A case in which metric data are better analyzed by an ordinal model

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The data are from an unpublished study by Shannon Bailey and Dr.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:

Unfortunately, the histograms of the data in the upper-right panels above show that a t distribution is a terrible description of the data. We cannot really interpret the parameters of the model very meaningfully when the model doesn't describe the data very well.

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:

Notice that responses are strongly limited to 0, 500, 1000, 1500, and 2000. There are very few response values between those multiples of 500. The data were therefore converted to ordinal values as follows:
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:

Notice in the upper-right panels above that the data are described very accurately by the model. We can therefore put some merit into the interpretation of the parameters. The effect size (lower-right panel above) has a magnitude of about 0.35, indicating that people assigned fines in the kennel about 1/3 standard deviation higher than in the shelter.



Thanks go to Shannon Bailey for bringing this to my attention and for sharing the data so I could make this blog post.

Bayesian meta-analysis provides high-precision estimate of cosmic impact event at ~12,800 years before 1950

A recently published article in PNAS reports a Bayesian meta-analysis of data from 354 samples from 23 geological sites on 4 continents, yielding a 95% credible interval from 12,835 to 12,735 years before 1950 for the cosmic impact event that formed the Younger Dryas boundary layer. The article, authored by J. P. Kennett with many co-authors, can be found at the PNAS web site.