A probabilistic formulation and statistical analysis of guttman scaling
TL;DR: In this paper, the latent or true nature of subjects is identified with a limited number of response patterns (the Guttman scale patterns), and the probability of an observed response pattern can be written as the sum of products of the true type multiplied by the chance of sufficient response error to cause the observed pattern to appear.
Abstract: By proposing that the latent or true nature of subjects is identified with a limited number of response patterns (the Guttman scale patterns), the probability of an observed response pattern can be written as the sum of products of the probability of the true type multiplied by the chance of sufficient response error to cause the observed pattern to appear. This model contains the proportions of the true types as parameters plus some misclassification probabilities as parameters. Maximum likelihood methods are used to make estimates and test the fit for some examples.
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TL;DR: In this article, the probability of latent class membership is functionally related to concomitant variables with known distribution, and a general procedure for imposing linear constraints on the parameter estimates is introduced.
Abstract: This article introduces and illustrates a new type of latent-class model in which the probability of latent-class membership is functionally related to concomitant variables with known distribution. The function (or so-called submodel) may be logistic, exponential, or another suitable form. Concomitant-variable models supplement latent-class models incorporating grouping by providing more parsimonious representations of data for some cases. Also, concomitant-variable models are useful when grouping models involve a greater number of parameters than can be meaningfully fit to existing data sets. Although parameter estimates may be calculated using standard iterative procedures such as the Newton—Raphson method, sample analyses presented here employ a derivative-free approach known as the simplex method. A general procedure for imposing linear constraints on the parameter estimates is introduced. A data set involving arithmetic test items in a mastery testing context is used to illustrate fitting a...
458 citations
Cites methods from "A probabilistic formulation and sta..."
...psychological scaling models derived from the notions of Guttman scales (e.g., Dayton and Macready 1976, 1980; Formann 1982; Goodman 1974, 1975; Proctor 1970)....
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TL;DR: A statistical model can be called a latent class (LC) or mixture model if it assumes that some of its parameters differ across unobserved subgroups, LCs, or mixture components as mentioned in this paper.
Abstract: A statistical model can be called a latent class (LC) or mixture model if it assumes that some of its parameters differ across unobserved subgroups, LCs, or mixture components. This rather general idea has several seemingly unrelated applications, the most important of which are clustering, scaling, density estimation, and random-effects modeling. This article describes simple LC models for clustering, restricted LC models for scaling, and mixture regression models for nonparametric random-effects modeling, as well as gives an overview of recent developments in the field of LC analysis. Moreover, attention is paid to topics such as maximum likelihood estimation, identification issues, model selection, and software.
431 citations
TL;DR: The concomitant latent class model for analyzing multivariate categorical outcome data is studied, and practical theory for reducing and identifying such models is developed.
Abstract: Quantifying human health and functioning poses significant challenges in many research areas. Commonly in the social and behavioral sciences and increasingly in epidemiologic research, multiple indicators are utilized as responses in lieu of an obvious single measure for an outcome of interest. In this article we study the concomitant latent class model for analyzing such multivariate categorical outcome data. We develop practical theory for reducing and identifying such models. We detail parameter and standard error fitting that parallels standard latent class methodology, thus supplementing the approach proposed by Dayton and Macready. We propose and study diagnostic strategies, exemplifying our methods using physical disability data from an ongoing gerontologic study. Throughout, the focus of our work is on applications for which a primary goal is to study the association between health or functioning and covariates.
425 citations
01 Jan 1997
TL;DR: 1.1 When you report results obtained with EM, you should refer to this manual as " Vermunt, J.K.
Abstract: 1 When you report results obtained with EM, you should refer to this manual as " Vermunt, J.K.
349 citations
Cites background from "A probabilistic formulation and sta..."
...3 An interesting type of application of equality restrictions on probabilities is the specification of probabilistic Guttman scales (Proctor, 1970; McCutcheon, 1987)....
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01 Jan 1965TL;DR: Algebra of Vectors and Matrices, Probability Theory, Tools and Techniques, and Continuous Probability Models.
Abstract: Algebra of Vectors and Matrices. Probability Theory, Tools and Techniques. Continuous Probability Models. The Theory of Least Squares and Analysis of Variance. Criteria and Methods of Estimation. Large Sample Theory and Methods. Theory of Statistical Inference. Multivariate Analysis. Publications of the Author. Author Index. Subject Index.
8,300 citations
TL;DR: The theory of least squares and analysis of variance has been studied in the literature for a long time, see as mentioned in this paper for a review of some of the most relevant works. But the main focus of this paper is on the analysis of variance.
Abstract: Algebra of Vectors and Matrices. Probability Theory, Tools and Techniques. Continuous Probability Models. The Theory of Least Squares and Analysis of Variance. Criteria and Methods of Estimation. Large Sample Theory and Methods. Theory of Statistical Inference. Multivariate Analysis. Publications of the Author. Author Index. Subject Index.
5,182 citations
TL;DR: In this paper, the problem of approximating one matrix by another of lower rank is formulated as a least-squares problem, and the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another.
Abstract: The mathematical problem of approximating one matrix by another of lower rank is closely related to the fundamental postulate of factor-theory. When formulated as a least-squares problem, the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another. The solution of the problem is simplified by first expressing the matrices in a canonic form. It is found that the problem always has a solution which is usually unique. Several conclusions can be drawn from the form of this solution. A hypothetical interpretation of the canonic components of a score matrix is discussed.
3,576 citations