Albert Satorra

A scaled difference chi-square test statistic for moment structure analysis

A Scaled Di erence Chi-square Test Statistic for Moment Structure Analysis Albert Satorra Universitat Pompeu Fabra and Peter M. Bentler University of California, Los Angeles August 3, 1999 Research supported by the Spanish DGES grant PB96-0300, and USPHS grants DA00017 and DA01070.

Complex Sample Data in Structural Equation Modeling

Large-scale surveys using complex sample designs are frequently carried out by government agencies. The statistical analysis technology available for such data is, however, limited in scope. This study investigates and further develops statistical methods that could be used in software for the analysis of data collected under complex sample designs. First, it identifies several recent methodological lines of inquiry which taken together provide a powerful and general statistical basis for a complex sample, structural equation modeling analysis. Second, it extends some of this research to new situations of interest. A Monte Carlo study that empirically evaluates these techniques on simulated data comparable to those in largescale complex surveys demonstrates that they work well in practice. Due to the generality of the approaches, the methods cover not only continuous normal variables but also continuous nonnormal variables and dichotomous variables. Two methods designed to take into account the complex sample structure were

Scaled and adjusted restricted tests in multi-sample analysis of moment structures

We extend to score, Wald and difference test statistics the scaled and adjusted corrections to goodness-of-fit test statistics developed in Satorra and Bentler (1988a,b). The theory is framed in the general context of multisample analysis of moment structures, under general conditions on the distribution of observable variables. Computational issues, as well as the relation of the scaled and corrected statistics to the asymptotic robust ones, is discussed. A Monte Carlo study illustrates the comparative performance in finite samples of corrected score test statistics.

Power of the likelihood ratio test in covariance structure analysis

A procedure for computing the power of the likelihood ratio test used in the context of covariance structure analysis is derived. The procedure uses statistics associated with the standard output of the computer programs commonly used and assumes that a specific alternative value of the parameter vector is specified. Using the noncentral Chi-square distribution, the power of the test is approximated by the asymptotic one for a sequence of local alternatives. The procedure is illustrated by an example. A Monte Carlo experiment also shows how good the approximation is for a specific case.

Testing Structural Equation Models or Detection of Misspecifications

Assessing the correctness of a structural equation model is essential to avoid drawing incorrect conclusions from empirical research. In the past, the chi-square test was recommended for assessing the correctness of the model but this test has been criticized because of its sensitivity to sample size. As a reaction, an abundance of fit indexes have been developed. The result of these developments is that structural equation modeling packages are now producing a large list of fit measures. One would think that this progression has led to a clear understanding of evaluating models with respect to model misspecifications. In this article we question the validity of approaches for model evaluation based on overall goodness-of-fit indexes. The argument against such usage is that they do not provide an adequate indication of the “size” of the models misspecification. That is, they vary dramatically with the values of incidental parameters that are unrelated with the misspecification in the model. This is illustrated using simple but fundamental models. As an alternative method of model evaluation, we suggest using the expected parameter change in combination with the modification index (MI) and the power of the MI test.

Principles and Practice of Scaled Difference Chi-Square Testing

We highlight critical conceptual and statistical issues and how to resolve them in conducting Satorra–Bentler (SB) scaled difference chi-square tests. Concerning the original (Satorra & Bentler, 2001) and new (Satorra & Bentler, 2010) scaled difference tests, a fundamental difference exists in how to compute properly a models scaling correction factor (c), depending on the particular structural equation modeling software used. Because of how LISREL 8 defines the SB scaled chi-square, LISREL users should compute c for each model by dividing the models normal theory weighted least-squares (NTWLS) chi-square by its SB chi-square, to recover c accurately with both tests. EQS and Mplus users, in contrast, should divide the models maximum likelihood (ML) chi-square by its SB chi-square to recover c. Because ML estimation does not minimize the NTWLS chi-square, however, it can produce a negative difference in nested NTWLS chi-square values. Thus, we recommend the standard practice of testing the scaled difference in ML chi-square values for models M 1 and M 0 (after properly recovering c for each model), to avoid an inadmissible test numerator. We illustrate the difference in computations across software programs for the original and new scaled tests and provide LISREL, EQS, and Mplus syntax in both single- and multiple-group form for specifying the model M 10 that is involved in the new test.

Alternative test criteria in covariance structure analysis: A unified approach

In the context of covariance structure analysis, a unified approach to the asymptotic theory of alternative test criteria for testing parametric restrictions is provided. The discussion develops within a general framework that distinguishes whether or not the fitting function is asymptotically optimal, and allows the null and alternative hypothesis to be only approximations of the true model. Also, the equivalent of the information matrix, and the asymptotic covariance matrix of the vector of summary statistics, are allowed to be singular. When the fitting function is not asymptotically optimal, test statistics which have asymptotically a chi-square distribution are developed as a natural generalization of more classical ones. Issues relevant for power analysis, and the asymptotic theory of a testing related statistic, are also investigated.

Model conditions for asymptotic robustness in the analysis of linear relations

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Technical aspects of Muthén's liscomp approach to estimation of latent variable relations with a comprehensive measurement model

Muthén (1984) formulated a general model and estimation procedure for structural equation modeling with a mixture of dichotomous, ordered categorical, and continuous measures of latent variables. A general three-stage procedure was developed to obtain estimates, standard errors, and a chi-square measure of fit for a given structural model. While the last step uses generalized least-squares estimation to fit a structural model, the first two steps involve the computation of the statistics used in this model fitting. A key component in the procedure was the development of a GLS weight matrix corresponding to the asymptotic covariance matrix of the sample statistics computed in the first two stages. This paper extends the description of the asymptotics involved and shows how the Muthén formulas can be derived. The emphasis is placed on showing the asymptotic normality of the estimates obtained in the first and second stage and the validity of the weight matrix used in the GLS estimation of the third stage.

Robustness issues in structural equation modeling: a review of recent developments

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 of z. (It should be noted, however, that (1) is sensitive to structural misspecification.) A typical assumption used for purpose (2), is the assumption that the vector z of 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 of z. For instance, ADF-WLS is asymptotically optimal under a variety of distributions of z, 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 when z is non-normal provided certain model conditions hold (the conditions of Theorem 1). That is, in a specific application with z non-normally distributed, the assumption that z is 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. Roughly speaking, such additional model conditions amount to strengthen the usual assumption of uncorrelation among some random constituents of the model to the assumption of stochastic independecen.