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If the common regression results as typically reported in an empirical primary study (sample size n, regression coefficients (Betas), t-statistic, p-value, R^2, adj. R^2) are given in the case of more than 2 independent variables, is there a way to derive the Pearson correlation coefficients between the dependent and all independent as well as the correlation coefficients between the independent variables from this data? If this is possible, how does it work?

I am not a mathematician, but the regression coefficients are connected to the partial correlations, so in my opinion, it should be possible to compute the correlation coefficients, or?

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  • $\begingroup$ Are the only known quantities the coefficients (and hence $p$), t-stats, p-values, $R^2$ and adjusted-$R^2$? (well, presumably $n$ as well, though we can work it out from those). Is anything else known? $\endgroup$ – Glen_b -Reinstate Monica May 3 '15 at 1:39
  • $\begingroup$ Sorry, of course the sample size n is given as well, but nothing else. Is this enough for working out the correlation coeff.? $\endgroup$ – jeffrey May 3 '15 at 5:16
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I believe this won't be possible in the general case.

While it might be possible to get to the partial correlations $r_{y,x_i|\mathbf{x}_(i)}$ (where $\mathbf{x}_(i)$ indicates all the x-variables aside from the $i$th one) from the available information, that would only seem to have $p$ degrees of freedom. There are more degrees of freedom (generally $p(p-1)/2$) in the correlation matrix.

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