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Assume I have one variable X that I experimentally manipulate, and then measure the corresponding values obtained for another variable, Y. Assume also that the two variables are both measured along the same scale (units).

Why is only regression - but not correlation - an appropriate tool to quantify the effect of X on Y?

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    $\begingroup$ They are primordially different conceptions, association and influence. Association is seen as symmetric (it is not about "effect"), influence is seen as directed. In regression, we typically perceive the predictor as error-free, and the predictand as model+error. In correlation, the model is on neither "side", it is bivariate, or, so to speak, on a side of some in-between "latent" variable, no special placement of error is typically indicated. In case of nonlinear association, X->Y and Y->X regressions may be quite different, while correlation (of a selected type) is one. $\endgroup$ – ttnphns Nov 11 '16 at 18:20
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    $\begingroup$ I have seen Fisher describe regression as finding the relation f(X) that maximizes the correlation between Y and f(X). $\endgroup$ – Sextus Empiricus Nov 14 '18 at 20:02
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Since you can estimate slope of simple linear regression using correlation coefficient

$$ \hat {\beta} = {\rm cor}(Y_i, X_i) \cdot \frac{ {\rm SD}(Y_i) }{ {\rm SD}(X_i) } $$

It is not true that there are cases when regression could be appropriate where correlation is not. The only such case where the statement could make sense is if you are talking about multivariate relations to account for, but still, you can use partial correlation as well in such cases.

As noted by whuber, regression is much more sophisticated model that gives you more information then correlation alone, but the difference is not about appropriateness, but about their utility and the fact that regression provides additional information.

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    $\begingroup$ I am concerned that your "not true..." assertion might be misunderstood. Regression is a different model and thereby supplies information that correlation cannot, including standard errors of estimate of the coefficients, their variance-covariance matrix, the estimated error variance, and various ways to diagnose departures from the assumed model. All one gets from a correlation coefficient is its value and a standard error for it. $\endgroup$ – whuber Nov 10 '16 at 22:46
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Correlation is a dimensionless quantity that is derived from the ratio of the covariance to individual variances; it gives the effect relative to the individual variations, not the absolute effect. The slope given in regression, on the other hand, is in terms of the units of the variables, and represents the absolute effect. If an increase of one unit of $X_1$ consistently increase $Y_1$ by .0001, while an increase of one unit of $X_2$ inconsistently increases $Y_2$ by 10 on average, the correlation in the first case is 1 but the effect size is tiny, while the correlation in the second case is small but the effect size is large. Hence, correlation is not a good tool to quantify the effect of $X$ on $Y$, unless you are interested in consistency more than magnitude.

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