Bayesian Semiparametric Structural Equation Models with Latent Variables

Structural equation models (SEMs) with latent variables are widely useful for sparse covariance structure modeling and for inferring relationships among latent variables. Bayesian SEMs are appealing in allowing for the incorporation of prior information and in providing exact posterior distributions of unknowns, including the latent variables. In this article, we propose a broad class of semiparametric Bayesian SEMs, which allow mixed categorical and continuous manifest variables while also allowing the latent variables to have unknown distributions. In order to include typical identifiability restrictions on the latent variable distributions, we rely on centered Dirichlet process (CDP) and CDP mixture (CDPM) models. The CDP will induce a latent class model with an unknown number of classes, while the CDPM will induce a latent trait model with unknown densities for the latent traits. A simple and efficient Markov chain Monte Carlo algorithm is developed for posterior computation, and the methods are illustrated using simulated examples, and several applications.     

Modeling Associations Among Multivariate Longitudinal Categorical Variables in Survey Data: A Semiparametric Bayesian Approach

This paper proposes a semiparametric Bayesian framework for the analysis of associations among multivariate longitudinal categorical variables in high-dimensional data settings. This type of data is frequent, especially in the social and behavioral sciences. A semiparametric hierarchical factor analysis model is developed in which the distributions of the factors are modeled nonparametrically through a dynamic hierarchical Dirichlet process prior. A Markov chain Monte Carlo algorithm is developed for fitting the model, and the methodology is exemplified through a study of the dynamics of public attitudes toward science and technology in the United States over the period 1992?C2001.     

Modeling Growth in Latent Variables Using a Piecewise Function

Latent growth curve models with piecewise functions for continuous repeated measures data have become increasingly popular and versatile tools for investigating individual behavior that exhibits distinct phases of development in observed variables. As an extension of this framework, this research study considers a piecewise function for describing segmented change of a latent construct over time where the latent construct is itself measured by multiple indicators gathered at each measurement occasion. The time of transition from one phase to another is not known a priori and thus is a parameter to be estimated. Utility of the model is highlighted in 2 ways. First, a small Monte Carlo simulation is executed to show the ability of the model to recover true (known) growth parameters, including the location of the point of transition (or knot), under different manipulated conditions. Second, an empirical example using longitudinal reading data is fitted via maximum likelihood and results discussed. Mplus (Version 6.1) code is provided in Appendix C to aid in making this class of models accessible to practitioners.     

A Bayesian Semiparametric Latent Variable Model for Mixed Responses

In this paper we introduce a latent variable model (LVM) for mixed ordinal and continuous responses, where covariate effects on the continuous latent variables are modelled through a flexible semiparametric Gaussian regression model. We extend existing LVMs with the usual linear covariate effects by including nonparametric components for nonlinear effects of continuous covariates and interactions with other covariates as well as spatial effects. Full Bayesian modelling is based on penalized spline and Markov random field priors and is performed by computationally efficient Markov chain Monte Carlo (MCMC) methods. We apply our approach to a German social science survey which motivated our methodological development. We thank the editor and the referees for their constructive and helpful comments, leading to substantial improvements of a first version, and Sven Steinert for computational assistance. Partial financial support from the SFB 386 “Statistical Analysis of Discrete Structures” is also acknowledged.     

The Latent Structure of Substance Use Among American Indian Adolescents: An Example Using Categorical Variables

Researchers frequently have only categorical data to analyze and cannot, for theoretical or methodological reasons, assume that the observed variables are discrete representations of an underlying continuous variable. We present latent class analysis as an alternative method of measuring latent variables in these circumstances. Latent class analysis does not require the assumptions of factor analyses about the nature of manifest and latent variables, but does allow the use of more precise model selection than techniques such as cluster analysis. We modeled the lifetime substance use of American Indian youth. The latent class model of American Indian teenagers' substance use had four classes: Abstaining, Predominantly Alcohol, Predominantly Alcohol and Marijuana, and Plural Substance. We then demonstrated the usefulness of this latent variable by using it to differentiate levels of several variables in a manner consistent with Social Cognitive Theory.     

