Estimating multiple classification latent class models

This paper presents a new class of models for persons-by-items data. The essential new feature of this class is the representation of the persons: every person is represented by its membership tomultiple latent classes, each of which belongs to onelatent classification. The models can be considered as a formalization of the hypothesis that the responses come about in a process that involves the application of a number ofmental operations. Two algorithms for maximum likelihood (ML) and maximum a posteriori (MAP) estimation are described. They both make use of the tractability of the complete data likelihood to maximize the observed data likelihood. Properties of the MAP estimators (i.e., uniqueness and goodness-of-recovery) and the existence of asymptotic standard errors were examined in a simulation study. Then, one of these models is applied to the responses to a set of fraction addition problems. Finally, the models are compared to some related models in the literature.

[1]  J. Wolfowitz,et al.  Introduction to the Theory of Statistics. , 1951 .

[2]  Paul F. Lazarsfeld,et al.  Latent Structure Analysis. , 1969 .

[3]  Franklin A. Graybill,et al.  Introduction to the Theory of Statistics, 3rd ed. , 1974 .

[4]  L. A. Goodman.  The Analysis of Systems of Qualitative Variables When Some of the Variables Are Unobservable. Part I-A Modified Latent Structure Approach , 1974, American Journal of Sociology.

[5]  V. Smith,et al.  Multivariate analysis: Theory and practice , 1974 .

[6]  Stephen E. Fienberg,et al.  Discrete Multivariate Analysis: Theory and Practice , 1976 .

[7]  M. R. Novick,et al.  Statistical methods for educational and psychological research , 1976 .

[8]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[9]  B. Muthén.  Contributions to factor analysis of dichotomous variables , 1978 .

[10]  Susan E. Whitely,et al.  Multicomponent latent trait models for ability tests , 1980 .

[11]  Philip E. Gill,et al.  Practical optimization , 1981 .

[12]  R. D. Bock,et al.  Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm , 1981 .

[13]  T. Louis.  Finding the Observed Information Matrix When Using the EM Algorithm , 1982 .

[14]  R. D. Bock,et al.  Marginal maximum likelihood estimation of item parameters , 1982 .

[15]  D. Rubin,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[16]  Donald B. Rubin,et al.  Comment : A noniterative sampling/importance resampling alternative to the data augmentation algorithm for creating a few imputations when fractions of missing information are modest : The SIR Algorithm , 1987 .

[17]  J. Loehlin.  Latent variable models , 1987 .

[18]  Carl P. M. Rijkes,et al.  Loglinear multidimensional IRT models for polytomously scored items , 1988 .

[19]  P. Boeck,et al.  Hierarchical classes: Model and data analysis , 1988 .

[20]  Jacques A. Hagenaars,et al.  Categorical Longitudinal Data. , 1991 .

[21]  J. Hagenaars.  Loglinear Models with Latent Variables , 1993 .

[22]  Discrete latent variable models , 1994 .

[23]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[24]  E. Maris.  Psychometric latent response models , 1995 .

[25]  Iven Van Mechelen,et al.  The conjunctive model of hierarchical classes , 1995 .

[26]  I. Mechelen,et al.  Probability matrix decomposition models , 1996 .

[27]  D. S. Sivia,et al.  Data Analysis , 1996, Encyclopedia of Evolutionary Psychological Science.