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There are a certain number of houses spread randomly across a city. We have addresses and GPS coordinates for each house.

How can we select a geographically representative, fixed-size sample out of the houses in the selection?

To illustrate with an example, say we have about 100 houses on a map like the one below:

enter image description here

Each house has been identified as a likely supporter in a political campaign, and ideally we visit all of the houses and solicit support. However, it isn't possible to visit all of the houses; we can visit 25 of the houses. By spreading the 25 visits as evenly across the map as possible, we can leverage the fact that neighbors talk to each other.

In other words, how do we select a subset of the GPS coordinates so the entire set of coordinates is as well represented as possible?

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In order to be able to calculate estimates based on the typical design-based survey sampling paradigm (that is, using estimators based on the Horvitz-Thompson estimator), it's important to have a known probability of selection of each unit. For the estimate of error in the survey, it's important to have known joint probabilities of selection as well.

However, often if we want to spread out our sample relative to some measure, we ease up on the "known joint probabilities of selection" requirement and we perform Systematic Sampling. It's usually assumed fair to use the same variance formulas as in Simple Random Sampling, even though they aren't strictly correct.

So, in your case I'd:

  1. Order the units in the frame by west to east (or by road position and house number: put houses that you wouldn't want to select together next to each other in the list)
  2. Choose a random start number (1, 2, 3, or 4). The house at this position in the frame is the first unit sampled.
  3. Skip down 4 houses in the list repeatedly to select your sample.

Once this is done, I'd analyse the survey using the Horvitz Thompson Estimator.

One extra note: in cases where there are huge areas to cover - prohibiting uniformly sampling across the whole area, we do Cluster Sampling which might also be of interest if you worry about costs.

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  • $\begingroup$ If I were to use cluster sampling, how would I define the clusters? $\endgroup$
    – quantif
    Aug 29 '17 at 11:45
  • $\begingroup$ Usually clusters are defined by convenience - for example by suburb or city block. In a way clusters are the opposite of strata: whereas strata are optimally designed to be as uniform as possible, clusters are best designed so each cluster is as much representative of the entire population as possible. $\endgroup$
    – RoryT
    Aug 29 '17 at 11:57

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