Sequential Models for Rating Scales

The recent discussion of the Success (Glas & Verhelst 1991) and Failure (Linacre 1991) models (Beyond Partial Credit, RMT 1991 5,2 155) is encouraging because it indicates a widening perspective on rating scales. These innovations, however, are more in the area of interpretation and application than in statistical theory.

The Success and Failure models are examples of Upward and Downward sequential models. The origins of sequential modelling of ordinal response data are not clear. The earliest reference appears to be Mantel (1966) who describes the Upward and Downward linear logit models. An example of an Upward ordinal response variable is the number of children in a family. Here an intuitively appealing selection mechanism is sequential, in the sense that, beginning with no children, the couple can decide to have the second child only after it already has the first one, and so on for the third and subsequent children. The Upward and Downward models are known in contingency table literature as continuation ratio models. The Andrich (1978)/Masters (1982) rating scale models are examples of adjacent logit models.

Kutylowski (1992) gives a systematic treatment of sequential models for ordinal response data. He uses simple extensions of the standard framework of generalized linear and non-linear models to formulate the commonalities and differences among various sequential models.

There appears to be no general formal criterion helping to choose between Upward and Downward models for any particular data set. The Upward model is likely to be more realistic in many applications where the highest category would be inappropriate as the first step in the sequence. Cox (1988) treated reaction to a prescribed drug (poor, fair, good, excellent) as ordinal responses categories on an Upward scale, but Fienberg (1980) treated "amount of education" (completed secondary school, some secondary school, elementary school) as ordinal response categories on a Downward scale.

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

Cox C 1988. Multinomial regression models based on continuation ratios. Statistics in medicine 7, 435-441.

Fienberg SE 1980. The analysis of cross-classified data. 2nd Ed. (1st Ed: 1977). MIT Press.

Glas CAW & Verhelst ND 1991. Using the Rasch model for dichotomous data for analyzing polytomous responses. Measurement & Research Dept Report 91-3. Arnhem, The Netherlands: CITO

Kutylowski, A.J. (1992). Sequential models for ordinal response data. In: van der Heijden, P.G.M., Jansen, W., Francis, B., and Seber, G.U.H. (eds.): Statistical Modelling. Amsterdam: North Holland.

Linacre JM 1991. Structured rating scales. Sixth International Objective Measurement Workshop. April. Chicago. ERIC TM 016615

Linacre JM 1991. Beyond Partial Credit: Rasch Success and Failure Models. Rasch Measurement Transactions, 5:2 p. 155

Mantel N 1966. Models for complex contingency tables and polychotomous dosable response curves. Biometrics 22,1 83-95.

Masters GN 1982. A Rasch model for partial credit scoring. Psychometrika 47 149-174

Sequential Models for Rating Scales, A Kutylowski … Rasch Measurement Transactions, 1991, 5:3 p. 161

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