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The present paper aims to investigate some semi-separation axioms relating to the Alexandroff one point compactification (Alexandroff compactification, for short) of the digital plane with the Marcus-Wyse ( M W -, for brevity) topology. The Alexandroff compactification of the M W -topological plane is called the infinite M W -topological sphere up to homeomorphism. We first prove that under the M W -topology on Z 2 the connectedness of X (⊂ Z 2) with X ♯ ≥ 2 implies the semi-openness of X. Besides, for the infinite M W -topological sphere, we introduce a new condition for the hereditary property of the compactness of it. In addition, we investigate some conditions preserving the semi-openness or semi-closedness of a subset of the M W -topological plane in the process of an Alexandroff compactification. Finally, we prove that the infinite M W -topological sphere is a semi-regular space; thus, it is a semi- T 3 -space because it is a semi- T 1 -space. Hence we finally conclude that an Alexandroff compactification of the M W -topological plane preserves the semi- T 3 separation axiom. [ABSTRACT FROM AUTHOR]

Analysis of ordered binary and unordered binary data has received considerable attention in social and psychological research. This article introduces a Bayesian approach, which has several nice features in practical applications, for analyzing nonlinear structural equation models with dichotomous data. We demonstrate how to use the software WinBUGS and R2WinBUGS to obtain Bayesian estimates of the unknown parameters, estimates of latent variables, and the Deviance Information Criterion for model comparison. An illustrative example with an artificial data set is provided. Finally, simulation studies are conducted, not only to reveal the empirical performance of the Bayesian approach, but also to show that incorrectly treating binary data as ordinal, and vice versa, would produce misleading results. (Contains 3 figures and 8 tables.)

Local influence analysis is an important statistical method for studying the sensitivity of a proposed model to model inputs. One of its important issues is related to the appropriate choice of a perturbation vector. In this paper, we develop a general method to select an appropriate perturbation vector and a second-order local influence measure to address this issue in the context of latent variable models. An application to nonlinear structural equation models is considered. Six perturbation schemes are investigated, including three schemes under which simultaneous perturbations are made on components of latent vectors to assess the influence of these components and pinpoint the influential ones. The proposed procedure is illustrated by artificial examples and a simulation study as well as a real example.

In longitudinal studies, investigators often measure multiple variables at multiple time points and are interested in investigating individual differences in patterns of change on those variables. Furthermore, in behavioral, social, psychological, and medical research, investigators often deal with latent variables that cannot be observed directly and should be measured by 2 or more manifest variables. Longitudinal latent variables occur when the corresponding manifest variables are measured at multiple time points. Our primary interests are in studying the dynamic change of longitudinal latent variables and exploring the possible interactive effect among the latent variables. Much of the existing research in longitudinal studies focuses on studying change in a single observed variable at different time points. In this article, we propose a novel latent curve model (LCM) for studying the dynamic change of multivariate manifest and latent variables and their linear and interaction relationships. The proposed LCM has the following useful features: First, it can handle multivariate variables for exploring the dynamic change of their relationships, whereas conventional LCMs usually consider change in a univariate variable. Second, it accommodates both first- and second-order latent variables and their interactions to explore how changes in latent attributes interact to produce a joint effect on the growth of an outcome variable. Third, it accommodates both continuous and ordered categorical data, and missing data. (Contains 6 figures and 2 tables.)

In this paper, normal/independent distributions, including but not limited to the multivariate t distribution, the multivariate contaminated distribution, and the multivariate slash distribution, are used to develop a robust Bayesian approach for analyzing structural equation models with complete or missing data. In the context of a nonlinear structural equation model with fixed covariates, robust Bayesian methods are developed for estimation and model comparison. Results from simulation studies are reported to reveal the characteristics of estimation. The methods are illustrated by using a real data set obtained from diabetes patients.

Structural equation models are widely appreciated in behavioral, social, and psychological research to model relations between latent constructs and manifest variables, and to control for measurement errors. Most applications of structural equation models are based on fully observed data that are independently distributed. However, hierarchical data with a correlated structure are common in behavioral research, and very often, missing data are encountered. In this article, we propose a 2-level structural equation model for analyzing hierarchical data with missing entries, and describe a Bayesian approach for estimation and model comparison. We show how to use WinBUGS software to get the solution conveniently. The proposed methodologies are illustrated through a simulation study, and a real application in relation to organizational and management research concerning the study of the interrelationships of the latent constructs about job satisfaction, job responsibility, and life satisfaction for citizens in 43 countries. (Contains 3 figures and 3 tables.)