# Poisson regression with interaction

Im am trying to identify the effect of an antiviral compound. Therefore, I count the amount of vRNA spots per cell (=dependent variable) which is modelled as a logistic poisson regression. Each cell is a different observation. The compound works on two levels:

1. reduces the basal transcription of the virus -> reduction in vRNA spots per cell
2. reduces reactivation -> reduction of vRNA spots when the cells are previously reactivated

I've tested the compound at 5 different concentrations (3.12;6.25;12.5;25;50). The model contains two indicator variables, the concentration of the compound(categorical: 5 different concentrations) and reactivation (yes/no). There was an interaction between concentration and reactivation so the model can be summarised as follows:

Logit P = β0 + β1 Concentration + β2 reactivation + β3 (concentration x reactivation)

Based on the above parameters I made the following conclusions and I would like to know if these interpretations are correct:

1. When the cells are not treated or reactivated, we expect the outcome to be exp(1.892) or 6.63.
2. The odds ratio for cells that are treated with a concentration of 6.25 compound changes with exp(-0.624) or 0.54 -> this equals a 46% decrease in odds ratio compared to cells not treated with compound. Similar, the odds ratio for cells treated with 25.0 compound changes with exp(-1.952) or 0.14; equivalent to a 85% decrease.
3. The odds ratio for cells that are reactivated (but not treated with any compound) changes with exp(1.6) or 4.95 compared to the cells that are not reactivated; this equals a 395% increase.
4. Finally, the interpretation of cells that are both reactivated and treated seems more difficult to interpret due to the interaction factor. I want to know how the compound affects reactivation (compared to the reactivated cells without compound). I reasoned that I need to multiply exp(-0.624)*exp(0.2175) which is 0.66. Thus for the cells that are reactivated, we observe a 33% decrease in odds ratio of cells treated with 6.25 compound compared to no treated cells. However, I think it is also possible to calculate how reactivation of cells affects the outcome within one compound concentration. But since the compound affects both vRNA spots in unreactivated cells and reactivated cells I am less intereseted in this effect since it will be difficult to interpret.

I appreciate all feedback and comments.

Julie

• It is almost impossible (for me anyway) to read the output you posted as a picture. It would be helpful if you can replace it with the actual text output. If you put 3 backticks () at the start and end, then the system will try to format it nicely May 22, 2021 at 20:31
• Hi Robert, the table loses its formatting but I replaced the table by a better quality image. May 23, 2021 at 9:58
• OK, that's better. I've posted an answer anyway :) May 23, 2021 at 10:06

A few points.

Your model says Logit P = . It's not logit, it's log. Logit usually arises from logistic regression where we are modelling a binary outcome.

When the cells are not treated or reactivated, we expect the outcome to be exp(1.892) or 6.63.

Yes

The odds ratio for cells that are treated with a concentration of 6.25 compound changes with exp(-0.624) or 0.54 -> this equals a 46% decrease in odds ratio compared to cells not treated with compound. Similar, the odds ratio for cells treated with 25.0 compound changes with exp(-1.952) or 0.14; equivalent to a 85% decrease.

Not quite. When two variables interact, the interpretation of the main effects is conditional on the other variable being zero (or at it's reference level in the care of a categorical variable). So you need to add "when reactivation is zero" (if it is a continuous variable) or "when reactivation is ____" where ____ is the reference level for reactivation (if it is a binary variable, which I think it probably is)

The odds ratio for cells that are reactivated (but not treated with any compound) changes with exp(1.6) or 4.95 compared to the cells that are not reactivated; this equals a 395% increase.

Again not quite; for the same reason - it is conditional on concentration being at it's reference level.

Regarding the interactions, yes, you are on the right track but rather than exp(-0.624)*exp(0.2175) I would rather use exp(-0.624 + 0.2175)`. These result in the same value but the former is a bit more clear (for me). It's a very minor point.

• Thank you Robert for this clarification. May 23, 2021 at 12:16
• You're welcome. If this helps, please consider upvoting, and if it answers your question please consider marking it as the accepted answer. May 23, 2021 at 12:31
• Alternatively, I analysed the same data but I generated a poisson regression for the unreactivated cells and reactivated cells separately. As a result, I only have a main effect (which is concentration) for each of these datasets. The output is identical as the model in which I analyse all the cells (unreactivated and reactivated combined). Does this indicate that a the model considering both concentration and reactivation as main effects (with the additional interaction between both main effects) does not necessarily fit the data better than a main effect (concentration) only model? May 23, 2021 at 12:43
• Do you mean the estimates for concentration are the same in both models on the splot data? I would find that surprising because the output in your question here shows that there is an interaction, which means that some of the concentration estimates should differ by reactivation status. Also beware of splitting the data because you lose a lot of statistical power by doing so. May 23, 2021 at 12:49