Finite mixtures in confirmatory factor-analysis models

In this paper, various types of finite mixtures of confirmatory factor-analysis models are proposed for handling data heterogeneity. Under the proposed mixture approach, observations are assumed to be drawn from mixtures of distinct confirmatory factor-analysis models. But each observation does not need to be identified to a particular model prior to model fitting. Several classes of mixture models are proposed. These models differ by their unique representations of data heterogeneity. Three different sampling schemes for these mixture models are distinguished. A mixed type of the these three sampling schemes is considered throughout this article. The proposed mixture approach reduces to regular multiple-group confirmatory factor-analysis under a restrictive sampling scheme, in which the structural equation model for each observation is assumed to be known. By assuming a mixture of multivariate normals for the data, maximum likelihood estimation using the EM (Expectation-Maximization) algorithm and the AS (Approximate-Scoring) method are developed, respectively. Some mixture models were fitted to a real data set for illustrating the application of the theory. Although the EM algorithm and the AS method gave similar sets of parameter estimates, the AS method was found computationally more efficient than the EM algorithm. Some comments on applying the mixture approach to structural equation modeling are made.Note: This paper is one of the Psychometric Society's 1995 Dissertation Award papers.—EditorThis article is based on the dissertation of the author. The author would like to thank Peter Bentler, who was the dissertation chair, for guidance and encouragement of this work. Eric Holman, Robert Jennrich, Bengt Muthén, and Thomas Wickens, who served as the committee members for the dissertation, had been very supportive and helpful. Michael Browne is appreciated for discussing some important points about the use of the approximate information in the dissertation. Thanks also go to an anonymous associate editor, whose comments were very useful for the revision of an earlier version of this article.     

Shrinkage estimation of linear combinations of true scores

This paper is concerned with combining observed scores from sections of tests. It is demonstrated that in the presence of population information a linear combination of true scores can be estimated more efficiently than by the same linear combination of the observed scores. Three criteria for optimality are discussed, but they yield the same solution which can be described and motivated as a multivariate shrinkage estimator.Input from Eric Bradlow, Charles Lewis, and Linda Zeger is acknowledged. Research for this paper was funded by the Program Research Council (ETS). Suggestions of the Editor and of anonymous referees were instrumental in several improvements of the paper.     

The danger of conflating level-specific effects of control variables when primary interest lies in level-2 effects

In the multilevel modelling literature, methodologists widely acknowledge that a level-1 variable can have distinct within-cluster and between-cluster effects, and that failing to disaggregate these can yield a slope estimate that is an uninterpretable, conflated blend of the two. Methodologists have stated, however, that including conflated slopes of level-1 variables in a model is not problematic if substantive interest lies only in effects of level-2 predictors. Researchers commonly follow this advice and use methods that do not disaggregate effects of level-1 control variables (e.g., grand mean centering) when examining effects of level-2 predictors. The primary purpose of this paper is to show that this is a dangerous practice. When level-specific effects of level-1 variables differ, failing to disaggregate them can severely bias estimation of level-2 predictor slopes. We show mathematically why this is the case and highlight factors that can exacerbate such bias. We corroborate these findings with simulations and present an empirical example, showing how such distortions can severely alter substantive conclusions. We ultimately recommend that simply including the cluster mean of the level-1 variable as a control will alleviate the problem.     

“易”的世界观和思维模式

本文认为,《周易》的阴阳世界观注重的是阴阳间的关系,而这种关系又有两个重要状态,一个是“空无”,另一个是“中庸”。对于这两种状态的不同态度形成了古人不同的人生态度和理想追求。同时,本文还认为,《周易》所运用的是取象比类式的象数推演体系,它不同于概念式的演绎体系。人类生活有很多重要内容不能通过概念式的语言去思维和表达,而这些“不可言说”的内容却往往可以用“象征语言”和象数思维来表达。     

Measurement invariance of the Satisfaction With Life Scale (SWLS) by gender and age in Angola

Subjective well-being is a research arena that has grown almost exponentially: over the last 20 years, the number of publications on subjective well-being has increased approximately 16-fold (Diener, 2009). The cognitive aspect of subjective well-being or life satisfaction is referred to a conscious cognitive judgment of life (Diener, Emmons, Larsen, & Griffin 1985), in which person’s quality of life is globally assessed (Shin & Johnson, 1978). The Satisfaction With Life Scale (SWLS, Diener et al., 1985) is the most widely used instrument for its measurement. A reliable, valid and invariant measurement is critical for meaningful comparisons. The aim of this study is to examine the configural, metric and scalar invariance across age in the Portuguese version of the SWLS with a sample of 5630 Angolans. A standard measurement invariance procedure has been applied both across gender and age. Results shown that scalar invariance of the SWLS held across gender and age. This strong invariance allowed for meaningful latent mean comparisons. There were latent mean differences due to gender and age, but while gender differences were modest, the age differences were larger. Results are discussed in their relation with existing literature.     

