Showing papers on "Latent variable model published in 1993"

Bayesian analysis of binary and polychotomous response data

Abstract: A vast literature in statistics, biometrics, and econometrics is concerned with the analysis of binary and polychotomous response data. The classical approach fits a categorical response regression model using maximum likelihood, and inferences about the model are based on the associated asymptotic theory. The accuracy of classical confidence statements is questionable for small sample sizes. In this article, exact Bayesian methods for modeling categorical response data are developed using the idea of data augmentation. The general approach can be summarized as follows. The probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data. Values of the latent data can be simulated from suitable truncated normal distributions. If the latent data are known, then the posterior distribution of the parameters can be computed using standard results for normal linear models. Draws from this posterior are used to sample new latent data, and t...

The use of causal indicators in covariance structure models: some practical issues.

Abstract: In conventional representations of covariance structure models, indicators are defined as linear functions of latent variables, plus error. In an alternative representation, constructs can be defined as linear functions of their indicators, called causal indicators, plus an error term. Such constructs are not latent variables but composite variables, and they have no indicators in the conventional sense. The presence of composite variables in a model can, in some situations, result in problems with identification of model parameters. Also, the use of causal indicators can produce models that imply zero correlation among many measured variables, a problem resolved only by the inclusion of a potentially large number of additional parameters. These phenomena are demonstrated with an example, and general principles underlying them are discussed. Remedies are described so as to allow for the evaluation of models that contain causal indicators.

Loglinear Models with Latent Variables

Abstract: Introduction The Loglinear Model The Latent Class Model Loglinear Modeling with Latent Variables Internalizing External Variables Causal Models with Latent Variables A Modified LISREL Approach Latent Variable Models for Longitudinal Data Problems and New Developments

Modelling Covariances and Latent Variables Using EQS

Abstract: The Ideas of Covariance and Covariance Structure. Writing a Simple EQS Program. Statistical Modelling in EQS. Confirmatory Factor Analysis Models. Multitrait-Multimethod and Multiple Cause-Multiple Indicator Models. Models for Longitudinal Data. Simultaneous Analysis of Two or More Groups. Practical Problems.

Goodness-of-Fit Testing for Latent Class Models.

Abstract: Latent class models with sparse contingency tables can present problems for model comparison and selection, because under these conditions the distributions of goodness-of-fit indices are often unknown. This causes inaccuracies both in hypothesis testing and in model comparisons based on normed indices. In order to assess the extent of this problem, we carried out a simulation investigating the distributions of the likelihood ratio statistic G(2), the Pearson statistic ⊃(2), and a new goodness-of-fit index suggested by Read and Cressie (1988). There were substantial deviations between the expectation of the chi-squared distribution and the means of the G(2) and Read and Cressie distributions. In general, the mean of the distribution of a statistic was closer to the expectation of the chi-squared distribution when the average cell expectation was large, there were fewer indicator items, and the latent class measurement parameters were less extreme. It was found that the mean of the χ(2) distribution is generally closer to the expectation of the chi-squared distribution than are the means of the other two indices we examined, but the standard deviation of the χ(2) distribution is considerably larger than that of the other two indices and larger than the standard deviation of the chi-squared distribution. We argue that a possible solution is to forgo reliance on theoretical distributions for expectations and quantiles of goodness-of-fit statistics. Instead, Monte Carlo sampling (Noreen, 1989) can be used to arrive at an empirical central or noncentral distribution.

Multiple perspectives on family relationships: a latent variables model.

Abstract: Many scholars are skeptical of family member reports on their interpersonal relationships. Familial reports are assumed to be biased by social desirability as well as other factors. In this study, a latent variables modeling approach was employed to evaluate rater reliability and bias in mother, father, and child ratings of parent-child negativity. Results based on 78 clinical families demonstrate that family member ratings contain a significant "true score" component that correlates with observer ratings of parental behavior. The presence of systematic rater effects is also demonstrated. The latent variables approach, which provides statistical control for rater effects, is recommended for the analysis of this type of data.

