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For Poisson regression, the assumption is that Y has a Poisson distribution. Is the same assumption true for Quasi-Poisson regression?

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NO. The quasi-Poisson **is not a distribution* at all, it is an estimation method. There is no distribution model that leads to the quasi-Poisson estimating equations, but still it is found to be useful because it has good asymptotic properties, and is a way to get around the often unreasonable property of the Poisson distribution that the variance equals the mean.

For more information, for instance Quasi-likelihood/Quasi Poisson or look through this list.

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