Abstract: Sample Size Planning for Latent Curve Models

When designing a study that uses structural equation modeling (SEM), an important task is to decide an appropriate sample size. Historically, this task is approached from the power analytic perspective, where the goal is to obtain sufficient power to reject a false null hypothesis. However, hypothesis testing only tells if a population effect is zero and fails to address the question about the population effect size. Moreover, significance tests in the SEM context often reject the null hypothesis too easily, and therefore the problem in practice is having too much power instead of not enough power.

An alternative means to infer the population effect is forming confidence intervals (CIs). A CI is more informative than hypothesis testing because a CI provides a range of plausible values for the population effect size of interest. Given the close relationship between CI and sample size, the sample size for an SEM study can be planned with the goal to obtain sufficiently narrow CIs for the population model parameters of interest.

Latent curve models (LCMs) is an application of SEM with mean structure to studying change over time. The sample size planning method for LCM from the CI perspective is based on maximum likelihood and expected information matrix. Given a sample, to form a CI for the model parameter of interest in LCM, it requires the sample covariance matrix S, sample mean vector , and sample size N. Therefore, the width (w) of the resulting CI can be considered a function of S, , and N. Inverting the CI formation process gives the sample size planning process. The inverted process requires a proxy for the population covariance matrix Σ, population mean vector μ, and the desired width ω as input, and it returns N as output. The specification of the input information for sample size planning needs to be performed based on a systematic literature review. In the context of covariance structure analysis, Lai and Kelley (2011) discussed several practical methods to facilitate specifying Σ and ω for the sample size planning procedure.     

Abstract: Adequacy of Semiparametric Approximations for Growth Models with Nonnormal Random Effects

Abstract

Conventional growth models assume that the random effects describing individual trajectories are conditionally normal. In practice, this assumption may often be unrealistic. As an alternative, Nagin (2005) Nagin, D. 2005. Group-based modeling of development, Cambridge: Harvard University Press. [Crossref] , [Google Scholar] suggested a semiparametric group-based approach (SPGA) which approximates an unknown, continuous distribution of individual trajectories with a mixture of group trajectories.

Prior simulations (Brame, Nagin, &; Wasserman, 2006 Brame, R., Nagin, D. and Wasserman, L. 2006. Exploring some analytical characteristics of finite mixture models.. Journal of Quantitative Criminology, 22: 31–59. [Crossref], [Web of Science ®] , [Google Scholar]; Nagin, 2005 Nagin, D. 2005. Group-based modeling of development, Cambridge: Harvard University Press. [Crossref] , [Google Scholar]) indicated that SPGA could generate nearly-unbiased estimates of means and variances of a nonnormal distribution of individual trajectories, as functions of group-trajectory estimates. However, these studies used few random effects—usually only a random intercept. Based on the analytical relationship between SPGA and adaptive quadrature, we hypothesized that SPGA's ability to approximate (a) random effect variances/covariances and (b) effects of time-invariant predictors of growth should deteriorate as the dimensionality of the random effects distribution increases. We expected this problem to be mitigated by correlations among the random effects (highly correlated random effects functioning as fewer dimensions) and sample size (larger N supporting more groups).

We tested these hypotheses via simulation, varying the number of random effects (1, 2, or 3), correlation among the random effects (0 or .6), and N (250, 500). Results indicated that, as the number of random effects increased, SPGA approximations remained acceptable for fixed effects, but became increasingly negatively biased for random effect variances. Whereas correlated random effects and larger N reduced this underestimation, correlated random effects sometimes distorted recovery of predictor effects. To illustrate this underestimation, Figure 1 depicts SPGA's approximation of the intercept variance from a three correlated random effect generating model (N < eqid1 > 500). These results suggest SPGA approximations are inadequate for the nonnormal, high-dimensional distributions of individual trajectories often seen in practice.

Abstract: Adequacy of Semiparametric Approximations for Growth Models with Nonnormal Random Effects

All authors

Sonya K. Sterba, Ruth E. Mathiowetz &; Daniel J. Bauer

https://doi.org/10.1080/00273170802640509

Published online:

18 December 2008

FIGURE 1 SPGA-approximated intercept variance from a three correlated random effect generating model. Notes. The dashed horizontal lines denote + 10% bias. The solid horizontal line denotes the population-generating parameter value; * denotes the best-BIC selected number of groups. The vertical bars denote 90% confidence intervals.     

9.

