Rasch models for partial-credit scoring are discussed and a multidimensional version of the model is formu lated. A model may be specified in which consecutive item responses depend on an underlying latent trait. In the multidimensional partial-credit model, different re sponses may be explained by different latent traits. Data from van Kuyk's (1988) size concept test and the Raven Progressive Matrices test were analyzed. Maximum likelihood estimation and goodness-of-fit testing are dis cussed and applied to these datasets. Goodness-of-fit statistics show that for both tests, multidimensional par tial-credit models were more appropriate than the unidi mensional partial-credit model.
Andersen, E.B. (1973a). Conditional inference and models for measuring . Unpublished doctoral dissertation, Mentalhygiejnisk Forskningsinstitut , Copenhagen.
2.
Andersen, E.B. (1973b). Conditional inference and multiple choice questionnaires . British Journal of Mathematical and Statistical Psychology , 26, 31-44.
Andersen, E.B. (1980). Discrete statistical models with social science applications. Amsterdam: North Holland .
5.
Andrich, D. (1978a). A rating scale formulation for ordered response categories. Psychometrika, 43, 561-573.
6.
Andrich, D. (1978b). Scaling attitude items constructed and scored in the Likert tradition. Educational and Psychological Measurement , 38, 665-680.
7.
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In F. M. Lord & M. R. Novick, Statistical theories of mental test scores (pp. 397-479). Reading MA: Addison-Wesley.
Cochran, W. (1952). The X2 test of goodness-of-fit. Annals of Mathematical Statistics, 23, 315-345.
10.
Cochran, W. (1954). Some methods for strengthening the common X2 test . Biometrics, 10, 256-266.
11.
Cox, M.A.A., & Placket, R.L. (1980). Small samples in contingency tables. Biometrika, 67, 1-13.
12.
Deming, W.E., & Stephan, F.F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Annals of Mathematical Statistics, 11, 427-444.
13.
Fischer, G.H. (1974). Einführung in die Theorie psychologischer Tests [Introduction to the theory of psychological tests]. Bern : Huber.
14.
Fischer, G.H. (1995). Derivations of the Rasch model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 15-38). New York: Springer-Verlag .
15.
Glas, C.A.W. (1988). The derivation of some tests for the Rasch model from the multinomial distribution. Psychometrika, 53, 525-546.
16.
Glas, C.A.W., & Verhelst, N.D. (1989). Extensions of the partial credit model. Psychometrika, 54, 635-660.
17.
Haberman, S.J. (1977). Log-linear models and frequency tables with small expected cell counts. Annals of Statistics, 5, 1148-1169.
18.
Haberman, S.J. (1979). Analysis of qualitative data: New developments (Vol. 2). New York: Academic Press.
19.
Kelderman, H. (1992). Computing maximum likelihood estimates of loglinear models from marginal sums with special attention to loglinear item response theory. Psychometrika, 57, 437-450.
20.
Kelderman, H. (1995). The polytomous Rasch model within the class of generalized linear symmetry models. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 307-324). New York: Springer-Verlag.
21.
Kelderman, H., & Steen, R. (1993). LOGIMO I: A program for loglinear item response theory modeling [Computer program manual]. Groningen, The Netherlands: IEC ProGAMMA.
22.
Koehler, K.J. (1977). Goodness-of-fit statistics for large sparse multinomials. Unpublished doctoral dissertation, University of Minnesota , Minneapolis, School of Statistics.
23.
Lancaster, H.O. (1961). Significance tests in discrete distributions. Journal of the American Statistical Association,56, 223-234.
24.
Larnz, K. (1978). Small-sample comparisons of exact levels for chi-square statistics. Journal of the American Statistical Association,73, 412-419.
25.
Lehman, E.L. (1983). The theory of point estimation. New York: Wiley.
26.
Lord, F.M. (1980). Applications of item response theory to practical testing problems. Hillsdale NJ: Erlbaum.
27.
Lord, F.M., & Novick, M.R. (1968). Statistical theories of mental test scores. Reading MA: Addison-Wesley.
28.
Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149-174.
29.
Masters, G.N. (1987). Measurement models for ordered response categories . In R. Langeheine & J. Rost (Eds.), Latent trait and latent class models (pp. 11-29). New York: Plenum .
30.
Molenaar, I.W. (1995). Some background for item response theory and the Rasch model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 3-14). New York: Springer-Verlag.
31.
Muraki, E. (1990). Fitting a polytomous item response model to Likert-type data. Applied Psychological Measurement, 14, 59-71.
32.
Neyman, J., & Scott, E.L. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 1-32.
33.
Rao, C.R. (1973). Linear statistical inference and its applications (2nd ed.). New York: Wiley.
34.
Rasch, G. (1960/80). Probabilistic models for some intelligence and attainment tests. (Copenhagen, Danish Institute for Educational Research). Expanded edition ( 1980), with foreword and afterword by B. D. Wright. Chicago : The University of Chicago Press.
35.
Rasch, G. (1961). On the general laws and meaning of measurement in psychology. In J. Neyman (Ed.), Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability (Vol. 5; pp. 321-333). Berkeley: University of California Press.
36.
Rasch, G. (1977). On specific objectivity: An attempt at formalizing the request for generality and validity of scientific statements. Danish Yearbook of Philosophy, 17, 58-94.
37.
Raven, J., Raven, J.C., & Court, J.H. (1991). Manual for Raven's Progressive Matrices and Vocabulary Scales (section 1): General overview. Oxford: Oxford Psychologists Press.
38.
Roskam, E.E., & Jansen, P.G.W. (1984). A new derivation of the Rasch model. In E. Degreef & J. Van Buggenhaut (Eds.), Trends in mathematical psychology (pp. 293-308). Amsterdam: Elsevier Science Publishers.
39.
Rost, J. (1988). Measuring attitudes with a threshold model drawing on a traditional scaling concept. Applied Psychological Measurement , 12, 397-409.
40.
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement , No. 17.
41.
Sternberg, R.J. (1990). Metaphors of mind: Conceptions of the nature of intelligence. Cambridge: Cambridge University Press.
42.
van den Wollenberg, A.L. (1979). The Rasch model and time limit tests. Unpublished doctoral thesis, Katholieke Universiteit Nijmegen.
43.
van denWollenberg, A.L. (1982). Two new test statistics for the Rasch model. Psychometrika, 47, 123-140.
44.
van Kuyk, J.J. (1988). Verwerven van grootte-begrippen [Acquiring size concepts]. Pedagogische Studièn, 65, 1-10.
45.
Vodegel Matzen, L. (1994). Performance on Raven's Progressive Matrices: What makes the difference? Unpublished doctoral dissertation, University of Amsterdam, Faculty ofPsychology.
