I have performed correlation analysis between y and seven different variables (x1,x2,x3,x4,x5,x6,x7). In some cases, result shows zero order correlation is positive and partial correlation is negative (for ex. y & x3). How can I explain this result? My results are:

control variables----------zero order corr (partial corr) between y & x1,x2.....x7

x2,x3,x4,x5,x6,x7----------y & x1 = 0.88 (0.41)

x1,x3,x4,x5,x6,x7----------y & x2 = 0.85 (0.08*)

x1,x2,x4,x5,x6,x7----------y & x3 = 0.77 (-0.19)

x1,x2,x3,x5,x6,x7----------y & x4 = 0.44 (0.23)

x1,x2,x3,x4,x6,x7----------y & x5 = 0.38 (-0.02*)

x1,x2,x3,x4,x5,x7----------y & x6 = -0.23 (-0.13)

x1,x2,x3,x4,x5,x6----------y & x7 = -0.57 (-0.19)

  • indicates insignificant partial correlation.

Thank you.


Let me use your $y$ and $x_3$ values. If you look only at the direct effect of $x_3$ on $y$, you will see a strong positive correlation. However, this ignores the other variables that may/may-not be influencing $y$.

If you account for the other variables, you have to account for the influence those variables have on $y$ and on $x_3$. In this case, you see that $x_3$ actually has a negative relationship WHEN the effect of the other variables has been "partialled" out. Conceptually, if you take the residuals from predicting $y$ form everything by $x_3$ and the residuals from predicting $x_3$ with everything (but $y$, of course), these residuals have a negative correlation.

One additional way to think about this is from a causal perspective (so extreme caution must be taken in presenting such an interpretation). $x_3$ may have some influence on the other variables (often called multicollinearity in this context). From the causal perspective, this means if you change $x_3$, you change the other variables, too. Now, when you model the TOTAL effects from $x_3$ on $y$, you have to account for the direct effect (which appears to be negative) and the indirect effect of $x_3$ influencing the other variables which in turn influence $y$. When you combine the negative direct impact with the indirect effects, you end up with an overall "positive" impact.

Hope this helps.

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  • $\begingroup$ Hi Gregg, It helps a lot. But still I have confusion. As you explained x3 may have some influence on the other variables (often called multicollinearity in this context). So correlation between x1 & x3 is 0.79, x2 & x3 is 0.94, x4 & x3 is 0.36, x5 & x3 is 0.32, x6 & x3 is -0.40 and x7 & x3 is -0.66. Basically these x1,x2....x7 are climatic variables which shows impact on evaporation (y). If I explain direct correlation, climatic variable x3 have large impact on evaporation, on the others hand partial correlation shows x3 have very low impact on evaporation. Please throw some lights. $\endgroup$ – SONY Apr 3 '18 at 6:49
  • $\begingroup$ Think of it this way: x3 impacts BOTH evaporation AND the other variables affecting evaporation. The direct effect from x3 may be small, but if you aggregate the indirect effects from x3 to x_i to y, you obtain the larger impact you see if you were to examine just x3 & y. $\endgroup$ – Gregg H Apr 3 '18 at 13:30

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