Stoel Reinoud D, Garre Francisca Galindo, Dolan Conor, van den Wittenboer Godfried
Department of Education, University of Amsterdam, Amsterdam, Netherlands.
Psychol Methods. 2006 Dec;11(4):439-55. doi: 10.1037/1082-989X.11.4.439.
The authors show how the use of inequality constraints on parameters in structural equation models may affect the distribution of the likelihood ratio test. Inequality constraints are implicitly used in the testing of commonly applied structural equation models, such as the common factor model, the autoregressive model, and the latent growth curve model, although this is not commonly acknowledged. Such constraints are the result of the null hypothesis in which the parameter value or values are placed on the boundary of the parameter space. For instance, this occurs in testing whether the variance of a growth parameter is significantly different from 0. It is shown that in these cases, the asymptotic distribution of the chi-square difference cannot be treated as that of a central chi-square-distributed random variable with degrees of freedom equal to the number of constraints. The correct distribution for testing 1 or a few parameters at a time is inferred for the 3 structural equation models mentioned above. Subsequently, the authors describe and illustrate the steps that one should take to obtain this distribution. An important message is that using the correct distribution may lead to appreciably greater statistical power.
作者展示了结构方程模型中对参数使用不等式约束如何影响似然比检验的分布。不等式约束在常用结构方程模型（如公共因子模型、自回归模型和潜在增长曲线模型）的检验中被隐性使用，尽管这一点并未得到普遍认可。此类约束是原假设的结果，在原假设中参数值被置于参数空间的边界上。例如，在检验增长参数的方差是否显著不同于0时就会出现这种情况。结果表明，在这些情况下，卡方差异的渐近分布不能被视为自由度等于约束数量的中心卡方分布随机变量的分布。针对上述3种结构方程模型，推断出了一次检验1个或几个参数时的正确分布。随后，作者描述并说明了获取该分布应采取的步骤。一个重要的信息是，使用正确的分布可能会显著提高统计功效。
