# Negative Coefficients for Interaction Term in Dichotomous Variables

I've come across an issue while analyzing some data and haven't been able to find a similar question on the site.

I'm working with data concerning the impact of certain genes and smoking on the concentration of a particular drug in the blood. My variables are dichotomous (gen1 = 1/0, gen2 = 1/0, smoking = 1/0). The genes, individually, as well as the act of smoking, all decrease the concentration of the drug in the blood. However, in my model, these estimates yield negative values, specifically around -14 and -17.

In an effort to better understand my data, I attempted to explore the interaction between gene1 and smoking. Upon executing this interaction in R, I ended up with the following:

gen1[1] = -14.4
smoke[1] = -17.3
gen1[1]:smoke[1] = 15.4


I'm finding it challenging to interpret these results. While it makes sense that the multiplication of two negative values yields a positive one, I'm struggling to understand how the interaction between two factors, which both individually decrease concentration, could lead to an increase in concentration when interacted.

There is a fantastic example by @Underminer of a case where it could be possible , but in my case it would be really costly to invest in researching without being completely sure.

Below is a snapshot of my data:

ID  gen1    smoke   drug_concentration
1   1.0     1.0     80
2   0.0     1.0     85
3   1.0     0.0     90
4   0.0     0.0     95


In some posts on the site I find that there are professionals who indicate that the data may have a large skew factor. In my case this is correct, it has -3.34669. But the solution they give is to center the data. Being dichotomous, I can't center them because it doesn't make sense in the application that someone has 0.30 of a gene... That is, either you have that allele or you don't.

I'm beginning to think I may be misunderstanding the statistical interpretation here and I would appreciate any assistance.

Edit1: Plot via sjPlot package (plot_model):

Plot via ggEffect:

• Always easier to interpret complex models if you plot the model output.
– mkt
Jul 21, 2023 at 8:14
• I'll edit my question then. Thank you @mkt. Jul 21, 2023 at 9:17
• That's good, but there are better visualisations. Plot drug concentration as a function of gen1, and use two different lines and colours for the 2 levels of smoking. Like this: i.sstatic.net/hg2Wp.png
– mkt
Jul 21, 2023 at 9:37
• If you use R, the ggeffects package makes this quite easy. As a side note, this kind of plot would be useful to do with the raw data as well (violin plots, mean and confidence intervals, or even mean and all the raw data if you don't have many points). But since your question is about interpreting coefficients, plotting the model output would be most useful to you.
– mkt
Jul 21, 2023 at 9:42
• Done @mkt, thank you for your suggestions. Jul 21, 2023 at 12:13

## 2 Answers

I like the plotting ideas given by @mkt in the comments. But, in addition, you can figure out the numbers. You didn't tell us the coefficient for gene 2 (and why not look at that interaction with smoking) but, ignoring that for now we have

gen1[1] = -14.4 smoke[1] = -17.3 gen1[1]:smoke[1] = 15.4

So, the predicted value for people who smoke and have gene 1 is -14.4 - 17.3 + 15.4 = -16.3. If there was no interaction, it would be -31.7. That is, the effect of gene 1 is lower in people who smoke, and the effect of smoking is lower in people who have gene 1.

• Oh, I understand. I get your point Peter, the coefficient of the second gene is also negative, but much lower, -2,. Thank you for your answer, Do you think my other plot helps? Jul 21, 2023 at 12:13
• Viewing and analising the plot I did as suggested mkt, your point is valid so I'm going to select your answer as correct. Thank you again Peter. Jul 21, 2023 at 12:17
• Thanks. @mkt 's plot is much more useful than yours, I think. Jul 21, 2023 at 12:19

Try using effects coding (-1 / +1) instead of (0 / 1) This will have no effect on significance, or coefficients for your main effects, but it will change the intercept. But it will change the values of your interaction data column and thus change the coefficients, and make them more interpretable.