Abstract: Inference and Interval Estimation for Indirect Effects With Latent Variable Models     

Carl F. Falk  Jeremy C. Biesanz 《Multivariate behavioral research》2013,48(6)

Models specifying indirect effects (or mediation) and structural equation modeling are both popular in the social sciences. Yet relatively little research has compared methods that test for indirect effects among latent variables and provided precise estimates of the effectiveness of different methods.

This simulation study provides an extensive comparison of methods for constructing confidence intervals and for making inferences about indirect effects with latent variables. We compared the percentile (PC) bootstrap, bias-corrected (BC) bootstrap, bias-corrected accelerated (BC _a ) bootstrap, likelihood-based confidence intervals (Neale & Miller, 1997), partial posterior predictive (Biesanz, Falk, and Savalei, 2010), and joint significance tests based on Wald tests or likelihood ratio tests. All models included three reflective latent variables representing the independent, dependent, and mediating variables. The design included the following fully crossed conditions: (a) sample size: 100, 200, and 500; (b) number of indicators per latent variable: 3 versus 5; (c) reliability per set of indicators: .7 versus .9; (d) and 16 different path combinations for the indirect effect (α = 0, .14, .39, or .59; and β = 0, .14, .39, or .59). Simulations were performed using a WestGrid cluster of 1680 3.06GHz Intel Xeon processors running R and OpenMx.

Results based on 1,000 replications per cell and 2,000 resamples per bootstrap method indicated that the BC and BC _a bootstrap methods have inflated Type I error rates. Likelihood-based confidence intervals and the PC bootstrap emerged as methods that adequately control Type I error and have good coverage rates.     

10.

Gender and Grade Differences in Latent Ability Variables     

Kjell Härnqvist 《Scandinavian journal of psychology》1997,38(1):55-62

Scores in ability tests administered to students in grades 4–9 were simultaneously factor-analyzed within twelve gender by grade groups. Gender and grade differences in means and variances were studied for five latent ability factors according to a hierarchical model and compared with means and variances in the observed scores.
Girls had higher means than boys in a general ability factor (G), in a residual general speed factor (Gs') and in a residual factor of numerical facility (N'). Boys were higher in a residual vocabulary factor (V') and most of all in a residual spatial visualization factor (Vz'). Boys had larger variances than girls in N' and Gs'. In general the differences in means and variances were in the same direction for the closest corresponding observed scores, but some striking discrepancies between latent and observed scores were found. As a rule, the discrepancies were due to the complexity of the tests where one factor could compensate for another.
In the discussion it was pointed out that some of the differences found were likely to have changed between the testing in the late 1950s and the present. Nevertheless the demonstration of the divergence between analyses of latent vs. observed scores remains valid.     

11.

Abstract: Permutation Tests of Coupled Latent Differential Equations     

Yueqin Hu  Steven M. Boker 《Multivariate behavioral research》2013,48(1)

     

12.

Latent Growth Modeling for Logistic Response Functions     

Jaehwa Choi  Jeffrey R. Harring  Gregory R. Hancock 《Multivariate behavioral research》2013,48(5):620-645

Throughout much of the social and behavioral sciences, latent growth modeling (latent curve analysis) has become an important tool for understanding individuals' longitudinal change. Although nonlinear variations of latent growth models appear in the methodological and applied literature, a notable exclusion is the treatment of growth following logistic (sigmoidal; S-shape) response functions. Such trajectories are assumed in a variety of psychological and educational settings where learning occurs over time, and yet applications using the logistic model in growth modeling methodology have been sparse. The logistic function, in particular, may not be utilized as often because software options remain limited. In this article we show how a specialized version of the logistic function can be modeled using conventional structural equation modeling software. The specialization is a reparameterization of the logistic function whose new parameters correspond to scientifically interesting characteristics of the growth process. In addition to providing an example using simulated data, we show how this nonlinear functional form can be fit using transformed subject-level data obtained through a learning task from an air traffic controller simulation experiment. LISREL syntax for the empirical example is provided.     

13.