A Hyperbolic Cosine Latent Trait Model For Unfolding Dichotomous Single-Stimulus Responses

Abstract: Social-psychological variables are typically measured using either cumulative or unfolding response processes. In the former, the greater the location of a person relative to the location of a stimulus on the continuum, the greater the proba bility of a positive response; in the latter, the closer the location of the person to the location of the statement, irrespective of direction, the greater the probability of a positive response. Formal probability models for these processes are, respec tively, monotonically increasing and single-peaked as a function of the location of the person relative to the location of the statement. In general, these models have been considered to be independent of each other. However, if statements constructed on the basis of a cumulative model have three ordered response categories, the response function within the statement for the middle category is in fact single-peaked. Using this observation, a unidimen sional model for responses to statements that have an unfolding structure was constructed from the cumulative Rasch model for ordered response categories. A location and unit of measurement parameter exist for each statement. A joint maxi mum likelihood estimation procedure was inves tigated. Analysis of a small simulation study and a small real dataset showed that the model is readily applicable.

The Asymptotic Posterior Normality of the Latent Trait in an IRT Model.

Abstract: It has long been part of the item response theory (IRT) folklore that under the usual empirical Bayes unidimensional IRT modeling approach, the posterior distribution of examinee ability given test response is approximately normal for a long test. Under very general and nonrestrictive nonparametric assumptions, we make this claim rigorous for a broad class of latent models.

Structured latent curve models

Abstract: Latent curve models are equivalent to factor analysis models in which common factor means are not assumed to be zero. The data model therefore generates a structure for the manifest variable mean vector as well as for the manifest variable covariance matrix. As in unrestricted factor analysis, there is a rotation problem in latent curve analysis. This problem may be avoided if a structure is imposed on the factor matrix. A method for doing this was employed in Browne and Du Toit (1991). The present paper presents theoretical details of this method for obtaining structured latent curve models for learning data. It is shown that use of a first order Taylor expansion about a monotonic target function generates a restricted factor matrix that has properties that allow meaningful interpretation of its columns as latent curves. Possible monotonic mean curves are discussed and details are given of associated factor matrices whose elements are functions of a small number of parameters. Models for the error covariance matrix are also considered. Joint latent curve and factor analysis models are suggested. These models are suitable for situations where both learning scores and scores on concomitant variables are available. A practical example is presented. Theory derived in Browne (1990) concerning the robustness of asymptotic properties of normal theory minimum discrepancy methods is applied to investigate the asymptotic robustness of maximum multivariate normal likelihood methods for the present models.

A latent trait finite mixture model for the analysis of rating agreement.

Abstract: This article presents a latent distribution model for the analysis of agreement on dichotomous or ordered category ratings. The model includes parameters that characterize bias, category definitions, and measurement error for each rater or test. Parameter estimates can be used to evaluate rater performance and to improve classification or measurement with use of multiple ratings. A simple maximum likelihood estimation procedure is described. Two examples illustrate the approach. Although considered in the context of analyzing rater agreement, the model provides a general approach for mixture analysis using two or more ordered-caregory measures.

Multitrait-Multimethod Analyses: Inferring Each Trait-Method Combination With Multiple Indicators

Abstract: The Campbell and Fiske (1959) guidelines are used extensively for examining multitrait-multimethod data. Although their logic and heuristic value are widely accepted, the guidelines are criticized for being based on correlations among observed variables instead of correlations among latent constructs. Therefore, researchers have developed latent variable models for multitrait- multimethod data that provide information more closely related to the Campbell-Fiske guidelines. In these latent variable approaches, a single scale score, which is often an average of multiple items used to infer the trait being measured, is typically used to represent each trait-method combination. When individual items (or item parcels) are used as multiple indicators of that scale, however, each trait-method combination can be represented as a latent construct, and the Campbell-Fiske guidelines can be applied to correlations among latent constructs, thereby removing a major objection to their use. Furthermore, latent variable ap...

