Describing a visual relationship statistically I have a dataset here that describes the relationship between problem_complexity and solution_complexity in a work process.
library(RCurl)
data <- getURL("https://gist.githubusercontent.com/aronlindberg/a8a6644a639c35ceb9d9/raw/5b7c13bbcd439dacbca70e07c96b6c33804273d2/data.csv")
object <- read.csv(text = data)
qplot(object$solution_complexity, object$problem_complexity)

...and when we plot it it looks like below:

I am trying to capture the relationship in the data. To me it seems that it could either be an increasing relationship (blue line) or a quadratic relationship (green line). However, even if the relationship is visually discernible, as the values on the Y-axis increase, they become more sparse, and therefore the broad range of variation at low Y-values overwhelm these other observations.
How can I characterize these relationships, despite the weak effect size?
I've considered using bootstrap and segmented regression, as a positive linear relationship becomes more clear if one only considers the upper portions of the data (on the y-axis).
 A: This follows my suggestion of treating problem complexity as predictor and treating it on a logarithmic scale, if only because otherwise what you see will be dominated by the effects of about 10 wild points. 
The results are not convincing of a well-determined relationship. Here the smoother is fairly arbitrary but the scatter doesn't encourage variants. 
 
Bottom line: Disappointing or surprising though it may be, there isn't a convincing relationship between these variables. 
Further thoughts: 


*

*If you have other variables, perhaps more structure can be seen within. 

*Measurement questions may be part of the problem: notably the solution complexity varies over about a four-fold range, while the problem complexity varies about fifty-fold. So the complexity measures may not be easily comparable. 
A: [Note: the question has changed markedly since this answer was posted]
If the idea to test that particular hypothesis was generated by the plot, then a test done on the same data -- i.e. a data-generated hypothesis -- doesn't have the nominal properties (i.e. your p-values will be nonsense).
If your aim is to test whether an assumption of constant variance is tenable, a hypothesis test isn't the right tool (generally constant variance isn't exactly true, so in big samples you'll reject even if the change in variance is tiny, and in small samples you won't reject even when the change in variance is huge).
The problem is essentially one of effect size, not significance.
In some cases there is an a priori reason to expect spread related to mean and then you might consider a GLM or a transformation or any of a number of other possibilities (depending on circumstances).
If you have no a prior reason to expect a spread-related-to-mean in some simple way but also no a priori reason to think you should have constant variance and you're interested in consistent estimation of standard errors of coefficients, you could (if sample sizes are not small) look at heteroscedasticity-consistent standard errors

That said, you can test for increasing spread; it's just unlikely to apply to what seems to be the situation.
