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 Goodness of fit with Poisson regression

Prism can compute goodness-of-fit of Poission in four ways, selectable in the Diagnostics tab.

## Pseudo R-Squared

It is not possible to compute R2 with Poisson regression models. Instead, Prism reports the pseudo R2. You can interpret it as you do a regular R2. This is the simplest goodness-of-fit measure to understand, so we recommend it.

Pseudo R2is computed from log-likelihoods of three models: LLo, the log-likelihood of horizontal-line model; LLfit, the log-likelihood of the model you chose; and LLmax, the maximum log-likelihood possible, which would occur when the actual responses exactly equal the predicted responses so the curve goes through every point and all the residuals equal 0.0. The equation that computes the pseudo R2 is:

R2 = (LLfit - LLo) / (LLmax - LLo)

## The deviance is twice the difference between the maximum possible log-likelihood (see above) and the log-likelihood of the fitted model. The formula for the deviance is D=2(LLmax - LLfit). This is also called G2.

### Dispersion ratio

When data are sampled from the Poisson distribution, the variance equals the mean. Prism can report the variance-to-mean ratio (VMR), called the dispersion ratio. Prism reports the degree of overdispersion with a value phi. . If phi is much greater than 1.0, then the actual variance of points around the curve is greater than the mean, and the Poisson model may not be appropriate. This is called overdispersion. Some programs offer extensions to Poisson regression to deal with overdispersion, but Prism does not (let us know if you need this).

## AICc

The AICc is useful only if you separately fit the same data to three or more models. You can then use the AICc to choose between them. But note that it only makes sense to compare AICc between fits when the only difference is the model you chose. If the data  are not identical between fits, then any comparison of AICc values would be meaningless. It is also essential that the weighting, or regression method, be the same for all the fits. If you use Poisson regression for one fit, you need to use it for all.