For recent US Presidential elections, in each state wealthier voters tend to be more likely to vote Republican, yet states that are wealthier in an aggregate sense are more likely to have more Democrat than Republican votes (Gelman and Hill 2007, Section 14.2.). Sketch a plot that illustrates how this instance of Simpson’s paradox could occur. Specify a GLMM with random effects for states that could be used to analyze data for a sample of voters using their state of residence, their household income, and their vote in an election.

Explain how the model could be generalized to allow the income effect to vary by state, to reflect that Republican-leaning states tend to have stronger associations between income and vote.