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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.

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    $\begingroup$ You might want to take a look at (or citations of) Gotway, Carol A., and Linda J. Young. "Combining incompatible spatial data." Journal of the American Statistical Association 97.458 (2002): 632-648. $\endgroup$ – Jonathan Lisic Mar 29 '17 at 16:58
  • $\begingroup$ @JonathanLisic Thanks for the reference, I just gave the relevant parts a read. The closest part seemed to be section 5.5, where they talked about using a covariate to transform the data between the two 'spatial supports'. In this case, I'm using households to do that. I'm not a statistician so I'm doing my best to parse what was recommended, but it seemed like instead of assigning individual values to each household, I should reprsent their probability using a Poisson distribution. Am I on the right track? I'll track down some of the newer papers on this now. $\endgroup$ – Brideau Mar 29 '17 at 19:16
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    $\begingroup$ You can actually get bounds on your allocated numbers, see stats.stackexchange.com/a/15803/1036. $\endgroup$ – Andy W Mar 29 '17 at 19:27
  • $\begingroup$ @AndyW Thanks! It seems like this would apply to the case where you are breaking down a single area into multiple smaller areas, but it might not apply when moving between areas of similar size that do not necessarily overlap. $\endgroup$ – Brideau Mar 29 '17 at 20:12
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    $\begingroup$ If terminology helps for your searching, the case where you are shifting spatial divisions around without resizing is called the "zoning" variant of the MAUP. But, do you have any reference that suggests MAUP applies to probabilities like these? I thought it only was a problem for correlation. $\endgroup$ – J Kelly Mar 30 '17 at 1:15
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The best answer I've found to this so far is from the paper Dasymetric Mapping Using High Resolution Address Point Datasets by Paul Zandbergen. His approach calculates the error empirically, and is based on migrating population numbers between differing polygons, which would involve less error since there's no question about whether a person is a person (in contrast to whether a person voted a certiain way).

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