C36. Conditional versus marginal dependence

Topic: Marginal models, and two responses at once · more questions on this topic
C37 ›

How can responses be independent conditional on a random effect but correlated marginally?

Worked answer (click to show)

Definitions/notation. Conditional independence fixes the shared random effect, whereas marginalisation averages over it. For Y_{ij}=m_{ij}+b_i+\varepsilon_{ij} with fixed m_{ij}, independent errors and b_i independent of all errors, \operatorname{Cov}(Y_{ij},Y_{ik})=\operatorname{Var}(b_i) for j\ne k: the unobserved shared intercept induces marginal correlation.