ABSTRACT
The two sets of probabilities are equivalent, but simple models for the cumulative probabilities are likely to have better properties for ordinal response scales than equally simple models based on the category probabilities. Measurement scales can be classified at a number of levels. Bivariate responses are perhaps the simplest instances of compound measurement scales. In applications, the distinction between nominal and ordinal scales is usually but not always clear. This chapter discusses that measurement scales of a slightly different type where the categories are ordered, but in a stronger or more rigid sense. Genuine interval scales having these three properties are rare in practice because, although properties 1 and 2 may be satisfied, it is rare to find a response scale having well-determined cardinal scores attached to the categories. The idea of representing ordered categories as contiguous intervals on a continuous scale goes back at least to K. Pearson, who investigated coat-colour inheritance in thoroughbred horses.
