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Say I have two demographic variables, job$\in{j_1, j_2}$ and lives$\_$alone$\in${YES, NO}. I also have N books $B_1$, $\ldots$, $B_N$, each labelled with a genre that I know in advance, out of {fiction,non-fiction}.

In an experiment, I ask representatives of each instance of job, and each instance of lives$\_$alone, to label the books (off course, they do not know the true labels). That is, I have 4 treatment groups: job=$j_1$ and lives$\_$alone=YES, job=$j_1$ and lives$\_$alone=NO, job=$j_2$ and lives$\_$alone=YES, job=$j_2$ and lives$\_$alone=NO. So, in the end, each book has 4 labels, one coming from each treatment group.

At this point, I can compute metrics such as accuracy, precision, recall, etc., to get an idea about which group is best at correctly labelling the books. I could also do statistical tests to see if the difference in e.g. accuracy between two groups is significant.

However, where I am stuck is figuring out if there is any interaction between the two variables, job and lives$\_$alone. How do I begin to make a statement about how the two variables interact?

Thank you!

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An interaction means that the effect of one variable depends on the level of the other. You can examine whether the difference between live_alone = YES and live_alone = NO differs between those with job=$j_1$ and those with job=$j_2$. If there is a difference between these differences, then you have an interaction. Of course, you need to statistically test for the presence of an interaction, which regression or ANOVA can do.

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