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    $\begingroup$ The indications of strongly skewed conditional distributions suggest this is not a good approach. When you also observe that the skewness of the sample size distribution will cause the few largest sample sizes to control the appearance of a trend in the regression, you will see why others are recommending preliminary transformations of the data. $\endgroup$ Commented Sep 5, 2017 at 18:33
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    $\begingroup$ I am not guessing or speculating: the plot in the question clearly shows these characteristics. Also see the plots created by R Greg Stacey, which--by applying the suggested log-log transformations--demonstrates what they accomplish. $\endgroup$ Commented Sep 5, 2017 at 20:23
  • $\begingroup$ I just found the data and did the study myself - please see updated answer. $\endgroup$ Commented Sep 5, 2017 at 20:39
  • $\begingroup$ Your study has succumbed to the two problems I noted: the appearance of "no correlation" derives in no small part to the skewed conditional responses and the leverage for the high regressor values. In particular, neither the fitted line nor its error bands are trustworthy. $\endgroup$ Commented Sep 5, 2017 at 20:48
  • $\begingroup$ Please see the plot I just added; I hope I am not missing anything in this last iteration. $\endgroup$ Commented Sep 5, 2017 at 20:48