Count data with variance smaller than mean

[Elin Vimefall]

Hi

I try to analyze the number of income sources for households.
Since this is a count data I thought about using a poission of negative binomial model.
However; I turns out my variance is smaller than the mean.
What is the appropriate model to use in this situation?

Here are some statistics from my data:
Observations: 7001
Mean 3.02
Standard deviation: 1.15
Variance: 1.31
Skewness 0.39

I would really appreciate your help on this.

//Elin Vimefall

[Maarten Buis]

Could you tell a bit more?

I suspect that very few people have 0 income sources, that part of the population tends to die out (starve) pretty quickly.

Depending on how detailed the income sources were asked and the country being investigated it is possible that a substantial share of the population has just 1 income source.

In that case you want a model that excludes the possibility of a 0 and treats the 1s as a special case. But that is all speculation on my part.

[Elin Vimefall]

The max number of income sources are 8, and the min 0.
24 Households (out of 7001) have 0 income sources. However, this must be do to misreporting (since we include subsistence farming)

[Nick Cox]

My bias, which is far from original, is that Poisson regression is conventionally poorly named and in any case I wouldn't bail out just because marginal mean is not equal to marginal variance. The great appeal of Poisson regression is that a logarithmic link function often, indeed perhaps usually, works better for non-negative responses, how they behave, and the kind of predictions or fits that we need.

Bill Gould's blog post http://blog.stata.com/tag/poisson-regression/ is canonical here.

That said, your problems with zeros and with measurement bias might cause problems with any model.

[Marcos Almeida]

Elin, maybe you should think about performing a generalized poisson model if you are really sure you bumped into a (rare) case of underdispersion.

I recently got an interesting book on the subject: Joseph Hilbe, Modeling Count Data, Cambridge University Press, 2014.

Here, the link to Stata Bookstore. http://www.stata.com/bookstore/modeling-count-data/

But if you are mostly interested in performing the analysis under a generalized poisson model, you could also check the link on "supplements": it's the author's page, where you find further information and commands useful for performing the task: http://works.bepress.com/joseph_hilbe/


Best,

Marcos

[Elin Vimefall]

However when I run the Poisson (with ,robust)
The Deviance goodness-of-fit and the Pearson goodness-of-fit tells me that it is inappropirate.
Should I still use it?

[Elin Vimefall]

Thanks all for your comment.

Just to be sure: Marcos if I do sum y, detail and get a mean of 3,0 and a varaince of 1.3.
Doesn't that meen that it is underdispersion?

I will try to find the book, but unfortantly I do have a deadline...

[Marcos Almeida]

Hi, Ellin,

You can also type "search gpoisson" in Stata and get some helpful information.

Best,

Marcos

[Elin Vimefall]

Thank you all, you saved my day! (or at least a part of my paper)
I think gpoisson is exactly what I need.


Best

/Elin VImefall