ciuupi2 (version 1.0.1)

find_rho: Find rho

Description

Find the correlation rho for given \(n\) by \(p\) design matrix X and given \(p\)-vectors a and c

Usage

find_rho(X, a, c)

Arguments

X

The \(n\) by \(p\) design matrix

a

A vector used to specify the parameter of interest

c

A vector used to specify the parameter about which we have uncertain prior information

Value

The value of the correlation rho.

\(X\), <code>a</code> and <code>c</code> for a particular example

Consider the same 2 x 2 factorial example as that described in Section 4 of Kabaila and Giri (2009), except that the number of replicates is 3 instead of 20. In this case, \(X\) is a 12 x 4 matrix, \(\beta\) is an unknown parameter 4-vector and \(\epsilon\) is a random 12-vector with components that are independent and identically normally distributed with zero mean and unknown variance. In other words, the length of the response vector \(y\) is \(n\) = 12 and the length of the parameter vector \(\beta\) is \(p\) = 4, so that \(m = n - p\) = 8. The parameter of interest is \(\theta = \) a' \(\beta\), where the column vector a = (0, 2, 0, -2). Also, the parameter \(\tau = \) c' \(\beta\), where the column vector c = (0, 0, 0, 1). The uncertain prior information is that \(\tau = \) t, where t = 0.

The design matrix \(X\) and the vectors a and c (denoted in R by a.vec and c.vec, respectively) are entered into R using the commands in the following example.

Details

Suppose that $$y = X \beta + \epsilon$$ where \(y\) is a random \(n\)-vector of responses, \(X\) is a known \(n\) by \(p\) matrix with linearly independent columns, \(\beta\) is an unknown parameter \(p\)-vector and \(\epsilon\) is a random \(n\)-vector with components that are independent and identically normally distributed with zero mean and unknown variance. The parameter of interest is \(\theta = \) a' \(\beta\). The uncertain prior information is that \(\tau = \) c' \(\beta\) takes the value t, where a and c are specified linearly independent nonzero \(p\)-vectors and t is a specified number. rho is the known correlation between the least squares estimators of \(\theta\) and \(\tau\). It is determined by the \(n\) by \(p\) design matrix X and the \(p\)-vectors a and c.

References

Kabaila, P. and Giri, R. (2009). Confidence intervals in regression utilizing prior information. Journal of Statistical Planning and Inference, 139, 3419-3429.

Examples

# NOT RUN {
col1 <- rep(1,4)
col2 <- c(-1, 1, -1, 1)
col3 <- c(-1, -1, 1, 1)
col4 <- c(1, -1, -1, 1)
X.single.rep <- cbind(col1, col2, col3, col4)
X <- rbind(X.single.rep, X.single.rep, X.single.rep)
a.vec <- c(0, 2, 0, -2)
c.vec <- c(0, 0, 0, 1)

# Find the value of rho
rho <- find_rho(X, a=a.vec, c=c.vec)
rho

# The value of rho is -0.7071068

# }