Analysis of Covariance Structures under Elliptical Distributions

Shapiro, Alexander; Browne, Michael W.

doi:10.1080/01621459.1987.10478544

Cited by 103 publications

(64 citation statements)

References 21 publications

(26 reference statements)

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“…For elliptical distributions, Browne (1984), Kano (1992), Shapiro and Browne (1987), and Satorra and Bentler (1994) have developed corrections to the normal likelihood ratio statistic. (To avoid technicalities, we simply remark that if a set of variables follows an elliptical distribution, all its marginal distributions are symmetric with the same kurtosis.…”

Section: Multivariate Normalitymentioning

confidence: 99%

Principles and practice in reporting structural equation analyses.

McDonald

2002

Psychological Methods

3,639

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Principles for reporting analyses using structural equation modeling are reviewed, with the goal of supplying readers with complete and accurate information. It is recommended that every report give a detailed justification of the model used, along with plausible alternatives and an account of identifiability. Nonnormality and missing data problems should also be addressed. A complete set of parameters and their standard errors is desirable, and it will often be convenient to supply the correlation matrix and discrepancies, as well as goodness-of-fit indices, so that readers can exercise independent critical judgment. A survey of fairly representative studies compares recent practice with the principles of reporting recommended here.Structural equation modeling (SEM), also known as path analysis with latent variables, is now a regularly used method for representing dependency (arguably "causal") relations in multivariate data in the behavioral and social sciences. Following the seminal work of Jöreskog (1973), a number of models for linear structural relations have been developed (Bentler & Weeks, 1980;Lohmoller, 1981;McDonald, 1978), and work continues on these. Commercial statistical packages include LISREL (Jöreskog & Sör-bom, 1989, 1996, EQS (Bentler, 1985(Bentler, , 1995, CALIS (Hartmann, 1992), MPLUS (Muthén & Muthén, 1998), RAMONA (Browne, Mels, & Cowan, 1994), SEPATH (Steiger, 1995), and AMOS (Arbuckle, 1997). Available freeware includes COSAN (Fraser & McDonald, 1988) and Mx (Neale, 1997). McArdle and McDonald (1984) proved that different matrix formulations of a path model with latent variables are essentially equivalent. Programs such as those listed supply essentially the same basic information, with minor variations in the details supplied. Thus, the eight parameter LISREL model, which arose out of the work of Keesling and Wiley (see Wiley, 1973) and was subsequently developed to its current state by Jöreskog (see Jöreskog and Sör-bom, 1996), the four-matrix model of Lohmoller (1981), the three-matrix EQS model of Bentler and Weeks (1980), and the two-matrix RAM model (see McArdle & McDonald, 1984) rearrange the same set of parameters. Not surprisingly-and perhaps not regrettably-user guides and texts on this topic are not in agreement in their recommendations about the style of presentation of results (e.g., see Bollen, 1989;Loehlin, 1992;Long, 1983aLong, , 1983b. There is even less agreement in the form of the results actually reported in articles on applications.It would be immodest for any journal article to offer a code of practice for the presentation of SEM results. It could also be counterproductive. (We note that for a long time there was a uniformly accepted convention for the publication of analysis of variance, or ANOVA results: the standard ANOVA table and the table of cell means. The near-disappearance of this from journals is regrettable.) Sound guidelines for the reporting of SEM results have been offered previously

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Section: Multivariate Normalitymentioning

confidence: 99%

Principles and practice in reporting structural equation analyses.

McDonald

2002

Psychological Methods

3,639

2,447

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“…For example, it holds for the factor analysis model (2.2). By the above discussion we have the following results, which in somewhat different forms were obtained in Tyler (1983) and Browne (1982Browne ( , 1984, and in the present form in Shapiro and Browne (1987). Recall that the condition "σ 0 ∈ T ", used in the above theorem, holds if the set Ξ 0 is positively homogeneous, which in turn is implied by condition (C) (invariance under a constant scaling factor).…”

Section: Elliptical Distributionsmentioning

confidence: 80%

“…Recall that the condition "σ 0 ∈ T ", used in the above theorem, holds if the set Ξ 0 is positively homogeneous, which in turn is implied by condition (C) (invariance under a constant scaling factor). We also have the following result about asymptotic robustness of the MDF estimators (Shapiro and Browne, 1987). THEOREM 6.2.…”

Section: Elliptical Distributionsmentioning

confidence: 99%

11 Statistical Inference of Moment Structures

Shapiro¹

2007

Handbook of Latent Variable and Related Models

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The following material is focused on the statistical inference of moment structure models. Although the theory is presented in terms of general moment structures, the main emphasis is on the analysis of covariance structures. Identifiability and the minimum discrepancy function (MDF) approach to statistical analysis (estimation) of such models are discussed. Topics of the large samples theory, in particular, consistency, asymptotic normality of MDF estimators and asymptotic chi-squaredness of MDF test statistics are addressed. Finally, some results addressing asymptotic robustness of the normal theory based MDF statistical inference in the analysis of covariance structures are presented.

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“…asymptotically has a noncentral chi-square distribution with df = p(p + 1)/2 − q degrees of freedom and noncentrality parameter (1 + κ) −1 δ, where Shapiro & Browne, 1987). Therefore, similar to (16), we can use the corrected normal distribution approximation of the distribution of T M L with mean nF * M L + (1 + κ)df and variance…”

Section: Non-normal Distributionsmentioning

confidence: 99%

Normal Versus Noncentral Chi-square Asymptotics of Misspecified Models

Chun

Shapiro

2009

Multivariate Behavioral Research

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The noncentral chi-square approximation of the distribution of the likelihood ratio (LR) test statistic is a critical part of the methodology in structural equations modeling (SEM).Recently, it was argued by some authors that in certain situations normal distributions may give a better approximation of the distribution of the LR test statistic. The main goal of this paper is to evaluate the validity of employing these distributions in practice.Monte Carlo simulation results indicate that the noncentral chi-square distribution describes behavior of the LR test statistic well under small, moderate and even severe misspecifications regardless of the sample size (as long as it is sufficiently large), while the normal distribution, with a bias correction, gives a slightly better approximation for extremely severe misspecifications. However, neither the noncentral chi-square distribution nor the theoretical normal distributions give a reasonable approximation of the LR test statistics under extremely severe misspecifications. Of course, extremely misspecified models are not of much practical interest.

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Analysis of Covariance Structures under Elliptical Distributions

Cited by 103 publications

References 21 publications

Principles and practice in reporting structural equation analyses.

Principles and practice in reporting structural equation analyses.

11 Statistical Inference of Moment Structures

Normal Versus Noncentral Chi-square Asymptotics of Misspecified Models

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