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Let me set the stage.

  • We are dealing with two variables; $A$ and $B$.
  • We can easily obtain $A(x)$ for a specific data point $x$.
  • $B(x)$, on the other hand, is very difficult to know.
  • We know Pearson's Correlation Coefficient for the $(A,B)$ couple (referred to as $r(A,B)$ from now on).

Given only $A(x)$ and $r(A,B)$, can we reliably obtain an interval or an approximation for $B(x)$?

Gathering data points for $B$ is very difficult, let alone data points presenting both A and B. $r(A,B)$ was obtained via extensive research, which I unfortunately cannot replicate. Using linear regression to predict $B$ based on $A$ is thus not an option.

I figured that since Pearson's Correlation Coefficient describes how linear a relationship between two variables is, there might be a way to interpret it as an inequality; given $A(x)$ and $r(A,B)$, $B(x)$ can be found in this range.

Statistics are far from my domain of expertise. I tried to go back to the mathematical definition of the coefficient, thinking that I could trace the variance back to an inequality of some kind through standard deviation, but this didn't lead me anywhere.

I also thought that maybe a geometrical approach could help, in a similar vein to this, where we use the Cauchy-Schwarz inequality and the link between correlation and cosine to deduce an interval for $Corr(A,C)$ knowing $Corr(A,B)$ and $Corr(B,C)$, but I am also out of my depth on this one.

Any help would be greatly appreciated. Thank you for reading this far! :)

EDIT: Changed formatting. Here is some additional information:

  • Obtaining a small number of observations $(A(x),B(x))$ is manageable.
  • The standard deviation of $B$ is not known.
  • Both A and B range from 0 to 100.
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  • $\begingroup$ Your restrictions rule out being able to do anything, because you supply no information that could be used to estimate an unknown additive constant for $B.$ This problem would be overcome by having even one paired observation $(A(x),B(x)).$ $\endgroup$ Commented Apr 19, 2021 at 17:04
  • $\begingroup$ That sounds reasonable enough. What would this enable? $\endgroup$ Commented Apr 19, 2021 at 17:06
  • $\begingroup$ It would permit you to do regression, which is what you need. I'm curious how you know $r(A,B),$ though: is this estimated from other data? Predicted by theory? Established in some indirect way? BTW, you also need some estimate of the variance of $B,$ but that can be accomplished by having (at least) two paired observations. $\endgroup$ Commented Apr 19, 2021 at 17:08
  • $\begingroup$ Do you know the standard deviation of $B$? $\endgroup$ Commented Apr 19, 2021 at 17:11
  • $\begingroup$ The correlation coefficient was obtained by a third party which had the means to conduct a large-scale experiment. Only the result of said experiment is available to me. That being said, having a few observations is possible, but not many (I would say 5 maximum?). However, if I understand your suggestion correctly, performing a linear regression with only a single observation would be fairly imprecise. Would knowing the correlation coefficient help at all? $\endgroup$ Commented Apr 19, 2021 at 17:14

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