Proportional profiles and latent structure

The identity of problem and solution in Lazarsfeld's latent structure analysis and Cattell's proportional profiles is pointed out. Anderson's latent structure solution is adapted to proportional profiles to yield a possible solution for the communality and rotational problems in factor analysis. A numerical example of the latter is provided.This paper was initiated at the University of North Carolina and completed at the Center for Advanced Study in the Behavioral Sciences.     

An empirical comparison of restricted and general latent distance analysis

Latent distance analysis provides a probability model for the non-perfect Guttman scale; the restricted latent distance structure is simpler to compute than the general structure. Since no sampling theory for latent structure analysis is available, the advantages of the general structure cannot be expressed formally. The two structures are compared in terms of their fit to fifteen sets of empirical data. The computation schemes used are summarized.     

Extending latent class solutions to other variables

A procedure is derived for estimating the latent parameters of items not initially included in a latent class solution, on the assumption that the relations between those additional items and the original ones are accounted for by the same latent structure. A chi-square test is proposed for evaluating the relatedness of the additional items to the latent structure. The extension is generalized to associate continuous outside variables with the original solution, and this is accompanied by the suggestion that a simple analysis of variance be used to assess the results. Finally, this latent structure extension is compared with the Dwyer extension of factor analysis.I am grateful to J. J. Mellinger for clarifying discussions of the statistical portions of this paper.Opinions expressed herein are the author's, not the Army's.     

A note on the derivation of the general latent class model

A general formulation of the latent structure principle is suggested, from which it is possible to derive Lazarsfeld's accounting equations in their most general form. The basic equations of Gibson's latent profile model can thence be derived in a single step.     

Taxometric analysis as a general strategy for distinguishing categorical from dimensional latent structure

Statistical analyses investigating latent structure can be divided into those that estimate structural model parameters and those that detect the structural model type. The most basic distinction among structure types is between categorical (discrete) and dimensional (continuous) models. It is a common, and potentially misleading, practice to apply some method for estimating a latent structural model such as factor analysis without first verifying that the latent structure type assumed by that method applies to the data. The taxometric method was developed specifically to distinguish between dimensional and 2-class models. This study evaluated the taxometric method as a means of identifying categorical structures in general. We assessed the ability of the taxometric method to distinguish between dimensional (1-class) and categorical (2-5 classes) latent structures and to estimate the number of classes in categorical datasets. Based on 50,000 Monte Carlo datasets (10,000 per structure type), and using the comparison curve fit index averaged across 3 taxometric procedures (Mean Above Minus Below A Cut, Maximum Covariance, and Latent Mode Factor Analysis) as the criterion for latent structure, the taxometric method was found superior to finite mixture modeling for distinguishing between dimensional and categorical models. A multistep iterative process of applying taxometric procedures to the data often failed to identify the number of classes in the categorical datasets accurately, however. It is concluded that the taxometric method may be an effective approach to distinguishing between dimensional and categorical structure but that other latent modeling procedures may be more effective for specifying the model.     

Planning, motivation, and evaluation in orientation to the future: A latent structure analysis

The paper presents a model of orientation to the future in terms of three processes; motivation, planning, and evaluation. To test this model a total of 154 11-, 15- and 18-year-old adolescents were interviewed about their goals and hopes for the future. Seven observed variables were estimated and a model including three latent constructs was tested using the LISREL VI computer program. The planning construct consisted of the amount of knowledge, the complexity of plans and their level of realization. The evaluation construct included internality, estimation of the likelihood that the goals would be realized and an overall emotional evaluation of the future. The motivation construct consisted of one observed variable, extension. Confirmatory factor analysis showed that the model fitted the data, thus providing support for the theoretical model.     

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.     

Generalized latent variable models with non‐linear effects

Until recently, item response models such as the factor analysis model for metric responses, the two‐parameter logistic model for binary responses and the multinomial model for nominal responses considered only the main effects of latent variables without allowing for interaction or polynomial latent variable effects. However, non‐linear relationships among the latent variables might be necessary in real applications. Methods for fitting models with non‐linear latent terms have been developed mainly under the structural equation modelling approach. In this paper, we consider a latent variable model framework for mixed responses (metric and categorical) that allows inclusion of both non‐linear latent and covariate effects. The model parameters are estimated using full maximum likelihood based on a hybrid integration–maximization algorithm. Finally, a method for obtaining factor scores based on multiple imputation is proposed here for the non‐linear model.     

Factor analysis with (mixed) observed and latent variables in the exponential family

We develop a general approach to factor analysis that involves observed and latent variables that are assumed to be distributed in the exponential family. This gives rise to a number of factor models not considered previously and enables the study of latent variables in an integrated methodological framework, rather than as a collection of seemingly unrelated special cases. The framework accommodates a great variety of different measurement scales and accommodates cases where different latent variables have different distributions. The models are estimated with the method of simulated likelihood, which allows for higher dimensional factor solutions to be estimated than heretofore. The models are illustrated on synthetic data. We investigate their performance when the distribution of the latent variables is mis-specified and when part of the observations are missing. We study the properties of the simulation estimators relative to maximum likelihood estimation with numerical integration. We provide an empirical application to the analysis of attitudes.     

