Generalized latent variable models with non‐linear effects

Until recently, item response models such as the factor analysis model for metric responses, the two‐parameter logistic model for binary responses and the multinomial model for nominal responses considered only the main effects of latent variables without allowing for interaction or polynomial latent variable effects. However, non‐linear relationships among the latent variables might be necessary in real applications. Methods for fitting models with non‐linear latent terms have been developed mainly under the structural equation modelling approach. In this paper, we consider a latent variable model framework for mixed responses (metric and categorical) that allows inclusion of both non‐linear latent and covariate effects. The model parameters are estimated using full maximum likelihood based on a hybrid integration–maximization algorithm. Finally, a method for obtaining factor scores based on multiple imputation is proposed here for the non‐linear model.     

Drug awareness in adolescents attending a mental health service: analysis of longitudinal data

One of the procedures used most recently with longitudinal data is linear mixed models. In the context of health research the increasing number of studies that now use these models bears witness to the growing interest in this type of analysis. This paper describes the application of linear mixed models to a longitudinal study of a sample of Spanish adolescents attending a mental health service, the aim being to investigate their knowledge about the consumption of alcohol and other drugs. More specifically, the main objective was to compare the efficacy of a motivational interviewing programme with a standard approach to drug awareness. The models used to analyse the overall indicator of drug awareness were as follows: (a) unconditional linear growth curve model; (b) growth model with subject-associated variables; and (c) individual curve model with predictive variables. The results showed that awareness increased over time and that the variable 'schooling years' explained part of the between-subjects variation. The effect of motivational interviewing was also significant.     

Indeterminacy Problems in the Lisrel Model

The latent variables and errors of the Lisrel model are indeterminate even when the parameters of the model are perfectly identified. The reason for the indeterminacy is that the Lisrel model gives a solution in terms of estimation of latent variables by means of observed variables. The indeterminacy is relevant also in practice; the minimum correlation between equivalent latent variables, is often negative in empirical examples. The degree of indeterminacy of the latent variables depends on the data. The average minimum correlation is a linear combination of the eigenvalues of the correlation matrix of solutions and it is always included in weak bounds which depend on the same eigenvalues.     

Regression among factor scores

Structural equation models with latent variables are sometimes estimated using an intuitive three-step approach, here denoted factor score regression. Consider a structural equation model composed of an explanatory latent variable and a response latent variable related by a structural parameter of scientific interest. In this simple example estimation of the structural parameter proceeds as follows: First, common factor models areseparately estimated for each latent variable. Second, factor scores areseparately assigned to each latent variable, based on the estimates. Third, ordinary linear regression analysis is performed among the factor scores producing an estimate for the structural parameter. We investigate the asymptotic and finite sample performance of different factor score regression methods for structural equation models with latent variables. It is demonstrated that the conventional approach to factor score regression performs very badly. Revised factor score regression, using Regression factor scores for the explanatory latent variables and Bartlett scores for the response latent variables, produces consistent estimators for all parameters.     

Predicting group-level outcome variables from variables measured at the individual level: a latent variable multilevel model

In multilevel modeling, one often distinguishes between macro-micro and micro-macro situations. In a macro-micro multilevel situation, a dependent variable measured at the lower level is predicted or explained by variables measured at that lower or a higher level. In a micro-macro multilevel situation, a dependent variable defined at the higher group level is predicted or explained on the basis of independent variables measured at the lower individual level. Up until now, multilevel methodology has mainly focused on macro-micro multilevel situations. In this article, a latent variable model is proposed for analyzing data from micro-macro situations. It is shown that regression analyses carried out at the aggregated level result in biased parameter estimates. A method that uses the best linear unbiased predictors of the group means is shown to yield unbiased estimates of the parameters.     

Ordinary Least Squares Estimation of Parameters in Exploratory Factor Analysis With Ordinal Data

Exploratory factor analysis (EFA) is often conducted with ordinal data (e.g., items with 5-point responses) in the social and behavioral sciences. These ordinal variables are often treated as if they were continuous in practice. An alternative strategy is to assume that a normally distributed continuous variable underlies each ordinal variable. The EFA model is specified for these underlying continuous variables rather than the observed ordinal variables. Although these underlying continuous variables are not observed directly, their correlations can be estimated from the ordinal variables. These correlations are referred to as polychoric correlations. This article is concerned with ordinary least squares (OLS) estimation of parameters in EFA with polychoric correlations. Standard errors and confidence intervals for rotated factor loadings and factor correlations are presented. OLS estimates and the associated standard error estimates and confidence intervals are illustrated using personality trait ratings from 228 college students. Statistical properties of the proposed procedure are explored using a Monte Carlo study. The empirical illustration and the Monte Carlo study showed that (a) OLS estimation of EFA is feasible with large models, (b) point estimates of rotated factor loadings are unbiased, (c) point estimates of factor correlations are slightly negatively biased with small samples, and (d) standard error estimates and confidence intervals perform satisfactorily at moderately large samples.     

