Matrix Algebra and Linear Models

Week 1 - Matrix Algebra

See edX

Suppose we are analyzing a set of 4 samples. The first two samples are from a treatment group A and the second two samples are from a treatment group B. This design can be represented with a model matrix like so:

X <- matrix(c(1,1,1,1,0,0,1,1), nrow=4)
rownames(X) <- c("a", "a", "b", "b")
X

##   [,1] [,2]
## a    1    0
## a    1    0
## b    1    1
## b    1    1

Suppose that the fitted parameters for a linear model give us:

beta <- c(5, 2)

What is the fitted value for the A samples? (The fitted Y values.)

X.a <- X[rownames(X) == "a",]
X.a %*% beta

##   [,1]
## a    5
## a    5

What is the fitted value for the B samples? (The fitted Y values.)

X.b <- X[rownames(X) == "b",]
X.b %*% beta

##   [,1]
## b    7
## b    7

Suppose now we are comparing two treatments B and C to a control group A, each with two samples. This design can be represented with a model matrix like so:

X <- matrix(c(1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,1), nrow=6)
rownames(X) <- c("a","a","b","b","c","c")

Suppose that the fitted values for the linear model are given by:

beta <- c(10,3,-3)

What is the fitted value for the B samples?

X.b = X[rownames(X) == "b",]
X.b %*% beta

##   [,1]
## b   13
## b   13

What is the fitted value for the C samples?

X.c = X[rownames(X) == "c",]
X.c %*% beta

##   [,1]
## c    7
## c    7