Bayesian analysis of structural equation models with mixed exponential family and ordered categorical data

Structural equation models are very popular for studying relationships among observed and latent variables. However, the existing theory and computer packages are developed mainly under the assumption of normality, and hence cannot be satisfactorily applied to non‐normal and ordered categorical data that are common in behavioural, social and psychological research. In this paper, we develop a Bayesian approach to the analysis of structural equation models in which the manifest variables are ordered categorical and/or from an exponential family. In this framework, models with a mixture of binomial, ordered categorical and normal variables can be analysed. Bayesian estimates of the unknown parameters are obtained by a computational procedure that combines the Gibbs sampler and the Metropolis–Hastings algorithm. Some goodness‐of‐fit statistics are proposed to evaluate the fit of the posited model. The methodology is illustrated by results obtained from a simulation study and analysis of a real data set about non‐adherence of hypertension patients in a medical treatment scheme.     

Local influence analysis of nonlinear structural equation models

By regarding the latent random vectors as hypothetical missing data and based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm, we investigate assessment of local influence of various perturbation schemes in a nonlinear structural equation model. The basic building blocks of local influence analysis are computed via observations of the latent variables generated by the Metropolis-Hastings algorithm, while the diagnostic measures are obtained via the conformal normal curvature. Seven perturbation schemes, including some perturbation schemes on latent vectors, are investigated. The proposed procedure is illustrated by a simulation study and a real example. Acknowledgment: This research is fully supported by a grant (CUHK 4243/02H) from the Research Grant Council of the Hong Kong Special Administration Region. The authors are indebted to ICPSR and the relevant funding agency for allowing use of their data, and to the Editor and reviewers for their valuable comments for improving the paper.     

Similar mechanisms underlie curvature comparison by static and dynamic touch.

In four experiments, we tested whether haptic comparison of curvature ranging from -4/m to +4/m is qualitatively the same for static and for dynamic touch. In Experiments 1 and 3, we tested whether static and dynamic curvature discrimination are based on height differences, attitude (slope) differences, curvature differences, or a combination of these geometrical variables. It was found that both static and dynamic hepatic curvature discrimination are based on attitude differences. In Experiments 2 and 4, we tested whether this mechanism leads to errors in the comparison of stimuli with different lengths for static and dynamic touch, respectively. If the judgments are based on attitude differences, subjects will make systematic errors in these comparisons. In both experiments, we found that subjects compared the curvatures of strips of the same length vertically, whereas they made systematic errors if they were required to compare the curvatures of strips of different lengths. Longer stimuli were judged to be more curved than shorter stimuli with the same curvature. We conclude that similar mechanisms underlie static and dynamic haptic curvature comparison. Moreover, additional data comparison showed that static and dynamic curvature comparison is not only qualitatively, but also quantitatively similar.     

Similar mechanisms underlie curvature comparison by static and dynamic touch

In four experiments, we tested whether haptic comparison of curvature ranging from ?41m to +41m is qualitatively the same for static and for dynamic touch. In Experiments 1 and 3, we tested whether static and dynamic curvature discrimination are based on height differences, attitude (slope) differences, curvature differences, or a combination of these geometrical variables. It was found that both static and dynamic haptic curvature discrimination are based on attitude differences. In Experiments 2 and 4, we tested whether this mechanism leads to errors in the comparison of stimuli with different lengths for static and dynamic touch, respectively. If the judgments are based on attitude differences, subjects will make systematic errors in these comparisons. In both experiments, we found that subjects compared the curvatures of strips of the same length veridically, whereas they made systematic errors if they were required to compare the curvatures of strips of different lengths. Longer stimuli were judged to be more curved than shorter stimuli with the same curvature. We conclude that similar mechanisms underlie static and dynamic haptic curvature comparison. Moreover, additional data comparison showed that static and dynamic curvature comparison is not only qualitatively, but also quantitatively similar.     

Non‐ignorable missingness item response theory models for choice effects in examinee‐selected items

Examinee‐selected item (ESI) design, in which examinees are required to respond to a fixed number of items in a given set, always yields incomplete data (i.e., when only the selected items are answered, data are missing for the others) that are likely non‐ignorable in likelihood inference. Standard item response theory (IRT) models become infeasible when ESI data are missing not at random (MNAR). To solve this problem, the authors propose a two‐dimensional IRT model that posits one unidimensional IRT model for observed data and another for nominal selection patterns. The two latent variables are assumed to follow a bivariate normal distribution. In this study, the mirt freeware package was adopted to estimate parameters. The authors conduct an experiment to demonstrate that ESI data are often non‐ignorable and to determine how to apply the new model to the data collected. Two follow‐up simulation studies are conducted to assess the parameter recovery of the new model and the consequences for parameter estimation of ignoring MNAR data. The results of the two simulation studies indicate good parameter recovery of the new model and poor parameter recovery when non‐ignorable missing data were mistakenly treated as ignorable.     

