Showing papers on "Latent variable model published in 1997"

Basics of structural equation modeling

Abstract: PART ONE: BACKGROUND What Does It Mean to Model Hypothesized Causal Processes with Nonexperimental Data? History and Logic of Structural Equation Modeling PART TWO: BASIC APPROACHES TO MODELING WITH SINGLE OBSERVED MEASURES OF THEORETICAL VARIABLES The Basics Path Analysis and Partitioning of Variance Effects of Collinearity on Regression and Path Analysis Effects of Random and Nonrandom Error on Path Models Recursive and Longitudinal Models Where Causality Goes in More Than One Direction and Where Data Are Collected Over Time PART THREE: FACTOR ANALYSIS AND PATH MODELING Introducing the Logic of Factor Analysis and Multiple Indicators to Path Modeling PART FOUR: LATENT VARIABLE STRUCTURAL EQUATION MODELS Putting It All Together Latent Variable Structural Equation Modeling Using Latent Variable Structural Equation Modeling to Examine Plausability of Models Logic of Alternative Models and Significance Tests Variations on the Basic Latent Variable Structural Equation Model Wrapping up

General Longitudinal Modeling of Individual Differences in Experimental Designs: A Latent Variable Framework for Analysis and Power Estimation

Abstract: The generality of latent variable modeling of individual differences in development over time is demonstrated with a particular emphasis on randomized intervention studies. First, a brief overview is given of biostatistica l and psychometric approaches to repeated measures analysis. Second, the generality of the psychometric approach is indicated by some nonstandard models. Third, a multiple-population analysis approach is proposed for the estimation of treatment effects. The approach clearly describes the treatment effect as development that differs from normative, control-group development. This framework allows for interactions between treatment and initial status in their effects on development. Finally, an approach for the estimation of power to detect treatment effects in this framework is demonstrated. Illustrations of power calculations are carried out with artificial data, varying the sample sizes, number of timepoints, and treatment effect sizes. Real data are used to illustrate analysis strategies and power calculations. Further modeling extensions are discussed.

Finite-Mixture Structural Equation Models for Response-Based Segmentation and Unobserved Heterogeneity