Models for pedestrian gap acceptance behaviour analysis at unprotected mid-block crosswalks under mixed traffic conditions

Pedestrian safety is an important aspect while crossing the road and it can be explained by pedestrian gap acceptance behaviour. The statistical models such as multiple linear regression (MLR) is often used to model linear relationships between dependent variable (viz., pedestrian gap acceptance behaviour) and independent variables, due to their ability to quantitatively predict the effect of various factors on the dependent variable. However such linear models cannot consider the effect of several variables on the output variable, due to primary assumptions of normality, linear, homoscedasticity and multicollinearity. In this regard, the non-linear models based on the artificial neural network (ANN), which are free from assumptions of linear models, can be easily employed for obtaining the effect of several input variables on the pedestrian accepted gap size. However, researchers have rarely applied ANN modelling technique for predicting the pedestrian gap acceptance behaviour, as the pedestrian gap acceptance behaviour depends on several pedestrian, traffic and vehicular characteristics. The ANN based models would be quite useful in establishing relationship between these factors on the pedestrian gap acceptance behaviour at midblock crosswalks under mixed traffic conditions. In this direction, the present study adopts both MLR as well as ANN with different pedestrian, traffic and vehicular characteristics to assess the significant contributing factors for pedestrians’ gap acceptance behaviour at unprotected mid-block crosswalks under mixed traffic conditions. For this purpose, a video graphic survey was conducted at a six lane divided road at unprotected mid-block crossing in Mumbai, India. The data such as pedestrian (gender and age), vehicular, traffic and pedestrian behavioural characteristics were extracted to model pedestrian accepted gaps. The model results show that pedestrian rolling behaviour has a significant effect on pedestrian accepted gap size. The model results concluded that ANN has a better prediction with possibility to consider the effect of more number of variables on the pedestrian gap acceptance behaviour as compared to the MLR model under mixed traffic conditions. However, the quantification of significant contributing variables on pedestrian accepted gap size is easy by MLR model as compared to the ANN technique. So, both models have their own significant role in pedestrian gap acceptance analysis. The developed models may be useful to enhance the existing mid-block crosswalk facilities or planning new facilities by more accurate prediction of the pedestrian gap acceptance behaviour considering the influence of various factors under mixed traffic conditions.     

Robust regression: Testing global hypotheses about the slopes when there is multicollinearity or heteroscedasticity

A well-known concern regarding the usual linear regression model is multicollinearity. As the strength of the association among the independent variables increases, the squared standard error of regression estimators tends to increase, which can seriously impact power. This paper examines heteroscedastic methods for dealing with this issue when testing the hypothesis that all of the slope parameters are equal to zero via a robust ridge estimator that guards against outliers among the dependent variable. Included are results related to leverage points, meaning outliers among the independent variables. In various situations, the proposed method increases power substantially.     

The Houdini Transformation: True,but Illusory

Molenaar (2003, 2011) showed that a common factor model could be transformed into an equivalent model without factors, involving only observed variables and residual errors. He called this invertible transformation the Houdini transformation. His derivation involved concepts from time series and state space theory. This article verifies the Houdini transformation on a general latent variable model using algebraic methods. The results show that the Houdini transformation is illusory in the sense that the Houdini-transformed model remains a latent variable model. Contrary to common knowledge, a model that is a path model with only observed variables and residual errors may, in fact, be a latent variable model.     

