Conventional and Optimal Interval Scores for Ordinal Variables

Sociologists are often confronted with a discrepancy between the level of measurement required by the linear statistical model and that characteristic of most variables of sociological interest. This problem is typically resolved by the use of conventional scores which assign equal interval scores to the categories of ordinal variables subject only to a monotonicity constraint. Alternatively, it is often possible to obtain optimal interval scores which maximize the average interitem correlation within a set of variables. However, since optimal scores can be seen as simply nonlinear transformations of conventional scores, these optimal scores can increase the correlations between the variables only to the extent that the conventional scores yield nonlinear relationships among the variables. Therefore, it is possible to assess the optimality of conventional scores for a particular set of variables by comparing the squared correlation coefficients with the monotonic correlation ratios for those variables. Differences between the correlation coefficients obtained from conventional and optimal scores are limited, even in the presence of nonlinear relationships among the conventionally scored variables, by the "quasi-invariance" property of the correlation coefficient under nonlinear transformations of the variables.