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Exercises

1. 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)
   

   Use the matrix multiplication operator, %*%, in R to answer the following questions:

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

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

3. 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")
    X
   

   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?

4. What is the fitted value for the C samples?