The Metric Quality of Ordered Categorical Data

An index of metric quality for ordered categorical scales is proposed. The index is defined as the extent to which the ordered categorical measurement is linearly related to an underlying interval scaled measurement of the property. The index, operationalized as the squared correlation between the ordered categorical scale and an underlying normally distributed variable, is about 0.9 or higher provided the scale is well designed and the number of levels in the scale is 5 or higher. For any given number of levels, the index is maximized by designing the scale in such a way that the frequency distribution resembles the shape of the underlying distribution. The index of metric quality can be considerably enhanced by averaging several ordered categorical scales. We show that up to a point, errors in measurement, actually improve the metric quality and predictive power of the average of two or more scales. The proposed indices for the raw scale and for the average are potentially useful, for instance, in defining limits on explanatory power (coefficient of determination) in multiple regression models with ordered categorical dependent variables and in examining the appropriateness of using parametric tests such as t and F with ordered categorical data.