The Book of Why: The New Science of Cause and Effect

by Judea Pearl

So far most of our examples of Simpson’s reversal and paradox have involved binary variables: a patient either got Drug D or didn’t and either had a heart attack or didn’t. However, the reversal can also occur with continuous variables and is perhaps easier to understand in that case because we can draw a picture. Consider a study that measures weekly exercise and cholesterol levels in various age groups. When we plot hours of exercise on the x-axis and cholesterol on the y-axis, as in Figure 6.6(a), we see in each age group a downward trend, indicating perhaps that exercise reduces cholesterol. On the other hand, if we use the same scatter plot but don’t segregate the data by age, as in Figure 6.6(b), then we see a pronounced upward trend, indicating that the more people exercise, the higher their cholesterol becomes. Once again we seem to have a BBG drug situation, where Exercise is the drug: it seems to have a beneficial effect in each age group but a harmful effect on the population as a whole. To decide whether Exercise is beneficial or harmful, as always, we need to consult the story behind the data. The data show that older people in our population exercise more. Because it seems more likely that Age causes Exercise rather than vice versa, and since Age may have a causal effect on Cholesterol, we conclude that Age may be a confounder of Exercise and Cholesterol. So we should control for Age. In other words, we should look at the age-segregated data and conclude that...