In our last entry, we discussed one way for getting better estimates of constituency opinion using characteristics of the respondents — multilevel regression and post-stratification or Mr. P.
Another way of producing better estimates (where better means `better than the lousy estimates we get from direct estimation’) is to use characteristics of the constituency. In particular, we’re going to use a basic maxim of political geography (well, geography in general) called Tobler’s Law, which states that things which are closer together are more alike than things that are further apart.
Let’s suppose that instead of surveying people about political matters, we’re carrying out some geological surveys for a mining concern — or metal-detecting on a beach, or some such activity. Suppose we want to know the volume of gold found in each square kilometre.
Now, geological soundings are expensive, requiring copious amounts of explosives or lots of spadework. So we want to get a good estimate, but we can’t dig up the entire beach, or blast the entire peninsula.
Suppose that one work team carries out some soundings in a particular area, and finds a better-than-average concentration of gold. We have a hunch that this patch would merit excavation.
But how can we be sure that the levels of gold there are really worth it?
We can firm up with hunch by `borrowing strength’ from observations in neighbouring plots. If our plot, like the button for 5 in your mobile phone’s keypad, has eight neighbours, then we can look at the responses in those eight other plots. If those plots also show higher-than-average levels of gold, we can be more confident than the true figure in the plot we started out with is also higher than average.
In the models that we use, the degree to which we can borrow strength from neighbouring constituencies to `firm up’ our estimates is going to depend on the variance parameter in a conditional autoregressive distribution — but to simplify greatly, we’re going to borrow strength from neighbouring areas only to the extent that doing it that way helps us explain the pattern of responses we get in each individual area.
The more we can explain observations in a particular area by the areas around them, and the less we need to invoke particularities of this or that area, the better this technique works.
Oh, and while other people have labelled this technique `local smoothing’, the name we give to it is (including) spatially correlated random effects, or SCRE (pronounced scree). Hopefully, that both explains the title of this post, and the rather geological example.
In future posts, we’ll look at both how much Tobler’s law holds in our particular case, and how SCRE can be combined with Mr. P.