In one of my research projects, I found a negative correlation between $X$ (a regressor of interest) and $Y$ (an outcome variable). In the regression model, I found that the coefficient of $X$ is positive after controlling for other variables in the regression model.

What could be the explanation of the negative correlation in the unadjusted model? I expected that a negative correlation would yield a negative regression coefficient in the multivariate model.

  • $\begingroup$ I would prefer holding this question open since it pertains to a specific phenomenon about what happens to the sign of variables after multivariate adjustment. I agree based on the problem description there's a lack of understanding about how the estimated effect changes... both geometrically and practically. But the changing of direction of effect refers to a specific phenomenon in regression analysis which was not mentioned in the excellent answer from @gung in the duplicate post. $\endgroup$ – AdamO Jul 10 '15 at 19:51

The correlation and the bivariate least squares regression slope are always of the same sign since:

$$\beta_{OLS} = \rho \frac{\mbox{sd} (Y) }{\mbox{sd}(X)}$$

Where the $\rho$ designates the correlation between $X$ and $Y$ and the $\mbox{sd}(\cdot)$ operator is the root of the sample variance which is always positive.

When you control for other variables in a multivariate regression model, the sign of this effect-of-interest may change. The reason for this may be because of spurious data structure and correlation between factors. However, if the control variable is a confounder: meaning that it is causally related to the outcome of interest and bears some association with your regressor of interest ($X$), then this is an example of Simpson's Paradox.

  • $\begingroup$ Adam, could you expand a bit with some examples? $\endgroup$ – Antoni Parellada Jul 10 '15 at 18:21
  • 1
    $\begingroup$ @AntoniParellada From the wiki which I already linked: "One of the best-known real-life examples of Simpson's paradox occurred when the University of California, Berkeley was sued for bias against women who had applied for admission to graduate schools there... [researchers] concluded that women tended to apply to competitive departments with low rates of admission even among qualified applicants (such as in the English Department), whereas men tended to apply to less-competitive departments with high rates of admission among the qualified applicants (such as in engineering and chemistry)." $\endgroup$ – AdamO Jul 10 '15 at 19:44

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