Rating scale analysis with latent class models

A general approach for analyzing rating data with latent class models is described, which parallels rating models in the framework of latent trait theory. A general rating model as well as a two-parameter model with location and dispersion parameters, analogous to Andrich's Dislocmodel are derived, including parameter estimation via the EM-algorithm. Two examples illustrate the application of the models and their statisticalcontrol. Model restrictions through equality constrains are discussed and multiparameter generalizations are outlined.     

Higher-order latent trait models for cognitive diagnosis

Higher-order latent traits are proposed for specifying the joint distribution of binary attributes in models for cognitive diagnosis. This approach results in a parsimonious model for the joint distribution of a high-dimensional attribute vector that is natural in many situations when specific cognitive information is sought but a less informative item response model would be a reasonable alternative. This approach stems from viewing the attributes as the specific knowledge required for examination performance, and modeling these attributes as arising from a broadly-defined latent trait resembling theϑ of item response models. In this way a relatively simple model for the joint distribution of the attributes results, which is based on a plausible model for the relationship between general aptitude and specific knowledge. Markov chain Monte Carlo algorithms for parameter estimation are given for selected response distributions, and simulation results are presented to examine the performance of the algorithm as well as the sensitivity of classification to model misspecification. An analysis of fraction subtraction data is provided as an example. This research was funded by National Institute of Health grant R01 CA81068. We would like to thank William Stout and Sarah Hartz for many useful discussions, three anonymous reviewers for helpful comments and suggestions, and Kikumi Tatsuoka and Curtis Tatsuoka for generously sharing data.     

Constrained latent class analysis: Simultaneous classification and scaling of discrete choice data

A reparameterization of a latent class model is presented to simultaneously classify and scale nominal and ordered categorical choice data. Latent class-specific probabilities are constrained to be equal to the preference probabilities from a probabilistic ideal-point or vector model that yields a graphical, multidimensional representation of the classification results. In addition, background variables can be incorporated as an aid to interpreting the latent class-specific response probabilities. The analyses of synthetic and real data sets illustrate the proposed method.The authors thank Yosiho Takane, the editor and referees for their valuable suggestions. Authors are listed in reverse alphabetical order.     

A comparison of latent trait and latent class analyses of Likert-type data

This paper brings together and compares two developments in the analysis of Likert attitude scales. The first is the generalization of latent class models to ordered response categories. The second is the introduction of latent trait models with multiplicative parameter structures for the analysis of rating scales. Key similarities and differences between these two methods are described and illustrated by applying a latent trait model and a latent class model to the analysis of a set of life satisfaction data. The way in which the latent trait model defines a unit of measurement, takes into account the order of the response categories, and scales the latent classes, is discussed. While the latent class model provides better fit to these data, this is achieved at the cost of a logically inconsistent assignment of individuals to latent classes.The author wishes to thank Clifford C. Clogg, Otis Dudley Duncan and Benjamin D. Wright for their helpful comments on an earlier version of this paper.     

Defining a Family of Cognitive Diagnosis Models Using Log-Linear Models with Latent Variables

This paper uses log-linear models with latent variables (Hagenaars, in Loglinear Models with Latent Variables, 1993) to define a family of cognitive diagnosis models. In doing so, the relationship between many common models is explicitly defined and discussed. In addition, because the log-linear model with latent variables is a general model for cognitive diagnosis, new alternatives to modeling the functional relationship between attribute mastery and the probability of a correct response are discussed.     

Multicomponent latent trait models for ability tests

A major research direction for ability measurement has been to identify the information-processes that are involved in solving test items through mathematical modeling of item difficulty. However, this research has had limited impact on ability measurement, since person parameters are not included in the process models. The current paper presents some multicomponent latent trait models for reproducing test performance from both item and person parameters on processing components. Components are identified from item subtasks, in which performance is a logistic function (i.e., Rasch model) of person and item parameters, and then are combined according to a mathematical model of processing on the composite item.The author would like to thank David Thissen for his invaluable insights concerning this model and an anonymous reviewer for his suggestion about the sample space for the model.This research was partially supported by National Institute of Education grant number NIE-6-7-0156 to Susan E. Whitely, principal investigator. However the opinions expressed herein do not necessarily reflect the position or policy of the National Institute of Education, and no official endorsement by the National Institute of Education should be referred. Part of this paper was presented at the annual meeting of thePsychometric Society, Monterey, California: June, 1979.     

