# A variable contributes less to R2 value in the multiple regression model than alone in the linear regression model

I would like to ask a basic question. A multiple regression model, y ~ x1 + x2 + x3, has R2 value of 0.152. Now I have a new variable z, and its linear regression model, y ~ z, has R2 value of 0.051. However, when I put z in the multiple regression model, y ~ x1 + x2 + x3 + z, then R2 value is 0.153. Why is it not 0.152 + 0.051?

Remember that $R^2$ is the percent of variability in $y$ that is explained by your independent variables. Based on your data, this means that $z$ on its own explains about $5.1\%$ of the variability in $y$ but that it only explains $0.01\%$ of the variability in $y$ once $x_1$, $x_2$, and $x_3$ have already been taken into account.
Depending on the statistical software you use, this might be different. It will depend on which "sums of squares" each software uses. R, for example, uses sequential sums of squares, which means that the first independent variable $x_1$ in will explain some of that variability in $y$, then the next independent variable $x_2$ may explain some more variability in $y$ or it might not once $x_1$ is already taken into account. Thus the $R^2$ when adding in a new variable $z$ will not necessarily increase by the same $R^2$ value when you merely regress $y$ on $z$.
• It's probably worth pointing out that this effect is due to the fact that $z$ is correlated with one or more of the other explanatory variables. In other words, once you're already taking the other variables into account, only the component of $z$ which is orthogonal to all of the other variables contributes to explaining variability in $y$. OP, please accept the answer provided by @MattBrems if you find it acceptable. – Josh Dec 10 '15 at 22:43