Showing papers on "Latent variable model published in 2005"

Latent curve models : a structural equation perspective

Abstract: Preface. 1 Introduction. 1.1 Conceptualization and Analysis of Trajectories. 1.2 Three Initial Questions About Trajectories. 1.3 Brief History of Latent Curve Models. 1.4 Organization of the Remainder of the Book. 2 Unconditional Latent Curve Model. 2.1 Repeated Measures. 2.2 General Model and Assumptions. 2.3 Identification. 2.4 Case-By-Case Approach. 2.5 Structural Equation Model Approach. 2.6 Alternative Approaches to the SEM. 2.7 Conclusions. Appendix 2A: Test Statistics, Nonnormality, and Statistical Power. 3 Missing Data and Alternative Metrics of Time. 3.1 Missing Data. 3.2 Missing Data and Alternative Metrics of Time. 3.3 Conclusions. 4 Nonlinear Trajectories and the Coding of Time. 4.1 Modeling Nonlinear Functions of Time. 4.2 Nonlinear Curve Fitting: Estimated Factor Loadings. 4.3 Piecewise Linear Trajectory Models. 4.4 Alternative Parametric Functions. 4.5 Linear Transformations of the Metric of Time. 4.6 Conclusions. Appendix 4A: Identification of Quadratic and Piecewise Latent Curve Models. 4A.1 Quadratic LCM. 4A.2 Piecewise LCM. 5 Conditional Latent Curve Models. 5.1 Conditional Model and Assumptions. 5.2 Identification. 5.3 Structural Equation Modeling Approach. 5.4 Interpretation of Conditional Model Estimates. 5.5 Empirical Example. 5.6 Conclusions. 6 The Analysis of Groups. 6.1 Dummy Variable Approach. 6.2 Multiple-Group Analysis. 6.3 Unknown Group Membership. 6.4 Conclusions. Appendix 6A: Case-by-Case Approach to Analysis of Various Groups. 6A.1 Dummy Variable Method. 6A.2 Multiple-Group Analysis. 6A.3 Unknown Group Membership. 6A.4 Appendix Summary. 7 Multivariate Latent Curve Models. 7.1 Time-Invariant Covariates. 7.2 Time-Varying Covariates. 7.3 Simultaneous Inclusion of Time-Invariant and Time-Varying Covariates. 7.4 Multivariate Latent Curve Models. 7.5 Autoregressive Latent Trajectory Model. 7.6 General Equation for All Models. 7.7 Implied Moment Matrices. 7.8 Conclusions. 8 Extensions of Latent Curve Models. 8.1 Dichotomous and Ordinal Repeated Measures. 8.2 Repeated Latent Variables with Multiple Indicators. 8.3 Latent Covariates. 8.4 Conclusions. References. Author Index. Subject Index.

The Problem of Measurement Model Misspecification in Behavioral and Organizational Research and Some Recommended Solutions.

Abstract: The purpose of this study was to review the distinction between formative- and reflective-indicator measurement models, articulate a set of criteria for deciding whether measures are formative or reflective, illustrate some commonly researched constructs that have formative indicators, empirically test the effects of measurement model misspecification using a Monte Carlo simulation, and recommend new scale development procedures for latent constructs with formative indicators. Results of the Monte Carlo simulation indicated that measurement model misspecification can inflate unstandardized structural parameter estimates by as much as 400% or deflate them by as much as 80% and lead to Type I or Type II errors of inference, depending on whether the exogenous or the endogenous latent construct is misspecified. Implications of this research are discussed.

Investigating population heterogeneity with factor mixture models.

Abstract: Sources of population heterogeneity may or may not be observed. If the sources of heterogeneity are observed (e.g., gender), the sample can be split into groups and the data analyzed with methods for multiple groups. If the sources of population heterogeneity are unobserved, the data can be analyzed with latent class models. Factor mixture models are a combination of latent class and common factor models and can be used to explore unobserved population heterogeneity. Observed sources of heterogeneity can be included as covariates. The different ways to incorporate covariates correspond to different conceptual interpretations. These are discussed in detail. Characteristics of factor mixture modeling are described in comparison to other methods designed for data stemming from heterogeneous populations. A step-by-step analysis of a subset of data from the Longitudinal Survey of American Youth illustrates how factor mixture models can be applied in an exploratory fashion to data collected at a single time point.

