Wai-Yin Poon

Conformal normal curvature and assessment of local influence

In 1986, R. D. Cook proposed differential geometry to assess local influence of minor perturbations of statistical models. We construct a conformally invariant curvature, the conformal normal curvature, for the same purpose. This curvature provides a measure of local influence ranging from 0 to 1, with objective bench-marks to judge largeness. We study various approaches to using the conformal normal curvature and the relationships between these approaches.

Structural equation models with continuous and polytomous variables

A two-stage procedure is developed for analyzing structural equation models with continuous and polytomous variables. At the first stage, the maximum likelihood estimates of the thresholds, polychoric covariances and variances, and polyserial covariances are simultaneously obtained with the help of an appropriate transformation that significantly simplifies the computation. An asymptotic covariance matrix of the estiates is also computed. At the second stage, the parameters in the structural covariance model are obtained via the generalized least squares approach. Basic statistical properties of the estimates are derived and some illustrative examples and a small simulation study are reported.

Full maximum likelihood analysis of structural equation models with polytomous variables

This paper is concerned with the analysis of structural equation models with polytomous variables. Identification conditions for the basic model are discussed. Theory for the full simultaneous maximum likelihood estimation of the thresholds and the covariance structure parameters is developed. An example is presented to illustrate the method.

A three-stage estimation procedure for structural equation models with polytomous variables

This paper is concerned with the analysis of structural equation models with polytomous variables. A computationally efficient three-stage estimator of the thresholds and the covariance structure parameters, based on partition maximum likelihood and generalized least squares estimation, is proposed. An example is presented to illustrate the method.

Comparison of Approaches in Estimating Interaction and Quadratic Effects of Latent Variables

Various approaches using the maximum likelihood (ML) option of the LISREL program and products of indicators have been proposed to analyze structural equation models with non-linear latent effects on the basis of Kenny and Judds formulation. Recently, some methods based on the Bayesian approach and the exact ML approaches have been developed. This article reviews, elaborates and compares several approaches for analyzing nonlinear models with interaction and/or quadratic effects. A total of four approaches are examined, including the product indicator ML approaches proposed by Jaccard and Wan (1995) and Joreskog and Yang (1996), a Bayesian approach and an exact ML approach. The empirical performances of these approaches are assessed using simulation studies in terms of their capabilities in producing reliable parameter and standard error estimates. It is found that whilst the Bayesian and the exact ML approaches produce satisfactory results in all the settings under consideration, and are in general very reliable; the product indicator ML approaches can only produce reasonable results in simple models with large sample sizes.

The Comparison of Single Item Constructs by Relative Mean and Relative Variance

This article discusses the use of an approach advocated in the psychometrics and statistics literature for analyzing square contingency tables with ordered and comparable categories. Assuming that observed ordinal categorical variables are manifestations of underlying continuous variables, a model is formulated to compare the variables by studying the relative location and the relative dispersion of the underlying continuous variables. The formulation, interpretation, and analysis of the model are discussed, and the implementation of the proposed procedure using easily accessible software is addressed. The proposed approach is then compared with a widely adopted simple method that treats the ordinal measures as if they were interval scales. Analyses of real data and simulation results show that the simple method can be misleading and that the proposed approach is preferable in detecting variable differences.

Maximum likelihood estimation of polyserial correlations

This paper considers a multivariate normal model with one of the component variables observable only in polytomous form. The maximum likelihood approach is used for estimation of the parameters in the model. The Newton-Raphson algorithm is implemented to obtain the solution of the problem. Examples based on real and simulated data are reported.

Simultaneous analysis of multivariate polytomous variates in several groups

The frequencies ofm independentp-way contingency tables are analyzed by a model that assumes that the ordinal categorical data in each ofm groups are generated from a latent continuous multivariate normal distribution. The parameters of these multivariate distributions and of the relations between the ordinal and latent variables are estimated by maximum likelihood. Goodness-of-fit statistics based on the likelihood ratio criterion and the Pearsonian chisquare are provided to test the hypothesis that the proposed model is correct, that is, it fits the observed sample data. Hypotheses on the invariance of means, variances, and polychoric correlations of the latent variables across populations are tested by Wald statistics. The method is illustrated on an example involving data on three five-point ordinal scales obtained from male and female samples.

Generalized multimode latent variable models: implementation by standard programs

Abstract Three-mode models in factor analysis have not been used very frequently duein part to their mathematical, statistical, and computational complexity. It is shown that standardly-available computer programs such as LISREL and EQS can be used to estimate and test such models. The models are generalized to permit more complex measurement structures, as well as to allow linear structural regressions among the latent variables. These generalized multimode models can be similarly easily computationally implemented. An example is used to illustrate the ideas.

A distribution free approach for analysis of two-level structural equation model

Abstract A distribution free method is developed to analyze the two-level structural equation model. A two stage estimation procedure is employed. Asymptotic properties such as the distribution of the estimates and the goodness-of-fit test statistic for evaluating the model adequacy are derived. Computational aspects and estimation under constraints are discussed. Results of a simulation study are reported.