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I'm using a Poisson regression to model count data (number of orders). My observation lengths are different, so I tried to include an offset variable in the model. The problem is that when I estimate length variable coefficient instead of fixing it to 1, model's quality becomes substantially better. But I'm not sure if it is theoretically justified which would be great. Is it a way to explain this?

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  • $\begingroup$ If it is a log-linear model, then the offset should be log(length). Also, how are you determining model quality? $\endgroup$ – alex keil Apr 18 '14 at 3:33
  • $\begingroup$ @Alexkeil yeah, I'm using a log of length. since my main goal is prediction qualiyy I use different measures like RMSE, MAE, few correlation coefficients $\endgroup$ – Evgenii Nikitin Apr 18 '14 at 5:43
  • $\begingroup$ well, with exposure as a free variable insted of as an offset, the model is more general so the fit is more flexible! Maybe this model flexibility is used to catch some other model problem ... $\endgroup$ – kjetil b halvorsen May 22 '14 at 17:34
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First make sure that the length really is the exposure. For example, if you were counting bacteria on a Petri dish the surface area would be a better measure of exposure than the radius of the Petri dish. In your case, this is very unlikely to be the case since the length of the interval sounds like a good exposure measure for the number of orders.

The other issue is whether the underlying poisson process is heterogeneous or not. If the intensity of the number of orders vary through out the day, then fitting a single poisson distribution will capture average behaviour. By itself, this would not cause a good measure of exposure to have a coefficient greater than one. If, however, you did not select the interval lengths randomly, then the interval length may be correlated with the intensity of the process. For example, if this data was recorded by hand by employees, they will record less often during busy times, so long times between recordings (a long exposure length) will tend to coincide with high traffic.

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  • $\begingroup$ Thank you for your answer. You're right, first concern is not applicable because my data has economic nature. The dependent variable is a number of orders for DRTV ads. So, the only possible exposure is the length after the ad and before the next one. Then, you're right, in fact, the time of day is one of the most important factors in my model. And what would you suggest? Building 48 Poisson regression models for different half-hour time slots doesn't seem to be a goos idea. $\endgroup$ – Evgenii Nikitin Apr 18 '14 at 5:45
  • $\begingroup$ What software are you using? I know that another way to do an offset is by dividing the dependent variable by the exposure and then applying weights to the regression to correct the variance. If the software implements the offset by just dividing the dependent variable by the exposure then a variance adjustment is needed. $\endgroup$ – Bruno Apr 18 '14 at 6:07
  • $\begingroup$ Stata. I'm not sure, but my guess is that it just divides the variable by the exposure. How do I do the variance adjustment? $\endgroup$ – Evgenii Nikitin Apr 18 '14 at 6:14
  • $\begingroup$ I think Stata does not just divide. If you are curious, you would use the aweight option with length as the weight variable, but this is not it. Did you use exposure() or offset() to specify the offset? $\endgroup$ – Bruno Apr 18 '14 at 22:51

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