Memory for curvature of objects: Haptic touch vs. vision

The present study examined the role of vision and haptics in memory for stimulus objects that vary along the dimension of curvature. Experiment 1 measured haptic‐haptic (T‐T) and haptic‐visual (T‐V) discrimination of curvature in a short‐term memory paradigm, using 30‐second retention intervals containing five different interpolated tasks. Results showed poorest performance when the interpolated tasks required spatial processing or movement, thereby suggesting that haptic information about shape is encoded in a spatial‐motor representation. Experiment 2 compared visual‐visual (V‐V) and visual‐haptic (V‐T) short‐term memory, again using 30‐second delay intervals. The results of the ANOVA failed to show a significant effect of intervening activity. Intra‐modal visual performance and cross‐modal performance were similar. Comparing the four modality conditions (inter‐modal V‐T, T‐V; intra‐modal V‐V, T‐T, by combining the data of Experiments 1 and 2), in a global analysis, showed a reliable interaction between intervening activity and experiment (modality). Although there appears to be a general tendency for spatial and movement activities to exert the most deleterious effects overall, the patterns are not identical when the initial stimulus is encoded haptically (Experiment 1) and visually (Experiment 2).     

On the explaining‐away phenomenon in multivariate latent variable models

Many probabilistic models for psychological and educational measurements contain latent variables. Well‐known examples are factor analysis, item response theory, and latent class model families. We discuss what is referred to as the ‘explaining‐away’ phenomenon in the context of such latent variable models. This phenomenon can occur when multiple latent variables are related to the same observed variable, and can elicit seemingly counterintuitive conditional dependencies between latent variables given observed variables. We illustrate the implications of explaining away for a number of well‐known latent variable models by using both theoretical and real data examples.     

Modern Regression Methods that can Substantially Increase Power and Provide a more Accurate Understanding of Associations

During the last half century, hundreds of papers published in statistical journals have documented general conditions where reliance on least squares regression and Pearson's correlation can result in missing even strong associations between variables. Moreover, highly misleading conclusions can be made, even when the sample size is large. There are, in fact, several fundamental concerns related to non‐normality, outliers, heteroscedasticity, and curvature that can result in missing a strong association. Simultaneously, a vast array of new methods has been derived for effectively dealing with these concerns. The paper (i) reviews why least squares regression and classic inferential methods can fail, (ii) provides an overview of the many modern strategies for dealing with known problems, including some recent advances, and (iii) illustrates that modern robust methods can make a practical difference in our understanding of data. Included are some general recommendations regarding how modern methods might be used. Copyright © 2011 John Wiley & Sons, Ltd.     

The on effect of changes in curve geometry magnitude estimates of road-like perspective curvature

This study investigated the perception of curvature of perspectively viewed road-like curves in two-dimensional display. The effects of the geometric variables curve radius and curve angle, as well as the influence of experimental instructions, were assessed in a factorial laboratory experiment. Subjects estimated the magnitude of curvature of a series of stimulus curves in relation to a standard view. The results showed that decreasing the curve angle led to perceptual flattening of these curves. Reductions in curve radius resulted in a paradoxical decrease in perceived curvature. Judgments of curvature were the same for uninstructed subjects and those instructed in the meaning of physical curvature. These findings are in contrast to those expected from curve geometry and highlight an illusion in perceived curvature potentially harmful for curve negotiation on the road.     

Asymptotic biases of the unrotated/rotated solutions in principal component analysis

Asymptotic biases of the parameter estimates in principal component analysis with substantial misspecification are derived. The solutions for unstandardized and standardized observed variables are considered with and without orthogonal and oblique rotations. The distribution of observed variables can be non‐normal as long as the finite fourth‐order moments of the observed variables exist. When multivariate normality holds for the observed variables, substantial reduction of the amount of computation can be achieved. Numerical examples with simulations are given, with some discussion on the tendency of the biases to reduce the absolute values of parameter estimates.     

