Re: standardising weights in Poisson models?

klange wrote: 
Hi all,

I have had a colleague explain the following problem to me. I hope it's
clear and that someone can offer some suggestions. If anyone has any
questions then I'll do my best to find out more info and clarify.

They are looking at incidence rates of a certain kind of injury, broken
down by age categories. They obviously have data on the number of
injuries (#inj) and the population size (#popn) for each age group, for
each of 9 years. In addition they have 'age standardised population
weights' (wght). These weights account for the fact that the age
distribution in the population changes over time (and I believe are
based on data from the last national census).

The usual way they use this information is to calculate #inj / #popn *
wght for each age-by-year combination. These weighted rates are then
all summed together to give an overall incidence rate.

They now would like to use this data in a Poisson regression model so
they can look at the rates over time and investigate possible trends
that may be present. The question is how to incorporate the
age-standardised weights into the Poisson model?

I'm afraid I don't know enough about how the weights are calculated to
know what is appropriate so I'm hoping that this will sound familiar to
someone who deals with this kind of data. I also may be using confusing
terminology but am happy to try clarify anything if needed.

Thanks for any suggestions or references to articles where this kind of
analysis has been carried out.

Is the term "weights" just a distraction? I don't work in this field either, but from your description, you can write down the expected number of individuals that would be observed to have the injury (E_it) in age class i at time t as

E_it = p_it * N_it / w_it

where N_it is the population size, w_it is the weight (I assume that N_it / w_it is an estimate of the actual population size). p_it is the probability that an individual has an injury in the time period: this is the only factor that is not observed, and is equivalent to the incidence rate. It's this that you want to model further.

In a Poisson regression, the model is linear on the log scale, so you have

log(E_it) = log(p_it) + log(N_it / w_it)

where log(N_it / w_it) is a known constant. You can simply include it in your analysis as an offset term.

HTH

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FIN-00014 University of Helsinki
Finland

Telephone: +358-9-191 51479
Mobile: +358 50 599 0540
Fax:  +358-9-191 51400
WWW:  http://www.RNI.Helsinki.FI/~boh/
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.

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