The Metric Quality of Ordered Categorical Data

We quantify the information loss incurred by categorizing an unobserved continuous variable (X) into an ordered categorical scale (Z). The continuous variable is conceptualized as true score (.) (which varies across individuals) plus random error (.), with both components assumed to be normally distributed. The index of metric quality is operationalized as r2(Z,.)/r2(X,.), where r2, the squared correlation coefficient, is a descriptive measure of the power of X or Z to predict . . The index is useful in defining limits on explanatory power (population R2) in multiple regression models in which an ordered categorical variable is regressed against a set of predictors. The index can also be used to correct correlations for the effects of ordered categorical measurement. The index of metric quality is extended to the case when several ordered categorical scales are averaged as in the multi-item measurement of a construct. We prove theoretically that as long as the error variance is "large," the index of metric quality for the average Z of ordered categorical scales goes to 1 as the number of scales becomes "large." The index for averaged data is useful in answering questions such as whether the measurement of a construct by averaging three 5-point scales is better or worse than the measurement obtained by averaging five 3-point scales. The results indicate that the loss of information by marketing researchers' ad hoc use of Z as opposed to the more refined X is small (<10%) when the Z scale is well designed with at least five categories. The loss would be even smaller when a multi-item based Z is employed.