Abstract: Two endemic problems face researchers in the social sciences e.g., Marketing, Economics, Psychology, and Finance: unobserved heterogeneity and measurement error in data. Structural equation modeling is a powerful tool for dealing with these difficulties using a simultaneous equation framework with unobserved constructs and manifest indicators which are error-prone. When estimating structural equation models, however, researchers frequently treat the data as if they were collected from a single population Muthen [Muthen, Bengt O. 1989. Latent variable modeling in heterogeneous populations. Psychometrika54 557--585.]. This assumption of homogeneity is often unrealistic. For example, in multidimensional expectancy value models, consumers from different market segments can have different belief structures Bagozzi [Bagozzi, Richard P. 1982. A field investigation of causal relations among cognitions, affect, intentions, and behavior. J. Marketing Res.19 562--584.]. Research in satisfaction suggests that consumer decision processes vary across segments Day [Day, Ralph L. 1977. Extending the concept of consumer satisfaction. W. D. Perreault, ed. Advances in Consumer Research, Vol. 4. Association for Consumer Research, Atlanta, 149--154.]. This paper shows that aggregate analysis which ignores heterogeneity in structural equation models produces misleading results and that traditional fit statistics are not useful for detecting unobserved heterogeneity in the data. Furthermore, sequential analyses that first form groups using cluster analysis and then apply multigroup structural equation modeling are not satisfactory. We develop a general finite mixture structural equation model that simultaneously treats heterogeneity and forms market segments in the context of a specified model structure where all the observed variables are measured with error. The model is considerably more general than cluster analysis, multigroup confirmatory factor analysis, and multigroup structural equation modeling. In particular, the model subsumes several specialized models including finite mixture simultaneous equation models, finite mixture confirmatory factor analysis, and finite mixture second-order factor analysis. The finite mixture structural equation model should be of interest to academics in a wide range of disciplines e.g., Consumer Behavior, Marketing, Economics, Finance, Psychology, and Sociology where unobserved heterogeneity and measurement error are problematic. In addition, the model should be of interest to market researchers and product managers for two reasons. First, the model allows the manager to perform response-based segmentation using a consumer decision process model, while explicitly allowing for both measurement and structural error. Second, the model allows managers to detect unobserved moderating factors which account for heterogeneity. Once managers have identified the moderating factors, they can link segment membership to observable individual-level characteristics e.g., socioeconomic and demographic variables and improve marketing policy. We applied the finite mixture structural equation model to a direct marketing study of customer satisfaction and estimated a large model with 8 unobserved constructs and 23 manifest indicators. The results show that there are three consumer segments that vary considerably in terms of the importance they attach to the various dimensions of satisfaction. In contrast, aggregate analysis is misleading because it incorrectly suggests that except for price all dimensions of satisfaction are significant for all consumers. Methodologically, the finite mixture model is robust; that is, the parameter estimates are stable under double cross-validation and the method can be used to test large models. Furthermore, the double cross-validation results show that the finite mixture model is superior to sequential data analysis strategies in terms of goodness-of-fit and interpretability. We performed four simulation experiments to test the robustness of the algorithm using both recursive and nonrecursive model specifications. Specifically, we examined the robustness of different model selection criteria e.g., CAIC, BIC, and GFI in choosing the correct number of clusters for exactly identified and overidentified models assuming that the distributional form is correctly specified. We also examined the effect of distributional misspecification i.e., departures from multivariate normality on model performance. The results show that when the data are heterogeneous, the standard goodness-of-fit statistics for the aggregate model are not useful for detecting heterogeneity. Furthermore, parameter recovery is poor. For the finite mixture model, however, the BIC and CAIC criteria perform well in detecting heterogeneity and in identifying the true number of segments. In particular, parameter recovery for both the measurement and structural models is highly satisfactory. The finite mixture method is robust to distributional misspecification; in addition, the method significantly outperforms aggregate and sequential data analysis methods when the form of heterogeneity is misspecified i.e., the true model has random coefficients. Researchers and practitioners should only use the mixture methodology when substantive theory supports the structural equation model, a priori segmentation is infeasible, and theory suggests that the data are heterogeneous and belong to a finite number of unobserved groups. We expect these conditions to hold in many social science applications and, in particular, market segmentation studies. Future research should focus on large-scale simulation studies to test the structural equation mixture model using a wide range of models and statistical distributions. Theoretical research should extend the model by allowing the mixing proportions to depend on prior information and/or subject-specific variables. Finally, in order to provide a fuller treatment of heterogeneity, we need to develop a general random coefficient structural equation model. Such a model is presently unavailable in the statistical and psychometric literatures.

Latent Variable Modeling of Longitudinal and Multilevel Data

Abstract: An overview is given of modeling of longitudinal and multilevel data using a latent variable framework. Particular emphasis is placed on growth modeling. A latent variable model is presented for three-level data, where the modeling of the longitudinal part of the data imposes both a covariance and a mean structure. Examples are discussed where repeated observations are made on students sampled within classrooms and schools.

Latent Variable Models for Mixed Discrete and Continuous Outcomes

Abstract: We propose a latent variable model for mixed discrete and continuous outcomes. The model accommodates any mixture of outcomes from an exponential family and allows for arbitrary covariate effects, as well as direct modelling of covariates on the latent variable. An EM algorithm is proposed for parameter estimation and estimates of the latent variables are produced as a by-product of the analysis. A generalized likelihood ratio test can be used to test the significance of covariates affecting the latent outcomes. This method is applied to birth defects data, where the outcomes of interest are continuous measures of size and binary indicators of minor physical anomalies. Infants who were exposed in utero to anticonvulsant medications are compared with controls.

Multilevel item response models: An approach to errors in variables regression

Abstract: In this article we show how certain analytic problems that arise when one attempts to use latent variables as outcomes in regression analyses can be addressed by taking a multilevel perspective on item response modeling. Under a multilevel, or hierarchical, perspective we cast the item response model as a within-student model and the student population distribution as a between-student model. Taking this perspective leads naturally to an extension of the student population model to include a range of studentlevel variables, and it invites the possibility of further extending the models to additional levels so that multilevel models can be applied with latent outcome variables. In the two-level case, the model that we employ is formally equivalent to the plausible value procedures that are used as part of the National Assessment of Educational Progress (NAEP), but we present the method for a different class of measurement models, and we use a simultaneous estimation method rather than two-step estimation. In our application of the models to the appropriate treatment of measurement error in the dependent variable of a between-student regression, we also illustrate the adequacy of some approximate procedures that are used in NAEP.

