Berkson's Paradox

Berkson's paradox is the appearance of a spurious association after conditioning on a selection rule. Here, two Gaussian variables can look negatively correlated after conditioning on a combined threshold such as aX + bY > T. It is also a case of the explaining away effect.

selected cases rejected cases fitted line (selected) selection boundary

Population correlation: r = 0.00
Selected correlation: r = 0.00
Selected cases: 0 (0%)

Parameters

Selection rule

True correlation rho

Coefficient a

Coefficient b

Threshold tx

Threshold ty

Threshold T

Band width delta

Sample size n

Suggested exploration: Set rho to about 0.20. Compare X > tx with aX + bY > T. As you raise the threshold, the first typically stays positive, while the second can turn negative. Then try |aX + bY - T| < delta to make the trade-off especially vivid.