How to model number of days in the last week smoking cigarettes (0 to 7 - 'U' shaped)? I am currently analysing data where the outcome variable is 'U' shaped. The outcome variable asks 'how many of the last seven days have you smoked'. Most responses to this fall in the first (none) and last (all seven) categories. Because of this I do not think a count data model is appropriate.
What would be a good approach to modelling this variable?
 A: You might want to take a look at two-part (aka hurdle) count data models. A good place to start is Chapter 17 of Cameron and Trivedi's Microeconometrics using Stata. In fact, your smoking example is the one they use to motivate this. Essentially, you have one model to determine if a person takes up smoking, and then another one that determines how much if they decide to do it.
Another good source for overdispersed hurdle count data is Farbmacher (2011) SJ paper (scroll down to find it). Overdispersion happens when the (conditional) variance of your outcome exceeds the (conditional) mean, which is often the case with data like this.  
A: Think about the construct of interest
I'd think about the construct you are trying to measure. As Macro mentioned, it may be that your variable is largely reflecting the fact that people are either smokers or not smokers. If they are smokers, they will tend to smoke every day of the week, and if they are not smokers, they wont.
There might also be a third category of casual or occasional smokers. That said, your single item measure might not be the best way of discriminating between these three categories. So, if you are interested in the distinction between regular and casual smokers, then I'd look at incorporating some other indicators of casual smoking.
If you are interested in frequency or intensity of smoking, then your item is poor at measuring that. You would be better off asking about average frequency of smoking per day or some similar question.
General recommendations
Thus, I'd consider thinking more deeply about what you want to measure. But if you're stuck with the data you have, you might want to do one of a few different things:


*

*Recode the variable to none or one or more and predict using binary logistic regression.

*Recode the variable to none, one to six, and 7 and predict using multinomial logistic regression.

*Do no recoding and predict the variable using something like an ordered probit or ordered logistic regression.

