An analysis of simple counting methods for ordering incomplete ordinal data

Measurement in the social sciences often involves an attempt to completely order a set of entities on the basis of an underlying attribute. However, limitations of the measurement process often prevent complete empirical determination of the desired ordering. Nevertheless, the ordinal data obtained from the measurement process can be used in attempting to recover or construct more of the underlying order than is provided by the data. Previous research (Fishburn and Gehrlein, 1974a) has shown that a simple one-stage construction method, referred to as the cardinal rule, is fairly effective in correctly identifying ordered pairs in the underlying linear order that are not identified by the measurement process. The present paper re-examines the cardinal rule from the perspective of construction methods based on simple counting measures derived from the data, and argues that it is the best one-stage method in this class when a natural monotonicity assumption holds for the measurement process. The paper then examines two-stage construction rules that are based on the cardinal rule and the simple counting measures. It is shown that one of the two-stage rules gives better overall results than does the cardinal rule by itself.