Non‐linear structural equation models with correlated continuous and discrete data

Structural equation models (SEMs) have been widely applied to examine interrelationships among latent and observed variables in social and psychological research. Motivated by the fact that correlated discrete variables are frequently encountered in practical applications, a non‐linear SEM that accommodates covariates, and mixed continuous, ordered, and unordered categorical variables is proposed. Maximum likelihood methods for estimation and model comparison are discussed. One real‐life data set about cardiovascular disease is used to illustrate the methodologies.     

On the Construct Validity of Integrity Tests: Individual and Situational Factors as Predictors of Test Performance

Although integrity tests are widely applied in screening job applicants, there is a need for research for examining the construct validity of these tests. In the present study, a theoretical model examining the causes of destructive behavior in organizational settings was used to develop background data measures of individual and situational variables that might be related to integrity test scores. Subsequently, 692 undergraduates were asked to complete these background data scales along with (a) two overt integrity tests – the Reid Report and the Personnel Selection Inventory, and (b) two personality‐based measures – the delinquency and socialization scales of the California Psychological Inventory. When scores of these measures were correlated with and regressed on the background data scales, it was found that relevant individual variables, such as narcissism and power motives, and relevant situational variables, such as alienation and exposure to negative peer groups, were related to scores on both types of integrity tests. However, a stronger pattern of validity evidence was obtained for the personality‐based measures and, in all cases, situational variables were found to be better predictors than individual variables. The implications of these findings for the validity of inferences drawn from overt and personality‐based integrity tests are discussed.     

A unified approach to multiple‐set canonical correlation analysis and principal components analysis

Multiple‐set canonical correlation analysis and principal components analysis are popular data reduction techniques in various fields, including psychology. Both techniques aim to extract a series of weighted composites or components of observed variables for the purpose of data reduction. However, their objectives of performing data reduction are different. Multiple‐set canonical correlation analysis focuses on describing the association among several sets of variables through data reduction, whereas principal components analysis concentrates on explaining the maximum variance of a single set of variables. In this paper, we provide a unified framework that combines these seemingly incompatible techniques. The proposed approach embraces the two techniques as special cases. More importantly, it permits a compromise between the techniques in yielding solutions. For instance, we may obtain components in such a way that they maximize the association among multiple data sets, while also accounting for the variance of each data set. We develop a single optimization function for parameter estimation, which is a weighted sum of two criteria for multiple‐set canonical correlation analysis and principal components analysis. We minimize this function analytically. We conduct simulation studies to investigate the performance of the proposed approach based on synthetic data. We also apply the approach for the analysis of functional neuroimaging data to illustrate its empirical usefulness.     

Latent variable models for multivariate longitudinal ordinal responses

The paper proposes a full information maximum likelihood estimation method for modelling multivariate longitudinal ordinal variables. Two latent variable models are proposed that account for dependencies among items within time and between time. One model fits item‐specific random effects which account for the between time points correlations and the second model uses a common factor. The relationships between the time‐dependent latent variables are modelled with a non‐stationary autoregressive model. The proposed models are fitted to a real data set.     

On local influence analysis of full information item factor models

The full information item factor (FIIF) model is very useful for analyzing relations of dichotomous variables. In this article, we present a feasible procedure to assess local influence of minor perturbations for identifying influence aspects of the FIIF model. The development is based on a Q-displacement function which is closely related with the Monte Carlo EM algorithm in the ML estimation. In the E-step of this algorithm, the conditional expectations are approximated by sample means of observations simulated by the Gibbs sampler from the appropriate conditional distributions. It turns out that these observations can be utilized for computing the building blocks of the proposed diagnostic measures. The diagnoses are based on the conformal normal curvature that can be computed easily. A number of interesting perturbation schemes are considered. The methodology is illustrated with two real examples.The research is fully supported by a grant (CUHK 4356/00H) from the Research Grant Council of the Hong Kong Special Administration Region. The authors are thankful to the Editor, Associate Editor, anonymous reviewers, and W.Y. Poon for valuable comments for improving the paper, and to ICPSR and the relevant founding agency for allowing us to use of their data. The assistance of Michael Leung and Esther Tam is gratefully acknowledged.     

