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While analysing the effect of environmental data on the activity of an animal species (the latter given as count data) I am fitting negative binomial GLMs with one predictor using the MASS library in R. Unfortunatley, the data set is very small (n=7 to 9).

In some cases, the theta value in glm.nb gets very large (accompanied with the warning "iteration limit reached"), possibly indicating that there's no overdispersion and a poisson GLM might be a better choice. Using a poisson GLM, however, a residual deviance of e.g. 150 on 7 degrees of freedom indicates that there actually is overdispersion - or did I miss something?

Using a quasi-poisson GLM works, but I would like to retain ML-based measures such as AIC and Vuong test for model comparison. Any suggestions for alternative approaches are greatly appreciated!

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It doesn't necessarily mean that there is overdispersion (though it could), just that a saturated model may be a better fit. If you only have 7-9 observations, it will be very difficult to accurately test for overdispersion unless you have some values that are just way out there under a Poisson assumption.

Another option you might look into is using the Poisson model but using a transformed value of your predictor rather than a linear fit on the raw variable. If it looks like the larger values of the predictor are where the Y-values are off more, you could try using something like a squared value of the predictor, or if it's the opposite then maybe a log-transform of the predictor.

Thinking about overdispersion in a count model is always a good idea, but it does introduce complexity into the model. With so few data points, your best approach might be to keep it as simple as possible.

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  • $\begingroup$ Sound statistics with small data sets are always difficult. So you'd recommend to stay with a Poisson GLM, possibly transforming the predictors if necessary? I've forgot to supply some data: x <- c(10.4,6.6,7.3,11,14.6,13.2,10.4,8.4,1.8); y <- c(0,1,10,34,52,58,8,19,4) $\endgroup$ – Djypvatn Aug 14 '13 at 10:52
  • $\begingroup$ Ya, you could consider squaring the x-value. With that data, you're definitely going to have something that looks like overdispersion or a lack of fit because you've got that one data point (10,0) at the start. So I wouldn't worry too much about that. Even with that, the Poisson model fitting y~x^2 gives a statistically significant intercept and coefficient so you can be pretty confident that the inference about the coefficient is relatively sound. Again, with only 8 data points you can only worry so much about the overdispersion. $\endgroup$ – Mike Nute Aug 15 '13 at 4:55

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