Fitting a Polytomous Item Response Model to Likert-Type Data

Abstract

This study examined the application of the MML-EM algorithm to the parameter estimation problems of the normal ogive and logistic polytomous response models for Likert-type items. A rating-scale model was devel oped based on Samejima's (1969) graded response model. The graded response model includes a separate slope parameter for each item and an item response parameter. In the rating-scale model, the item re sponse parameter is resolved into two parameters: the item location parameter, and the category threshold parameter characterizing the boundary between re sponse categories. For a Likert-type questionnaire, where a single scale is employed to elicit different re sponses to the items, this item response model is ex pected to be more useful for analysis because the item parameters can be estimated separately from the threshold parameters associated with the points on a single Likert scale. The advantages of this type of model are shown by analyzing simulated data and data from the General Social Surveys. Index terms: EM algorithm, General Social Surveys, graded response model, item response model, Likert scale, marginal maximum likelihood, polytomous item response model, rating-scale model.

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References

Andrich, D. (1978). A rating formulation for ordered response categories . Psychometrika, 43, 561-573.

Bimbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In F. M. Lord and M. R. Novick, Statistical theories of mental test scores. Reading MA: Addison-Wesley.

Bock, R.D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37, 29-51.

Bock, R.D. , & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika , 46, 443-459.

Bock, R.D. , Gibbons, R. , & Muraki, E. (1988). Full-information item factor analysis. Applied Psychological Measurement, 12, 261-280.

Bock, R.D. , & Jones, L.V. (1968). The measurement and prediction of judgment and choice. San Francisco: Holden-Day .

Bock, R.D. , & Lieberman, M. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179-197.

Bock, R.D. , Muraki, E. , & Pfeiffenberger, W. (1988). Item pool maintenance in the presence of item parameter drift. Journal of Educational Measurement, 25, 275-285.

Clogg, C.C. (1979). Some latent structure models for the analysis of Likert-type data. Social Science Research, 8, 287-301.

10.

Cooper, C. , & Breneman, B. (1988). Writing achievement of California eighth graders: A first look. Sacramento CA: State Department of Education.

11.

Davis, J.A. (1988). General Social Surveys, 1972-1988: Cumulative codebook. Chicago: National Opinion Research Center.

12.

Dempster, A.P. , Laird, N.M. , & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1-38.

13.

Edwards, A.L. , & Thurstone, L.L. (1952). An interval consistency check for scale values determined by the method of successive intervals and the method of graded dichotomies. Psychometrika, 17, 169-180.

14.

Kendall, M.G. , & Stuart, A. (1973). The advanced theory of statistics (Vol. 2). New York: Hafner .

15.

Likert, R. (1932). A technique for the measurement of attitudes. Archives of Psychology, No. 140.

16.

Lord, F.M. (1980). Application of item response theory to practical testing problems. Hillsdale NJ: Erlbaum.

17.

Lord, F.M. , & Novick, M.R. (1968). Statistical theories of mental test scores. Reading MA: Addison-Wesley.

18.

Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149-174.

19.

Mislevy, R.J. (1986). Bayes modal estimation in item response models . Psychometrika, 51, 177-195.

20.

Muraki, E. (1983). Marginal maximum likelihood estimation for three-parameter polychotomous item response models: Application of an EM algorithm. Doctoral dissertation, University of Chicago.

21.

Muraki, E. (1985, March). Full-information factor canalysis for polychotomous item response. Paper presented at the annual meeting of the American Educational Research Association, Chicago .

22.

Muraki, E. (1989). Maintaining consistency of performance assessment scales over several years. Paper presented at the 19th Annual Assessment Conference, Boulder CO

23.

Muraki, E. , & Bock, R.D. (1988). PARSCALE: Parameter scaling of rating data. Mooresville IN: Scientific Software, Inc.

24.

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research. [Expanded edition, University of Chicago Press, 1980.]

25.

Rost, J. (1988). Measuring attitudes with a threshold model drawing on a traditional scaling concept. Applied Psychological Measurement , 12, 397-409.

26.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement , No. 17.

27.

Samejima, F. (1972). A general model for free-response data. Psychometrika Monograph, No. 18.

28.

Stroud, A.H. , & Secrest, D. (1966). Gaussian quadrature formulas. Englewood Cliffs NJ: Prentice-Hall.

29.

Thissen, D. (1982). Marginal maximum likelihood estimation for the one parameter logistic model. Psychometrika, 47, 175-186.

30.

Thissen, D. (1986). MULTILOG Version 4.0 user's guide. Mooresville IN: Scientific Software, Inc.

31.

Thurstone, L.L. (1947). Multiple factor analysis. Chicago: University of Chicago Press.

32.

Weaver, C.N. , Alais, R.M. , Franz, R.S. , & Carpenter, J.B. (1986, April). U.S. public attitudes and trade relations with eight other countries: Further tests for relationship . Paper presented at the 22nd Interventional Atlantic Economic Conference, St. Thomas, Virgin Islands.