Estimating latent distributions in recurrent choice data

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.     

Latent class models for nonmonotone dichotomous items

Starting from perfectly discriminating nonmonotone dichotomous items, a class of probabilistic models with or without response errors and with or without intrinsically unscalable respondents is described. All these models can be understood as simply restricted latent class analysis. Thus, the estimation and identifiability of the parameters (class sizes and item latent probabilities) as well as the chi-squared goodness-of-fit tests (Pearson and likelihood-ratio) are free of the problems. The applicability of the proposed variants of latent class models is demonstrated on real attitudinal data.This research was supported by the Kulturamt der Stadt Wien, Magistratsabteilung 7.The author wishes to thank the editor, Ivo W. Molenaar, as well as Clifford C. Clogg and the anonymous reviewers for their valuable comments on the earlier drafts of this paper.     

Psychometric latent response models

In this paper, some psychometric models will be presented that belong to the larger class oflatent response models (LRMs). First, LRMs are introduced by means of an application in the field ofcomponential item response theory (Embretson, 1980, 1984). Second, a general definition of LRMs (not specific for the psychometric subclass) is given. Third, some more psychometric LRMs, and examples of how they can be applied, are presented. Fourth, a method for obtaining maximum likelihood (ML) and some maximum a posteriori (MAP) estimates of the parameters of LRMs is presented. This method is then applied to theconjunctive Rasch model. Fifth and last, an application of the conjunctive Rasch model is presented. This model was applied to responses to typical verbal ability items (open synonym items).This paper presents theoretical and empirical results of a research project supported by the Research Council [Onderzoeksraad] of the University of Leuven (grant number 89-9) to Paul De Boeck and Luc Delbeke.     

The use of latent class models for assessing prerequisite relations and transference among traits

This paper presents a strategy for pairwise assessment which may be used to evaluate the nature of both prerequisite and transference relations existing among a set of traits. This strategy is appropriate for use both within a confirmatory context, in which an attempt is made to establish the validity of some specified set of relations among traits, as well as within an exploratory context, in which a search is made for unconjectured prerequisite and transference relations existing between pairs of traits. Both uses of this strategy are based on a variety of latent class models which are representative of various possible relational states existing between pairs of traits. Thus, the nature of trait relations may be investigated through the use of statistical assessments of both absolute and relative fit attained by these models. An application is presented to exemplify how this strategy may be used within the exploratory context.The author is obliged to Clifford Clogg, C. Mitchell, Dayton, and William Schafer for helpful comments made regarding a previous draft of this paper as well as to Mary Papageorgiou who provided access to the data which was considered in this study.     

Estimating latent distributions

Consider vectors of item responses obtained from a sample of subjects from a population in which ability is distributed with densityg(), where the are unknown parameters. Assuming the responses depend on through a fully specified item response model, this paper presents maximum likelihood equations for the estimation of the population parameters directly from the observed responses; i.e., without estimating an ability parameter for each subject. Also provided are asymptotic standard errors and tests of fit, computing approximations, and details of four special cases: a non-parametric approximation, a normal solution, a resolution of normal components, and a beta-binomial solution.The author would like to thank R. Darrell Bock for his comments, suggestions, and encouragement during the course of this work.     

Generalized latent trait models

In this paper we discuss a general model framework within which manifest variables with different distributions in the exponential family can be analyzed with a latent trait model. A unified maximum likelihood method for estimating the parameters of the generalized latent trait model will be presented. We discuss in addition the scoring of individuals on the latent dimensions. The general framework presented allows, not only the analysis of manifest variables all of one type but also the simultaneous analysis of a collection of variables with different distributions. The approach used analyzes the data as they are by making assumptions about the distribution of the manifest variables directly.     

On the estimation of parameters in latent structure analysis

In this note, we describe the iterative procedure introduced earlier by Goodman to calculate the maximum likelihood estimates of the parameters in latent structure analysis, and we provide here a simple and direct proof of the fact that the parameter estimates obtained with the iterative procedure cannot lie outside the allowed interval. Formann recently stated that Goodman's algorithm can yield parameter estimates that lie outside the allowed interval, and we prove in the present note that Formann's contention is incorrect.This research was supported in part by Research Contract No. NSF SOC 76-80389 from the Division of the Social Sciences of the National Science Foundation. The author is indebted to C. C. Clogg for helpful comments and for the numerical results reported here (see, e.g., Table 1).     