Bayesian analysis of mixtures in structural equation models with non‐ignorable missing data

Structural equation models (SEMs) have become widely used to determine the interrelationships between latent and observed variables in social, psychological, and behavioural sciences. As heterogeneous data are very common in practical research in these fields, the analysis of mixture models has received a lot of attention in the literature. An important issue in the analysis of mixture SEMs is the presence of missing data, in particular of data missing with a non‐ignorable mechanism. However, only a limited amount of work has been done in analysing mixture SEMs with non‐ignorable missing data. The main objective of this paper is to develop a Bayesian approach for analysing mixture SEMs with an unknown number of components and non‐ignorable missing data. A simulation study shows that Bayesian estimates obtained by the proposed Markov chain Monte Carlo methods are accurate and the Bayes factor computed via a path sampling procedure is useful for identifying the correct number of components, selecting an appropriate missingness mechanism, and investigating various effects of latent variables in the mixture SEMs. A real data set on a study of job satisfaction is used to demonstrate the methodology.     

Prevalence estimation of school bullying with the Olweus Bully/Victim Questionnaire

The key aim of the present research was to study the “functionality” of two global variables in the Olweus Bully/Victim Questionnaire and to examine the appropriateness of different cutoff points of these variables for prevalence estimation. Several empirical and conceptual analyses strongly attested to the functionality of the two selected variables in terms of construct validity and selected measurement properties. Similarly, a number of analyses indicated that (having been bullied/having bullied other students) “2 or 3 times a month” was a reasonable and useful lower‐bound cutoff point. With this cutoff point, “involved” students, victims, and bullies differed very markedly and in clearly different ways from “non‐involved” students in conceptually related variables. Prevalence estimates derived in this way can be conveniently obtained, have a reasonably well‐defined meaning, can be easily understood by users, and can be reproduced unambiguously by different researchers/administrators and at different times. An important background for the article is the fact that several common methods, including peer nominations, are not well suited for prevalence estimation. Prevalence data for victims, bullies, and bully‐victims are also presented. All data were derived from the New Bergen Project Against Bullying, comprising a sample of 5,171 students from 37 schools in the town community of Bergen, Norway. At the time of the data collection, the spring of 1997, the 2,544 girls and 2,627 boys were in grades 5 through 9, with modal ages of 11 through 15 years. Aggr. Behav. 29:239–268, 2003. © 2003 Wiley‐Liss, Inc.     

Simple and multiple P‐splines regression with shape constraints

In many research areas, especially within social and behavioural sciences, the relationship between predictor and criterion variables is often assumed to have a particular shape, such as monotone, single‐peaked or U‐shaped. Such assumptions can be transformed into (local or global) constraints on the sign of the nth‐order derivative of the functional form. To check for such assumptions, we present a non‐parametric regression method, P‐splines regression, with additional asymmetric discrete penalties enforcing the constraints. We show that the corresponding loss function is convex and present a Newton–Raphson algorithm to optimize. Constrained P‐splines are illustrated with an application on monotonicity‐constrained regression with both one and two predictor variables, using data from research on the cognitive development of children.     

Stochastic EM for estimating the parameters of a multilevel IRT model

An item response theory (IRT) model is used as a measurement error model for the dependent variable of a multilevel model. The dependent variable is latent but can be measured indirectly by using tests or questionnaires. The advantage of using latent scores as dependent variables of a multilevel model is that it offers the possibility of modelling response variation and measurement error and separating the influence of item difficulty and ability level. The two‐parameter normal ogive model is used for the IRT model. It is shown that the stochastic EM algorithm can be used to estimate the parameters which are close to the maximum likelihood estimates. This algorithm is easily implemented. The estimation procedure will be compared to an implementation of the Gibbs sampler in a Bayesian framework. Examples using real data are given.     

Asymptotic efficiency of the pseudo‐maximum likelihood estimator in multi‐group factor models with pooled data

A multi‐group factor model is suitable for data originating from different strata. However, it often requires a relatively large sample size to avoid numerical issues such as non‐convergence and non‐positive definite covariance matrices. An alternative is to pool data from different groups in which a single‐group factor model is fitted to the pooled data using maximum likelihood. In this paper, properties of pseudo‐maximum likelihood (PML) estimators for pooled data are studied. The pooled data are assumed to be normally distributed from a single group. The resulting asymptotic efficiency of the PML estimators of factor loadings is compared with that of the multi‐group maximum likelihood estimators. The effect of pooling is investigated through a two‐group factor model. The variances of factor loadings for the pooled data are underestimated under the normal theory when error variances in the smaller group are larger. Underestimation is due to dependence between the pooled factors and pooled error terms. Small‐sample properties of the PML estimators are also investigated using a Monte Carlo study.