59 Chapter 4 Diagnostic Opportunities with the Rasch Model for Ordered Response Categories David Andrich, John H.A.L. de Jong and Barry E. Sheridan Murdoch University, Western Australia CITO, The Netherlands Edith Cowan University, Western Australia 1. Introduction Items with ordered response categories are used in social and other sciences when ideally measurements would be used, but no measuring instrument is available. They are similar to measurements in that they partition a latent unidimensional continuum into adjacent intervals. In elementary treatments of ordered response categories, they are in fact treated as measurements in that successive categories are simply assigned successive integers. This assumes that the distances between the thresholds which define the intervals are identical. In more advanced treatments, statistical models which parameterize the thresholds and values of objects of classification are applied. Central to the understanding of what constitutes more or less of the property on the continuum is the definition of the successive categories which reflect successively more of the property. However, there is no guarantee that the categories will operate in the way intended. Therefore, this ordering must be treated as an hypothesis about the data and it is important that the statistical model applied has the property that it can reject this hypothesis. That is, the ordering must be a property of the data themselves, not simply the model. The significance of this feature was articulated by Ronald Fisher (1958) who proposed a least squares method for analyzing such data. Following a demonstration of his procedure on a small example involving 12 serological readings classified into 5 levels of reaction, he wrote: It will be observed that the numerical values . . . lie . . . in the proper order for increasing reaction. This is not a consequence of the procedure by which they have been obtained, but a property of the data examined (Fisher, 1958, p. 294). Even though it was none other than Fisher who made the above point, it seems to have been generally ignored in the analysis of ordered categorical data, both in the choice of models and in exploiting model which can reveal disordering in the data. This paper illustrates, with two examples, how the Rasch model for ordered categories can be used to reveal whether or not ordered categories operate as intended, and the diagnostic opportunities this provides.