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I understand, in a correlation, r just signifies the relationship between two variables, and we CANNOT deduce that there is a causal relationship in the case of a correlation.

By contrast, we use a linear Regression analysis to predict y from x, using the equation y= mx+c. Can we discuss that there is a causal effect in this case?

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You are correct by saying you cannot deduce causal relationships when there is a statistically significant test of r=0, r being the Pearson correlation coefficient.

Statistical testing of the least squares regression slope (what you call m) is equivalent to tests of Pearson correlation coefficient: if one is non-zero, the other is non-zero. Tests of these hypotheses are asymptotically equivalent.

A necessary but insufficient condition to infer causality: you must either 1) randomize a cohort of participants to receive an experimental treatment or 2) control for confounding in a pseudoexperimental design using a multivariate model (estimating partial correlation or adjusted regression coefficients).

Suggested reading: Causality by Pearl 2nd ed, Causal Inference by Hernan, Robbins, Causal Diagrams for Epidemiologic Research by Greenland, Robins, Pearl.

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By contrast, we use a linear Regression analysis to predict y from x, using the equation y= mx+c. Can we discuss that there is a causal effect as well?

No. Consider the univariate regression where your coefficient m is intimately related to the correlation coefficient r.

You can study causality using linear regression, but in and of itself it doesn't say anything about causality. You can run and use regression when there's no causal relationship.

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  • $\begingroup$ As I understand, m is the rate of change (steeper slope of the graph), and r tests the relationship between the two var. I understand that we cannot infer causality in the case of Regression unless conditions (i) or (ii) are satisfied. My hypothesis predicts that users derive higher level of satisfaction (y) with beautiful products (x). How do I ideally interpret the following: y=2+ 0.22x. Larger (or smaller) values of y are associated with larger (or smaller) values of x.Can I deduce that "users derive higher level of satisfaction from greater product aesthetics?" $\endgroup$ – Vyas Sep 1 '17 at 16:25
  • $\begingroup$ Or is it wiser to use the word "impact" or "influence" when can are interpreting regression or correlation results. For example, is it proper to say that beautiful products have an impact on level of satisfaction? $\endgroup$ – Vyas Sep 1 '17 at 16:35
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    $\begingroup$ Vyas: Sure, why not? And then you could just switch the variables in your regression and thereby conclude, with equally firm logic, that level of satisfaction has an impact on the beauty of products. That will enable you to (mis)use statistics to conclude that anything causes anything else. $\endgroup$ – whuber Sep 1 '17 at 17:00
  • $\begingroup$ @Vyas, you cannot say anything about causal relationship from regression outputs. That's just a sad fact. There's no way around this, and using "influence" or "impact" is just using euphemisms, doesn't cut it. Establishing causality is difficult in what you're trying to do. The terms "predictor" and "dependent" variable are artifacts of the history, the regressions were first used in the context where the causality was out of the question, at least the direction of it. In your case, unless you already know the direction of the causality from other research the regression won't help you $\endgroup$ – Aksakal Sep 1 '17 at 17:19
  • $\begingroup$ @Vyas, if you already know that x causes y, then yes, you can use the regression to measure the strength of the causal impact or to otherwise study its properties. How would you know? Maybe it's obvious in your field of research, maybe there are other facts and considerations that leas you to believe that x causes y. However, even in this case I doubt that the univariate regression would be convincing. A simple factor such as a price may impact both x and y in your equation. If I buy 10,000\$ watch for 100\$ on the auction, I'll be aestethically satisfied $\endgroup$ – Aksakal Sep 1 '17 at 17:21
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First of all, as Aksakal pointed out as well, regression coefficients and (partial) correlation coefficients are linked, with the latter basically being a normalized version of the former.

Having noted that, you reason (I think) that because we can use a regression to "predict" y from x, this implies a kind of causality flowing from x to y. The problem is that prediction says nothing about causality. For example, the height of the sun in the sky is causally related to how light it is outside. Clearly, the sun being higher in the sky causes more illumination, and not the other way around. Yet, we can use our observation of how light it is outside to predict the height of the sun in the sky. I guess this can seem a bit misleading since the word "prediction" has a connotation of one event (causally) preceding another, but in this context the term is used more broadly to mean something similar to "inference".

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Say your x variable is the dollar value of goods imported from year 2007 - 2017, and your y variable is private health spending for a person in year 2007 - 2017 as well. You plot x, y, and see a clear increasing and linear trend between x and y, and you can definitely fit a regression model to get the m. But there is no causal relationship in this case, rather a lurking variable (the common year). The association between x and y are just due to the fact that both imports and health spending grew rapidly in those years.

Reference: Introduction to the practice of statistics, chapter 2.4.

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