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Let's say I have an explanatory variable and a response variable that represents counts. I want to see if the explanatory variable can predicts counts. I'm aware the response variable is overdispersed. What I probably should do is analyse the data with a generalised linear model using the negative binomial distribution. But let's say I ignore the overdispersion and analyse the data using poisson regression.

What would be consequences of analysing these data with a poisson distribution? Would consequences be the same as ignoring heteroscedasticity in linear regression therefore incorrect standard errors/P-values?

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  • $\begingroup$ unless there are multiple observations that share exactly the same sets of predictions, overdispersion is unidentifiable in binary data. $\endgroup$ – Ben Bolker Mar 30 '14 at 17:57
  • $\begingroup$ Question edited. I'm interested in whether the same issues that apply in linear regression also apply in generalised linear models $\endgroup$ – luciano Mar 30 '14 at 18:16
  • $\begingroup$ Ben: regarding overdispersion in binary data, you might want to respond to this question: stats.stackexchange.com/questions/91597/… $\endgroup$ – luciano Mar 30 '14 at 18:21
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The problem is in some sense "worse" than heteroskedasticity in linear models. At least there your variance estimate will in some sense be a kind of 'average' value in the data.

What happens is that with a Poisson model, the variance is set equal to the mean. In an overdispersed model the variance is large than the mean. All your standard errors for parameter estimates will be based on the 'variance=mean' assumption.

One alternative is to fit a quasi-Poisson regression; it will scale the parameter variance estimates for the variation in the data (by using a variance of $\phi \mu$ with $\phi$ being able to be larger than 1).

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  • $\begingroup$ If you are going down the quasi-likelihood path with a Poisson, why not just use a normal approximation and a regular linear regression framework with appropriate transformation of the response (e.g., logarithmic ) $\endgroup$ – Rider_X Mar 31 '14 at 2:22
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    $\begingroup$ @Rider_X Several reasons: (1) you actually have a model for the mean response, and no bias issues with backtransformation of a fit. (2) generally greater similarity of output to what one will get with negative binomial (the NB model is what I would do, but the OP seems to hesitate to do that, yet is willing to contemplate the Poisson, so I thought they might entertain something close to but better than the Poisson) (3) relative convenience. $\endgroup$ – Glen_b Mar 31 '14 at 2:30

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