Model Selection Information Criteria for Non-Nested Latent Class Models

Abstract: Latent class models have been developed for assessment of hierarchic relations in scaling and behavioral analysis. This article investigated the use of three model selection information criteria-Akaike AIC, Schwarz SIC, and Bozdogan CAIC-for non-nested models. In general, SIC and CAIC were superior to AIC for relatively simple models, whereas AIC was superior for more complex models, although accuracy was often quite low for such models. In addition, some effects were detected for error rates in the models. Latent class analysis (LCA) has found important applications in the behavioral sciences as a method for studying hierarchic relations among responses to sets of categorical variables. Typical applications include analysis of attitude items designed to represent Guttman scales, assessment of linear prerequisite relations among learning tasks in mathematics education, and assessment of developmental stages in concept learning. Modern estimation procedures for hierarchic models of the types introduced by Lazarsfeld (1950) and Lazarsfeld and Henry (1968) were first provided by Proctor (1970). Proctor postulated a Guttman scale with each respondent belonging to one latent, true scale type.

A Generalized Partial Credit Model

Abstract: A generalized partial credit model (GPCM) was formulated by Muraki (1992) based on Masters’ (1982, this volume) partial credit model (PCM) by relaxing the assumption of uniform discriminating power of test items. However, the difference between these models is not only the parameterization of item characteristics but also the basic assumption about the latent variable. An item response model is viewed here as a member of a family of latent variable models which also includes the linear or nonlinear factor analysis model, the latent class model, and the latent profile model (Bartholomew, 1987).

Expanding test–retest designs to include developmental time-lag components.

Abstract: Test-retest data can reflect systematic changes over varying intervals of time in a "time-lag" design. This article shows how latent growth models with planned incomplete data can be used to separate psychometric components of developmental interest, including internal consistency reliability, test-practice effects, factor stability, factor growth, and state fluctuation. Practical analyses are proposed using a structural equation model for longitudinal data on multiple groups with different test-retest intervals. This approach is illustrated using 2 sets of data collected from students measured on the Woodcock-Johnson—Revised Memory and Reading scales. The results show how alternative time-lag models can be fitted and interpreted with univariate, bivariate, and multivariate data. Benefits, limitations, and extensions of this structural time-lag approach are discussed.

Latent Variable Modeling of Diagnostic Accuracy

Abstract: Latent class analysis has been applied in medical research to assessing the sensitivity and specificity of diagnostic tests/diagnosticians. In these applications, a dichotomous latent variable corresponding to the unobserved true disease status of the patients is assumed. Associations among multiple diagnostic tests are attributed to the unobserved heterogeneity induced by the latent variable, and inferences for the sensitivities and specificities of the diagnostic tests are made possible even though the true disease status is unknown. However, a shortcoming of this approach to analyses of diagnostic tests is that the standard assumption of conditional independence among the diagnostic tests given a latent class is contraindicated by the data in some applications. In the present paper, models incorporating dependence among the diagnostic tests given a latent class are proposed. The models are parameterized so that the sensitivities and specificities of the diagnostic tests are simple functions of model parameters, and the usual latent class model obtains as a special case. Marginal models are used to account for the dependencies within each latent class. An accelerated EM gradient algorithm is demonstrated to obtain maximum likelihood estimates of the parameters of interest, as well as estimates of the precision of the estimates.

Latent Variable Modeling and Applications to Causality

Abstract: Causality and Path Models- Embedding Common factors in a Path Model- Measurement, Causation and Local Independence in Latent Variable Models- On the Identifiability of Nonparametric Structural Models- Estimating the Causal effects of Time Varying Endogeneous Treatments by G-Estimation of Structural Nested Models- Latent Variables- Model as Instruments, with Applications to Moment Structure Analysis- Bias and Mean Square Error of the Maximum Likelihood Estimators of the Parameters of the Intraclass Correlation Model- Latent Variable Growth Modeling with Multilevel Data- High-Dimensional Full-Information Item Factor Analysis- Dynamic Factor Models for the Analysis of Ordered Categorical Panel data- Model Fitting Procedures for Nonlinear Factor Analysis Using the Errors-in-Variables Parameterization- Multivariate Regression with Errors in Variables: Issues on Asymptotic Robustness- Non-Iterative fitting of the Direct Product Model for Multitrait-Multimethod Correlation Matrices- An EM Algorithm for ML Factor Analysis with Missing Data- Optimal Conditionally Unbiased Equivariant Factor Score Estimators

