# 2x2 Anova with simple generated data

I have generated data to understand a (repeated measures) 2x2 Anova better, however the results leave me baffled. The data is really simple: I am assuming I have 30 subjects, I am assuming that there are two medical tests, one in body region A, and one in body region B. I have created now the data as follows:

• Test1, region A: normal vector with mean 0.5 and standardeviation of 0.2
• Test1, region B: normal vector with mean -0.5 and standardeviation of 0.2

• Test2, region A: normal vector with mean -0.5 and standardeviation of 0.2

• Test2, region B: normal vector with mean 0.5 and standardeviation of 0.2

This is quite observations we really have: Test1 shows excitation of some substance in region A but inhibition in region B, while Test2 shows excitation of some other substance in region B but inhibition in region A.

Now the results of the ANOVA does show significant interaction of test and body region, however there is no significant main effect of region (p=0.86) - how can that be...? For both tests, the regions have clearly different measurement values. Help appreciated

This is sometimes called a cross-over interaction, and if you plot the data you will immediately understand why: (I generated normally distributed data according to your specified values of mean and SD)

The plot clearly shows that the effect of "region" is in different directions depending on "test". Moreover, the effects not only are in different directions, but they also have the same size. Hence, if you look at the effect of "region" averaged over the two levels of "test" the difference A vs. B would disappear. So the reason why you don't find a significant main effect of "region" (or of "test") is that is not there!

Edit: A main effect of "region" would indicate that there is evidence for a systematic difference between measurements taken in region A and B, regardless of the test used (test 1 vs. test 2). Plotting the same data from the above figure without considering the factor "test" makes it clear that there is no evidence for such difference in the data. • Aha, so do I understand correctly that one basically compares 1. average region over test1 to 2. average region over test2? And these averages would be basically 0 in both cases?
– user24544
Jan 10, 2018 at 13:07
• Yes, the main effect of a factor is computed by integrating over all the levels of the other factors or predictors. If $A_1$, $A_2$ and $B_1$ and $B_2$ are the values of test 1 and 2 in region A and B, respectively, a significant main effect of "region" would indicate that $\frac{{\left( {{A_1} + {A_2}} \right)}}{2}$ is different from $\frac{{\left( {{B_1} + {B_2}} \right)}}{2}$ Jan 10, 2018 at 14:12