A maximum likelihood method for latent class regression involving a censored dependent variable

Abstract: The standard tobit or censored regression model is typically utilized for regression analysis when the dependent variable is censored. This model is generalized by developing a conditional mixture, maximum likelihood method for latent class censored regression. The proposed method simultaneously estimates separate regression functions and subject membership in K latent classes or groups given a censored dependent variable for a cross-section of subjects. Maximum likelihood estimates are obtained using an EM algorithm. The proposed method is illustrated via a consumer psychology application.

Simultaneous Microeconometric Models with Censored or Qualitative Dependent Variables

Abstract: Publisher Summary This chapter defines two classes of simultaneous limited dependent variable regression models. The distinction between these two classes depends on whether the structural economic model is simultaneous in the latent or observed dependent variables. This distinction corresponds closely to whether or not the censoring mechanism itself acts as a constraint on individual agents' behavior. Type I models, which form the first class, are defined to be simultaneous in the underlying latent dependent variables. As a result, there exists an explicit and unique reduced form in the latent dependent variables under the usual identification conditions. In Type I models, individual behavior is completely described by the latent variable model and the censoring process simply acts as a constraint on the information available to the econometrician. Type II models form a general class in which the nonlinearity implicit in the censoring or discrete grouping process prevents an explicit solution for the reduced form. In Type II models, the observability rule also constrains the agent's choice set. For this second class of discrete or censored models a further coherency condition is required, this condition imposes restrictions that guarantee the existence of a unique but implicit reduced form for the observable endogenous variables. The chapter focuses on conditional maximum likelihood estimation. Estimation procedures may be applied across a wide variety of popular models and provide a useful basis for comparison and inference in such models.

Local homogeneity in latent trait models. A characterization of the homogeneous monotone IRT model

Abstract: The stochastic subject formulation of latent trait models contends that, within a given subject, the event of obtaining a certain response pattern may be probabilistic. Ordinary latent trait models do not imply that these within-subject probabilities are identical to the conditional probabilities specified by the model. The latter condition is called local homogeneity. It is shown that local homgeneity is equivalent to subpopulation invariance of the model. In case of the monotone IRT model, local homogeneity implies absence of item bias, absence of item specific traits, and the possibility to join overlapping subtests. The following characterization theorem is proved: the homogeneous monotone IRT model holds for a finite or countable item pool if and only if the pool is experimentally independent and pairwise nonnegative association holds in every positive subpopulation.

Modeling Social and Psychological Determinants of Exercise Behaviors via Structural Equation Systems

Abstract: Recent advances in structural modeling techniques allow for the testing of complex models representing social and behavioral processes. However, most reported applications in sport and physical activity have been limited to simple models involving variables measured at a single point in time. Therefore, the purpose of this article is to demonstrate the use of both cross-sectional and longitudinal latent variable modeling techniques by examining the relationships among efficacy cognitions, social support, and the exercise behaviors of sedentary adults. Results revealed a good fit for the respecified model, suggesting the existence of a lagged feedback mechanism in which exercise behaviors influenced residual change in social support. In turn, efficacy cognitions appeared to serve a mediational function in the synchronous relationship between social support and exercise behavior. Findings are discussed in terms of the utility of structural equation modeling techniques for understanding the complex ...

Analysis of contingency tables by correspondence models subject to order constraints

Abstract: Inferential correspondence analysis, which has gained much attention in recent years, is applied here to contingency tables with ordered categories. To reflect such order, the parameters of the underlying correspondence models are constrained to follow the order induced by the categories of the analyzed table. A reparameterization of the correspondence model in terms of a latent variable model is presented. This allows a simple and straightforward use of the EM algorithm to obtain efficient order-restricted estimates. A goodness-of-fit test is also discussed, and an example is analyzed. A small Monte Carlo example is presented.

