Measuring ordinal association in situations that contain tied scores

The construal of ties is critical for assessing the association between two variables. Ties should be excluded when the investigator's data-collection procedure forces ties to occur (e.g., a J-place rating scale is used to rate K items, with J < K; a criterion variable contains fewer than K possible outcomes per item). Four measures arising from excluding or including ties on 2 ordinal variables are Goodman & Kruskal's G, Somer's dyx, Kim's dyx, and Wilson's e. In contrast to measures having variance-accounted-for interpretations, probabilistic interpretations developed here can be applied meaningfully both to ordinal-scaled variables and to stronger scales. Recommendations are offered for which measure to use in various situations.