Compare variable across two groups conditionally on a variable that has a strong correlation with the variable

Question

I have the following scatter plot. I want to do a statistical test to asses something along the following lines: "are the red dots on average higher/lower, given the y-coordinate". But I don't really know how I can do this in a solid way. I thought of fitting a regression trough the black dots and then compare how many red dots are higher/lower than the curve. But since many red dots are together in the lower left, I think this won't work that well (small imprecision in this zone will have huge effect).

Another alternative could be to do this: for each red dot: take the interpolation of the closest black dots on each size. And then compare weather this with a Bernouilli experiment with p=0.5 to get a kind of p-value. The problem with this is that is hard to asses how much independent trails there are since, a black dot can appear in multiple comparisons with red dots. So I don't what to do.

Are there standard ways to answer this question with this data set?

Answer 1

This is exactly what a linear model can do, you estimate a "slope" coefficient for both variables red and black, and then simply compare the two coefficients to see which one is "steeper". Additionally you can check whether the slops are significantly different from 0 and if they are different from each other.