Berkson Selection and Collider Bias

Worked Example: Negative Association from Selection

Consider two independent traits, A and B, both with probability 0.5. We select individuals who have Either A or B (or both).

1. The original probability of A is P(A) = 0.5. The original probability of B is P(B) = 0.5.
2. Since they are independent, the joint probability is P(A and B) = P(A) × P(B) = 0.25.
3. The probability of having neither A nor B is P(not A and not B) = 0.5 × 0.5 = 0.25.
4. The probability of being selected (Either A or B) is P(S) = 1 - P(not A and not B) = 1 - 0.25 = 0.75.
5. In this selected population, the new probability of A is P(A | S) = P(A and S) / P(S) = 0.5 / 0.75 ≈ 0.667.
6. The new probability of B is P(B | S) = 0.5 / 0.75 ≈ 0.667.
7. The new joint probability is P(A and B | S) = P(A and B) / P(S) = 0.25 / 0.75 ≈ 0.333.
8. The covariance in the selected population is E[AB] - E[A]E[B] = 0.333 - (0.667 × 0.667) ≈ -0.111.

Despite being completely independent originally, A and B are now negatively correlated in the selected population. If we know someone has trait A, they are less likely to have trait B than the general population.

Interactive Experiment

Use the Berkson Selection interactive module to explore how selection rules affect the association between independent traits.

1. Set both Probability A and Probability B to 0.5.
2. Change the Selection rule to Either and observe the negative covariance in the selected population table.
3. Change the Selection rule to Exactly one. The negative correlation becomes even stronger (perfectly negative), as having one trait guarantees you don't have the other.
4. Change the Selection rule to None. Notice that the covariance remains 0, as no selection bias is introduced.
5. Try setting the probabilities to 0. Notice that if the selection rule makes it impossible to select anyone (e.g., probabilities are 0 but rule is "Either"), the conditional probabilities become unavailable.

Limitations

This model demonstrates Berkson's bias with simple, independent Bernoulli trials. Real-world scenarios often involve continuous variables or complex causal structures where the traits might not be completely independent to begin with. The model assumes perfect selection based strictly on the chosen rule, ignoring measurement error or partial observability of the traits, which could dampen or alter the observed correlation.

References

Berkson's bias is a foundational concept in epidemiology and causal inference, illustrating how spurious associations can arise from study design rather than true causality.

• Collider scope: when selection bias can substantially influence observed associations (PMC3237868)