[justk]

Sorry, I edited my post several times and finally choose a short form with links other sources. If you fix state=1 then there are no more random variables so the R^2 doesn't have any meaning. Just for fun, what the model should predict for state = 0.5?, that corresponds to a person that is 50% in the red state and 50% in the blue state, I think a mixed model is appropriated here when the state variable is discrete, so that each value of the state variable represents a different part of the population, the other model should be used when people move a lot and change frequently the state where they vote in, but in that case you should have to consider the fluctuations in the total population in each state at the time of voting.

[vcdimension]

@justk The R^2 value of 0.01 calculated on that webpage uses both states, not just one: the variance of the predicted values across both states is 0.55^3+0.45^3 - (0.55^2+0.45^2) ≃ 0.497 ≃ 0.5 I don't think it makes sense to use a mixed model in this case since the variance is the same for each state. A mixed model is used when the observations have some structured heteroskedasticity, i.e. different variances for different values of the independent variables.