ROBUSTNESS ISSUES IN STRUCTURAL EQUATION MODELING - A REVIEW OF RECENT DEVELOPMENTS
ROBUSTNESS ISSUES IN STRUCTURAL EQUATION MODELING - A REVIEW OF RECENT DEVELOPMENTS
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DOI：
10.1007/bf00152011
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发表时间：
1990-11-01
影响因子：
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通讯作者：
SATORRA, A
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文献类型：
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作者：
SATORRA, A
结论在结构方程模型中，统计学家需要进行假设，以便（1）保证对感兴趣的参数的估计是一致的，（2）评价估计的精度和检验统计量的显著性水平。关于目的（1），典型类型的分析（ML和WLS）对违反分布假设具有鲁棒性;即，估计值保持一致，或任何类型的WLS分析和z的分布。(It然而，应该注意，（1）对结构错误指定很敏感。）用于目的（2）的一个典型假设是，可观测向量z服从多元正态分布。与目的（2）有关，分布错误可能会对效率以及检验统计的功效产生影响（见Satorra，1989 a）;也就是说，对于给定的z的特定分布，某些估计方法可能比其他方法更精确。例如，ADF-WLS在各种z分布下是渐近最优的，而NT-WLS的渐近最优性可能在数据为非正态时丢失。违反分布假设可能会导致目的（2）。然而，最近的理论，如第7节和第8节中所描述的理论，表明在正态性假设下导出的检验统计量的估计量和渐近零分布的渐近方差可能是正确的，即使在某些模型条件成立的情况下（定理1的条件）。也就是说，在非正态分布的特定应用中，正态分布的假设起到了"工作装置"的作用，有助于计算感兴趣的统计数据的正确分布。这对应于第7节和第8节中所称的渐近稳健性。对于实践中考虑的大多数模型，用独立性假设代替假设不相关意味着达到渐近稳健性的性质;在这种情况下，为了评估感兴趣的统计量的渐近行为，即使对于非正态数据，Γ的NT形式也会产生正确的结果。无论使用何种拟合准则，这个鲁棒性结果都适用。在处理渐近鲁棒性问题时，"不相关"和"独立性"之间的区别变得至关重要。模型变量之间的统计独立性保证了感兴趣的统计量的分布是非正态变量的渐近分布;因此适用于Γ的NT形式。作为这种区别明显的一个例子，考虑一个带有异方差扰动项的简单回归模型。这里的干扰项与回归量无关，但方差随回归量的值而变化。关于表明ADF-WLS防止误差异方差性而ML通常会失败的研究，见Mooijaart和Satorra（1987）。在回归分析中，检测异方差性的常用方法是查看残差图。据推测，在结构方程模型中，区分不相关性和独立性的需要将迫使研究人员回到行数据，以便进行类似类型的“残差”检查。总之，重要的考虑是使用适当的公式计算估计和检验统计量的抽样变异性，而不要求估计过程在某种意义上是“最好的”。我们已经看到，这样的计算可以正确地进行使用错误的假设相对于可观察变量的向量的分布，提供了一些额外的模型条件举行。
ConclusionsIn structural equation modeling the statistician needs assumptions inorder (1) to guarantee that the estimates are consistent for the parameters of interest, and (2) to evaluate precision of the estimates and significance level of test statistics. With respect to purpose (1), the typical type of analyses (ML and WLS) are robust against violation of distributional assumptions; i.e., estimates remain consistent or any type of WLS analysis and distribution ofz. (It should be noted, however, that (1) is sensitive to structural misspecification.) A typical assumption used for purpose (2), is the assumption that the vectorzof observable follows a multivariate normal distribution.In relation to purpose (2), distributional misspecification may have consequences for efficiency, as well as power of test statistics (see Satorra, 1989a); that is, some estimation methods may bemore precise than others for a given specific distribution ofz. For instance, ADF-WLS is asymptotically optimal under a variety of distributions ofz, while the asymptotic optimality of NT-WLS may be lost when the data is non-normalViolation of a distributional assumption may have consequences for purpose (2). However, recent theory, such as the one described in Sections 7 and 8, showes that asymptotic variances of estimates and asympttic null distributions of test statistics derived under the normality assumption may be correct even whenzis non-normal provided certain model conditions hold (the conditions of Theorem 1). That is, in a specific application withznon-normally distributed, the assumption thatzis normal play the role of a “working device” that facilitates calculation of the correct distribution of statistics of interest. This corresponds to what in Section 7 and 8 has been called asymptotic robustness.For most of the models considered in practice, replacing the assyumption uncorrelation for the assumption of independence implised reaching the properties of asymptotic robustness; in that case, in order to evaluate the asymptotic behavior of statistics of interest, a NT form for Γ produces correct results even for non-normal data. This robustness result applies regardless of the type of fitting criterion used.Distinction between “uncorrelation’ and ‘independence’ becomes crucial when dealing with the asymptotic robustness issue. Statistical independence among variables of the model guarantee that the distribution of statistics of interest are asymptotically distribution-free of the non-normal variables; thus a NT form for Γ applies. As an example of where such distinction is apparent, consider a simple regression model with a heteroskedastic disturbance term. Here the disturbance term is uncorrelated with the regressor, but the variance varies with the value of the regressor. For a study showing that ADF-WLS protects against heteroskedasticity of erros, while ML wil generally fail, see Mooijaart and Satorra (1987).In regresion analysis the usual method for detecting heteroskedasticity is by looking at residual plots. Presumably, alsi in structural equation modeling, the need to distinguish between uncorrelation and independence will force the researcher to go back to the row data in order to do a similar type of “residuals’ inspection.In concluding, an importance consideration is to compute sampling variability for estimates and test statistics using appropriate formulae, without requiring that the estimation procedure be the ‘best’ in some sense. We have seen that such computations can be carried out correctly using the wrong assumptions with respect to the distribution of the vector of observable variables, provided some additional model conditions hold …