Distinguishing Caregiver Strain from Psychological Distress: Modeling the Relationships Among Child, Family, and Caregiver Variables     

Ana Maria Brannan  Craig Anne Heflinger 《Journal of child and family studies》2001,10(4):405-418

Although both have been used in studies of the impact of mental illness on the family, the constructs of caregiver strain (often referred to as burden of care) and psychological distress have not been clearly distinguished. The vagueness surrounding these constructs, and the lack of a cohesive conceptual framework for understanding how they relate, leads to contradictory interpretations of results. This compromises the building of the knowledge base needed to develop and evaluate interventions to support families as they struggle to meet the needs of their children with emotional and behavioral challenges. We utilized the ABCX Model as a framework for understanding caregiver strain and its relationship to psychological distress. Structural equations modeling was used to test the hypothesized relationship between caregiver strain and psychological distress, as well as the role of key child and family variables. These included child symptoms, stressful life events, social support, family functioning, and material resources. Our findings indicated that caregiver strain and psychological distress, although related, have distinct correlates and different implications in the family context.     

14.

New Methods for Comparing Groups     

Rand R. Wilcox 《Current directions in psychological science》2005,14(5):272-275

Abstract— A commonly used method for comparing groups of individuals is the analysis of variance (ANOVA) F test. When the assumptions underlying the derivation of this test are true, its power, meaning its probability of detecting true differences among the groups, competes well with all other methods that might be used. But when these assumptions are false, its power can be relatively low. Many new statistical methods have been proposed—ones that are aimed at achieving about the same amount of power when the assumptions of the F test are true but which have the potential of high power in situations where the F test performs poorly. A brief summary of some relevant issues and recent developments is provided. Some related issues are discussed and implications for future research are described.     

15.

A Bayesian Modeling Approach for Generalized Semiparametric Structural Equation Models     

Xin-Yuan Song  Zhao-Hua Lu  Jing-Heng Cai  Edward Hak-Sing Ip 《Psychometrika》2013,78(4):624-647

In behavioral, biomedical, and psychological studies, structural equation models (SEMs) have been widely used for assessing relationships between latent variables. Regression-type structural models based on parametric functions are often used for such purposes. In many applications, however, parametric SEMs are not adequate to capture subtle patterns in the functions over the entire range of the predictor variable. A different but equally important limitation of traditional parametric SEMs is that they are not designed to handle mixed data types—continuous, count, ordered, and unordered categorical. This paper develops a generalized semiparametric SEM that is able to handle mixed data types and to simultaneously model different functional relationships among latent variables. A structural equation of the proposed SEM is formulated using a series of unspecified smooth functions. The Bayesian P-splines approach and Markov chain Monte Carlo methods are developed to estimate the smooth functions and the unknown parameters. Moreover, we examine the relative benefits of semiparametric modeling over parametric modeling using a Bayesian model-comparison statistic, called the complete deviance information criterion (DIC). The performance of the developed methodology is evaluated using a simulation study. To illustrate the method, we used a data set derived from the National Longitudinal Survey of Youth.     

16.

Abstract: Information Processing Among Creative Students     

ALICE J. CORKILL  JOHN A. GLOVER 《创造性行为杂志》1989,23(1):68-69

     

17.

Abstract: Local Influence and Robust Methods for Mediation Models     

Jiyun Zu  Ke-Hai Yuan 《Multivariate behavioral research》2013,48(4)

Mediation analysis investigates how certain variables mediate the effect of predictors on outcome variables. Existing studies of mediation models have been limited to normal theory maximum likelihood (ML) or least squares with normally distributed data. Because real data in the social and behavioral sciences are seldom normally distributed and often contain outliers, classical methods can result in biased and inefficient estimates, which lead to inaccurate or unreliable test of the meditated effect. The authors propose two approaches for better mediation analysis. One is to identify cases that strongly affect test results of mediation using local influence methods and robust methods. The other is to use robust methods for parameter estimation, and then test the mediated effect based on the robust estimates. Analytic details of both local influence and robust methods particular for mediation models were provided and one real data example was given. We first used local influence and robust methods to identify influential cases. Then, for the original data and the data with the identified influential cases removed, the mediated effect was tested using two estimation methods: normal theory ML and the robust method, crossing two tests of mediation: the Sobel (1982) Sobel, M. E. 1982. “Asymptotic confidence intervals for indirect effects in structural equation models”. In Sociological methodology, Edited by: Leinhardt, S. 290–312. Washington, DC: American Sociological Association. [Crossref] , [Google Scholar] test using information-based standard error (z _I ) and sandwich-type standard error (z _SW ). Results show that local influence and robust methods rank the influence of cases similarly, while the robust method is more objective. The widely used z _I statistic is inflated when the distribution is heavy-tailed. Compared to normal theory ML, the robust method provides estimates with smaller standard errors and more reliable test.     

