I apologize if my question is exceedingly simple. Imagine, for example, I am studying a paper which explains that two variables correlate with a certain Pearson's r (no p-value, no confidence interval, no sample size provided).

Can I estimate the value of one of the two variables if I know the other one? How?

I found some examples but all dealt with the knowledge of all the data, not just the final Pearson's r.

(I use R)

Edit: added one more question

Can I estimate the value of one of the two variables if I know the other one and their means and standard deviations? How?

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    $\begingroup$ you can estimate the p value and confidence interval if you know the correlation and sample size. $\endgroup$ – TPM Jul 9 '18 at 15:37
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    $\begingroup$ Even if the relationship is linear, you'd need more than just the correlation; you'd need the means and standard deviations of both variables. $\endgroup$ – Glen_b Jul 9 '18 at 15:40
  • $\begingroup$ Thank you for your comment TPM. Let me be clearer: I do not want to estimate p-value, confidence interval or sample size. Instead, I would like to "convert" a value from one variable to the other one. Thank you for your comment Glen_b. So, simply speaking, I can't do what I would like to do? $\endgroup$ – statisticianwannabe Jul 9 '18 at 15:40
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    $\begingroup$ Ah, sorry for the confusion. Unfortunately, then, the answer is no. You would need to know the means and standard deviations of the variables to do this. $\endgroup$ – TPM Jul 9 '18 at 15:48
  • $\begingroup$ Thank you for you comments. I added another question so if you want to elaborate a bit and make an answer, I would be glad to upvote them $\endgroup$ – statisticianwannabe Jul 9 '18 at 16:02

In simple linear regression, using the model $Y_i = \beta_0 + \beta_1x_i + e_i,$ where $e_i$ are independently $\mathsf{Norm}(0,\sigma),$ one can find estimates of $\beta_0,$ $\beta_1,$ and $\sigma$ knowing $n, \bar x, \bar Y, S_x, S_y,$ and $r.$ Most of the formulas can be found in any elementary text that covers simple linear regression. For example, the slope is estimated as $\hat \beta_1 = rS_y/S_x.$ From there, $\hat \beta_0$ can be found by noting that the regression line must pass through $(\bar x, \bar Y).$

With $\hat \beta_0$ and $\hat \beta_1,$ one can find the point estimate $\hat Y_i$ corresponding to $x_i.$ Knowing $\hat \sigma^2 = S_{Y|x}^2,$ one can find a confidence interval corresponding to that point estimate.

Some additional formulas for confidence and prediction intervals also require $$S_{Y|x}^2 = \frac{n-1}{n-2}S_Y^2(1 - r^2),$$ where, by definition, $S_{Y|x}^2 = \frac{1}{n-2}\sum_{i=1}^n(\hat Y_i - Y_i)^2.$

However, 'regression diagnostics' (including residual plots and identification of influential points) are important to understanding whether the assumptions of the regression model are valid. Most of these cannot be done without access to the data.

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