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.
Best Answer
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.