Latent Variable Modeling of Longitudinal and Multilevel Substance Use Data

Abstract: This article demonstrates the use of a general model for latent variable growth analysis which takes into account cluster sampling. Multilevel Latent Growth Modeling (MLGR4) was used to analyze longitudinal and multilevel data for adolescent and parent substance use measured at four annual time points. An associative LGM model was tested for alcohol, marijuana, and cigarette use with a sample of 435 families. Hypotheses concerning the shape of the growth curve and the extent of individual differences in the common trajectory over time were tested. The effects of marital and family status and socio-economic status on family levels of substance use were also examined. Findings are discussed in terms of family-level substance use and similarities in developmental trajectories across substances, and the impact of contextual factors on family levels of substance use and development.

Latent variable models of functional somatic distress.

Abstract: Latent variable models of functional somatic symptoms were estimated for a sample of 686 family medicine patients. Symptom items from the NIMH Diagnostic Interview Schedule were selected to approximate diagnoses of fibromyalgia syndrome (FMS), chronic fatigue syndrome (CFS), and irritable bowel syndrome (IBS). Confirmatory factor analysis demonstrated that hypothesized latent variables of somatic depression, somatic anxiety, FM-like, CF-like, and IB-like syndromes fit the observed covariations better than models hypothesizing fewer latent variables. Results offer tentative confirmation of functional somatic syndromes as discrete entities and suggest that relaxing the diagnostic criteria for somatization may identify individuals with distress limited to a single functional system.

A multidimensional item response model: Constrained latent class analysis using the gibbs sampler and posterior predictive checks

Abstract: In this paper it will be shown that a certain class of constrained latent class models may be interpreted as a special case of nonparametric multidimensional item response models. The parameters of this latent class model will be estimated using an application of the Gibbs sampler. It will be illustrated that the Gibbs sampler is an excellent tool if inequality constraints have to be taken into consideration when making inferences. Model fit will be investigated using posterior predictive checks. Checks for manifest monotonicity, the agreement between the observed and expected conditional association structure, marginal local homogeneity, and the number of latent classes will be presented.

INLR, implicit non-linear latent variable regression

Abstract: A simple way to develop non-linear PLS models is presented, INLR (implicit non-linear latent variable regression). The paper shows that by simply added squared x-variables x2a, both the square and cross terms of the latent variables are implicitly included in the resulting PLS model. This approach works when X itself is well modelled by a projection model T*PT. Hence, if a latent structure is present in X, it is not necessary to include the cross terms of the X-variables in the polynomial expansion. Analogously, with cubic non-linearities, expanding X with cubic terms x3a is sufficient. INLR is attractive in that all essential features of PLS are preserved i.e. (a) it can handle many noisy and collinear variables, (b) it is stable and gives reliable results and (c) all PLS plots and diagnostics still apply. The principles of INLR are outlined and illustrated with three chemical examples where INLR improved the modelling and predictions compared with ordinary linear PLS. © 1997 John Wiley & Sons, Ltd.

Semiparametric Estimation of Location and Other Discrete Choice Moments

Abstract: Latent variable discrete choice model estimation and interpretation depend on the density function of the latent variable's unobserved random component. This paper provides a simple semiparametric estimator of the moments of this density. The results can be used as starting values for parametric estimators, to estimate the appropriate location and scaling for semiparametric estimators, for specification testing including tests of latent error skewness and kurtosis, and to estimate coefficients of discrete explanatory variables in the model.

Latent Variable Growth Modeling with Multilevel Data

Abstract: Growth modeling of multilevel data is presented within a latent variable framework that allows analysis with conventional structural equation modeling software. Latent variable modeling of growth considers a vector of observations over time for an individual, reducing the two-level problem to a one-level problem Analogous to this, three-level data on students, time points, and schools can be modeled by a two-level growth model. An interesting feature of this two-level model is that contrary to recent applications of multilevel latent variable modeling, a mean structure is imposed in addition to the covariance structure. An example using educational achievement data illustrates the methodology.

