# When to use offset() in negative binomial/poisson GLMs in R

I'm trying to detect relationships between species abundances (counts) and time (years) for many species using either Negative Binomial or Poisson regressions (depending on degree of dispersion). Sampling time (minutes) is not the same for all collections so my questions are:

1) What is the best way of determining when to use negative binomial vs. Poisson?

2) Is this an appropriate instance to include sampling time in an offset term? In most cases, sampling occurs for 10 minutes, but it is sometimes 15 or 20 minutes.

Any suggestions or advice would be appreciated.

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1) What is the best way of determining when to use neg. binom. vs. poisson?

A common way (not necessarily the best --- what's 'best' depends on your criteria for bestness) to decide this would be to see if there's overdispersion in a Poisson model (e.g. by looking at the residual deviance.

For example, look at summary(glm(count~spray,InsectSprays,family=poisson)) - this has a residual deviance of 98.33 for 66 df. That's about 50% larger than we'd expect, so it's probably big enough that it could matter for your inference.

[If you want a formal test, pchisq(98.33,66,lower.tail=FALSE), but formal testing of assumptions is generally answering the wrong question.]

So I'd be inclined to consider a negative binomial for that case.

More generally, if you're not reasonably confident that the Poisson makes sense, you could simply use negative binomial as a default, since it encompasses the Poisson as a limiting case.

2) Is this an appropriate instance to include sampling time in an offset term?

Yes, that's appropriate, and it would be my first instinct to include sampling time as an offset (rather than a predictor), since count would be expected to simply be proportional to the length of the sampling interval.

• I would consider adding log(sampling_time) as a predictor and seeing whether its coefficient is very different from 1 (proportionality). I can imagine, and have seen, plenty of situations where this is not true, e.g. increase in count decelerates as sampling time increases ... – Ben Bolker Dec 17 '14 at 3:14
• @BenBolker It's certainly possible that the coefficient isn't reasonably consistent with 1; if I was less than confident in it, I'd be more inclined to use a diagnostic tool rather than formal test, but you could certainly do a test. If I were to test it, I'd even consider putting it in both places (assuming it will let you without making a dummy copy or something) and see whether the remaining effect differed from 0. – Glen_b Dec 17 '14 at 3:31
• At least in R, you can do that. I believe there's an example in the "owls" project at groups.nceas.ucsb.edu/non-linear-modeling/projects – Ben Bolker Dec 17 '14 at 3:36