Manassa Mauler - 1 year ago 65
R Question

# How do I get regression coefficients from a variance covariance matrix in R?

I want to work out a multiple regression example all the way through using matrix algebra to calculate the regression coefficients.

``````#create vectors -- these will be our columns
y <- c(3,3,2,4,4,5,2,3,5,3)
x1 <- c(2,2,4,3,4,4,5,3,3,5)
x2 <- c(3,3,4,4,3,3,4,2,4,4)

#create matrix from vectors
M <- cbind(y,x1,x2)
k <- ncol(M) #number of variables
n <- nrow(M) #number of subjects

#create means for each column
M_mean <- matrix(data=1, nrow=n) %*% cbind(mean(y),mean(x1),mean(x2)); M_mean

#creates a difference matrix which gives deviation scores
D <- M - M_mean; D

#creates the covariance matrix, the sum of squares are in the diagonal and the sum of cross products are in the off diagonals.
C <-  t(D) %*% D; C
``````

I can see what the final values should be (-.19, -.01) and what the matrices before this calculation look like.

``````E<-matrix(c(10.5,3,3,4.4),nrow=2,ncol=2)
F<-matrix(c(-2,-.6),nrow=2,ncol=1)
``````

But I'm not sure how to create these from the variance-covariance matrix to get the coefficients using matrix algebra.

Hope you can help.

I can see that you are doing centred regression:

The answer by sandipan is not quite what you want, as it goes through the usual normal equation to estimate:

There is already a thread on the latter: Solving normal equation gives different coefficients from using `lm`? Here I focus on the former.

Actually you are already quite close. You have obtained the mixed covariance `C`:

``````#      y   x1   x2
#y  10.4 -2.0 -0.6
#x1 -2.0 10.5  3.0
#x2 -0.6  3.0  4.4
``````

From your definition of `E` and `F`, you know you need sub-matrices to proceed. In fact, you can do matrix subsetting rather than manually imputing:

``````E <- C[2:3, 2:3]

#     x1  x2
#x1 10.5 3.0
#x2  3.0 4.4

F <- C[2:3, 1, drop = FALSE]  ## note the `drop = FALSE`

#      y
#x1 -2.0
#x2 -0.6
``````

Then the estimate is just , and you can do in R (read `?solve`):

``````c(solve(E, F))  ## use `c` to collapse matrix into a vector
# [1] -0.188172043 -0.008064516
``````

Other suggestions

• you can find column means by `colMeans`, instead of a matrix multiplication (read `?colMeans`);
• you can perform centring by using `sweep` (read `?sweep`);
• use `crossprod(D)` than `t(D) %*% D` (read `?crossprod`).

Here is a session I would do:

``````y <- c(3,3,2,4,4,5,2,3,5,3)
x1 <- c(2,2,4,3,4,4,5,3,3,5)
x2 <- c(3,3,4,4,3,3,4,2,4,4)

M <- cbind(y,x1,x2)
M_mean <- colMeans(M)
D <- sweep(M, 2, M_mean)
C <- crossprod(D)

E <- C[2:3, 2:3]
F <- C[2:3, 1, drop = FALSE]
c(solve(E, F))
# [1] -0.188172043 -0.008064516
``````
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