Suppose I have a questionnaire and I ask respondents how often they eat at McDonalds:
- Less than once a month
- At least once a month but less than once a week
- 1-3 times a week
- More than 3 times a week
I then correlate these answers with whether the respondents are wearing brown shoes.
- Brown 65 -- not brown 38
- Brown 32 -- not brown 62
- Brown 17 -- not brown 53
- Brown 10 -- not brown 48
- Brown 9 -- not brown 6
The thing I can't get my head around is this: If a respondent picks #5, he (statistically) has a higher probability of wearing brown shoes than not. But, in a sense, his response subsumes responses 2-4, and if you accumulate their statistics (ie, "eats at McDonalds sometimes") he has a higher probability of not wearing brown shoes.
Now I realize that there are a bunch of caveats here -- sampling error in the stats, etc. But is there ever a valid argument for "rolling up" the stats (so that the values used in inference for #5 eg, would consist of the sums of the 2-5 values, or some other scheme), or is this concept just a product of my twisted mind?
(Note that I'm not talking about "collapsing" the stats into fewer observations, which I assume would be perfectly valid, but rather adjusting the probabilities that are used in inference, based on the knowledge that the various possible mutually-exclusive observations are on a sliding scale.)