A person-fit index for polytomous rasch models,latent class models,and their mixture generalizations

A normally distributed person-fit index is proposed for detecting aberrant response patterns in latent class models and mixture distribution IRT models for dichotomous and polytomous data.This article extends previous work on the null distribution of person-fit indices for the dichotomous Rasch model to a number of models for categorical data. A comparison of two different approaches to handle the skewness of the person-fit index distribution is included.Major parts of this paper were written while the first author worked at the Institute for Science Education, Kiel, Germany. Any opinions expressed in this paper are those of the authors and not necessarily of Educational Testing Service. The results presented in this paper were improved by valuable comments from J. Rost, K. Yamamoto, N.D. Verhelst, E. Bedrick and two anonymous reviewers.     

Two-Stage maximum likelihood estimation in the misspecified restricted latent class model

The maximum likelihood classification rule is a standard method to classify examinee attribute profiles in cognitive diagnosis models (CDMs). Its asymptotic behaviour is well understood when the model is assumed to be correct, but has not been explored in the case of misspecified latent class models. This paper investigates the asymptotic behaviour of a two-stage maximum likelihood classifier under a misspecified CDM. The analysis is conducted in a general restricted latent class model framework addressing all types of CDMs. Sufficient conditions are proposed under which a consistent classification can be obtained by using a misspecified model. Discussions are also provided on the inconsistency of classification under certain model misspecification scenarios. Simulation studies and a real data application are conducted to illustrate these results. Our findings can provide some guidelines as to when a misspecified simple model or a general model can be used to provide a good classification result.     

The EM algorithm for latent class analysis with equality constraints

The EM algorithm is a popular iterative method for estimating parameters in the latent class model where at each step the unknown parameters can be estimated simply as weighted sums of some latent proportions. The algorithm may also be used when some parameters are constrained to equal given constants or each other. It is shown that in the general case with equality constraints, the EM algorithm is not simple to apply because a nonlinear equation has to be solved. This problem arises, mainly, when equality constrints are defined over probabilities indifferent combinations of variables and latent classes. A simple condition is given in which, although probabilities in different variable-latent class combinations are constrained to be equal, the EM algorithm is still simple to apply.The authors are grateful to the Editor and the anonymous reviewers for their helpful comments on an earlier draft of this paper. C. C. Clogg and R. Luijkx are also acknowledged for verifying our results with their computer programs MLLSA and LCAG, respectively.     

Using latent class analysis to identify aggressors and victims of peer harassment

This study used latent class analysis (LCA) to identify and classify individuals into aggressor and victim latent classes. Participants were over 2,000 sixth grade students who completed peer nomination procedures that identified students who had reputations as perpetrators and/or victims of physical, verbal, or relational harassment. Results showed five latent classes. Consistent with previous research, LCA identified latent classes of victims, aggressors, and socially adjusted students. However, rather than a single aggressive-victim subgroup, LCA identified latent classes of highly-victimized aggressive-victims and highly-aggressive aggressive-victims. Comparisons showed differences in mean profiles and classification criteria between LCA and traditional dichotomization approaches. Adjustment outcomes showed that highly-victimized aggressive-victims generally experienced greater negative psychological and social adjustment outcomes than highly-aggressive aggressive-victims. Implications of these findings for better assessment of victim and aggressor subgroups were discussed.     

A note on parameter estimation for Lazarsfeld's latent class analysis

As the literature indicates, no method is presently available which takes explicitly into account that the parameters of Lazarsfeld's latent class analysis are defined as probabilities and are therefore restricted to the interval [0, 1]. In the present paper an appropriate transform on the parameters is performed in order to satisfy this constraint, and the estimation of the transformed parameters according to the maximum likelihood principle is outlined. In the sequel, a numerical example is given for which the basis solution and the usual maximum likelihood method failed. The different results are compared and the advantages of the proposed method discussed.     

Continuous and discrete latent structure models for item response data

Relations are examined between latent trait and latent class models for item response data. Conditions are given for the two-latent class and two-parameter normal ogive models to agree, and relations between their item parameters are presented. Generalizationss are then made to continuous models with more than one latent trait and discrete models with more than two latent classes, and methods are presented for relating latent class models to factor models for dichotomized variables. Results are illustrated using data from the Law School Admission Test, previously analyzed by several authors.