## Tuesday, February 19, 2013

### When Discrete Choice Becomes a Rating Scale: Constant Sum Allocation

Why limit our discrete choice task to next purchase when we can ask about next ten purchases?  It does not seem appropriate to restrict choice modeling to one selection only when repeat purchases from the same choice set are made by the same individual buying different products at different times.  Similarly, a purchasing agent or a company buyer will make multiple purchases over time for different people.  Why not use choice modeling for such multiple purchases?

Everyone seems to be doing it, although they might use different names, calling it a constant sum, a chip allocation, or simply shares.  For example, the R package ChoiceModelR allows the dependent variable to be a proportion or share.  Statistical Innovations' Latent Gold Choice software permits constant sum data.  Sawtooth Software prefers to call it chip allocation in its CBC/HB system because one can "normalize" whenever numbers have been assigned to the alternatives before analyzing the data.

A specific example might be helpful.  Suppose that we were conducting a discrete choice study varying the size and price of six different coffee menu items, we might use the following directions.
"Please assume that every week day you buy your coffee from the same small vendor offering only six possible selections.   I will give you a menu listing six different items plus the option of getting your coffee somewhere else.  I would like you to tell me how many of the different alternatives you would select over the next two weeks?  It is as if you had 10 chips to allocate across the seven alternatives.  If you would buy the same coffee every day, you would place 10 on that one alternative.  If every day you would get your coffee somewhere else, you would place 10 on the 'Get Somewhere Else' alternative.  You are free to allocate the 10 chips across the seven alternatives in any way you wish as long as it shows what you would buy or not buy over the next 10 days."
On the surface, it makes sense to treat the choice exercise as yielding not one choice but ten separate choices.  It is as if the respondent made ten independent purchases, one each day over a two week period.  That is, we could pretend that the respondent actually saw 10 different choice sets, all with the same attribute levels, and made 10 separate choices.  You do not need to analyze the data in this manner, but it is probably the most straightforward way of thinking about the task and the resulting choice data.  Thus, the data remain essentially the same regardless of whether you analyze the numbers as replicate weights (Latent Gold Choice) or introduce a "total task weight" (Sawtooth CBC/HB).

If you have read my last post on incorporating preference construction into the choice modeling process, you may have already guessed that people are probably not very good at predicting their future behavior.  Diversification bias is one of the problems respondents encounter.  When individuals are asked to decide what they intend to consume over the course of several time periods in the future, their selections are more varied than what they actually will select when observed over the same time periods.  Thus, going to a grocery store once a week and making all your purchases for an entire week of dinners will produce more variety than deciding what you are in the mood for each evening and making separate trips to the store.  Fortunately, we know a great deal about how we simulate future events and predict our preferences for the outcomes of those simulations.  As retrospection is remembering the past, prospection is experiencing the future.  Unsurprisingly, systematic errors limit what we can learn about actual future behavior from today's intentions.