I am currently working at a GIS (Geographical Information Systems)-related project where we have a bunch of pieces of land. I want to study whether the shape of each piece influences a target variable.
For that matter, I calculated the ratio $Q:=\frac{4\pi A}{P^2}$ for each of the regions (where $A$ stands for its area and $P$ for its perimeter) The idea is that high values of $Q$ represent circle-like shapes (with a lot of area for a given perimeter) while low values represent "thin and stretched rectangles"
The $4\pi$ factor is there so that the values of $Q$ have a $(0,1)$ range, but there is no real need for that. I chose to divide by $P^2$ so that $Q$ is adimensional and does not depend on the size of the region (I already have other variable that take size into account) The real regions I'm working with come in a huge variety of shapes (as they come from Geographical data) The total amount of regions is close to $700$.
So the problem I have is the following: There is a pretty strong ($-0.34$ in a sample of about $700$) correlation between area and $Q$ Given how $Q$ is conceived, shouldn't it be close to $0$? Is $Q$ an actual good measure of how "circle-like" my area is? Are there any sound alternatives?
As @user pointed out in his comment, it is possible that there is an actual correlation between $Q$ and region size, with smaller regions being more "regular" (higher $Q$) Can you think of a possible explanation why that may be the case? The "regions" are connected chunks of forest. @quarague made a very good point regarding the map resolution, so I tried to check it that's the case.
I am not sure if this plot reflects that, since you can find a wide range of values of $Q$ for small regions, while the rarity seems to be "big regions with circle-like shapes". It could also be, though, that the map resolution erases "some" of the irregularities