On Ordinal Prediction Problems

One limitation in building empirically testable models in sociology is that many familiar statistical techniques such as least-squares regression analysis require interval-level measurements while sociologists often have only ordinal-level measurements. Fortunately there do exist statistical techniques that use ordinal measurements. In this article we consider several types of prediction procedures which use ordinal data, as input, including individual ordinal prediction procedures and pairwise ordinal prediction procedures. The latter were studied by Wilson (1971) who claimed that ordinal variables could neither be used to build empirically testable models nor to state substantive propositions rigorously. But his claims are weakened because (1) he states, but fails to prove, that a particular loss function is the only one that can be used in pairwise ordinal prediction procedures, (2) he ignores alternative types of ordinal prediction procedures, and (3) his main mathematical theorem is in error. We consider his arguments, salvage his theorem, and display a similar theorem for individual ordinal prediction procedures. We argue that, for the three reasons mentioned, Wilson's results do not show that it is unprofitable to use ordinal variables in prediction procedures. Finally we consider a generalized individual ordinal prediction procedure in which one ordinal variable is only useless for predicting a second ordinal variable if the two variables are statistically independent.