Bivariate modelling of clustered continuous and ordered categorical outcomes.

Abstract: Simultaneous observation of continuous and ordered categorical outcomes for each subject is common in biomedical research but multivariate analysis of the data is complicated by the multiple data types. Here we construct a model for the joint distribution of bivariate continuous and ordinal outcomes by applying the concept of latent variables to a multivariate normal distribution. The approach is then extended to allow for clustering of the bivariate outcomes. The model can be parameterized in a way that allows writing the joint distribution as a product of a standard random effects model for the continuous variable and a correlated cumulative probit model for the ordinal outcome. This factorization suggests convenient parameter estimation using estimating equations. Foetal weight and malformation data from a developmental toxicity experiment illustrate the results.

Latent Variable Models for Teratogenesis Using Multiple Binary Outcomes

Abstract: Multiple outcomes are commonly measured in the study of birth defects. The reason is that most teratogens do not cause a single, uniquely defined defect, but rather result in a range of effects, including major malformations, minor anomalies, and deficiencies in birth weight, length and head circumference. The spectrum of effects associated with a particular teratogen is sometimes described as a “syndrome.” In this article we develop a latent variable model to characterize exposure effects on multiple binary outcomes. Not only does the method allow comparisons of control and exposed infants with respect to multiple outcomes, but it also provides a measure of the “severity” of each child's condition. Data from a study of the teratogenic effects of anticonvulsants illustrate our results.

A characterization of monotone unidimensional latent variable models

Abstract: Recently, the problem of characterizing monotone unidimensional latent variable models for binary repeated measures was studied by Ellis and van den Wollenberg and by Junker. We generalize their work with a de Finetti-like characterization of the distribution of repeated measures $\rm X = (X_1, X_2, \dots)$ that can be represented with mixtures of likelihoods of independent but not identically distributed random variables, where the data satisfy a stochastic ordering property with respect to the mixing variable. The random variables $X_j$ may be arbitrary real-valued random variables. We show that the distribution of X can be given a monotone unidimensional latent variable representation that is useful in the sense of Junker if and only if this distribution satisfies conditional association (CA) and a vanishing conditional dependence (VCD) condition, which asserts that finite subsets of the variables in X become independent as we condition on a larger and larger segment of the remaining variables in X. It is also interesting that the mixture representation is in a certain ordinal sense unique, when CA and VCD hold. The characterization theorem extends and simplifies the main result of Junker and generalizes methods of Ellis and van den Wollenberg to a much broader class of models. Exchangeable sequences of binary random variables also satisfy both CA and VCD, as do exchangeable sequences arising as location mixtures. In the same way that de Finetti's theorem provides a path toward justifying standard i.i.d.-mixture components in hierarchical models on the basis of our intuitions about the exchangeability of observations, this theorem justifies one-dimensional latent variable components in hierarchical models, in terms of our intuitions about positive association and redundancy between observations. Because these conditions are on the joint distribution of the observable data X, they may also be used to construct asymptotically power-1 tests for unidimensional latent variable models.

Tail-Measurability in Monotone Latent Variable Models.

Abstract: We consider latent variable models for an infinite sequence (or universe) of manifest (observable) variables that may be discrete, continuous or some combination of these. The main theorem is a general characterization by empirical conditions of when it is possible to construct latent variable models that satisfy unidimensionality, monotonicity, conditional independence, andtail-measurability. Tail-measurability means that the latent variable can be estimated consistently from the sequence of manifest variables even though an arbitrary finite subsequence has been removed. The characterizing,necessary and sufficient, conditions that the manifest variables must satisfy for these models are conditional association and vanishing conditional dependence (as one conditions upon successively more other manifest variables). Our main theorem considerably generalizes and sharpens earlier results of Ellis and van den Wollenberg (1993), Holland and Rosenbaum (1986), and Junker (1993). It is also related to the work of Stout (1990). The main theorem is preceded by many results for latent variable modelsin general—not necessarily unidimensional and monotone. They pertain to the uniqueness of latent variables and are connected with the conditional independence theorem of Suppes and Zanotti (1981). We discuss new definitions of the concepts of “true-score” and “subpopulation,” which generalize these notions from the “stochastic subject,” “random sampling,” and “domain sampling” formulations of latent variable models (e.g., Holland, 1990; Lord & Novick, 1968). These definitions do not require the a priori specification of a latent variable model.

