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I got a negative correlation value between two variables. Results of running Multiple hierarchical linear regression on the same variables indicated a positive beta coefficient between them. Explain with logic and literature reference how this is possible

kjetil b halvorsen
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Avid Researcher
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    Say you have two clusters separated in such a manner that within these clusters the correlation between the two variables (examined separately in the clusters) is positive, but the centers of the clusters are situation such that the line between the centers is negative slope. That would lead to the observation you are seeing. Don't need any literature but if you want to search, use: "Simpson's Paradox" as your term. https://stats.stackexchange.com/search?q=Simpsons+paradox – DWin Mar 31 '19 at 16:38
  • Actually I am about to write my discussion section for research and my supervisor was surprised at the reversal of signs. This sent me in to a panic that the results may be wrong. But I have since learnt that multiple hierarchical regression controls variables and correlation does not. I would be grateful for a link to a relevant article or research on this – Avid Researcher Mar 31 '19 at 16:43
  • As I said searching on Simpsons Paradox is the right way to proceed. I'm sure you can find Simpsons original article is you really need a citation to the statistical literature, – DWin Mar 31 '19 at 16:44
  • Also, I have read that Simpson's Paradox applies when the relationship was initially positive and then becomes negative. In my case, a positive Pearson correlation was obtained first that turned negative in multiple hierarchical linear regression – Avid Researcher Mar 31 '19 at 16:50
  • Are you sure that this is accounted for by Simpson's Paradox and not the fact that correlation and Multiple regression operate differently? – Avid Researcher Mar 31 '19 at 16:51
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    Yes, I'm sure. You (and your supervisor) really do need to crack open a stats book. The paradox is not restricted to reversal of signs only when relationship is "initially positive" sine the converse relations is easily explained ... conversely. And correlation and regression results always agree in sign since they are just the same analysis scaled differently. – DWin Mar 31 '19 at 16:54
  • I would like to bring your attention to "Multiple hierarchical linear regression" that is different from simple linear regression as predictor variables are arranged in blocks or models – Avid Researcher Mar 31 '19 at 16:57
  • Can you support your statement that correlation and regression "always" agree in sign with evidence? They are not the same analyses either – Avid Researcher Mar 31 '19 at 16:59
  • https://en.wikipedia.org/wiki/Partial_correlation The partial rho will always be the same sign as the beta coefficient in corresponding models. https://stats.stackexchange.com/questions/44279/example-where-a-simple-correlation-coefficient-has-a-sign-opposite-to-that-of-th/44290#44290 – DWin Mar 31 '19 at 17:01
  • Again, Partial correlation is not my analysis. It is one-tail pearson product moment correlation coefficient compared with multiple regression. – Avid Researcher Mar 31 '19 at 17:07
  • I'm saying the partial correlation coefficient would be the same sign as the beta coefficient in a multiple regression model. AND I'm saying that a simple two-way correlation coefficient being different than a multiple regression beta coefficient is not anything surprising because of the above fact. And I'm saying this is such a well-known fact that almost any stats book that handles multiple regression will cover this (and will probably not cite Simpsons original paper from 1951 although you can find the citation in the Wikipedia article.).) – DWin Mar 31 '19 at 17:12
  • Thank you. So, this does not need to be justified with simpson's paradox and can be accounted for by the above reasoning alone? – Avid Researcher Mar 31 '19 at 17:17
  • Simpsons paradox is isomorphic to the analysis above. – DWin Mar 31 '19 at 17:18
  • Thank you for the helpful insight – Avid Researcher Mar 31 '19 at 17:21
  • I would like to know whether this paradox raises a question on the study's effectiveness. Can it be observed as a weakness or limitation? Are there any ways to overcome it? – Avid Researcher Mar 31 '19 at 17:27
  • You have offered nothing as far as scientific background. Surely you cannot believe statistics equals science? – DWin Mar 31 '19 at 19:50

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