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    $\begingroup$ You mention that you should use the log transformation of area (as the Poisson model uses the log link) in your answer but your code doesn't do the transformation. I don't think offset applies the transformation by default but it has been a while since I've used offset. $\endgroup$ Commented Aug 31, 2016 at 15:34
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    $\begingroup$ Note that offset() does not apply the transformation by default; it merely forces the coefficient to be 1. See, eg, here. $\endgroup$ Commented Aug 31, 2016 at 17:49
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    $\begingroup$ Thanks for the comments. So should I change my data back to counts rather than densities and include area as a separate variable? I have also been advised on another forum to do a gamma or inverse Gaussian model and change my zero values to 0.00001 if I were to keep the data as densities, do you think that would also be appropriate? $\endgroup$ Commented Sep 1, 2016 at 7:40
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    $\begingroup$ @JDL It is not right to keep the non-integer density as the response once log(area) is included as an offset. The log link-function and the log(area) offset amounts to the a priori reasonable assumtion that $E(\mathrm{count}) = \exp(\beta^T x)\mathrm{area} = \exp(\beta^T x + \log(\mathrm{area}))$, that is, direct proportionality the expected number of fish and the size of each area. This implies that the expected density $E(\mathrm{count}/\mathrm{area}) = E(\mathrm{count})/\mathrm{area} = \exp(\beta^T x)$ is independent of log(area). $\endgroup$ Commented Sep 1, 2016 at 11:36
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    $\begingroup$ You cannot use the densities as the response. You must use the original counts as your response. Including the offset will automatically make the count response equivalent to densities in the correct manner. $\endgroup$ Commented Sep 1, 2016 at 14:51