Mean Comparison: Manifest Variable Versus Latent Variable

Yuan, Ke‐Hai; Bentler, Peter M.

doi:10.1007/s11336-004-1181-x

Cited by 27 publications

(18 citation statements)

References 48 publications

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“…So only the mean structure is misspecified in this model. But the misspecification will also affect the estimation of the covariance structure [16]. Table 1 lists the correspondence of effect size, root mean square error of approximation…”

Section: Empirical Comparisonmentioning

confidence: 99%

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Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses

Yuan

Hayashi

Bentler

2007

Journal of Multivariate Analysis

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The normal distribution based likelihood ratio (LR) statistic is widely used in structural equation modeling. Under a sequence of local alternative hypotheses, this statistic has been shown to asymptotically follow a noncentral chi-square distribution. In practice, the population mean vector and covariance matrix as well as the model and sample size are always fixed. It is hard to justify the validity of the noncentral chi-square distribution for the resulting LR statistic even when data are normally distributed and sample size is large. By extending results in the literature, this paper develops normal distributions to describe the behavior of the LR statistic for mean and covariance structure analysis. A sequence of local alternative hypotheses is not necessary for the proposed distributions to be asymptotically valid. When the effect size is medium and above or when the model is not trivially misspecified, empirical results indicate that a refined normal distribution describes the behavior of the LR statistic better than the commonly used noncentral chi-square distribution, as measured by the Kolmogorov-Smirnov distance. Quantile-quantile plots are also provided to better understand the different distributions.

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Section: Empirical Comparisonmentioning

confidence: 99%

“…T 1 = √n(T LR /n − 11 )/ˆ16 ; T 2 = √ n(T LR /n − 11 )/ˆ 11 ; T 3 = √ n(T LR /n − 21 )/ˆ 16 , where 21 = 11 + tr(U 2 )/n and tr(U 2 ) is given inTable 1; T 4 = √ n(T LR /n − 21 )/ˆ 11 . Because the calculation of tr(U 2 ) involves second derivatives, we might approximate tr(U 2 ) by the degrees of freedom b − a.…”

mentioning

confidence: 99%

Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses

Yuan

Hayashi

Bentler

2007

Journal of Multivariate Analysis

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“…. , 4) and ∆α 2 , we assume that the following Taylor series expansion holds with the three-time differentiability ofθ with respect to u in the neighborhood of τ:…”

Section: Asymptotic Expansions Of the Distributions Of The Parameter mentioning

confidence: 99%

“…In order to derive α 1 and α 3 in (4.3), it is also assumed that t has the following Taylor series expansion: (4) ; n 0 ). …”

Section: Asymptotic Expansion Of the Distribution Of The Pivotal Statmentioning

confidence: 99%

“…On the other hand, in mean and covariance structure analysis with structural means, the parameters common to or specific in means and covariances are estimated using their sample counterparts simultaneously. Typical models with structural means are those with nonzero means of latent variables (e.g., factors), which are of interest especially when samples from several populations are available to confirm different factor means [1][2][3][4]. An advantage for the model with common parameters in means and covariances is found in the added accuracy of their estimators over those using only sample covariances [5,6].…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic expansions in mean and covariance structure analysis

Ogasawara

2009

Journal of Multivariate Analysis

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a b s t r a c tAsymptotic expansions of the distributions of parameter estimators in mean and covariance structures are derived. The parameters may be common to, or specific in means and covariances of observable variables. The means are possibly structured by the common/specific parameters. First, the distributions of the parameter estimators standardized by the population asymptotic standard errors are expanded using the single-and the two-term Edgeworth expansions. In practice, the pivotal statistic or the Studentized estimator with the asymptotically distribution-free standard error is of interest. An asymptotic distribution of the pivotal statistic is also derived by the Cornish-Fisher expansion. Simulations are performed for a factor analysis model with nonzero factor means to see the accuracy of the asymptotic expansions in finite samples.

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