A dynamic generalization of the Rasch model

In the present paper a model for describing dynamic processes is constructed by combining the common Rasch model with the concept of structurally incomplete designs. This is accomplished by mapping each item on a collection of virtual items, one of which is assumed to be presented to the respondent dependent on the preceding responses and/or the feedback obtained. It is shown that, in the case of subject control, no unique conditional maximum likelihood (CML) estimates exist, whereas marginal maximum likelihood (MML) proves a suitable estimation procedure. A hierarchical family of dynamic models is presented, and it is shown how to test special cases against more general ones. Furthermore, it is shown that the model presented is a generalization of a class of mathematical learning models, known as Luce's beta-model.

[1]  F. Lord.  A theory of test scores. , 1952 .

[2]  N. D. Verhelst,et al.  Extensions of the partial credit model , 1989 .

[3]  Merrill M. Flood,et al.  On Stochastic Learning Theory , 1952 .

[4]  J. Kiefer,et al.  CONSISTENCY OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN THE PRESENCE OF INFINITELY MANY INCIDENTAL PARAMETERS , 1956 .

[5]  William K. Estes,et al.  Research and Theory on the Learning of Probabilities , 1972 .

[6]  R. Duncan Luce,et al.  Individual Choice Behavior , 1959 .

[7]  Dean Follmann,et al.  Consistent estimation in the rasch model based on nonparametric margins , 1988 .

[8]  C. A. W. Glas,et al.  The Rasch Model and Multistage Testing , 1988 .

[9]  Eugene Galanter,et al.  Handbook of mathematical psychology: I. , 1963 .

[10]  Erling B. Andersen,et al.  Conditional Inference and Models for Measuring , 1974 .

[11]  Norman Verhelst,et al.  Maximum Likelihood Estimation in Generalized Rasch Models , 1986 .

[12]  N. Laird.  Nonparametric Maximum Likelihood Estimation of a Mixing Distribution , 1978 .

[13]  Gerhard H. Fischer,et al.  On the existence and uniqueness of maximum-likelihood estimates in the Rasch model , 1981 .

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

[15]  G. A. Ferguson,et al.  Item selection by the constant process , 1942 .

[16]  G. H. Fischer,et al.  Logistic latent trait models with linear constraints , 1983 .

[17]  J. Neyman,et al.  Consistent Estimates Based on Partially Consistent Observations , 1948 .

[18]  H. Kelderman,et al.  Loglinear Rasch model tests , 1984 .

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

[20]  W. Kempf.  Dynamic Models for the Measurement of "Traits" in Social Behavior , 1977 .

[21]  R. Mislevy.  Estimating latent distributions , 1984 .

[22]  David Thissen,et al.  Marginal maximum likelihood estimation for the one-parameter logistic model , 1982 .

[23]  R. R. Bush,et al.  A Mathematical Model for Simple Learning , 1951 .

[24]  Georg Rasch,et al.  Probabilistic Models for Some Intelligence and Attainment Tests , 1981, The SAGE Encyclopedia of Research Design.

[25]  D. Lawley,et al.  XXIII.—On Problems connected with Item Selection and Test Construction , 1943, Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences.

[26]  Robert J. Jannarone,et al.  Conjunctive item response theory kernels , 1986 .