Covariance structure analysis with heterogeneous kurtosis parameters

Kano, Yutaka; Berkane, Maia; Bentler, Peter M.

doi:10.1093/biomet/77.3.575

Cited by 40 publications

(22 citation statements)

References 28 publications

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“…In additional to the above three classes of statistics, other statistics such as the normal theory generalized least squares statistic T GLS and the heterogeneous kurtosis statistic T HK (Kano, Berkane, & Bentler, 1990) are also available in standard software (Bentler, 1995). These are related to the normal theory based statistics of the first class.…”

Section: Fit Indices Versus Test Statisticsmentioning

confidence: 99%

Fit Indices Versus Test Statistics

Yuan¹

2005

Multivariate Behavioral Research

353

224

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Model evaluation is one of the most important aspects of structural equation modeling (SEM). Many model fit indices have been developed. It is not an exaggeration to say that nearly every publication using the SEM methodology has reported at least one fit index. Most fit indices are defined through test statistics. Studies and interpretation of fit indices commonly assume that the test statistics follow either a central chi-square distribution or a noncentral chi-square distribution. Because few statistics in practice follow a chi-square distribution, we study properties of the commonly used fit indices when dropping the chi-square distribution assumptions. The study identifies two sensible statistics for evaluating fit indices involving degrees of freedom. We also propose linearly approximating the distribution of a fit index/statistic by a known distribution or the distribution of the same fit index/statistic under a set of different conditions. The conditions include the sample size, the distribution of the data as well as the base-statistic. Results indicate that, for commonly used fit indices evaluated at sensible statistics, both the slope and the intercept in the linear relationship change substantially when conditions change. A fit index that changes the least might be due to an artificial factor. Thus, the value of a fit index is not just a measure of model fit but also of other uncontrollable factors. A discussion with conclusions is given on how to properly use fit indices.In social and behavioral sciences, interesting attributes such as stress, social support, socio-economic status cannot be observed directly. They are measured by multiple indicators that are subject to measurement errors. By segregating measurement errors from the true scores of attributes, structural equation modeling (SEM), especially its special case of covariance structure analysis, provides a methodology for modeling the latent variables directly. Although there are many MULTIVARIATE BEHAVIORAL RESEARCH, 40(1),

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Section: Fit Indices Versus Test Statisticsmentioning

confidence: 99%

Fit Indices Versus Test Statistics

Yuan¹

2005

Multivariate Behavioral Research

353

224

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“…Heterogeneous kurtosis (HK) theory (Kano et al 1990) defines a still more general class of multivariate distributions that allows marginal distributions to have heterogeneous kurtosis parameters. The elliptical distribution is a special case of this class of distributions.…”

Section: Some Specific Test Statisticsmentioning

confidence: 99%

“…For example, a distributionfree test [see Equation (10) below] developed by Browne about 15 years ago (Browne 1982(Browne , 1984 is not available in any extant computer program, includ-ing Browne's own program RAMONA (Browne et al 1994). Similarly, a test based on heterogeneous kurtosis theory was published a half decade ago (Kano et al 1990), yet it has not been incorporated into any programs, including Bentler's EQS program. Several other valuable statistics are similarly unavailable.…”

Section: Introductionmentioning

confidence: 99%

COVARIANCE STRUCTURE ANALYSIS: Statistical Practice, Theory, and Directions

Bentler

Dudgeon²

1996

Annu. Rev. Psychol.

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490

310

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Although covariance structure analysis is used increasingly to analyze nonexperimental data, important statistical requirements for its proper use are frequently ignored. Valid conclusions about the adequacy of a model as an acceptable representation of data, which are based on goodness-of-fit test statistics and standard errors of parameter estimates, rely on the model estimation procedure being appropriate for the data. Using analogies to linear regression and anova, this review examines conditions under which conclusions drawn from various estimation methods will be correct and the consequences of ignoring these conditions. A distinction is made between estimation methods that are either correctly or incorrectly specified for the distribution of data being analyzed, and it is shown that valid conclusions are possible even under misspecification. A brief example illustrates the ideas. Internet access is given to a computer code for several methods that are not available in programs such as EQS or LISREL.

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“…A discrepancy function F = F[S, E(0)] is a measure of the discrepancy between S and E(8) evaluated at an estimator 6. Many discrepancy functions have been suggested in the literature (e.g., Bentler, 1983;Browne, 1982;J6reskog, 1969;Kano, Berkane, & Bentler, 1990), and the development here would hold for any of them. The most widely known function, the normal theory maximum-likelihood (ML) discrepancy function, is used here:…”

Section: Estimation With Gramian Constraintsmentioning

confidence: 65%

Gramian Matrices in Covariance Structure Models

Bentler

Jamshidian

1994

Applied Psychological Measurement

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Covariance structure models frequently contain out-of-range estimates that make no sense from either substantive or statistical points of view. Negative variance estimates are the most wellknown of these improper solutions, but correlations that are out of range also occur. Methods to minimize improper estimates have been accomplished by reparameterization and estimation under simple inequality constraints; but these solutions, discussed previously in this journal (Marsh, 1989), do not guarantee that the covariance matrices involved represent variances and covariances of real numbers, as required. A general approach to avoiding improper solutions in structural equation models is proposed. Although this approach does not resolve inadequacies in the data or theoretical model that may generate an improper solution, it solves the long-standing problem of obtaining proper estimates.One of the oldest, and most widely recognized, problems in the application of covariance structure analysis or structural equation models is the frequent occurrence of negative variance estimates, especially for unique (error) factors. For example, Bollen (1987) estimated a one-factor factor analysis model for three variables. Two of the unique variances were estimated as positive, but one was negative. Such estimates, often called Heywood cases (after Heywood, 1931), present serious problems, because the variances being estimated are supposed to represent the variances of persons' scores on real and not imaginary variables. Because real random variables must have non-negative variances, a negative variance estimate has no meaning, that is, it is inadmissible or improper. Yet, such variance estimates occur frequently in covariance structure analysis, especially when the sample size is small and a model has an insufficient number of indicators having high loadings on each factor (e.g., Anderson & Gerbing, 1984;Boomsma, 1985). ~~lhn9s (1987) improper estimate was eliminated when outlier cases were deleted.A recent discussion of this problem, and an evaluation of some suggested solutions, can be found in Dillon, I~ur~ar, ~ Mulani (1987). Their discussion, however, was incomplete. It did not deal with various other ways that &dquo;offending estimates'&dquo; can occur in covariance structure analysis. Rindskopf (1984) discussed several sources of this problem; Marsh (1989) also addressed the issue. Since this paper was completed, an excellent review of the topic was provided by Wothke (1993). The purpose of the present paper is to summarize the issues, and then to provide a solution to the fundamental problem that Wothke believes still remains: &dquo;There is a definite need for the development and teaching of a more general statistical solution to the estimation problems discussed her~99 (p. 288).Although negative variance estimates are the most widely known inadmissible estimates, correlations that are out of range occur frequently as well. Marsh (1989) reported a number of confirmatory multitrait-multimethod models that had factor correlatio...

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Covariance structure analysis with heterogeneous kurtosis parameters

Cited by 40 publications

References 28 publications

Fit Indices Versus Test Statistics

Fit Indices Versus Test Statistics

COVARIANCE STRUCTURE ANALYSIS: Statistical Practice, Theory, and Directions

Gramian Matrices in Covariance Structure Models

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