I'm confused about the interpretation of interaction terms in poisson regressions. Here's a hypothetical dataset
Id Group Condition Error rate 1 A X 1 1 A Y 2 2 B X 3 2 B Y 6
Imagine that individuals 1 and 2 are duplicated a number times (with random noise) to make two groups. If we analyse this data using a poisson regression (as it's count data) there will be no Group by Condition interaction (in both groups, error rates increase by 100% in condition Y, i.e. an Incidence Rate Ratio of 2). If we analyse this data using an ANOVA (and don't transform data) it is likely to yield a significant Group by Condition interaction (I've simulated this). While the poisson regression looks at proportional differences across conditions, the ANOVA is more sensitive to absolute differences.
It seems to me that because the poisson regression uses a multiplicative scale (via log transformation), an additive effect under the poisson regression will become multiplicative when the same data is analysed using OLS (or ANOVA).
So which interaction term is more "truthful"? Is it just a matter of interpretation? Or are there any a priori reasons for choosing one over the other. For example, could you argue that because count data exhibits ratio properties we should be analysing ratios and not intervals (according to Stevens)?
Thanks and hope this makes sense.