Here we consider some data that might have been smoothly distributed over a metric scale, but ended up being concentrated on only a few values. The usual treatment of the data as normally or t-distributed is not appropriate, and instead the data are binned and analyzed as ordinal.
The data are from an unpublished study by Shannon Bailey and Dr. Valerie Sims. In their experiment, people read a description of animal cruelty that
occurred either in a kennel (group 1, N=270) or in an animal shelter
(group 2, N=253). The people then responded with how big a fine they
thought should be assessed to the transgressor, on a continuous scale
from zero to 2,000 dollars. ($2,000 was the maximum allowed by state law at the time of the experiment.)
Because the responses are on a continuous scale, it seems reasonable to apply a model that describes the data as t-distributed, for which we estimate the means, scales, and normality. For details of the model, see Ch. 16 of DBDA2E or the article in JEP:General. The result is shown below:
Despite the fact that the response scale was continuous, the responses were spontaneously ordinal. A histogram of the data (collapsed across groups) is shown below:
0 - 50 --> 1
50 - 450 --> 2
450 - 550 --> 3
550 - 950 --> 4
and so forth.
The resulting ordinal data were then analyzed using the cumulative thresholded normal model described in Ch. 23.3 of DBDA2E. The results were as follows:
Thanks go to Shannon Bailey for bringing this to my attention and for sharing the data so I could make this blog post.