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?
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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