Using Parcels to Convert Path Analysis Models Into Latent Variable Models.

Abstract: The biasing effects of measurement error in path analysis models can be overcome by the use of latent variable models. In cases where path analysis is used in practice, it is often possible to use parcels as indicators of a latent variable. The purpose of the current study was to compare latent variable models in which parcels were used as indicators of the latent variables, path analysis models of the aggregated variables, and models in which reliability estimates were used to correct for measurement error in path analysis models. Results showed that point estimates of path coefficients were smallest for the path analysis models and largest for the latent variable models. It is concluded that, whenever possible, it is better to use a latent variable model in which parcels are used as indicators than a path analysis model using total scale scores.

Sampling Weights in Latent Variable Modeling

Abstract: This article reviews several basic statistical tools needed for modeling data with sampling weights that are implemented in Mplus Version 3. These tools are illustrated in simulation studies for several latent variable models including factor analysis with continuous and categorical indicators, latent class analysis, and growth models. The pseudomaximum likelihood estimation method is reviewed and illustrated with stratified cluster sampling. Additionally, the weighted least squares method for estimating structural equation models with categorical and continuous outcomes implemented in Mplus extended to incorporate sampling weights is also illustrated. The performance of several chi-square tests under unequal probability sampling is evaluated. Simulation studies compare the methods used in several statistical packages such as Mplus, HLM, SAS Proc Mixed, MLwiN, and the weighted sample statistics method used in other software packages.

A Probabilistic Interpretation of Canonical Correlation Analysis

Abstract: We give a probabilistic interpretation of canonical correlation (CCA) analysis as a latent variable model for two Gaussian random vectors. Our interpretation is similar to the probabilistic interpretation of principal component analysis (Tipping and Bishop, 1999, Roweis, 1998). In addition, we cast Fisher linear discriminant analysis (LDA) within the CCA framework.

Robust Bayesian mixture modelling

Abstract: Bayesian approaches to density estimation and clustering using mixture distributions allow the automatic determination of the number of components in the mixture. Previous treatments have focussed on mixtures having Gaussian components, but these are well known to be sensitive to outliers, which can lead to excessive sensitivity to small numbers of data points and consequent over-estimates of the number of components. In this paper we develop a Bayesian approach to mixture modelling based on Student-t distributions, which are heavier tailed than Gaussians and hence more robust. By expressing the Student-t distribution as a marginalization over additional latent variables we are able to derive a tractable variational inference algorithm for this model, which includes Gaussian mixtures as a special case. Results on a variety of real data sets demonstrate the improved robustness of our approach.

Discrete-Time Survival Mixture Analysis

Abstract: This article proposes a general latent variable approach to discrete-time survival analysis of nonrepeatable events such as onset of drug use. It is showvn how the survival analysis can beformulated as a generalized latent class analysis of event history indicators. The latent class analysis can use covariates and can be combined wvith the joint modeling of other outcomes such as repeated measures for a related process. It is shown that conventional discrete-time survival analysis corresponds to a single-class latent class analysis. Multiple-class extensions are proposed, including the special cases of a class of long-term survivors and classes defined by outcomes related to survivaL The estimation uses a general latent variable framework, including both categorical and continuous latent variables and incorporated in the Mplus program. Estimation is carried out using maximum likelihood via the EM algorithm. Two examples serve as illustrations. The first example concerns recidivism after incarceration in a randomized field experiment. The second example concerns school removal related to the development of aggressive behavior in the classroom.

Empirical and Conceptual Problems With Longitudinal Trait-State Models: Introducing a Trait-State-Occasion Model.