18.

Two-Step Estimation of Models Between Latent Classes and External Variables     

Zsuzsa Bakk  Jouni Kuha 《Psychometrika》2018,83(4):871-892

We consider models which combine latent class measurement models for categorical latent variables with structural regression models for the relationships between the latent classes and observed explanatory and response variables. We propose a two-step method of estimating such models. In its first step, the measurement model is estimated alone, and in the second step the parameters of this measurement model are held fixed when the structural model is estimated. Simulation studies and applied examples suggest that the two-step method is an attractive alternative to existing one-step and three-step methods. We derive estimated standard errors for the two-step estimates of the structural model which account for the uncertainty from both steps of the estimation, and show how the method can be implemented in existing software for latent variable modelling.     

19.

An Inequality for Correlations in Unidimensional Monotone Latent Variable Models for Binary Variables     

Jules L. Ellis 《Psychometrika》2014,79(2):303-316

It is shown that a unidimensional monotone latent variable model for binary items implies a restriction on the relative sizes of item correlations: The negative logarithm of the correlations satisfies the triangle inequality. This inequality is not implied by the condition that the correlations are nonnegative, the criterion that coefficient H exceeds 0.30, or manifest monotonicity. The inequality implies both a lower bound and an upper bound for each correlation between two items, based on the correlations of those two items with every possible third item. It is discussed how this can be used in Mokken’s (A theory and procedure of scale-analysis, Mouton, The Hague, 1971) scale analysis.     

20.

Comparing Latent Means Without Mean Structure Models: A Projection-Based Approach     

Lifang Deng  Ke-Hai Yuan 《Psychometrika》2016,81(3):802-829

The conventional setup for multi-group structural equation modeling requires a stringent condition of cross-group equality of intercepts before mean comparison with latent variables can be conducted. This article proposes a new setup that allows mean comparison without the need to estimate any mean structural model. By projecting the observed sample means onto the space of the common scores and the space orthogonal to that of the common scores, the new setup allows identifying and estimating the means of the common and specific factors, although, without replicate measures, variances of specific factors cannot be distinguished from those of measurement errors. Under the new setup, testing cross-group mean differences of the common scores is done independently from that of the specific factors. Such independent testing eliminates the requirement for cross-group equality of intercepts by the conventional setup in order to test cross-group equality of means of latent variables using chi-square-difference statistics. The most appealing piece of the new setup is a validity index for mean differences, defined as the percentage of the sum of the squared observed mean differences that is due to that of the mean differences of the common scores. By analyzing real data with two groups, the new setup is shown to offer more information than what is obtained under the conventional setup.     

Abstract: Inference and Interval Estimation for Indirect Effects With Latent Variable Models

Models specifying indirect effects (or mediation) and structural equation modeling are both popular in the social sciences. Yet relatively little research has compared methods that test for indirect effects among latent variables and provided precise estimates of the effectiveness of different methods.

This simulation study provides an extensive comparison of methods for constructing confidence intervals and for making inferences about indirect effects with latent variables. We compared the percentile (PC) bootstrap, bias-corrected (BC) bootstrap, bias-corrected accelerated (BC _a ) bootstrap, likelihood-based confidence intervals (Neale & Miller, 1997), partial posterior predictive (Biesanz, Falk, and Savalei, 2010), and joint significance tests based on Wald tests or likelihood ratio tests. All models included three reflective latent variables representing the independent, dependent, and mediating variables. The design included the following fully crossed conditions: (a) sample size: 100, 200, and 500; (b) number of indicators per latent variable: 3 versus 5; (c) reliability per set of indicators: .7 versus .9; (d) and 16 different path combinations for the indirect effect (α = 0, .14, .39, or .59; and β = 0, .14, .39, or .59). Simulations were performed using a WestGrid cluster of 1680 3.06GHz Intel Xeon processors running R and OpenMx.

Results based on 1,000 replications per cell and 2,000 resamples per bootstrap method indicated that the BC and BC _a bootstrap methods have inflated Type I error rates. Likelihood-based confidence intervals and the PC bootstrap emerged as methods that adequately control Type I error and have good coverage rates.     

