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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?

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    $\begingroup$ What are you trying to get at by using the number of days as the dependent variable? It seems (and your data seems to verify this) that people are either smokers or they are not so it could be appropriately viewed as a binary measurement. If you simply must use the number of days as the outcome, then you could use an ordinal regression model (e.g. the proportional odds model) but I'm not sure what added understanding that would give since your response distribution is basically binary. $\endgroup$
    – Macro
    Commented Feb 21, 2013 at 13:36
  • $\begingroup$ Just to clarify, you say "the outcome variable asks", do you mean "the outcome variable indicates"? I.e. the "outcome variable" is actually the variable predicted by the regression? $\endgroup$
    – Wayne
    Commented Feb 21, 2013 at 16:13
  • $\begingroup$ @Macro: It's "basically binary", but perhaps they are concerned with the middle outcomes -- which are tail events as it were. For example, maybe they are looking at smokers who are trying to quit and possible relapse triggers? (And perhaps once you relapse, it's highly likely that you'll stay in a relapsed state for a while.) Or perhaps they're looking at non-addicted smokers (who do exist), to see if events on certain days tend to trigger smoking. $\endgroup$
    – Wayne
    Commented Feb 21, 2013 at 16:18
  • $\begingroup$ Thank you for your help. Sorry- I realise my question does not make complete sense. The variable I am predicting is how many days a week a respondent says that they have smoked. From the data I can see that responses are most frequent at zero and seven, however, I think it would be inefficient if I make this variable binary. $\endgroup$
    – Becky
    Commented Feb 21, 2013 at 16:53
  • $\begingroup$ Answers here will be applicable to this case as well, How to model this odd-shaped distribution (almost a reverse-J). $\endgroup$
    – Andy W
    Commented Feb 21, 2013 at 18:05

2 Answers 2

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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.

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  • $\begingroup$ I would be concerned with the censoring at the top of the distribution as well. $\endgroup$
    – Andy W
    Commented Feb 21, 2013 at 18:11
  • $\begingroup$ @AndyW You might have to elaborate on this. I am not sure I understand what it would mean for someone to smoke for 8 or more days per week. Is some contexts, like when stadium demand exceeds seating capacity, this makes sense, but not here. $\endgroup$
    – dimitriy
    Commented Feb 21, 2013 at 18:17
  • $\begingroup$ On second thought after a bit more coffee, I see what you're getting at. The outcome is bounded above by 7, which violates the nonnegative integer count assumption, which is indeed a problem. I wonder if this can be recast as a two-part proportion model if you rescale the outcome to be fraction of the week. Maybe one can do this using frm from SSC. $\endgroup$
    – dimitriy
    Commented Feb 21, 2013 at 18:36
  • $\begingroup$ It doesn't really matter how you logically interpret it, there is an upper limit on the outcome that will likely not be approximated well with any count model. It is always possible if you have a really excellent exogenous predictor of when someone will smoke all of the last 7 days there won't be a problem with predictions, but I wouldn't assume that in the (vast) majority of situations. $\endgroup$
    – Andy W
    Commented Feb 21, 2013 at 18:36
  • $\begingroup$ I'm not following I wonder if this can be recast as a two-part proportion model if you rescale the outcome to be fraction of the week. Maybe one can do this using frm from SSC.. Off the cuff it might be reasonable to approach this an an ordinal regression problem as well. $\endgroup$
    – Andy W
    Commented Feb 21, 2013 at 19:40
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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.
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