Correlations with ordinal data

[Introduction] In most econometric analyses the data are uniquely defined except for a choice of units (e.g., physical quantities or value flows) and/or a location parameter (e.g., time). In some cases the cardinality of the data is less clear. For instance, building inspectors may rate various aspects of dwellings and neighborhoods on a one to five scale, the resulting indices being used in regressions explaining housing prices [Kain and Quigley (1970), King and Mieszkowski (1973)]. Battalio et al. (1973) recently pointed out that the sign of the correlation coefficient may not be invariant with respect to order preserving transformations of the ordinal variable and presented a numerical example. The exact conditions under which the sign of the sample correlation between a cardinal variable and an ordinal variable is 'identified' are given in section 2. The case of two ordinal variables is dealt with in section 3.