Two classes of models for magnitude estimation

One class of models assumes that presentation of a signal results in an internal representation as a random variable. Depending on whether the signal is close to or far from the preceding signal, the variance of the representation is smaller or larger. Responses are determined largely by this random variable; however, when the signal is close to the preceding one, the response is generated by modifying the representation multiplicatively by some function of the ratio of the previous response to its representation. Power and linear functions are explored. The form of the random variable is assumed to be that arising from either the timing or the counting model operating on a Poisson process. Detailed analyses are carried out successfully only for the timing model with neural sample sizes independent of intensity; however, the data require the sample to increase with intensity. The linear response function coupled with the constant sample size counting model appears somewhat viable, but detailed calculations are very difficult to carry out. The second class of models postulates a power function relation between magnitude estimates and signals intensity for which the exponent is a Gaussian distributed random variable and the unit is the product of two log normal random variables. Again we assume an attention band such that succesive stimuli that are widely separated in intensity lead to independent samples of the random variables while a variety of assumptions is explored for successive stimuli that are near each other in intensity. Although they each give rise to the qualitative features of the data, estimates of parameters are sufficiently inconsistent that we are led to reject all of the submodels studied.     

Latent growth curve analysis with dichotomous items: Comparing four approaches

A Monte Carlo study was used to compare four approaches to growth curve analysis of subjects assessed repeatedly with the same set of dichotomous items: A two‐step procedure first estimating latent trait measures using MULTILOG and then using a hierarchical linear model to examine the changing trajectories with the estimated abilities as the outcome variable; a structural equation model using modified weighted least squares (WLSMV) estimation; and two approaches in the framework of multilevel item response models, including a hierarchical generalized linear model using Laplace estimation, and Bayesian analysis using Markov chain Monte Carlo (MCMC). These four methods have similar power in detecting the average linear slope across time. MCMC and Laplace estimates perform relatively better on the bias of the average linear slope and corresponding standard error, as well as the item location parameters. For the variance of the random intercept, and the covariance between the random intercept and slope, all estimates are biased in most conditions. For the random slope variance, only Laplace estimates are unbiased when there are eight time points.     

The prediction of intelligence in preschool children using alternative models to regression

Statistical prediction of an outcome variable using multiple independent variables is a common practice in the social and behavioral sciences. For example, neuropsychologists are sometimes called upon to provide predictions of preinjury cognitive functioning for individuals who have suffered a traumatic brain injury. Typically, these predictions are made using standard multiple linear regression models with several demographic variables (e.g., gender, ethnicity, education level) as predictors. Prior research has shown conflicting evidence regarding the ability of such models to provide accurate predictions of outcome variables such as full-scale intelligence (FSIQ) test scores. The present study had two goals: (1) to demonstrate the utility of a set of alternative prediction methods that have been applied extensively in the natural sciences and business but have not been frequently explored in the social sciences and (2) to develop models that can be used to predict premorbid cognitive functioning in preschool children. Predictions of Stanford–Binet 5 FSIQ scores for preschool-aged children is used to compare the performance of a multiple regression model with several of these alternative methods. Results demonstrate that classification and regression treesprovided more accurate predictions of FSIQ scores than does the more traditional regression approach. Implications of these results are discussed.     

Studying Multivariate Change Using Multilevel Models and Latent Curve Models

In longitudinal research investigators often measure multiple variables at multiple points in time and are interested in investigating individual differences in patterns of change on those variables. In the vast majority of applications, researchers focus on studying change in one variable at a time. In this article we consider methods for studying relations1.1ips between patterns of change on different variables. We show how the multilevel modeling framework, which is often used to study univariate change, can be extended to the multivariate case to yield estimates of covariances of parameters representing aspects of change on different variables. We illustrate this approach using data from a study of physiological response to marital conflict in older married couples, showing a substantial correlation between rate of linear change on different stress-related hormones during conflict. We also consider how similar issues can be studied using extensions of latent curve models to the multivariate case, and we show how such models are related to multivariate multilevel models.     

Interpersonal deviance, organizational deviance, and their common correlates: a review and meta-analysis

Interpersonal deviance (ID) and organizational deviance (OD) are highly correlated (R. S. Dalal, 2005). This, together with other empirical and theoretical evidence, calls into question the separability of ID and OD. As a further investigation into their separability, relationships among ID, OD, and their common correlates were meta-analyzed. ID and OD were highly correlated (rho = .62) but had differential relationships with key Big Five variables and organizational citizenship behaviors, which lends support to the separability of ID and OD. Whether the R. J. Bennett and S. L. Robinson (2000) instrument was used moderated some relationships. ID and OD exhibited their strongest (negative) relationships with organizational citizenship, Agreeableness, Conscientiousness, and Emotional Stability. Correlations with organizational justice were small to moderate, and correlations with demographic variables were generally negligible.     

