# R$^2$ after adding one variable

An interesting question my friend told me: I have two simple linear regression models with R$^2$ = 0.1

Y ~ X1
Y ~ X2


If I combine these two models, Y ~ X1 + X2, is it possible that R$^2$ > 0.2? If so, can you provide an example?

• Why is that a surprise? Oct 17 '17 at 5:46

Yes.

This doesn't get your numbers on the nose, but the spirit is the same

df <- data.frame(
x1 = c(0, 0, 0, 1, 1, 1),
x2 = c(0, 1, 2, -1, 0, 1),
y = c(0, 2, 4, 2, 4, 6)
)


With model $R^2$ statistics

> summary(lm(y ~ x1, data=df))$r.squared  0.2727273 > summary(lm(y ~ x2, data=df))$r.squared
 0.2066116
> summary(lm(y ~ x1 + x2, data=df))$r.squared  1  • This example shows that the R$^2$for the sum of the Xs can be greater than the sum of the two R$^2$s. As you say you can probably manipulate the data values to make the original two R$^2$s =0.1 and make the R$2$for the sum greater than 0,2. But since the sum in your example is (0,1,2,0,1,2) and y=(0,2,4,2,4,0) how can the R$^2$=1? There is not perfect correlation. Is x1+x2 in the code actually the sum or is it the least squares linear combination of x1 and x2 including an intercept? Oct 17 '17 at 5:50 • y ~ x1 + x2y is not a sum but R code for: explain y linearly through x1, x2 and an intercept. This is a OLS model with an intercept, even though this intercept happens to be 0. It's$y\leftarrow 4\times x_1+2\times x_2 + 0 + \epsilon$. with$\epsilon=0\$. Oct 17 '17 at 6:11