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

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... ## 1 Answer 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$x_2$. 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.

• Thank you very much for this clear answer! Do you know if there exists a formula for the general case (for N covariates) ? – Philippe Remy Feb 26 '15 at 14:09