Abstract: The latent trait-state-error model (TSE) and the latent state-trait model with autoregression (LST-AR) represent creative structural equation methods for examining the longitudinal structure of psychological constructs. Application of these models has been somewhat limited by empirical or conceptual problems. In the present study, Monte Carlo analysis revealed that TSE models tend to generate improper solutions when N is too small, when waves are too few, and when occasion factor stability is either too large or too small. Mathematical analysis of the LST-AR model revealed its limitation to constructs that become more highly auto-correlated over time. The trait-state-occasion model has fewer empirical problems than does the TSE model and is more broadly applicable than is the LST-AR model.

The Stochastic Conditional Duration Model: A Latent Factor Model for the Analysis of Financial Durations

Abstract: We introduce a class of models for the analysis of durations, which we call stochastic conditional duration (SCD) models These models are based on the assumption that the durations are generated by a dynamic stochastic latent variable The model yields a wide range of shapes of hazard functions The estimation of the parameters is performed by quasi-maximum likelihood and using the Kalman filter The model is applied to trade, price and volume durations of stocks traded at NYSE We also investigate the relation between price durations, spread, trade intensity and volume

Residual Diagnostics for Growth Mixture Models: Examining the Impact of a Preventive Intervention on Multiple Trajectories of Aggressive Behavior

Abstract: Growth mixture modeling has become a prominent tool for studying the heterogeneity of developmental trajectories within a population. In this article we develop graphical diagnostics to detect misspecification in growth mixture models regarding the number of growth classes, growth trajectory means, and covariance structures. For each model misspecification, we propose a different type of empirical Bayes residual to quantify the departure. Our procedure begins by imputing multiple independent sets of growth classes for the sample. Then, from these so-called “pseudoclass” draws, we form diagnostic plots to examine the averaged empirical distributions of residuals in each such class. Our proposals draw on the property that each single set of pseudoclass adjusted residuals is asymptotically normal with known mean and (co)variance when the underlying model is correct. These methods are justified in simulation studies involving two classes of linear growth curves that also differ by their covariance structures....

A Hierarchical Bayesian Model for Learning Nonlinear Statistical Regularities in Nonstationary Natural Signals

Abstract: Capturing statistical regularities in complex, high-dimensional data is an important problem in machine learning and signal processing. Models such as principal component analysis (PCA) and independent component analysis (ICA) make few assumptions about the structure in the data and have good scaling properties, but they are limited to representing linear statistical regularities and assume that the distribution of the data is stationary. For many natural, complex signals, the latent variables often exhibit residual dependencies as well as nonstationary statistics. Here we present a hierarchical Bayesian model that is able to capture higher-order nonlinear structure and represent nonstationary data distributions. The model is a generalization of ICA in which the basis function coefficients are no longer assumed to be independent; instead, the dependencies in their magnitudes are captured by a set of density components. Each density component describes a common pattern of deviation from the marginal density of the pattern ensemble; in different combinations, they can describe nonstationary distributions. Adapting the model to image or audio data yields a nonlinear, distributed code for higher-order statistical regularities that reflect more abstract, invariant properties of the signal.

Solving and Testing for Regressor-Error (in)Dependence When no Instrumental Variables are Available: With New Evidence for the Effect of Education on Income

Abstract: This paper has two main contributions. Firstly, we introduce a new approach, the latent instrumental variables (LIV) method, to estimate regression coefficients consistently in a simple linear regression model where regressor-error correlations (endogeneity) are likely to be present. The LIV method utilizes a discrete latent variable model that accounts for dependencies between regressors and the error term. As a result, additional ‘valid’ observed instrumental variables are not required. Furthermore, we propose a specification test based on Hausman (1978) to test for these regressor-error correlations. A simulation study demonstrates that the LIV method yields consistent estimates and the proposed test-statistic has reasonable power over a wide range of regressor-error correlations and several distributions of the instruments. Secondly, the LIV method is used to re-visit the relationship between education and income based on previously published data. Data from three studies are re-analyzed. We examine the effect of education on income, where the variable ‘education’ is potentially endogenous due to omitted ‘ability’ or other causes. In all three applications, we find an upward bias in the OLS estimates of approximately 7%. Our conclusions agree closely with recent results obtained in studies with twins that find an upward bias in OLS of about 10% (Card, 1999). We also show that for each of the three datasets the classical IV estimates for the return to education point to biases in OLS that are not consistent in terms of size and magnitude. Our conclusion is that LIV estimates are preferable to the classical IV estimates in understanding the effects of education on income.

