I noticed something else: you may have lost track of expectations when translating between log-normal and normal.
The log of the median of a log-normal distribution is the same as the median of the normal distribution which you get by taking logs, and the same as the mean of that latter distribution. But the log of the mean of the log-normal distribution will be higher.
This affects what happens with your regressions. Assuming the initial estimates are point estimates, you end up with a distribution of possible values after regressing. With normal distributions, everything behaves nicely, the mean is equal to the median, and your calculations are correct.
With log-normal distributions, we normally want to make decisions based on expected value, which will be higher than the median I think you’ve produced. In particular, the log-normal will be wider (and the mean:median ratio higher) when there is less correlation between true values and estimates. This looks like it might reduce the importance of the r-value relative to your calculations, at least if you do care about expectations, and at the upper end of the range.
I’m not sure how this all cashes out. But wanted to flag it as something else which may significantly affect the answer!
I noticed something else: you may have lost track of expectations when translating between log-normal and normal.
The log of the median of a log-normal distribution is the same as the median of the normal distribution which you get by taking logs, and the same as the mean of that latter distribution. But the log of the mean of the log-normal distribution will be higher.
This affects what happens with your regressions. Assuming the initial estimates are point estimates, you end up with a distribution of possible values after regressing. With normal distributions, everything behaves nicely, the mean is equal to the median, and your calculations are correct.
With log-normal distributions, we normally want to make decisions based on expected value, which will be higher than the median I think you’ve produced. In particular, the log-normal will be wider (and the mean:median ratio higher) when there is less correlation between true values and estimates. This looks like it might reduce the importance of the r-value relative to your calculations, at least if you do care about expectations, and at the upper end of the range.
I’m not sure how this all cashes out. But wanted to flag it as something else which may significantly affect the answer!