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I have a sales data for a certain type of grocery products at stores' transaction-level (sales data gathered through cashiers' scanners). As you can imagine, for the most part, the number of units sold per transaction is rather small, which results in this non-normal distribution of sales:

enter image description here

I do have infrequent extremes (not shown here) that go into hundreds of units and (rarely) thousands.

Since the number of units sold is my dependent variable, I seek to find the best data distribution to fit the model. In the absence of normality, OLS is not an option, so I wonder if the use of a count model like Poisson, negative binomial or gamma could be justified here? (I compared the three in Stata, and the Gamma model fits best).

Any suggestions on the appropriateness of the gamma distribution for the transaction-level sales data? Other options I could consider?

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  • $\begingroup$ Have you tried to fit a NB distribution, shifting your var by 1? So you would be modelling "the number of units greater than 1". $\endgroup$
    – user101115
    Jan 19 '16 at 0:51
  • $\begingroup$ Can you please be more specific? $\endgroup$
    – Olga
    Jan 19 '16 at 1:21
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    $\begingroup$ I'd lean toward considering the log-series distribution and the negative binomial (indexed from 1 not 0) as potential models. You might also consider mixtures (e.g. mixtures of negative binomial or perhaps more simply, mixtures of geometric rv's). $\endgroup$
    – Glen_b
    Jan 19 '16 at 2:53
  • $\begingroup$ By saying "indexed 1", you mean, all transactions with zero units sold must be, for example, transformed into transactions with 1 unit sold? If so, in a model like this, how do you interpret the intercept? $\endgroup$
    – Olga
    Jan 19 '16 at 4:59
  • $\begingroup$ Are these sales of the EXACT SAME PRODUCT? Did you try plotting the distribution in log-log scale? And finally, could you accept some answer :)? $\endgroup$
    – famargar
    May 8 '17 at 13:10
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If I were you, I would probably try the Negative Binomial distribution first, which includes the Poisson model as a limiting case. The NB distribution is quite flexible as it has an extra parameter and so it is frequently used to counter overdispersion. By all means give it a try.

The reason I have excluded the Gamma distribution is that your response is not continuous to begin with. If you are bent on the Gamma model nevertheless, at least compare the AIC values with the NB model. I am not sure what you should be looking for in Stata but usually it is the lower the better.

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  • $\begingroup$ My AIC values are the lowest for Gamma (4) vs. all others mentioned (5-8). BIC is very large and negative. $\endgroup$
    – Olga
    Jan 19 '16 at 0:38
  • $\begingroup$ @Olga 'All others' include the NB model as well? $\endgroup$
    – JohnK
    Jan 19 '16 at 0:41
  • $\begingroup$ At this stage, I am exploring all options. I will try the negative binomial. One thing: can you tell me conceptually why you think my data is count? I feel that it doesn't fit a classic textbook definition of being count. Am I wrong? $\endgroup$
    – Olga
    Jan 19 '16 at 0:45
  • $\begingroup$ Yes, NB and Poisson. But the difference in AICs is miniscule. BIC is the worst with Gamma. $\endgroup$
    – Olga
    Jan 19 '16 at 0:46
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    $\begingroup$ @Olga You mentioned 'number of units sold per transaction'. In my book, that is a count variable. At the end of the day, however, it is all up to you. If you think Gamma is the best fit, go for it. $\endgroup$
    – JohnK
    Jan 19 '16 at 0:48
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I wonder if you tried fitting your data with something like a powerlaw distribution ($ax^{-b}$ with both $a$ and $b$ positive) or its discrete equivalent, Zeta or Yule-Simon distribution (thanks Andrey!). The options above worked in my real world case, which appears to be very similar to yours.

If you want to stick with the gamma, then I agree on the usage of the negative binomial rather than the Gamma - the binomial is basically the discrete equivalent of the Gamma distribution. A discrete is indeed more appropriate as you are counting "units" of stuff, thus your variable is an integer.

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    $\begingroup$ I think we can consider zeta or Yule-Simon distributions as discrete equivalents. $\endgroup$ Jan 4 '17 at 11:07

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