PLS regression, PLS path modeling and generalized Procrustean analysis: a combined approach for multiblock analysis

Abstract: A situation where J blocks of variables are observed on the same set of individuals is considered in this paper. A factor analysis logic is applied to tables instead of variables. The latent variables of each block should well explain their own block and, at the same time, the latent variables of same order should be as positively correlated as possible to improve interpretation. The paper first (1) reviews the main methods for multiblock analysis based on a criterion to be optimized, (2) describes the hierarchical PLS path modeling algorithm and (3) recalls that it allows one to recover some usual multiblock analysis methods. It is then supposed that the number of latent variables can be different from one block to another and that these latent variables are orthogonal. PLS regression and PLS path modeling are used for this situation. The relation between Horst's generalized canonical correlation analysis and generalized Procrustean analysis for this specific application is also studied. The approach is illustrated by an example from sensory analysis. Copyright © 2005 John Wiley & Sons, Ltd.

Nonparametric Bayesian Modeling for Multivariate Ordinal Data

Abstract: This article proposes a probability model for k-dimensional ordinal outcomes, that is, it considers inference for data recorded in k-dimensional contingency tables with ordinal factors. The proposed approach is based on full posterior inference, assuming a flexible underlying prior probability model for the contingency table cell probabilities. We use a variation of the traditional multivariate probit model, with latent scores that determine the observed data. In our model, a mixture of normals prior replaces the usual single multivariate normal model for the latent variables. By augmenting the prior model to a mixture of normals we generalize inference in two important ways. First, we allow for varying local dependence structure across the contingency table. Second, inference in ordinal multivariate probit models is plagued by problems related to the choice and resampling of cutoffs defined for these latent variables. We show how the proposed mixture model approach entirely removes these problems. We ill...

Factor Analysis with Categorical Indicators: A Comparison Between Traditional and Latent Class Approaches

Abstract: The linear factor analysis (FA) model is a popular tool for exploratory data analysis or, more precisely, for assessing the dimensionality of sets of items. Although it is well known that it is meant for continuous observed indicators, it is often used with dichotomous, ordinal, and other types of discrete variables, yielding results that might be incorrect. Not only parameter estimates may be biased, but also goodness-of-fit indices cannot be trusted. Magidson and Vermunt (2001) presented a nonlinear factor-analytic model based on latent class (LC) analysis that is especially suited for dealing with categorical indicators, such as dichotomous, ordinal, and nominal variables,

The performance of mixture models in heterogeneous closed population capture-recapture.

Abstract: SummaryDorazio and Royle (2003, Biometrics59, 351–364) investigated the behavior of three mixture models for closed population capture–recapture analysis in the presence of individual heterogeneity of capture probability. Their simulations were from the beta-binomial distribution, with analyses from the beta-binomial, the logit-normal, and the finite mixture (latent class) models. In this response, simulations from many different distributions give a broader picture of the relative value of the beta-binomial and the finite mixture models, and provide some preliminary insights into the situations in which these models are useful.