Rasch's model for reading speed with manifest explanatory variables

Abstract: In educational and psychological measurement we find the distinction between speed and power tests. Although most tests are partially speeded, the speed element is usually neglected. Here we consider a latent trait model developed by Rasch for the response time on a (set of) pure speed test(s), which is based on the assumption that the test response times are approximately gamma distributed, with known shape parameters and scale parameters depending on subject “ability” and test “difficulty” parameters. In our approach the subject parameters are treated as random variables having a common gamma distribution. From this, maximum marginal likelihood estimators are derived for the test difficulties and the parameters of the latent subject distribution. This basic model can be extended in a number of ways. Explanatory variables for the latent subject parameters and for the test parameters can be incorporated in the model. Our methods are illustrated by the analysis of a simulated and an empirical data set.

Business Conditions and Speculative Assets

Abstract: In this paper we examine the hypothesis that the predictable components of U.K. shares and bonds are related to business conditions. Financial market variables, such as maturity and default premia, are constructed in an attempt to capture different components of business-conditions risk. The hypothesis is investigated using multivariate regression analysis and a latent variable model. One of the main conclusions reached in this paper is that the time-varying component of U.K. share and bond excess returns tends to exhibit varying degrees of sensitivity to information on business conditions as captured ex ante by a number of financial variables.

System Identification by Dynamic Factor Models

Abstract: This paper concerns the modeling of stochastic processes by means of dynamic factor models. In such models the observed process is decomposed into a structured part called the latent process and a remainder that is called noise. The observed variables are treated in a symmetric way so that no distinction between inputs and outputs is required. This motivates the additional condition that the prior assumptions on the noise are symmetric in nature. One of the central questions in this paper is how uncertainty about the noise structure translates into nonuniqueness of the possible underlying latent processes. We investigate several possible noise specifications and analyze properties of the resulting class of observationally equivalent factor models. This concerns in particular the characterization of optimal models and properties of continuity and consistency.

Comparative social research with multi-sample latent class models

Abstract: Latent class analysis provides a powerful, flexible approach to the analysis of categoricallyscored data. Perhaps the most widespread use of the latent class model (LCM) is as a categorical data analog to factor analysis. For example, LCMs are often used as a data reduction technique to examine the latent structure of a joint distribution of a set of indicator items, often with an interest in „building clusters for qualitative data“ (Formann 1985, p.87) or as an explicit scaling and measurement model to investigate the relations between the categorical indicators and the underlying categorical latent variable(s) which the indicators are intended to measure (see e.g., Clogg and Sawyer, 1981; Croon, 1990; Heinen, 1996).

A Simultaneous Probit Model

Abstract: The usual formulation of probit models includes, as endogenous variables, both continuous latent variables and binary observable variables. The solution of simultaneous probit models involving only latent endogenous variables among the explanatory variables is straightforward, provided the equations are identified. In contrast, simultaneous probit models in which the binary endogenous variables appear among the explanatory variables raise problems of coherence.

A program implementing Mardia's multivariate normality test for use in structural equation modeling with latent variables

Abstract: A FORTRAN 77 program is presented that tests assumptions of multivariate normality in a data set Violation of the normality assumption, especially excessive kurtosis, suggests that alternative estimation techniques other than maximum likelihood should be used. Even if all the univariate distributions are normal, the joint distribution may depart substantially from multivariate normality. Consequently, testing variables individually may not be sufficient. This program is of use to those engaged in structural equation modeling with latent variables.

Simultaneous study of individual and group patterns of latent longitudinal change using structural equation modeling

Abstract: This article utilizes structural equation modeling for purposes of simultaneous study of individual and group latent change patterns on several longitudinally assessed variables The approach is based on a special case of the comprehensive latent curve analysis by Meredith and Tisak (1990) Substantively interesting aspects of individual and group growth curves, as well as the interrelations among their patterns, are parameterized at the latent ability level The method is illustrated on data from a two‐group study by Baltes, Dittmann‐Kohli, and Kliegl (1986)