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Why does Poisson Regression require an offset variable when we model rates instead of a count?

In a Poisson Distribution, lambda itself is a rate (e.g. cars per minute). It seems like the Poisson Regression should be naturally able to handle rates since its based on the Poisson Distribution which itself is based on a rate.

So why can't I just use the basic Poisson Regression and replace the count variable with the equivalent rate? Is there some special reason for this?

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  • $\begingroup$ What do you mean by 'modelling rates instead of a count'? $\endgroup$ Feb 28 at 15:48
  • $\begingroup$ Rate=Count/Exposure @GeorgeSavva $\endgroup$
    – Michael M
    Feb 28 at 16:07
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    $\begingroup$ You can also directly model rates, using the exposure as weight. The model coefficients will be identical to a model of with log(Exposure) as offset. $\endgroup$
    – Michael M
    Feb 28 at 16:08

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Lambda is the average number of events in a fixed interval (time or space). For example the distribution of chewing gums on each tile on the sidewalk - shamelessly stolen from Wikipedia ;)

enter image description here

Now what would you do when the tiles were different sizes (as some of them are in this snapshot)? You can associate your counts to a varying interval size via the offset specification. So the offset adjust your counts when the interval associated with your counts changes over your experiment.

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  • $\begingroup$ thank you for this cool example! but why cant I just replace the count response variable with a rate variable and carry on anyways? $\endgroup$ Feb 28 at 16:32
  • $\begingroup$ @pq44pq Consider 100 pieces of gum in 100 square meters of pavement vs 1 piece of gum on 1 square meter. These are the same rate but do not carry the same amount of information. $\endgroup$ Feb 28 at 16:34
  • $\begingroup$ @pq44pq have a look at Gung's answer here: stats.stackexchange.com/a/175365/32477 or ocram's answer here: stats.stackexchange.com/a/11183/32477 This goes into much more detail. $\endgroup$
    – Stefan
    Feb 28 at 16:37
  • $\begingroup$ thanks! I will read these answers... btw do either of you have any opinions about the previous question I asked? stats.stackexchange.com/questions/641228/… can random sampling be useful here? $\endgroup$ Feb 28 at 16:42

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