The Latent Process Decomposition of cDNA Microarray Data Sets

Abstract: We present a new computational technique (a software implementation, data sets, and supplementary information are available at http://www.enm.bris.ac.uk/lpd/) which enables the probabilistic analysis of cDNA microarray data and we demonstrate its effectiveness in identifying features of biomedical importance. A hierarchical Bayesian model, called Latent Process Decomposition (LPD), is introduced in which each sample in the data set is represented as a combinatorial mixture over a finite set of latent processes, which are expected to correspond to biological processes. Parameters in the model are estimated using efficient variational methods. This type of probabilistic model is most appropriate for the interpretation of measurement data generated by cDNA microarray technology. For determining informative substructure in such data sets, the proposed model has several important advantages over the standard use of dendrograms. First, the ability to objectively assess the optimal number of sample clusters. Second, the ability to represent samples and gene expression levels using a common set of latent variables (dendrograms cluster samples and gene expression values separately which amounts to two distinct reduced space representations). Third, in constrast to standard cluster models, observations are not assigned to a single cluster and, thus, for example, gene expression levels are modeled via combinations of the latent processes identified by the algorithm. We show this new method compares favorably with alternative cluster analysis methods. To illustrate its potential, we apply the proposed technique to several microarray data sets for cancer. For these data sets it successfully decomposes the data into known subtypes and indicates possible further taxonomic subdivision in addition to highlighting, in a wholly unsupervised manner, the importance of certain genes which are known to be medically significant. To illustrate its wider applicability, we also illustrate its performance on a microarray data set for yeast.

A Semiparametric Approach to Modeling Nonlinear Relations Among Latent Variables

Abstract: To date, finite mixtures of structural equation models (SEMMs) have been developed and applied almost exclusively for the purpose of providing model-based cluster analyses. This type of analysis constitutes a direct application of the model wherein the estimated component distributions of the latent classes are thought to represent the characteristics of distinct unobserved subgroups of the population. This article instead considers an indirect application of the SEMM in which the latent classes are estimated only in the service of more flexibly modeling the characteristics of the aggregate population as a whole. More specifically, the SEMM is used to semiparametrically model nonlinear latent variable regression functions. This approach is first developed analytically and then demonstrated empirically through analyses of simulated and real data.

Latent structure of depression in a community sample: a taxometric analysis.

Abstract: Background. The latent structure of depression was examined using taxometric analysis, a family of statistical procedures designed specifically to test whether a given construct is best conceptualized as a distinct category or a continuous dimension.Method. Data were derived from the Australian National Survey of Mental Health and Well-Being, a large epidemiological survey that measured the prevalence of the major DSM-IV and ICD-10 mental disorders. Two taxometric procedures, maximum covariance (MAXCOV) and mean above minus below a cut (MAMBAC), were carried out on a sample of 1933 community volunteers. Simulated categorical and dimensional datasets aided in the interpretation of the research data.Results. The results of the taxometric analyses in the subsample who endorsed at least one symptom of depression were consistent with a dimensional latent structure of depression.Conclusions. The findings of the current study suggest that depression, as measured in this subsample, is best conceptualized, measured and classified as a continuously distributed syndrome rather than as a discrete diagnostic entity. Incorporation of dimensional measurement into psychiatric classification systems remains a challenge for the future.

The longitudinal structure of the Children's Depression Inventory: testing a latent trait-state model.

Abstract: In a 6-wave longitudinal study, children (Grades 4-6, n = 648), adolescents (Grades 7-9, n = 1,489), and their parents completed child-adolescent or parent versions of the Children's Depression Inventory (CDI; M. Kovacs, 1981). Using structural equation modeling, the authors conducted latent trait-state analyses to distinguish between a stable trait dimension of depression (in which individual differences are stable over time) and an autoregressive dimension (in which individual differences are less stable over time). Children's CDIs reflected the autoregressive dimension more than a stable trait dimension, whereas parents' CDIs reflected a stable trait dimension more than an autoregressive dimension. Reports from adolescents and their parents reflected a stable trait dimension more than an autoregressive dimension of depressive symptoms. Results suggest that the longitudinal structure of the CDI varies considerably depending on the age of the target and the type of informant.

