Random Distribution of Cities

[Maximilian Bruder]

Dear Statalist Users,
I am trying to figure out a way of determining whether a number of hypothetical cities (who's coordinates are given) are randomly distributed throughout a country.
So far I have plotted them in a scatterplot and by eyeballing they look indeed randomly distributed. However, is there some formal way to test this?
Any feedback is much appreciated.

Cheers

[Nick Cox]

Yes; there are entire books on this. See e.g. http://www.amazon.com/Statistical-Sp.../dp/1466560231 and most monographs or texts on spatial statistics. The usual idea is that a random point pattern means a realisation of a Poisson distribution in the plane.

Frankly, these things are not, or rather do not appear to be, well supported in Stata. If this were really my problem, I would grit my teeth and learn more about R. (Nothing against R; it would just be a lot of work for me.) Actually, if it were really my problem, I would decide to write some Stata programs for point pattern analysis, but that is not going to happen in the foreseeable future.

[wbuchanan]

I think it would be difficult to justify an assumption of random distribution of cities (e.g., the probability of a city being located in the middle of the desert is not the same as a city that would be located near a flowing source of potable water). Looking solely at point locations in isolation of all other factors would probably increase your false positive rates. I would think you would also want to consider things like extreme temperatures (e.g., there's a reason people don't permanently reside in Antarctica), terrain features, proximity to water, proximity to other cities, access to the location (e.g., can people get to the location without the use of a TARDIS), and a host of other human factors. It may also be difficult to model this since cities are the result of shared behavior of humans (e.g., a bunch of humans move to a location and then want to improve their lives and start building infrastructure which also attracts more humans). Maybe you could provide a bit more about the context in which this came up to see if others may have advice that would be more tailored to your situation?

Also, if you're very interested in these types of spatial statistics you may want to check out QGIS (the open source alternative to ArcGIS).

[Nick Cox]

Billy: I know no more than is stated here about Maximilian's project.

But when this kind of thing was popular in urban geography long ago (interest reached a peak around 1969), there was never any serious disagreement that complete spatial randomness offered anything but a benchmark state, for all the reasons you give here and many more. (The lack of a historical or temporal dimension to the analysis is another sticky point.)

Now work on this seems concentrated within economics and I can't comment on the present state of thought.

[wbuchanan]

Nick Cox I suppose they could be asking about some feature of the city, but it struck me as a bizarre assumption that the location of cities would ever be considered random. I'm not even sure what purpose would be served by spatial randomness, although I'm now a bit intrigued at how the spatial statistics approach could potentially play out in the context of social network applications where cities would be nodes and the edges could be the straight line distances between them.

[Maximilian Bruder]

Thank you both for the feedback. I'm aware that the initial assumptions would not hold in real life. For an assignment I'm using "Distance a person lives from a community center" as an instrumental variable to determine whether participation in community center activities has a positive effect on a person's income. So for this to be a strong instrument, I would need the community centers to be randomly distributed throughout the country. I put cities in the title of the question because it is more straightforward, in case there were an "easy" solution.

[wbuchanan]

In that case it isn't completely clear that the distribution of cities would need to be completely random. For example, from Hoxby (2000):

In the results section, I demonstrate that streams fulfill the first condition for valid instrumental variables: that is, they are correlated with measures of choice. But, what about the second condition for valid instrumental variables-that streams are exogenous to school productivity? The condition is highly plausible. Such plausibility is important because it is impossible to fully test the second condition, but I do show some partial tests of it, including two overidentification tests and an examination of the covariances between streams and industrial composition and between streams and modern commuting times. (p. 1217)

Similar to your situation, the strategy was to use natural land barriers (on the basis of travel time) as an instrument. No assumption was made that the distribution of streams were random, only that the number of streams in a given area was uncorrelated with the error term in the outcome equation. I'm assuming from your response that your assumption is that participation in the community based activities increases social/political capital (e.g., build relationships with community members) which subsequently causes the subjects income to rise? You may also want to consider some estimate of housing density (e.g., with few available units it is more difficult to find a home that is closer to the community based centers and also to the location where I would assume many of the larger businesses exist).

Hope this helps

Hoxby, C. M. (2000). Does competition among public schools benefit students and taxpayers? The American Economic Review, 90(5) pp 1209-1238. Retrieved from: http://faculty.smu.edu/millimet/clas...pers/hoxby.pdf

[Maximilian Bruder]

It does help, thank you very much!