Rhobit Link Function

Computes the rhobit link transformation, including its inverse and the first two derivatives.

Usage

rhobitlink(theta, bminvalue = NULL, bmaxvalue = NULL,
           inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)

Arguments

theta

Numeric or character. See below for further details.

bminvalue, bmaxvalue

Optional boundary values, e.g., values of theta which are less than or equal to -1 can be replaced by bminvalue before computing the link function value. And values of theta which are greater than or equal to 1 can be replaced by bmaxvalue before computing the link function value. See Links.

inverse, deriv, short, tag

Details at Links.

Details

The rhobitlink link function is commonly used for parameters that lie between \(-1\) and \(1\). Numerical values of theta close to \(-1\) or \(1\) or out of range result in Inf, -Inf, NA or NaN.

Value

For deriv = 0, the rhobit of theta, i.e., log((1 + theta)/(1 - theta)) when inverse = FALSE, and if inverse = TRUE then (exp(theta) - 1)/(exp(theta) + 1).

For deriv = 1, then the function returns d eta / d theta as a function of theta if inverse = FALSE, else if inverse = TRUE then it returns the reciprocal.

Author

Thomas W. Yee

Note

Numerical instability may occur when theta is close to \(-1\) or \(1\). One way of overcoming this is to use bminvalue, etc.

The correlation parameter of a standard bivariate normal distribution lies between \(-1\) and \(1\), therefore this function can be used for modelling this parameter as a function of explanatory variables.

The link function rhobitlink is very similar to fisherzlink, e.g., just twice the value of fisherzlink.

Examples

theta <- seq(-0.99, 0.99, by = 0.01)
y <- rhobitlink(theta)
if (FALSE) { # \dontrun{
plot(theta, y, type = "l", ylab = "", main = "rhobitlink(theta)")
abline(v = 0, h = 0, lty = 2)
} # }

x <- c(seq(-1.02, -0.98, by = 0.01), seq(0.97, 1.02, by = 0.01))
rhobitlink(x)  # Has NAs
#>  [1]       NaN       NaN      -Inf -5.293305 -4.595120  4.184591  4.595120
#>  [8]  5.293305       Inf       NaN       NaN
rhobitlink(x, bminvalue = -1 + .Machine$double.eps,
              bmaxvalue =  1 - .Machine$double.eps)  # Has no NAs
#>  [1] -36.736801 -36.736801 -36.736801  -5.293305  -4.595120   4.184591
#>  [7]   4.595120   5.293305  36.736801  36.736801  36.736801