I have some data which I am using to show that there is no relationship between two variables. (Or only a weak one.) In a previous writeup, I included the scatterplot showing no visible relationship, as well as the Pearson's and Spearman's correlation coefficients, which were both low. One of the reviewers commented that the statistics in the paper were weak and we 'only showed some correlations'. Is there any better way to show that there is little/no relationship between these variables? I have transformed the data using every reasonable transformation (log, square root, exponentiation), and even averaged the values for each datapoint, though it seems to me that that is falling prey to the ecological fallacy. Nothing. Not even a hint of a pattern anywhere.
Obviously, I can't show everything I tried. I want to show that there is no (or at least little) relationship, because common wisdom is that there should be a strong one. The fact that there is not a strong relationship is a surprising result. I know that you can't prove the null hypothesis, but I would like to show as much as possible that any other options are unlikely.
How do I convincingly show that there is no relationship between the variables? (Other than hypothesis-testing my correlations, which I am planning to do, but that doesn't show that there is no relationship.)
Note that both variables have heavy-tailed distributions - does that make a difference to the answer?
Below is a scatterplot of my data.