Gender and Grade Differences in Latent Ability Variables

Scores in ability tests administered to students in grades 4–9 were simultaneously factor-analyzed within twelve gender by grade groups. Gender and grade differences in means and variances were studied for five latent ability factors according to a hierarchical model and compared with means and variances in the observed scores.
Girls had higher means than boys in a general ability factor (G), in a residual general speed factor (Gs') and in a residual factor of numerical facility (N'). Boys were higher in a residual vocabulary factor (V') and most of all in a residual spatial visualization factor (Vz'). Boys had larger variances than girls in N' and Gs'. In general the differences in means and variances were in the same direction for the closest corresponding observed scores, but some striking discrepancies between latent and observed scores were found. As a rule, the discrepancies were due to the complexity of the tests where one factor could compensate for another.
In the discussion it was pointed out that some of the differences found were likely to have changed between the testing in the late 1950s and the present. Nevertheless the demonstration of the divergence between analyses of latent vs. observed scores remains valid.     

Latent Growth Modeling for Logistic Response Functions

Throughout much of the social and behavioral sciences, latent growth modeling (latent curve analysis) has become an important tool for understanding individuals' longitudinal change. Although nonlinear variations of latent growth models appear in the methodological and applied literature, a notable exclusion is the treatment of growth following logistic (sigmoidal; S-shape) response functions. Such trajectories are assumed in a variety of psychological and educational settings where learning occurs over time, and yet applications using the logistic model in growth modeling methodology have been sparse. The logistic function, in particular, may not be utilized as often because software options remain limited. In this article we show how a specialized version of the logistic function can be modeled using conventional structural equation modeling software. The specialization is a reparameterization of the logistic function whose new parameters correspond to scientifically interesting characteristics of the growth process. In addition to providing an example using simulated data, we show how this nonlinear functional form can be fit using transformed subject-level data obtained through a learning task from an air traffic controller simulation experiment. LISREL syntax for the empirical example is provided.     

Distinguishing Caregiver Strain from Psychological Distress: Modeling the Relationships Among Child, Family, and Caregiver Variables

Although both have been used in studies of the impact of mental illness on the family, the constructs of caregiver strain (often referred to as burden of care) and psychological distress have not been clearly distinguished. The vagueness surrounding these constructs, and the lack of a cohesive conceptual framework for understanding how they relate, leads to contradictory interpretations of results. This compromises the building of the knowledge base needed to develop and evaluate interventions to support families as they struggle to meet the needs of their children with emotional and behavioral challenges. We utilized the ABCX Model as a framework for understanding caregiver strain and its relationship to psychological distress. Structural equations modeling was used to test the hypothesized relationship between caregiver strain and psychological distress, as well as the role of key child and family variables. These included child symptoms, stressful life events, social support, family functioning, and material resources. Our findings indicated that caregiver strain and psychological distress, although related, have distinct correlates and different implications in the family context.     

New Methods for Comparing Groups

Abstract— A commonly used method for comparing groups of individuals is the analysis of variance (ANOVA) F test. When the assumptions underlying the derivation of this test are true, its power, meaning its probability of detecting true differences among the groups, competes well with all other methods that might be used. But when these assumptions are false, its power can be relatively low. Many new statistical methods have been proposed—ones that are aimed at achieving about the same amount of power when the assumptions of the F test are true but which have the potential of high power in situations where the F test performs poorly. A brief summary of some relevant issues and recent developments is provided. Some related issues are discussed and implications for future research are described.     

A Bayesian Modeling Approach for Generalized Semiparametric Structural Equation Models

In behavioral, biomedical, and psychological studies, structural equation models (SEMs) have been widely used for assessing relationships between latent variables. Regression-type structural models based on parametric functions are often used for such purposes. In many applications, however, parametric SEMs are not adequate to capture subtle patterns in the functions over the entire range of the predictor variable. A different but equally important limitation of traditional parametric SEMs is that they are not designed to handle mixed data types—continuous, count, ordered, and unordered categorical. This paper develops a generalized semiparametric SEM that is able to handle mixed data types and to simultaneously model different functional relationships among latent variables. A structural equation of the proposed SEM is formulated using a series of unspecified smooth functions. The Bayesian P-splines approach and Markov chain Monte Carlo methods are developed to estimate the smooth functions and the unknown parameters. Moreover, we examine the relative benefits of semiparametric modeling over parametric modeling using a Bayesian model-comparison statistic, called the complete deviance information criterion (DIC). The performance of the developed methodology is evaluated using a simulation study. To illustrate the method, we used a data set derived from the National Longitudinal Survey of Youth.     

