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

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.     

The asymptotic posterior normality of the latent trait in an IRT model

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.This research was partially supported by Office of Naval Research Cognitive and Neural Sciences Grant N0014-J-90-1940, 442-1548, National Science Foundation Mathematics Grant NSF-DMS-91-01436, and the National Center for Supercomputing Applications. We wish to thank Kumar Joag-dev and Zhiliang Ying for enlightening suggestions concerning the proof of the basic result.The authors wish to thank Kumar Joag-Dev, Brian Junker, Bert Green, Paul Holland, Robert Mislevy, and especially Zhiliang Ying for their useful comments and discussions.     

Charting the process of change: A primer on survival analysis

Survival analysis is a powerful and useful technique for understanding qualitative change. This article provides a practical, nontechnical introduction to the use of survival analysis for social scientists. Important issues in using survival analysis are discussed, including research design, data preparation and management, and data analysis. Attendance data from a self-helf organization are used to illustrate common survival analysis tasks such as describing the overall survival and hazard functions, examining covariate effects, and modeling the form of the hazard function over time. An appendix that discusses the strengths and weaknesses of current survival analysis computer programs is included. Editor's note: Edward Seidman served as action editor for this article while serving as Associate Editor for Methodology.I thank William Davidson, Susan Englund, Bruce Rapkin, Kurt Ribisl, and three anonymous reviewers for their helpful comments on earlier drafts. The example data presented here were collected with the support of an NIMH grant (MH37390) awarded to Julian Rappaport and Ed Seidman.     

Category discrimination by pigeons using five polymorphous features

Eight pigeons were trained to discriminate between sets of color photographs of natural scenes. The scenes differed along five two-valued dimensions (site, weather, camera distance, camera orientation, and camera height), and all combinations of the feature values were used. One value of each dimension was designated as positive, and slides containing three or more positive feature values were members of the positive stimulus set. Thus, each feature had an equal, low, correlation with reinforcement, and all features had zero correlations with each other. Seven of the 8 pigeons learned this discrimination, and their responding came under the control of all five features. Within the positive and negative stimulus sets, response rates were higher to stimuli that contained more positive feature values. Once discrimination had been achieved, reversal training was given using a subset of the slides. In this subset, only a single feature was correlated with reinforcement. All pigeons learned this reversal successfully and generalized it to additional photographs with the same feature content. After reversal, the original reinforcement contingencies were reinstated, and training was continued using all the slides except those that had been used in reversal. Reversal generalized to these slides to some extent. Analysis of the response rates to individual slides showed that, compared with prereversal training, only the feature that had been subjected to reversal contingencies showed a reversed correlation with response rate. The remaining features showed the same correlation with response rate as they had before reversal training. Thus, reversal on some members of a category following category discrimination training led to generalization to stimuli within the category that were not involved in the reversal, but not to features that were not reversed. It is therefore inappropriate to refer to the pigeons as learning a concept.     

Three-term contingency patterns in mother-child verbal interactions during first-language acquisition

Selections from a large longitudinal data set of verbal interactions between a mother and her child are presented. Two sets of three-term contingency sequences that seemed to reflect maternal rewards and corrections were noted. Both the antecedents as well as the immediate consequences of maternal interventions are presented to explore training and learning processes. The observed frequencies of three-step sequences are compared to those expected based upon Markov-chain logic to substantiate the patterning of the interactions. Behavioral conceptualizations of the learning process are supported by these analyses, although their sufficiency is questioned. It is suggested that maternal rewards and corrections should be integrated with perceptual, cognitive, and social learning conceptualizations in a skill-learning approach to explain the complexity of language transmission and acquisition processes.     

Interpreting canonical correlation analysis through biplots of structure correlations and weights

This paper extends the biplot technique to canonical correlation analysis and redundancy analysis. The plot of structure correlations is shown to the optimal for displaying the pairwise correlations between the variables of the one set and those of the second. The link between multivariate regression and canonical correlation analysis/redundancy analysis is exploited for producing an optimal biplot that displays a matrix of regression coefficients. This plot can be made from the canonical weights of the predictors and the structure correlations of the criterion variables. An example is used to show how the proposed biplots may be interpreted.     

New lower and upper bounds for communality in factor analysis

We derive several relationships between communalities and the eigenvalues for ap ×p correlation matrix under the usual factor analysis model. For suitable choices ofj, _j (), where _j () is thej-th largest eigenvalue of , provides either a lower or an upper bound to the communalities for some of the variables. We show that for at least one variable, 1 - _p () improves on the use of squared mulitiple correlation coefficient as a lower bound.This research was done while the second author was at Tokyo Institute of Technology.     

Modeling item responses when different subjects employ different solution strategies

A model is presented for item responses when different subjects employ different strategies, but only responses, not choice of strategy, can be observed. Using substantive theory to differentiate the likelihoods of response vectors under a fixed set of strategies, we model response probabilities in terms of item parameters for each strategy, proportions of subjects employing each strategy, and distributions of subject proficiency within strategies. The probabilities that an individual subject employed the various strategies can then be obtained, along with a conditional estimate of proficiency under each. A conceptual example discusses response strategies for spatial rotation tasks, and a numerican example resolves a population of subjects into subpopulations of valid responders and random guessers.The first author's work was supported by Contract No. N00014-85-K-0683, project designation NR 150-539, from the Cognitive Science Program, Cognitive and Neural Sciences Division, Office of Naval Research. We are grateful to Murray Aitkin, Isaac Bejar, Neil Dorans, Frederiksen, and Marklyn Wingersky for their comments and suggestions, and to Alison Gooding, Maxine Kingston, Donna Lembeck, Joling Liang, and Kentaro Yamamoto for their assistance with Example 2.     

Finite mixtures in confirmatory factor-analysis models

In this paper, various types of finite mixtures of confirmatory factor-analysis models are proposed for handling data heterogeneity. Under the proposed mixture approach, observations are assumed to be drawn from mixtures of distinct confirmatory factor-analysis models. But each observation does not need to be identified to a particular model prior to model fitting. Several classes of mixture models are proposed. These models differ by their unique representations of data heterogeneity. Three different sampling schemes for these mixture models are distinguished. A mixed type of the these three sampling schemes is considered throughout this article. The proposed mixture approach reduces to regular multiple-group confirmatory factor-analysis under a restrictive sampling scheme, in which the structural equation model for each observation is assumed to be known. By assuming a mixture of multivariate normals for the data, maximum likelihood estimation using the EM (Expectation-Maximization) algorithm and the AS (Approximate-Scoring) method are developed, respectively. Some mixture models were fitted to a real data set for illustrating the application of the theory. Although the EM algorithm and the AS method gave similar sets of parameter estimates, the AS method was found computationally more efficient than the EM algorithm. Some comments on applying the mixture approach to structural equation modeling are made.Note: This paper is one of the Psychometric Society's 1995 Dissertation Award papers.—EditorThis article is based on the dissertation of the author. The author would like to thank Peter Bentler, who was the dissertation chair, for guidance and encouragement of this work. Eric Holman, Robert Jennrich, Bengt Muthén, and Thomas Wickens, who served as the committee members for the dissertation, had been very supportive and helpful. Michael Browne is appreciated for discussing some important points about the use of the approximate information in the dissertation. Thanks also go to an anonymous associate editor, whose comments were very useful for the revision of an earlier version of this article.