A second try at disordered thresholds

It has been suggested, with some justification, that I may be a little chauvinistic depending so heavily on a baseball analogy when pondering disordered thresholds. So for my friends in Australia, Cyprus, and the Czech Republic, I’ll try one based on apple pie.

Certified pie judges for the Minnesota State Fair are trained to evaluate each entry on the criteria in Table 1 and the results for pies, at least the ones entered into competitions, are unimodal, somewhat skewed to the left.

Table 1: Minnesota State Fair Pie Judging Rubric

Aspect	Points
Appearance	20
Color	10
Texture	20
Internal appearance	15
Aroma	10
Flavor	25
Total	100

We might suggest some tweaks to this process, but right now our assignment is to determine preferences of potential customers for our pie shop. All our pies would be 100s on the State Fair rubric so it won’t help. We could collect preference data from potential customers by giving away small taste samples at the fair and asking each taster to respond to a short five-category rating scale with categories suggested by our psychometric consultant.

My feeling about this pie is:

0	1	2	3	4
I’d rather have boiled liver	Can I have cake instead?	Almost as good as my mother’s	Among the best I’ve ever eaten	I could eat this right after a major feast!

The situation is hypothetical; the data are simulated from unimodal distributions with roughly equal means. On day one, thresholds 3 and 4 were reversed; on day two, thresholds 2 and 3 for some tasters were also reversed. None of that will stop me from interpreting the results. It is not shown in this summary of the data shown below, but the answer to our marketing question is pies made with apples were the clear winners. (To appropriate a comment that Rasch made about correlation coefficients, this result is population-dependent and therefore scientifically rather uninteresting.) Any problems that the data might have with the thresholds did not prevent us from reaching this conclusion rather comfortably. The most preferred pies received the highest scores in spite of our problematic category labels. Or at least that’s the story I will include with my invoice.

The numbers we observed for the categories are shown in Table 2. Right now we are only concerned with the categories, so this table is summed over the pies and the tasters.

Table 2: Results of Pie Preference Survey for Categories

Day	I’d rather have boiled liver	Can I have cake instead?	Almost as good as my mother’s	Among the best I’ve ever eaten	I could eat this right after a major feast!
One	10	250	785	83	321
Two	120	751	95	22	482

In this scenario, we have created at least two problems; first, the wording of the category descriptions may be causing some confusion. I hope those distinctions survive the cultural and language differences between the US and the UK. Second, the day two group is making an even cruder distinction among the pies; almost I like it or I don’t like it.

The category 4 was intended to capture the idea that this pie is so good that I will eat it even if I have already eaten myself to the point of pain. For some people that may not be different than this pie is among the best I’ve ever eaten, which is why relatively few chose category 3. Anything involving mothers is always problematic on a rating scale. Depending on your mother, “Almost as good as my mother’s” may be the highest possible rating; for others, it may be slightly above boiled liver. That suggests there may be a problem with the category descriptors that our psychometrician gave us, but the fit statistics would not object. And it doesn’t explain the difference between days one and two.

Day Two happened to be the day that apples were being judged in a separate arena, completely independently of the pie judging. Consequently every serious apple grower in Minnesota was at the fair. Rather than spreading across the five categories, more or less, this group tended to see pies as a dichotomy: those that were made with apples and those that weren’t. While the general population spread out reasonably well across the continuum, the apple growers were definitely bimodal in their preferences.

The day two anomaly is in the data, not the model or thresholds. The disordered thresholds that exposed the anomaly by imposing a strong model, but not reflected in the standard fit statistics, are an indication that we should think a little more about what we are doing. Almost certainly, we could improve on the wording of the category descriptions. But we might also want to separate apple orchard owners from other respondents to our survey. The same might also be true for banana growers but they don’t figure heavily in Minnesota horticulture. Once again, Rasch has shown us what is population-independent, i.e., the thresholds (and therefore scientifically interesting) and what is population-dependent, i.e., frequencies and preferences, (and therefore only interesting to marketers.)

These insights don’t tell us much about marketing pies better but I wouldn’t try to sell banana cream to apple growers and I would want to know how much of my potential market are apple growers. I am still at a loss to explain why anyone, even beef growers, would pick liver over anything involving sugar and butter.