Abstract: Local Influence and Robust Methods for Mediation Models

Mediation analysis investigates how certain variables mediate the effect of predictors on outcome variables. Existing studies of mediation models have been limited to normal theory maximum likelihood (ML) or least squares with normally distributed data. Because real data in the social and behavioral sciences are seldom normally distributed and often contain outliers, classical methods can result in biased and inefficient estimates, which lead to inaccurate or unreliable test of the meditated effect. The authors propose two approaches for better mediation analysis. One is to identify cases that strongly affect test results of mediation using local influence methods and robust methods. The other is to use robust methods for parameter estimation, and then test the mediated effect based on the robust estimates. Analytic details of both local influence and robust methods particular for mediation models were provided and one real data example was given. We first used local influence and robust methods to identify influential cases. Then, for the original data and the data with the identified influential cases removed, the mediated effect was tested using two estimation methods: normal theory ML and the robust method, crossing two tests of mediation: the Sobel (1982) Sobel, M. E. 1982. “Asymptotic confidence intervals for indirect effects in structural equation models”. In Sociological methodology, Edited by: Leinhardt, S. 290–312. Washington, DC: American Sociological Association. [Crossref] , [Google Scholar] test using information-based standard error (z _I ) and sandwich-type standard error (z _SW ). Results show that local influence and robust methods rank the influence of cases similarly, while the robust method is more objective. The widely used z _I statistic is inflated when the distribution is heavy-tailed. Compared to normal theory ML, the robust method provides estimates with smaller standard errors and more reliable test.     

Two-Step Estimation of Models Between Latent Classes and External Variables

We consider models which combine latent class measurement models for categorical latent variables with structural regression models for the relationships between the latent classes and observed explanatory and response variables. We propose a two-step method of estimating such models. In its first step, the measurement model is estimated alone, and in the second step the parameters of this measurement model are held fixed when the structural model is estimated. Simulation studies and applied examples suggest that the two-step method is an attractive alternative to existing one-step and three-step methods. We derive estimated standard errors for the two-step estimates of the structural model which account for the uncertainty from both steps of the estimation, and show how the method can be implemented in existing software for latent variable modelling.     

An Inequality for Correlations in Unidimensional Monotone Latent Variable Models for Binary Variables

It is shown that a unidimensional monotone latent variable model for binary items implies a restriction on the relative sizes of item correlations: The negative logarithm of the correlations satisfies the triangle inequality. This inequality is not implied by the condition that the correlations are nonnegative, the criterion that coefficient H exceeds 0.30, or manifest monotonicity. The inequality implies both a lower bound and an upper bound for each correlation between two items, based on the correlations of those two items with every possible third item. It is discussed how this can be used in Mokken’s (A theory and procedure of scale-analysis, Mouton, The Hague, 1971) scale analysis.     

Comparing Latent Means Without Mean Structure Models: A Projection-Based Approach

The conventional setup for multi-group structural equation modeling requires a stringent condition of cross-group equality of intercepts before mean comparison with latent variables can be conducted. This article proposes a new setup that allows mean comparison without the need to estimate any mean structural model. By projecting the observed sample means onto the space of the common scores and the space orthogonal to that of the common scores, the new setup allows identifying and estimating the means of the common and specific factors, although, without replicate measures, variances of specific factors cannot be distinguished from those of measurement errors. Under the new setup, testing cross-group mean differences of the common scores is done independently from that of the specific factors. Such independent testing eliminates the requirement for cross-group equality of intercepts by the conventional setup in order to test cross-group equality of means of latent variables using chi-square-difference statistics. The most appealing piece of the new setup is a validity index for mean differences, defined as the percentage of the sum of the squared observed mean differences that is due to that of the mean differences of the common scores. By analyzing real data with two groups, the new setup is shown to offer more information than what is obtained under the conventional setup.