# High Pearson correlation, but very low coefficient in multiple regression analysis?

I have been running a few linear regression models to test the absolute and relative effect of several independent variables related to spending/investment on different tools on one measure of performance that I want to increase. I first ranked the Pearson correlation of these independent variables/tools, with some having high correlation and some low ones. Then I ran a linear regression with the dependent and independent variables, it end up that most of ones with high Pearson's also have high coefficient (most of the time also high t-stat) in regression. However, a small number of tools have very high Pearson's but extremely low or non-existent coefficient, meaning their effect is almost not there to be seen - what can be some technical/statistical/mathematical explanation for it?

• Are you describing several different single-variable regressions or are you describing one regression with all the variables? Also are you getting the correlations independently? I am not clear from your description. – Doctorambient Dec 9 '16 at 19:39

They are correlated with other variables in the model. When you correlate those variables (say, $x$) with the response ($y$), you are looking at them in isolation. Thus, you are measuring their association, plus the association of all the variables they are correlated with (e.g., $z$). It can be the case that there is no actual correlation between the variable and the response ($r_{x,y}=0$), but high correlations between $z$ and $y$, and between $x$ and $z$. So when you run the univariate correlation, $x$ just acts as a proxy for $z$ and you get a high univariate correlation, but when you include $z$, you see that $x$ was irrelevant.
• I am inclined to think that an approach, assuming the poster is talking about separate regression models for each $x$, is to look at the formula: $\beta = r_{x,y} \cdot \frac{s_y}{s_x}$ and consider various values for $s_x$, but I am not sure if the definition of $r$ prohibits this by locking that value in some way. That might allow addressing the specific question asked, I think. – Doctorambient Dec 9 '16 at 21:23