I'm working on the problem of migrating voting data from one grouping, electoral divisions, to another, census divisions. I'm not sure if this is a common problem in statistics broadly, but in the spatial world it's often called the "modifiable areal unit problem", and I'm struggling to find a way to quantify the error.
For example, imagine we have a two-party system where people can either vote for Party A, Party B, or not vote at all. In a small neighbourhood, a street is divided so that there are 10 houses, split between two groups as follows, where the lines represent the grouping of houses (the H's) into election divisions:
EDiv 1 EDiv 2 ________________________ __________________________ H1 H2 H3 H4 H5 H6 H7 H8 H9 H10
We know that exactly 1 person lives in each house. We also know that in EDiv 1, there were 2 Party A votes, 1 Party B votes, and 2 non-voters. In EDiv 2, there was 1 Party A voter, 3 Party B voters, and 1 non-voter.
We're interested, however, in finding out vote distribution by census division. These, over the same houses as above, look like this:
CDiv 1 CDiv 2 CDiv 3 ____________ ______________________ ____________ H1 H2 H3 H4 H5 H6 H7 H8 H9 H10
The question is then in two parts: 1) how do you estimate the vote count total in the census divisions, and 2) how do you quantify the uncertainity.
My approach to the first part (which may be incorrect) was to perform calculations party-by-party. For example:
For Party A in EDiv 1, each house has a probability of voting for them of 2/5 = 0.4. In EDiv 2 it is 1/5 = 0.2. Then, when you group them by census divisions, you sum probabilities up, so CDiv 2 would likely have 0.4 + 0.4 + 0.2 + 0.2 = 1.2 votes for Party A.
Is this the correct approach, and if so, how do I quantify the error? It's also worth mentioning that the actual data I'm working with is at a very small scale (typically ~500 people per area), which helps to reduce the MAUP problem compared to larger spatial divisions.