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I have three product categories, $A,B,C$. Each category has two products, $0,1$. I provide a number of different kinds of choice situations, 1) the test subject is presented a single category and made to choose a product, 2) the test subject is presented with two categories and made to choose a product from two categories, and 3) the the test subject is presented with all three categories and made to choose a product from each. I believe that product choices depend on a number of measured covariates of the individual products, the product categories presented, and the choice in the other category (if such a choice is possible).

For example, let's say that we had a product category of vinegar, with two brands. The first brand is an expensive, balsamic vinegar. The second brand is an inexpensive, store brand, apple vinegar. Now, let's say we have two other product categories: salad greens and kitchen gloves, each containing an expensive, high quality brand and a cheap, generic brand. Even if a consumer chooses the expensive vinegar when asked to choose only from the vinegar category or from the vinegar and salad category, we might still expect that he would select the inexpensive vinegar if asked to choose products from the vinegar and kitchen glove categories. We might also expect that a person who chose the inexpensive vinegar, when asked to choose from the vinegar and salad green categories, will also choose the inexpensive salad greens.

This situation is similar to the "shopping basket" problems reviewed by P.B. Seetharaman, et. al. in "Models of Multi-Category Choice Behavior". However, the models I have seen consider the incidence of a product category as a function of the consumer, often as a stage model.

How would we estimate the coefficients of the measured covariates in the case when the chooser does not choose the categories they must select from.

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    $\begingroup$ The only awkward part seems to that you think product choices depend on products, their categories and the choice in the other category. Perhaps you'd like to elaborate a bit on this last dependency? Is it an ordered choice / framing effect? Or a cross-covariate constraint (e.g. get 1500 calories for as close to $10 as possible, or get one item from each major food group), or some other thing you have in mind. $\endgroup$ – conjugateprior Jul 15 '11 at 11:59
  • $\begingroup$ Does the above example clarify the problem? $\endgroup$ – fgregg Jul 20 '11 at 16:11
  • $\begingroup$ I don't share any of the intuitions about why 'we would expect' any of the things you say we would expect. Probably I don't take shopping seriously enough. So I'll try to back some out... $\endgroup$ – conjugateprior Jul 20 '11 at 18:21
  • $\begingroup$ Is it that salad greens and vinegar are more 'similar' in some way to each other than either is to gloves, which would lead the chooser to correlate the second step price decision on some sort of consistency grounds? This makes sense of the decision to choose jointly cheap salad and vinegar choices - they're red and yellow buses in the language of IIA violations. But what makes them choose expensive vinegar in the single choice situation? An independent but unstated budget constraint? $\endgroup$ – conjugateprior Jul 20 '11 at 18:21
  • $\begingroup$ One helpful observation may be that since the choice format is not determined by the chooser, you can, at least in principle, fit the correct model for all possible options, and determine the predicted choice probabilities by re-normalizing the chances of the remaining options (think of the Monte Hall problem) - provided you've got all the dependencies modelled. Which was why I asked about them. Might take some programming for the normalisation part of the estimation though. Best to start with the likelihood and work up. $\endgroup$ – conjugateprior Jul 20 '11 at 18:26
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Did you read this? http://www.jstor.org/pss/30038862 Edwards and Allenby seem to have the same basic setup as you, a multivariate probit, which you can find the code in the bayesm package.

It seems you should be able to evaluate the dependency by a test of if the probits are independent in the different scenarios by a liklihood ratio test on rho, just like the endogeneity tests people advocate. So run the seemingly unrelated multivariate probit, and do a likelihood ratio test on rho to see if the things impact each other. Here is an example of the test on rho in the SUR mv probit, about 2/3 of the way down: http://www.philender.com/courses/categorical/notes1/biprobit.html

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  • $\begingroup$ Thanks for this useful reference. In their article, Edwards and Allenby unfortunately collapse when a product choice is not offered and when a product is refused. $\endgroup$ – fgregg Jul 20 '11 at 16:11

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