A latent class unfolding model for analyzing single stimulus preference ratings

Abstract: A multidimensional unfolding model is developed that assumes that the subjects can be clustered into a small number of homogeneous groups or classes. The subjects that belong to the same group are represented by a single ideal point. Since it is not known in advance to which group of class a subject belongs, a mixture distribution model is formulated that can be considered as a latent class model for continuous single stimulus preference ratings. A GEM algorithm is described for estimating the parameters in the model. The M-step of the algorithm is based on a majorization procedure for updating the estimates of the spatial model parameters. A strategy for selecting the appropriate number of classes and the appropriate number of dimensions is proposed and fully illustrated on some artificial data. The latent class unfolding model is applied to political science data concerning party preferences from members of the Dutch Parliament. Finally, some possible extensions of the model are discussed.

An approximation for the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete

Abstract: Lord developed an approximation for the bias function for the maximum likelihood estimate in the context of the three-parameter logistic model. Using Taylor's expansion of the likelihood equation, he obtained an equation that includes the conditional expectation, given true ability, of the discrepancy between the maximum likelihood estimate and true ability. All terms of orders higher thann−1 are ignored wheren indicates the number of items. Lord assumed that all item and individual parameters are bounded, all item parameters are known or well-estimated, and the number of items is reasonably large. In the present paper, an approximation for the bias function of the maximum likelihood estimate of the latent trait, or ability, will be developed using the same assumptions for the more general case where item responses are discrete. This will include the dichotomous response level, for which the three-parameter logistic model has been discussed, the graded response level and the nominal response level. Some observations will be made for both dichotomous and graded response levels.

The Bias Function of the Maximum Likelihood Estimate of Ability for the Dichotomous Response Level.

Abstract: Samejima has recently given an approximation for the bias function for the maximum likelihood estimate of the latent trait in the general case where item responses are discrete, generalizing Lord's bias function in the three-parameter logistic model for the dichotomous response level. In the present paper, observations are made about the behavior of this bias function for the dichotomous response level in general, and also with respect to several widely used mathematical models. Some empirical examples are given.

Latent variable modeling of growth with missing data and multilevel data

Abstract: Latent variable modeling in psychometrics is connected with mainstream statistical theory in the areas of random coefficients, missing data, and clustered data. An educational achievement example points to the need for integrating these developments in a single analysis framework. It is shown that existing methods and software for latent variable modeling accomplish this.

Should “multiple imputations” be treated as “multiple indicators”?

Abstract: Rubin's “multiple imputation” approach to missing data creates synthetic data sets, in which each missing variable is replaced by a draw from its predictive distribution, conditional on the observed data. By construction, analyses of such filled-in data sets as if the imputations were true values have the correct expectations for population parameters. In a recent paper, Mislevy showed how this approach can be applied to estimate the distributions of latent variables from complex samples. Multiple imputations for a latent variable bear a surface similarity to classical “multiple indicators” of a latent variable, as might be addressed in structural equation modelling or hierarchical modelling of successive stages of random sampling. This note demonstrates with a simple example why analyzing “multiple imputations” as if they were “multiple indicators” does not generally yield correct results; they must instead be analyzed by means concordant with their construction.

Linking the Standard and Advanced forms of the Raven'S Progressive Matrices in both the Pencil-And-Paper and Computer-Adaptive-Testing Formats

Abstract: The concepts of adaptive testing, already used by Binet, and concepts of modern or latent trait theory, already extended from psychophysics by Thurstone, have been brought together by the advent of the computer, particularly the PC. With these developments, it is likely that to test specific abilities, special tests will be constructed for administration only by a computer and not in pencil and paper format. In the meantime, many existing paper-and-pencil tests, which usually are not administered adaptively, are being computerised. Among important tests in the educational and psychological literature, and therefore candidates for computerised administration, are the Raven's Progressive Matrices (RPM). Before some inferences can be made across different modes of presentation, however, it is necessary that the degree of consistency of the responses across the two modes be evaluated in a variety of circumstances and by a variety of processes. This paper describes use of the Rasch latent trait model to help (i) implement computerised administration of the standard and advanced forms of the RPM, (ii) compare the relative item difficulties of the computerised form with those from a pencil-and-paper administration of the same items to a different group of persons, and (iii) to convert scores between the advanced and standard forms of the RPM using the two modes of testing, and to compare these scores with the results of a traditional method of equating reported in the literature.

