Plotting graded data: a Tukey-ish approach

Graded data are those possessing an inherent order but falling short of a metric scale. Examples are opinions on a five-point scale, such as strongly disagree, disagree, neutral, agree, strongly agree. Graded data are, like ranked data, one kind of ordinal data. They are common in many fields, especially as the record of some considered judgment, but little attention seems to have been given to methods for their easy and effective graphical display. This presentation draws on suggestions made in various places by J.W. Tukey, using the principle that cumulative probabilities are a logical and practical way to represent graded data, which is after all the basis for many models for graded responses. Cumulative probability curves for different subsets of the data are found useful in initial description and exploration of the data. A Stata program ordplot offers various kinds of flexibility in showing such curves: (1) cumulating to the bottom, the middle and the top of each class; (2) complementary distribution functions (descending curves) may be shown as well as cumulative distribution functions (ascending curves); (3) logit, folded root (more generally, folded power), loglog, cloglog, normal (Gaussian), percent and raw scales are all allowed for cumulative probabilities; (4) for such scales, labels, lines and ticks may be in terms of the transformed units or in terms of probabilities or percents; (5) different scores may be assigned to grades on the fly. In practice, most datasets seem to reveal their basic structure either on raw or on logit scales. In some cases, the discrete response models fitted by previous authors appear, as a consequence, to be unnecessarily elaborate or indirect.