# Logarithmic offset + Scaling Counts + Multiplying by Scalar + Poisson Regression

I am working with a contingency table where I know that the population sizes from which the counts are derived are very different. From what I have been able to gather, an appropriate way to adjust for this is to use a logarithmic offset of my population sizes.

However...

(1) Is it appropriate to scale the counts based on the population size, such that a new value is ((old/population size) * old).

(2) If this is fine, is it ever appropriate to multiply values by a scalar, and then possibly round, prior to performing Poisson regression? My scaled counts end up quite low in some cases (<0.1). There is signal, as I may have counts such as 0.02, 0.1, 1.1 for a given column in the matrix. Obviously, though, rounding is out of the question.

Unfortunately, I do not have access to the data that the table was populated with. If I did, then I would anticipate that a zero-inflated model might be a good fit.

• Whether things are appropriate rather depends on what you are trying to do. Is this a potential Poisson regression problem? If so, then a dependent variable of a count and an offset of log(population size) is appropriate for smaller counts and a Binomial model for larger ones with N=population size and no offset. But I think we need a bit more detail on the task. – conjugateprior Jul 9 '15 at 8:44
• I have an 8x6 contingency table, with the objective to determine if there is a statistical difference for the 8 "outcomes" given counts for 6 "predictors". So, it seems that a Poisson Regression is in order, though I have seen it elsewhere to use a loglinear regression. It seems that the Poisson is a type of loglinear, as the link=log. This all said, how does what I am trying to do affect whether it is/is not appropriate to scale and multiply by a scalar??? Thank you. – BeyondApplied Jul 9 '15 at 19:54
• Well, if you have a model that assumes a count outcome then turning that into a real number would seem to be, prime facie, a bad idea. – conjugateprior Jul 10 '15 at 19:18
• Your scaling involves squaring counts and dividing them by population size. Why? – conjugateprior Jul 10 '15 at 19:22
• So let's see - you have 8 outcomes as rows, and for each of these, 6 predictors as columns, plus a population size for each row. And you wonder whether the 8 outcomes are 'the same' in some way. Is that a reasonable formulation of the question? I'm asking because without knowing what the question is it's hard to tell what the scaling is meant to be doing for you as you try to answer it, and therefore also hard to tell whether it's succeeding. – conjugateprior Jul 10 '15 at 19:31