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I'm working with spatial data and I have the following log-linear model for count data. Let $y \sim Poisson(\lambda_{i})$ such that $$ \log \lambda_{i} = \text{x}_i^\top\beta_{} + \epsilon_{i} $$ such that $\epsilon_{ij}$ is the spatial effect at site $i$.

I have counts for each site that I want to analyze as a response variables and other auxiliary counts that the literature says are related and may drive the counts up or down. My question is what's the best practice for count data as covariates, should I use

  • Raw counts;
  • Proportions (I know the total population at each site);
  • $z$-scores.
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The formal parameter for Poisson is $P(\mu)$, where $\mu=n \lambda$. The rate parameter you are listing $\lambda=\frac{\mu}{n}$ is the ratio of number of events (failures) divided by a denominator such as time, area, or sub-population size from which the events were based. Each row in Poisson regression usually represents an $i$th sub-population with some outcome failure(success) event of interest tallied in the numerator, divided by the sub-population size or total follow-up time (person-years) in the denominator. What makes the problem Poisson is that $n\lambda(1-\lambda)<5$. For example, a Binomial, $B(n,p)$, approximates Poisson whenever $npq<5$, where $q=1-p$.

See this answer for data setup to understand the rate parameter, $\lambda$.

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