Decoding facial blends of emotion: Visual field,attentional and hemispheric biases

Most clinical research assumes that modulation of facial expressions is lateralized predominantly across the right-left hemiface. However, social psychological research suggests that facial expressions are organized predominantly across the upper-lower face. Because humans learn to cognitively control facial expression for social purposes, the lower face may display a false emotion, typically a smile, to enable approach behavior. In contrast, the upper face may leak a person’s true feeling state by producing a brief facial blend of emotion, i.e. a different emotion on the upper versus lower face. Previous studies from our laboratory have shown that upper facial emotions are processed preferentially by the right hemisphere under conditions of directed attention if facial blends of emotion are presented tachistoscopically to the mid left and right visual fields. This paper explores how facial blends are processed within the four visual quadrants. The results, combined with our previous research, demonstrate that lower more so than upper facial emotions are perceived best when presented to the viewer’s left and right visual fields just above the horizontal axis. Upper facial emotions are perceived best when presented to the viewer’s left visual field just above the horizontal axis under conditions of directed attention. Thus, by gazing at a person’s left ear, which also avoids the social stigma of eye-to-eye contact, one’s ability to decode facial expressions should be enhanced.     

Perceiving vocal age and gender: An adaptation approach

Aftereffects of adaptation have revealed both independent and interactive coding of facial signals including identity and expression or gender and age. By contrast, interactive processing of non-linguistic features in voices has rarely been investigated. Here we studied bidirectional cross-categorical aftereffects of adaptation to vocal age and gender. Prolonged exposure to young (~ 20 yrs) or old (~ 70 yrs) male or female voices biased perception of subsequent test voices away from the adapting age (Exp. 1) and the adapting gender (Exp. 2). Relative to gender-congruent adaptor-test pairings, vocal age aftereffects (VAAEs) were reduced but remained significant when voice gender changed between adaptation and test. This suggests that the VAAE relies on both gender-specific and gender-independent age representations for male and female voices. By contrast, voice gender aftereffects (VGAEs) were not modulated by age-congruency of adaptor and test voices (Exp. 2). Instead, young voice adaptors generally induced larger VGAEs than old voice adaptors. This suggests that young voices are particularly efficient gender adaptors, likely reflecting more pronounced sexual dimorphism in these voices. In sum, our findings demonstrate how high-level processing of vocal age and gender is partially intertwined.     

Conditions for factor (in)determinacy in factor analysis

The subject of factor indeterminacy has a vast history in factor analysis (Guttman, 1955; Lederman, 1938; Wilson, 1928). It has lead to strong differences in opinion (Steiger, 1979). The current paper gives necessary and sufficient conditions for observability of factors in terms of the parameter matrices and a finite number of variables. Five conditions are given which rigorously define indeterminacy. It is shown that (un)observable factors are (in)determinate. Specifically, the indeterminacy proof by Guttman is extended to Heywood cases. The results are illustrated by two examples and implications for indeterminacy are discussed.     

Robust mean and covariance structure analysis through iteratively reweighted least squares

Robust schemes in regression are adapted to mean and covariance structure analysis, providing an iteratively reweighted least squares approach to robust structural equation modeling. Each case is properly weighted according to its distance, based on first and second order moments, from the structural model. A simple weighting function is adopted because of its flexibility with changing dimensions. The weight matrix is obtained from an adaptive way of using residuals. Test statistic and standard error estimators are given, based on iteratively reweighted least squares. The method reduces to a standard distribution-free methodology if all cases are equally weighted. Examples demonstrate the value of the robust procedure.The authors acknowledge the constructive comments of three referees and the Editor that lead to an improved version of the paper. This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017 and by the University of North Texas Faculty Research Grant Program.