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I would like to know the linear expression of weight and offset in terms of poisson regression in glm.

for instance for offset 

   glm( y ~ x + offset(of), data, family=poisson(link="log"))

the above glm model can be expressed as:

  log(y) = constant + beta*x + log(of).

My question is what is the equivalent expression for the following glm model:

glm(y ~ x, weight=weight, data, family=poisson(link="log")).

From my understand offset and weight are different and impact the estimates differently.

tk001
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1 Answers1

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I think your question may be prompted by the fact that Poisson log-linear glms can be used to model either counts or frequencies. If count is a vector of counts and of is a vector of positive offsets, then the following two fits are equivalent:

fit.count <- glm(count ~ x + offset(of), family=poisson)

and

frequency <- counts / of
fit.freq <- glm(frequency ~ x, family=poisson, weights=of)

Typically here of is the total exposure time. In the first version, the counts are modeled in the usual glm way, and the fact that the expected counts are proportional to exposure time is coded in the offset.

In the second version, we compute the frequency of counts per unit exposure time. This explicitly removes the dependence of the counts on exposure time but changes the variance. The weights argument then has to be used to reflect the fact that frequencies based on the longer exposure times are more precise.

Although the two response variables are on different scales, the two glm fits are completely equivalent in all important respects. They will yield the same coefficient estimates, the same deviances and the same p-values.

The equivalence of the two models has previously been pointed out by Alan Chalk on this forum: Can Weights and Offset lead to similar results in poisson regression?

This same trick can be used for any glm with an offset and a log-link. It can also be used for binomial glms where the response variable can be either the binomial count or the proportion of successes.

Gordon Smyth
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