# How do you deal with a “multiple choice” observation in Bayesian inference, when the choices are on a scale?

Suppose I have a questionnaire and I ask respondents how often they eat at McDonalds:

1. Never
2. Less than once a month
3. At least once a month but less than once a week
4. 1-3 times a week
5. More than 3 times a week

I then correlate these answers with whether the respondents are wearing brown shoes.

1. Brown 65 -- not brown 38
2. Brown 32 -- not brown 62
3. Brown 17 -- not brown 53
4. Brown 10 -- not brown 48
5. 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.)

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Not sure I see what Bayesian issues there are in this question. Maybe remove this? –  conjugateprior Jul 22 '11 at 9:23
I added Bayesian because I'm currently working on a Bayesian inference model, and that's the context of the question. (Plus I don't know stats well enough to pick any other keywords.) –  Daniel R Hicks Jul 22 '11 at 11:57

Are you being confused by the correlation / regression distinction perhaps? When you say 'I correlate these answers' with shoe colour, you could mean either

1. 'I compute the conditional probability of shoe colour given eating habits', or
2. 'I compute a measure of linear relatedness underlying the joint distribution of eating habits and shoe colour choices'.

Number 1 would be realised as a regression which, unlike a correlation, would completely ignore the dominance structure in the eating habits answers. Hence your mystification and interest in an data aggregation method that will embody this structure.

Number 2 is a job for a correlation coefficient. There are special, structure-respecting correlation coefficients that might be better for this sort of dominance-structured data (see the excellent tetrachoric and polychoric correlations page by John Uebersax, for an overview.)

Alternatively there are more explicit models in the latent trait / IRT family that will provide the raw materials for the various marginal and conditional distributional inferences that are, I think, the 'thing you can't get your head around'.

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I can comprehend (sort of) WHAT you say, but it gives me a queasy feeling because I can't quite comprehend WHY it's true. I'll happily look at your reference, though. Thanks. –  Daniel R Hicks Jul 22 '11 at 11:53
If it helps any I remembered that referring to it as dominance structure is not helpful if you've not met measurement theory. (Stochastic) monotonicity (think ordinal logit) is probably a more illuminating statistical term and more helpful if you're googling around. –  conjugateprior Jul 22 '11 at 13:07
CP: Thanks for your help. I think I understand things (about as well as I'm ever going to) now, and I have a software algorithm worked out. I'll give you the "answer". –  Daniel R Hicks Jul 22 '11 at 21:07

I don't think it's at all correct that the "5" response "subsumes responses 2-4." The answer options are, as you acknowledge at the end, all mutually exclusive. For example, "more than 3" is definitely not inclusive of "1 to 3." So all that's left, that I can make out at least, is the question as to whether to collapse categories. That's more of a domain-knowledge issue than a statistical one.

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I think he means there's a monotonicity structure to the question that should be reflected in a model of the answers. The thought is that, despite the unfortunate wording, the question is trying to discretize a daily rate of how often subjects eat at McDs by asking about thresholds. So if you eat at least 3 times a week then you also eat at least once a week, and at least once a month, etc. which seems reasonable - it's just an ordinal response. But he does, as I tried to suggest, have to treat it like a response, not a independent variable. –  conjugateprior Jul 22 '11 at 13:01