On Ordinal Prediction Problems

Abstract

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. The construction of empirically testable mathematical models is becoming increasingly important in sociological research. These models can yield precise and succinct information about complex social processes. But many popular statistical techniques such as leastsquares regression analysis require interval-level measurements while sociologists unfortunately often have only ordinal-level measurements.' Thus the temptation arises to treat ordinal variables as interval variables in order to apply familiar techniques. Debate has arisen over the appropriateness of using interval-level statistics with ordinal measurements. On the one hand Labowitz (1967; 1970) appeared to demonstrate empirically that the product-moment correlation is not greatly affected by random assignment (using a uniform distribution) of integer values to rank-order categories when the range of the values and the number of categories are both large. Similarly, Boyle (1970) and Leik and Gove (1969) demonstrate the utility of using the technique of causal modeling with ordinal variables. Finally, Good (1973) demonstrates that the correlation between two variables is not much affected by substituting powers of one of the variables for that variable unless the powers are very high. On the other hand, Mayer (1970; 1971), Vargo (1971), and Schweitzer and Schweitzer (1971) have argued mathematically that in extreme cases treating an ordinal variable as an interval variable can have disastrous consequences. In a recent article Wilson (1971) claims that ordinal variables cannot be used to build empirically testable models. The main purpose of the present article is to show that his arguments contain three flaws. First, although his comments appear to apply to the overall prob: The authors would like to acknowledge the helpful criticisms of the referees, and partial support from the Department of Health, Education and Welfare Grants ROI GM 18770 and R03 MH 21454. 1 For convenience we recall the definitions of nominal, ordinal, and interval scales of measurement. Nominal scales, or names, are such that only equality is relevant, and they are therefore invariant under all permutations. For ordinal scales or measurements, only inequalities and equalities are relevant, and they are therefore invariant under all (strictly) monotonic transformations. For interval scales, only ratios of differences. are relevant, and they are therefore invariant under all affine transformations. For a more extended discussion see, for example, Lea (1972).