I'm interested in learning how to develop a geographic approximation of some kind of epicenter based on the data from the John Snow Cholera outbreak. What statistical modeling could be used to solve such a problem without prior knowledge of where wells are located.
As a general problem, you would have available the time, location of known points, and the walking path of the observer. The method I'm looking for would use these three things to estimate the epicenter of the "outbreak".
Not to give a complete or authoritative answer, but just to stimulate ideas, I will report on a quick analysis I made for a lab exercise in a spatial stats course I was teaching ten years ago. The purpose was to see what effect an accurate accounting of likely travel pathways (on foot), compared to using Euclidean distances, would have on a relatively simple exploratory method: a kernel density estimate. Where would the peak (or peaks) of the density be relative to the pump whose handle Snow removed?
Using a fairly high-resolution raster representation (2946 rows by 3160 columns) of Snow's map (properly georeferenced), I digitized each of the hundreds of little black coffins shown on the map (finding 558 of them at 309 addresses), assigning each to the edge of the street corresponding to its address, and summarizing by address into a count at each location.
After some image processing to identify the streets and alleyways, I conducted a simple Gaussian diffusion limited to those areas (using repeated focal means in a GIS). This is the KDE.
The result speaks for itself--it scarcely even needs a legend to explain it. (The map shows many other pumps, but they all lie outside this view, which focuses on the areas of highest density.)
In [1,§3.2], David Freedman suggests an essentially negative answer to your question. That is, no (mere) statistical model or algorithm could solve John Snow's problem. Snow's problem was to develop a critical argument supporting his theory that cholera is a water-borne infectious disease, against the prevailing miasma theory of his day. (Chapter 3 in [1], titled “Statistical Models and Shoe Leather,” is also available in previously published form [2] here.)
In these few short pages [1, pp.47–53], much of which is an extended quote from John Snow himself, Freedman argues that "what Snow actually did in 1853–54 is even more interesting than the fable [of the Broad Street Pump]." As far as marshalling statistical evidence (other preliminaries such as index case identification, etc., are discussed besides), Snow exploited natural variation to effect a truly remarkable quasi-experiment.
It turns out that at an earlier time, there was a vigorous competition among water supply companies in London, and this resulted in spatial mixing of the water supply that was (in Snow's words) "of the most intimate kind."
The pipes of each Company go down all the streets, and into nearly all the courts and alleys. A few houses are supplied by one Company and a few by the other, according to the decision of the owner or occupier at that time when the Water Companies were in active competition.
...
As there is no difference whatever in the houses or the people receiving the supply of the two Water Companies, or in any of the physical conditions with which they are surrounded, it is obvious that no experiment could have been devised which would more thoroughly test the effect of water supply on the progress of cholera than this, which circumstances placed ready made before the observer.
—John Snow
Another critically important bit of 'natural variation' John Snow exploited in this quasi-experiment was that one water company had its water intake on the Thames downstream of sewage discharges, whereas the other had a few years before relocated its intake upstream. I'll let you guess which was which from John Snow's data table!
| Number of | Cholera | Deaths per
Company | houses | deaths | 10,000 houses
----------------------------------------------------------
Southwark & Vauxhall | 40,046 | 1263 | 315
Lambeth | 26,107 | 98 | 37
Rest of London | 256,423 | 1422 | 59
As Freedman notes witheringly,
As a piece of statistical technology, [the above table] is by no means remarkable. But the story it tells is very persuasive. The force of the argument results from the clarity of the prior reasoning, the bringing together of many different lines of evidence, and the amount of shoe leather Snow was willing to use to get the data. [1, p.51]
One further point of natural variation exploited by Snow occurred in the time dimension: the abovementioned water intake relocation occurred between two epidemics, enabling Snow to compare the same company's water with and without added sewage. (Thanks to Philip B. Stark, one author of [1], for this info via Twitter. See this online lecture of his.)
This matter also provides an instructive study in the contrast between deductivism and inductivism, as discussed in this answer.
Freedman D, Collier D, Sekhon JS, Stark PB. Statistical Models and Causal Inference: A Dialogue with the Social Sciences. Cambridge ; New York: Cambridge University Press; 2010.
Freedman DA. Statistical Models and Shoe Leather. Sociological Methodology. 1991;21:291-313. doi:10.2307/270939. Full text
