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I have fit a poisson regression model to my count data. The distribution of my dependent variable followed Poisson distribution and mean is almost equal to variance. But when I fit a Poisson regression model, some of goodness of fit test like deviance and pearson chi-square test showed that was not the case though my model fits reasonable well to the data and the profiles seem like the way its supposed to.

I read on one of the pages that deviance test is wrong about 94% of the time. Is there any other way to asses the validity of the model?

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In the case of overdispersion, you can use quasi-poisson models where the mean and variance can be decoupled. Regarding fit, I usually compare my model using Pearson’s residuals (which is what this post recommends too: Why goodness of fit via deviance and chisq is poor for poisson regression (glm)?).

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One of the most important things to check before going for a Poisson model is whether your data exhibit overdispersion.

Remember Poisson is very special in that it has one parameter. If the variance of your data is significantly greater than the mean, your model is misspecified.

Popular alternative is the negative binomial model, but think before you jump there.

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