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I have a large astronomical dataset, showing the OLS regression value $r$ between two continuous variables, and the number of observations $n$. My research supervisor has told me to include the value of $r^2n$ as part of my analysis of the dataset. Here's a quick example:

    Colour  &   |r|>0.5     &   n       &   r^2n > 1000
    --------------------------------------------------------------
    (U-V)   &   0.558545    &   11640   &   3631.36 
    (U-R)   &   0.567270    &   11632   &   3743.12 
    (U-I)   &   0.579188    &   11626   &   3900.04 
    (U-J)   &   0.536106    &   3709    &   1066 
    (U-K)   &   0.535644    &   3730    &   1070.19 
    (B-V)   &   0.580019    &   11884   &   3998.04
    (B-R)   &   0.574867    &   11763   &   3887.34 
    (B-I)   &   0.591288    &   11749   &   4107.7 

What's the significance or meaning behind this particular statistic?

EDIT: I'm trying to find a tightly fitting and highly correlated relationship between the combinations of magnitudes (colours) in the rightmost column, with the distance to the galaxy emitting that colour. I've got some very high $r$ $>0.8$ in some cases, but some of these have only a couple of thousand data points due to missing or invalid data for some sources. There may be up to $\sim 29000$ data points for any particular fit, so the more, the better. The only reason I can think of for including this statistic is that it's a way to filter out highly correlated variables (high $r$) that have only a few observations (low $n$).

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This is a non-standard statistic to report in regression analysis. It does not strike me as having any particularly useful properties that could not be done better with some other statistic, but perhaps your supervisor has some use for it in mind that we are not aware of. The statistic $R^2$ measures the proportion of the variability in the response variable that is attributable to variations in the explanatory variable, so if you multiply it by $n$ then I suppose you are now just extending this from a proportionality measure between zero and one to a measure from zero up to the number of data points. That does not strike me as being useful for anything, but since you have not specified the purpose and justification for this statistic proposed by your supervisor, it is hard to say.

Ultimately, the onus is on your supervisor to tell you what he wants you to do with this statistic and why he thinks it is useful. People on this forum can speculate on this, and can look at some of the properties of this statistic, but it is not sensible for us to speculate when you could just ask your supervisor what significance and meaning he claims that this statistic has. Once you have an explanation for how and why your supervisor proposes to use this statistic, you can then ask us if that is a sensible usage.

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  • $\begingroup$ Hmmm ok, thanks for that. I was worried that there was some obscure use that I'd missed in my rarefied undergrad statistics courses. I'll edit the question to include a little background on the data. $\endgroup$ – Jim421616 Feb 10 at 23:29
  • $\begingroup$ Since it has already been answered, rather than editing this question, it would be better to ask a new question (and link back to this one if you want). If you make a major edit to a question after it has been answered then the existing answers become obsolete and don't make sense. $\endgroup$ – Ben Feb 10 at 23:32
  • $\begingroup$ After rereading your answer, I realised that it might help if you knew a little about the data. I've upvoted your answer, as it helps me, but I still wonder if there's a rationale behind using $r^2n$. $\endgroup$ – Jim421616 Feb 10 at 23:34
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    $\begingroup$ As I said, I think the first step is to speak to your supervisor and find out what the claimed rationale actually is. It is not very useful for us to speculate on this -- the onus of proof is on the person asserting usefulness of a statistic. $\endgroup$ – Ben Feb 11 at 0:37

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