指数相互作用。通过比较用于生成数据的曲面和由每种方法构建的模型隐含曲面，计算出近似值的偏差估计和均方根误差。不出所料，参数方法比结构方程混合模型更有效。对于主效应模型，结构方程混合模型和参数方法的偏差相似。对于双线性相互作用，参数方法提供了几乎相同的结果，尽管结构方程混合模型方法的偏差略大一些。当参数方法假设为双线性相互作用且数据是从二次趋势或指数相互作用生成时，参数方法对真实曲面生成了有偏差的估计。对于所有非线性曲面，结构方程混合模型方法都以类似的低偏差水平近似真实数据生成曲面。例如，图1展示了双线性相互作用的真实曲面以及结构方程混合模型估计的平均曲面。结果表明，结构方程混合模型方法能够为潜在变量之间的各种非线性关系提供相对无偏差的近似值。
