acX_to_rho {ciuupi}

Computes the known correlation \rho between \widehat{\theta} and \widehat{\tau} from the p-vectors a and c and the design matrix X

Description

Computes the known correlation \rho between \widehat{\theta} and \widehat{\tau}. This correlation is computed from the p-vectors a and c and the n \times p design matrix X, with linearly independent columns, using the formula \rho=a^{\top}(X^{\top} X)^{-1} c /(v_{\theta} \, v_{\tau})^{1/2}, where v_{\theta} =a^{\top}(X^{\top} X)^{-1}a and v_{\tau} =c^{\top}(X^{\top} X)^{-1}c.

Usage

acX_to_rho(a, c, X)

Arguments

a

The p-vector a that specifies the parameter of interest \theta =a^{\top}\beta

c

The p-vector c used in the specification of the parameter \tau=c^{\top} \beta-t. The uncertain prior information is that \tau=0

X

The n \times p design matrix X, with linearly independent columns

Value

The known correlation \rho between \widehat{\theta} and \widehat{\tau}.

Examples

a <- c(0, 2, 0, -2)
c <- c(0, 0, 0, 1)
x1 <- c(-1, 1, -1, 1)
x2 <- c(-1, -1, 1, 1)
X <- cbind(rep(1, 4), x1, x2, x1*x2)
rho <- acX_to_rho(a, c, X)
print(rho)