Invariant Standardized Estimated Parameter Change for Model Modification in Covariance Structure Analysis.

Abstract: An estimated parameter change (EPC) has recently been introduced as another criterion to be considered in the process of model modification in covariance structure analysis. Kaplan (1989) provided a standardized version of this statistic (SEPC-K). It has been found that SEPC-K is only partially standardized; specifically, it is not invariant under different scalings of latent and measured variables. In this article, a new SEPC that is invariant to the original metrics of the measured and latent variables is suggested for use in model modification. A multivariate estimated parameter change (MEPC) which estimates changes for a set of fixed parameters to be freed simultaneously is also introduced. A standardized MEPC (SMEPC) is, furthermore, provided. Because there are now three different types of standardized solutions in structural modeling programs, general discussion of standardized solution in covariance structure analysis is provided. The inappropriate use of standardization for scale-specific models i...

Regression analysis of multilevel data with measurement error

Abstract: Multilevel data structures, such as data with individuals within groups, groups within areas, etc., arise in a variety of contexts. Regression models for such data allow for variation of the within-group regression coefficients. Since some of the explanatory variables may be observed subject to measurement error, it is desirable to define models that combine within-group correlation and measurement error. We discuss a general model formulation and a scoring algorithm for maximum likelihood estimation.

A latent class vector model for preference ratings

Abstract: A latent class formulation of the well-known vector model for preference data is presented. Assuming preference ratings as input data, the model simultaneously clusters the subjects into a small number of homogeneous groups (or latent classes) and constructs a joint geometric representation of the choice objects and the latent classes according to a vector model. The distributional assumptions on which the latent class approach is based are analogous to the distributional assumptions that are consistent with the common practice of fitting the vector model to preference data by least squares methods. An EM algorithm for fitting the latent class vector model is described as well as a procedure for selecting the appropriate number of classes and the appropriate number of dimensions. Some illustrative applications of the latent class vector model are presented and some possible extensions are discussed.

Estimating latent distributions in recurrent choice data

Abstract: This paper introduces a flexible class of stochastic mixture models for the analysis and interpretation of individual differences in recurrent choice and other types of count data. These choice models are derived by specifying elements of the choice process at the individual level. Probability distributions are introduced to describe variations in the choice process among individuals and to obtain a representation of the aggregate choice behavior. Due to the explicit consideration of random effect sources, the choice models are parsimonious and readily interpretable. An easy to implement EM algorithm is presented for parameter estimation. Two applications illustrate the proposed approach.

Multiple dimensions of affective and cognitive disturbance: Latent-variable models in a community sample.

Abstract: A clear understanding and confirmation of the structure of psychological distress has been hampered by different theoretical perspectives, ranges of measures, and methodologies. This study examined the latent construct structure of psychological distress as reflected in 27 self-report measures of psychological functioning from a community sample of 614 young adults

Latent Variables in Log-Linear Models of Repeated Observations

Abstract: .After a very brief introduction into log-linear modeling and an explanation of the latent class model as a log-linear model with latent variables, it is shown that even a small amount of unreliability may lead to quite misleading conclusions about the underlying processes of change .' Next, measurement models for indicators measured at several points in time are discussed . The main purpose of this discussion is to show how to disentangle 'true', latent changes and observed changes caused by unreliability of measure ments A main topic is the causal analysis of panel data . Extending Goodman's loglinear 'modified path analysis approach' to include latent variables, it is shown how to set up 'modified LISREL models' for the analyses of cross-sectional and longitudinal data . Finally, some other possible applications of the (categorical) latent variable approach are pointed out, along with some important new developments