Relation between $R^2$ and the covariate correlation matrix

Question

I'm quite new to Statistics and I'm facing a problem.

Is there any relation between $R^2$ and the correlation matrix of the covariates?

A short example is (case with 2 covariates) :

A7 ~ A1 + A2

cor(A7, A1) = 0.447777
cor(A7, A2) = -0.2116495
cor(A1, A2) = -0.6117879
summary(lm(A7 ~ A1 + A2)$r.squared = 0.2067062

Can we predict the $R^2$ with those values? I know that it's easy with one single covariate (square of the correlation but I don't know in the general case). Maybe it's not possible...

Answer 1

In a linear regression with two predictiors,

$$y \sim 1 + x_1 + x_2 + \epsilon,$$

you can use this formula to calculate $R^2$:

$$R^2 = \frac{r_{yx_1}^2 + r_{yx_2}^2 - 2 r_{yx_1} r_{yx_2} r_{x_1x_2}}{1 - r_{x_1x_2}^2},$$

where $r_{yx_1}$ is the correlation coefficient between $y$ and $x_1$, $r_{yx_2}$ is the correlation coefficient between $y$ and $x_2$, and $r_{x_1x_2}$ is the correlation coefficient between $x_1$ and $x_2$.

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An example in R:

# Create random numbers
set.seed(1)
y <- rnorm(100)
x1 <- y + rnorm(100, sd = 0.2)
x2 <- y - x1 + rnorm(100, sd = 0.3)

# Compute correlations
r_yx1 <- cor(y, x1)
r_yx1
# [1] 0.9779927
r_yx2 <- cor(y, x2)
r_yx2
# [1] 0.01581697    
r_x1x2 <- cor(x1, x2)
r_x1x2
# [1] -0.1003943

# Fit linear regression and extract R-squared
fit <- lm(y ~ x1 + x2)    
summary(fit)$r.squared
# [1] 0.9695985

# Calculate R-squared based on the correlation coefficients and the formula above    
(r_yx1 ^ 2 + r_yx2 ^ 2 - 2 * r_yx1 * r_yx2 * r_x1x2) / (1 - r_x1x2 ^ 2)
# [1] 0.9695985

As you can see, the analytical result matches the one of the model fit.