Dimensions and classes of psychosis in a population cohort: a four-class, four-dimension model of schizophrenia and affective psychoses

Abstract: Background. Classification of psychosis lacks a biological basis and current diagnostic categories may obscure underlying continuities. Data reduction methods of symptom profiles within a population-based cohort of people with a wide range of affective and non-affective psychoses may permit an empirical classification of psychosis. Method. OPCRIT (operational criteria) analysis was performed on 387 adults aged 18-65 years in an attempted ascertainment of all patients with psychosis from a geographical area with a stable population. The data were analysed firstly using principal components analysis with varimax rotation to identify factors, and secondly to establish latent classes. Information relating to key variables known to be of relevance in schizophrenia was coded blind to the establishment of the classes and dimensions. Results. Striking correspondence was obtained between the two methods. The four dimensions emerging were labelled 'depression', 'reality distortion', 'mania' and 'disorganization'. Latent classes identified were 'depression', 'bipolar', 'reality distortion/depression' and 'disorganization'. The latent classes corresponded well with DSM-III-R diagnoses, but also revealed groupings usually obscured by diagnostic boundaries. The latent classes differed on gender ratio, fertility, age of onset and self-harming behaviour, but not on substance misuse or season of birth. Conclusions. Both dimensional and categorical approaches are useful in tapping the latent constructs underlying psychosis. Broad agreement with other similar studies suggests such findings could represent discrete pathological conditions. The four classes described appear meaningful, and suggest that the term non-affective psychosis should be reserved for the disorganization class, which represents only a subgroup of those with schizophrenia.

Iterative Learning Control for Final Batch Product Quality Using Partial Least Squares Models

Abstract: A terminal iterative learning control (ILC) strategy for batch-to-batch and within-batch control of final product properties, based on empirical partial least squares (PLS) models, is presented. The strategy rejects persistent process disturbances and achieves new final product quality targets using an iterative procedure that works in the reduced space of a latent variable model rather than in the high dimensional space of the manipulated variable trajectories. Complete manipulated variable trajectory reconstruction is then achieved by exploiting the PLS model of the process. The approach is illustrated with a condensation polymerization example for the production of nylon.

Structural Equation Modeling: Categorical Variables

Abstract: In the behavioral sciences, response variables are often noncontinuous, common types being dichotomous, ordinal or nominal variables, counts and durations. Conventional structural equation models (SEMs) have thus been generalized to accommodate different kinds of responses. Keywords: structural equation model; categorical data; item response model; MIMIC model; generalized latent variable model

Using data augmentation to obtain standard errors and conduct hypothesis tests in latent class and latent transition analysis.

Abstract: Latent class analysis (LCA) provides a means of identifying a mixture of subgroups in a population measured by multiple categorical indicators. Latent transition analysis (LTA) is a type of LCA that facilitates addressing research questions concerning stage-sequential change over time in longitudinal data. Both approaches have been used with increasing frequency in the social sciences. The objective of this article is to illustrate data augmentation (DA), a Markov chain Monte Carlo procedure that can be used to obtain parameter estimates and standard errors for LCA and LTA models. By use of DA it is possible to construct hypothesis tests concerning not only standard model parameters but also combinations of parameters, affording tremendous flexibility. DA is demonstrated with an example involving tests of ethnic differences, gender differences, and an Ethnicity x Gender interaction in the development of adolescent problem behavior.

Conditions for the equivalence of the autoregressive latent trajectory model and a latent growth curve model with autoregressive disturbances

Abstract: Curran and Bollen combined two models for longitudinal panel data: the latent growth curve model and the autoregressive model. In their model, the autoregressive relationships are modeled between the observed variables. This is a different model than a latent growth curve model with autoregressive relationships between the disturbances. However when the autoregressive parameter p is invariant over time and lies between - 1 and 1, it can be shown that these models are algebraically equivalent. This result can be shown to generalize to the multivariate case. When the autoregressive parameters in the autoregressive latent trajectory model vary over time, the equivalence between the autoregressive latent trajectory model and a latent growth curve model with autoregressive disturbances no longer holds. However, a latent growth curve model with time-varying autoregressive parameters for the disturbances could be considered an interesting alternative to the autoregressive latent trajectory model with time-varying autoregressive parameters.