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I'm running multiple linear regression with 6 variables. For one of the variables D, the correlation coefficient between D and the response Y is - 0.34. But in the regression output, the coefficient for D is +8.9.

What is the best way to interpret D's influence on Y? Is it safe to assume that there's some confounding going on so I can ignore the fact that the correlation coefficient is negative and thus say increasing D will result in increasing Y?

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marked as duplicate by Michael Chernick, gung Sep 20 '18 at 18:13

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ I think you will find the information you need in the linked thread. Please read it. If it isn't what you want / you still have a question afterwards, come back here & edit your question to state what you learned & what you still need to know. Then we can provide the information you need without just duplicating material elsewhere that already didn't help you. $\endgroup$ – gung Sep 20 '18 at 18:13
  • $\begingroup$ The correlation coefficient r between D and Y summarizes the strength and direction of the relationship between D and Y for all subjects/items in your population represented by the ones in your study. The coefficient of D in the simple regression model relating Y to D only summarizes the same thing. You can compare their signs and they should agree. $\endgroup$ – Isabella Ghement Sep 20 '18 at 22:26
  • $\begingroup$ The coefficient of D in the multiple regression model describes a subset of the target population, not the entire population, so it can have a different sign than r. If your model were to include, say, Gender and Age in addition to D, then the coefficient of D in this multiple model describes the strength of the relationship between D and Y among the subset of subjects having the same Age (e.g., 40 years) and the same Gender (e.g., male). That relationship can potentially be different in sign and/or magnitude from the one you would obtain for all subjects regardless of their age and gender. $\endgroup$ – Isabella Ghement Sep 20 '18 at 22:28
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The correlation coefficient between D and Y is really measuring the correlation between "D and all the variables that are correlated with both D and Y". In other words, as D can both affect Y directly and be a proxy for the effect of other variables our measured correlation is not the "real" one.

It might be good to read up a bit more on omitted variable bias - this should help, p. 1-10 of this may help: https://eml.berkeley.edu/~dromer/papers/Econometrics%20Jan%202018.pdf.

But to the question at hand, if you think the covariates are reasonable things to adjust for (i.e. that you would expect that they should be correlated with both D and Y) then sure trust the coefficient more than the correlation.

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Assuming everything is going okay with the model—In a standard OLS regression you would interpret 8.9 as the change in predicted value of Y for every one-unit increase in D, holding the effects of the other covariates in the model constant.

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