A Scaling Procedure for Ordered Categorical Data

Abstract

The problem of analysing subjective measurements arises in many fields, particularly those concerning sensory testing or attitude scaling. Subjects are asked to rate their opinions on a scale consisting of, say, 6 or 7 ordered categories. A typical scale might be: 'Excellent, Very good, Good, Not very good, Poor, Very poor'. The statistical analysis of such data presents difficulties. Arbitrary scores of 0, 1, 2, 3, ... are often used and analysed by analysis of variance methods on the assumption either that the necessary criteria are satisfied or that the test is sufficiently robust for it not to matter. In our experience, however, simple integer scores often produce distributions for which such assumptions are unjustified. The data can, for instance, be very skew, because of bunching of observations towards one end of the rating scale. An optimal scoring procedure was first suggested by Fisher [1938]. It was optimal in the sense that the scores assigned to the scale categories naximised the ratio of the treatment sum of squares to the total sum of squares. Bradley, Katti and Coons [19621 have further developed this procedure, restraining the scores to lie in the same order as the scale categories; Fisher's method takes no account of scale order. But by maximizing the stated ratio, without regard to distribution assumptions, one is liable in significance testing to assume too small a probability, and thereby incorrectly reject the null hypothesis. A method based upon assumptions of normal distribution theory is preferable.