Structural Modeling of Measurement Error in Generalized Linear Models with Rasch Measures as Covariates

This paper proposes a structural analysis for generalized linear models when some explanatory variables are measured with error and the measurement error variance is a function of the true variables. The focus is on latent variables investigated on the basis of questionnaires and estimated using item response theory models. Latent variable estimates are then treated as observed measures of the true variables. This leads to a two-stage estimation procedure which constitutes an alternative to a joint model for the outcome variable and the responses given to the questionnaire. Simulation studies explore the effect of ignoring the true error structure and the performance of the proposed method. Two illustrative examples concern achievement data of university students. Particular attention is given to the Rasch model.     

Stochastic search item selection for factor analytic models

In this paper we implement a Markov chain Monte Carlo algorithm based on the stochastic search variable selection method of George and McCulloch (1993) for identifying promising subsets of manifest variables (items) for factor analysis models. The suggested algorithm is constructed by embedding in the usual factor analysis model a normal mixture prior for the model loadings with latent indicators used to identify not only which manifest variables should be included in the model but also how each manifest variable is associated with each factor. We further extend the suggested algorithm to allow for factor selection. We also develop a detailed procedure for the specification of the prior parameters values based on the practical significance of factor loadings using ideas from the original work of George and McCulloch (1993). A straightforward Gibbs sampler is used to simulate from the joint posterior distribution of all unknown parameters and the subset of variables with the highest posterior probability is selected. The proposed method is illustrated using real and simulated data sets.     

Models for the analysis of alternative sources of growth in correlated stochastic variables

An approach to the analysis of multivariate time series is presented in which linear structural relationships among multiple stochastic variables are investigated. A number of alternative structural models are considered for the case of two stochastic variables. Each model represents a possible hypothesis concerning the relationship of growth in one variable to growth in the second. Both symmetric and asymmetric models are considered. Extensions of two of the models to three variables are illustrated by means of a numerical example. Implications of the models for the problem of detecting change in multivariate time series are discussed.This paper is based in part on a paper read at the Psychometric Society Meetings, Princeton, N. J., Mar. 30–31, 1972. The research was supported by the Committe on Basic Research in Education, National Academy of Sciences, Grant No. OEG-0-9-140396-4497(010), Project No. 9-0396.     

Accounting for sensation seeking in route choice behavior with travel time information

The purpose of this paper is to demonstrate that latent variables, with the focus on sensation seeking concepts, incorporated in new technique of route choice modeling, improve our analyzing of route choice behavior with pre-trip travel time information. The application of a hybrid discrete choice model framework integrates a latent variable model and a route choice model by combining their measurement and structural equations. The model is estimated based on data from a laboratory experiment and a field study of a simple network. The results show that certain sensation seeking domains (e.g., thrill and adventure seeking) alongside traditional variables (e.g., travel time information) enrich our understanding and provide more insight into route choice behavior. Furthermore, observed personal variables, such as gender and marital status, may serve as causal indicators to sensation seeking variables.     

Measuring Five Dimensions of Religiosity Across Adolescence

This paper theorizes and tests a latent variable model of adolescent religiosity in which five dimensions of religiosity are interrelated: religious beliefs, religious exclusivity, external practice, private practice, and religious salience. Research often theorizes overlapping and independent influences of single items or dimensions of religiosity on outcomes such as adolescent sexual behavior, but rarely operationalizes the dimensions in a measurement model accounting for their associations with each other and across time. We use longitudinal structural equation modeling with latent variables to analyze data from two waves of the National Study of Youth and Religion. We test our hypothesized measurement model as compared to four alternate measurement models and find that our proposed model maintains superior fit. We then discuss the associations between the five dimensions of religiosity we measure and how these change over time. Our findings suggest how future research might better operationalize multiple dimensions of religiosity in studies of the influence of religion in adolescence.     

Binary recursive partitioning: Background,methods, and application to psychology

Binary recursive partitioning (BRP) is a computationally intensive statistical method that can be used in situations where linear models are often used. Instead of imposing many assumptions to arrive at a tractable statistical model, BRP simply seeks to accurately predict a response variable based on values of predictor variables. The method outputs a decision tree depicting the predictor variables that were related to the response variable, along with the nature of the variables' relationships. No significance tests are involved, and the tree's ‘goodness’ is judged based on its predictive accuracy. In this paper, we describe BRP methods in a detailed manner and illustrate their use in psychological research. We also provide R code for carrying out the methods.