That's not the same thing. A town in the middle of nowhere Appalachia is much more remote than sparse suburbs twenty miles from a city. Hell, you might find higher density in some remote towns than eg Farmville Iowa.
Plus, depending how you measure it, "population density" can be a nearly useless metric. County-level population density is spread widely over 3 orders of magnitude. The smallest county has 64 people and the largest has ten million.
My wife's job took us to a small rural city. By population size and density, we weren't that small. The metro area was 60k+ people. However, we were a 3 hours drive from any major city.
To me this panned out in two ways:
* There really wasn't a lot to do outside a 10 mile radius. Anything worth doing was 1 to 2+ hours drive (often more). That meant nearly everything we did was in the small city we lived in.
* Travel was frustrating. We essentially had to pay a 3 hour driving tax on every trip we took.
In other words, despite living in a "suburban" density city (with some urban density areas), it was really how our city was positioned relative to other cities that determined it's feeling. It wasn't large enough to explore internally.
Controlling for population density doesn't control for the impact of population density on the issue at hand, as these are two different factors which are related and which are named very similarly.
Consider this simplified example.
11 areas. 10 of these are isolated and have 10 people living in each area. The last area has 100,000 people living in it. Of those 10 areas, 9 have 0 incidents, and 1 has 1 incident. In the 100,000 area, there are 100 incidents.
Not controlling for the population:
The populated area has 100 incidents, the isolated areas have an average of 0.1 incidents. A 1,000 times difference. Populated area is much more dangerous.
Well that's obviously the wrong way to look at the data, so lets account for population:
Isolated areas have a rate of 1 per 100 population, populated areas have a rate of 1 per 1,000 population. A 10 times difference, in the opposite direction. So now we have established a link between being isolated and have more incidents, but we don't know why.
We still haven't controlled for the impact of population density on incident rates. We need much more data to solve this, as with the given information the result would be "Isolated areas have 10 times the risk of incidents" and then controlling for that factor we would see no more trends in our data. If we added thousands of more areas with different levels of population, calculate the per capita rate of incidents in each population, and then create an analysis of how populate density relates to incident rates, we could then control for both factors. The catch is that this last step is more difficult without enough data and often researchers aren't able to isolate single individual items to control for because they correlate too strongly with other issues.
