# [R] Matrix algebra in R to compute coefficients of a linear regression.

John Sorkin JSorkin at grecc.umaryland.edu
Sat Feb 18 14:36:11 CET 2012

```I am trying to use matrix algebra to get the beta coefficients from a simple bivariate linear regression, y=f(x).
The coefficients should be computable using the following matrix algebra: t(X)Y / t(x)X

I have pasted the code I wrote below. I clearly odes not work both because it returns a matrix rather than a vector containing two elements the beta for the intercept and the beta for x, and because the values produced by the matrix algebra are not the same as those returned by the linear regression. Can someone tell we where I have gone wrong, either in my use of matrix algebra in R, or perhaps at a more fundamental theoretical level?
Thanks,
John

# Define intercept, x and y.
int <- rep(1,100)
x   <- 1:100
y   <- 2*x + rnorm(100)

# Create a matrix to hold values.
data           <- matrix(nrow=100,ncol=3)
dimnames(data) <- list(NULL,c("int","x","y"))
data[,"int"] <- int
data[,"x"]   <- x
data[,"y"]   <- y
data

# Compute numerator.
num <-  cov(data)
num

# Compute denominator
denom <- solve(t(data) %*% data)
denom

# Compute betas, [t(X)Y]/[t(X)Y]
betaRon <-    num %*% denom
betaRon

# Get betas from regression so we can check
# values obtaned by matrix algebra.
fit0 <- lm(y~x)

John David Sorkin M.D., Ph.D.
Chief, Biostatistics and Informatics
University of Maryland School of Medicine Division of Gerontology
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