# Simpson’s paradox

### Mixed model’s and Simpson’s paradox

For a paper I am currently working on, I started to think about **Simpson’s paradox**, which wikipedia succinctly defines as

“a paradox in which a correlation (trend) present in different groups is reversed when the groups are combined. This result is often encountered in social-science […]”

The wikipedia page also gives a nice visual illustration. Here’s my own version of it. The plot shows 15 groups, each with 20 data points. The groups happen to order along the x-axis (“Pseudo distance from origin”) in a way that suggests a negative trend of the *Pseudo distance from origin* against the outcome (“Pseudo normalized phonological diversity”). However, this trend does not hold within groups. As a matter of fact, in this particular sample, most groups show the opposite of the global trend (10 out of 15 within-group slopes are clearly positive). If this data set is analyzed by an ordinary linear regression (which does not have access to the grouping structure), the result will be a significant negative slope for the *Pseudo distance from origin*. So, I got curious: what about linear mixed models?