Estimating latent correlations between repeated testings

A model for longitudinal latent structure analysis is proposed. We assume that test scores for a given mental or attitudinal test are observed for the same individuals at two different points in time. The purpose of the analysis is to fit a model that combines the values of the latent variable at the two time points in a two-dimensional latent density. The correlation coefficient between the two values of the latent variable can then be estimated. The theory and methods are illustrated by a Danish dataset concerning psychic vulnerability.     

Robustness of normal theory methods in the analysis of linear latent variate models

The structure of the covariance matrix of sample covariances under the class of linear latent variate models is derived using properties of cumulants. This is employed to provide a general framework for robustness of statistical inference in the analysis of covariance structures arising from linear latent variate models. Conditions for normal theory estimators and test statistics to retain each of their usual asymptotic properties under non-normality of latent variates are given. Factor analysis, LISREL and other models are discussed as examples.     

Career maturity: A latent means and covariance structure analysis of Daidalos

A latent means and covariance structure analysis was conducted to assess the construct validity and construct comparability in the measuring of career maturity across boys and girls. The career maturity inventory has been coined Daidalos. The sample consisted of 2,443 high school students recruited from one county in Norway. Of these, 1,132 were males, and 1,311 were females. The mean age of the participants was 17 years of age. Goodness-of-fit statistics provided support for a five-factor first-order model in which the factor loadings, factor covariances and item intercepts were invariant across groups. Additionally, ΔS-B χ² was shown to be non-significant for the testing of invariance of the measurement model. Three significant differences in latent means were detected, with boys scoring higher on negative attitudes toward school or dropout intentions, and girls scoring higher on career uncertainty and need for world-of-work information.     

Efficient estimation and local identification in latent class analysis

Estimators which are efficient in the sense of having minimum asymptotic variance are obtained for the structural parameters of Lazarsfeld's latent class model of latent structure analysis. Sufficient conditions for the local identification of the structural parameters are also presented. The writer wishes to acknowledge with appreciation the helpful advice of Professors Leonid Hurwicz and Jacob Bearman of the University of Minnesota. Dr. John Gurland of Iowa State College assisted the author in clarifying certain points in this article.     

Application of local linear embedding to nonlinear exploratory latent structure analysis

In this paper we discuss the use of a recent dimension reduction technique called Locally Linear Embedding, introduced by Roweis and Saul, for performing an exploratory latent structure analysis. The coordinate variables from the locally linear embedding describing the manifold on which the data reside serve as the latent variable scores. We propose the use of semiparametric penalized spline methods for reconstruction of the manifold equations that approximate the data space. We also discuss a crossvalidation strategy that can guide in selecting an appropriate number of latent variables. Synthetic as well as real data sets are used to illustrate the proposed approach. A nonlinear latent structure representation of a data set also serves as a data visualization tool.     

Application of the pi* goodness-of-fit index to latent structure models

Testing model fit for latent structure models (latent trait models and latent class models) is difficult because of the lack of goodness-of-fit statistics with a known distribution. This paper describes the application of the pi* goodness-of-fit statistic to latent structure models. The statistic pi* is based on the concept of latent classes and has a natural interpretation when applied to these models. This statistic assumes that the population is made up of several classes that follow a parametric model, and a residual class outside the model. The value of pi* is the population proportion in the residual class. This paper describes the estimation algorithms of pi* for latent trait and latent class models and an empirical example with a scale of study habits. There are two latent classes in these data: bad and regular students, which are related to the student s responsibility.     

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.     

The latent four-zone structure in classroom seating space

The purpose of this study was to extract a latent space structure of classrooms from students seating preferences, and to determine whether classroom division on the basis of latent space structure is useful in predicting actual seating behaviors. The actual seating positions of 151 female junior college students were observed in classrooms once a week for one semester; then students were asked to estimate their seating preference among 49 seating positions in an imaginary classroom. Students did have different preferences for seating positions. A factor analysis suggested that classroom seating space should be divided into four zones: rear, front, center, and sides. A multiple discriminant analysis suggested that the seating positions that students actually took reflected a four-zone structure. Classroom division based on the four-zone structure proved useful in understanding students' seating behaviors.     

A latent variable approach to executive control in healthy ageing

It is a well-established finding that the central executive is fractionated in at least three separable component processes: Updating, Shifting, and Inhibition of information (Miyake et al., 2000). However, the fractionation of the central executive among the elderly has been less well explored, and Miyake's et al. latent structure has not yet been integrated with other models that propose additional components, such as access to long-term information. Here we administered a battery of classic and newer neuropsychological tests of executive functions to 122 healthy individuals aged between 48 and 91 years. The test scores were subjected to a latent variable analysis (LISREL), and yielded four factors. The factor structure obtained was broadly consistent with Miyake et al.'s three-factor model. However, an additional factor, which was labeled 'efficiency of access to long-term memory', and a mediator factor ('speed of processing') were apparent in our structural equation analysis. Furthermore, the best model that described executive functioning in our sample of healthy elderly adults included a two-factor solution, thus indicating a possible mechanism of dedifferentiation, which involves larger correlations and interdependence of latent variables as a consequence of cognitive ageing. These results are discussed in the light of current models of prefrontal cortex functioning.     

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.