# Choosing alternatives to Poisson regression for overdispersed count data

I'm currently analyzing data from a series of behavioral experiments that all use the following measure. The participants in this experiment are asked to select clues that (fictitious) other people could use to help solve a series of 10 anagrams. The participants are led to believe that these other people will either gain or lose money, depending on their performance in solving the anagrams. The clues vary in how helpful they are. For example, for the anagram NUNGRIN, an anagram of RUNNING, three clues might be:

2. What you do in a marathon race (helpful)
3. Not always a healthy hobby (unhelpful)

To form the measure, I count the number of times (out of 10) a participant chooses an unhelpful clue for the other person. In the experiments, I'm using a variety of different manipulations to affect the helpfulness of the clues that people select.

Because the helpfulness / unhelpfulness measure is fairly strongly positively skewed (a large proportion of people always choose the 10 most helpful clues), and because the measure is a count variable, I've been using a Poisson Generalized Linear Model to analyze these data. However, when I did some more reading on Poisson regression, I discovered that because Poisson regression does not independently estimate the mean and variance of a distribution, it often underestimates the variance in a set of data. I started to investigate alternatives to Poisson regression, such as quasipoisson regression or negative binomial regression. However, I admit that I'm rather new to these kinds of models, so I'm coming here for advice.

Does anybody have any recommendations about which model to use for this kind of data? Are there any other considerations that I should be aware of (for example, is one particular model more powerful than another?)? What sort of diagnostics should I look at to determine if the model I select is handling my data appropriately?

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What about a robust variance/covariance estimator to relax the assumption that the variance is equal to the mean? – boscovich May 4 '12 at 16:31
Since they are count data and non-negative, what about quassi-poisson or a negative binomial regression model, that accounts for dispersion? – Arun May 4 '12 at 16:47
I've thought about using a quasi-poisson or negative binomial model, but what I don't understand is what sort of diagonistics to look at to assure myself that I'm modeling my data appropriately. Since there are several alternatives (quasi-poisson, negative binomial, and "zero-augmented" models), I'm also wondering whether there is a good way to choose between these alternatives. For example, is one method generally more powerful than the others? – Patrick S. Forscher May 4 '12 at 16:51
That depends on the data. Why not fit all of them to your data (Poisson, Negative binomial, zero-inflated Poisson and negative binomial, hurdle models for those in question) and compare them via say, AIC or BIC? See cran.r-project.org/web/packages/pscl/vignettes/countreg.pdf Then choose the one best suited for your data. You might also use quasi-likelihood models, but that's a matter of taste, I don't like them so much. – Momo May 4 '12 at 17:39
To check what distribution might be a good model for your response, you can use the vcd::distplot function. – Momo May 4 '12 at 17:41

Your outcome is the number of helpful clues out of 10, which is a binomial random variable. So you should analyze it with some sort of binomial regression, probably quasi-binomial to allow for overdispersion. Note that the Poisson and the misleadingly named negative binomial distributions are suited for unbounded count data.

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I mentioned the negative binomial because it is an overdispersed alternative to the Poisson which the poser suggested initially. Since each respondent has x/10 clues it could be binomial but that for each of the 10 clues there is a fixed probability pi for the ith respondent and the occurrences are independent. That may nit be the case. – Michael Chernick May 5 '12 at 0:36
beta-binomial is another possibility (beta-binomial is to binomial as negative binomial is to Poisson). betabin in the aod package will do it. – Ben Bolker May 5 '12 at 1:43
Thanks, Aniko, and thanks Ben for the package names. – Patrick S. Forscher May 5 '12 at 16:48

I too would recommend looking at the negative binomial if the possible outcomes were infinite like for the Poisson. You may want to consult one of Joe Hilbe's books. He has one on GEE and one on negative binomial regression which he contrasts with Poisson regression. But as was pointed out by Aniko there are only 10 clues so each respondent can only have 0, 1, 2, 3,..., 10 and hence neither Poisson nor negative exponential is appropriate.

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Good point by @Aniko. Another choice is Beta regression. There was a paper with the title "A Better Lemon Squeezer" that gave a lot of information on this method.

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But beta would be used to model a proportion and not a count variable on a finite set of integers. – Michael Chernick May 5 '12 at 0:43
It has wider uses, @MichaelChernick, see the article, which is quite good. – Peter Flom May 5 '12 at 11:08
@PeterFlom It also can't handle data on the interval [0,1], only (0,1). – colin Jul 8 at 17:47