Fisher's Z Link Function

Description

Computes the Fisher Z transformation, including its inverse and the first two derivatives.

Usage

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

Arguments

Details

The fisherz 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, 0.5 * log((1+theta)/(1-theta)) (same as atanh(theta)) when inverse = FALSE, and if inverse = TRUE then (exp(2*theta)-1)/(exp(2*theta)+1) (same as tanh(theta)).

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.

Here, all logarithms are natural logarithms, i.e., to base e.

Note

Numerical instability may occur when theta is close to -1 or 1. One way of overcoming this is to use, e.g., bminvalue.

The link function rhobit is very similar to fisherz, e.g., just twice the value of fisherz. This link function may be renamed to atanhlink in the near future.

Author(s)

Thomas W. Yee

References

McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London: Chapman & Hall.

See Also

Links, rhobit, atanh, logit.

Examples

theta <- seq(-0.99, 0.99, by = 0.01)
y <- fisherz(theta)
## Not run:  plot(theta, y, type = "l", las = 1, ylab = "",
   main = "fisherz(theta)", col = "blue")
abline(v = (-1):1, h = 0, lty = 2, col = "gray") 
## End(Not run)

x <- c(seq(-1.02, -0.98, by = 0.01), seq(0.97, 1.02, by = 0.01))
fisherz(x)  # Has NAs
fisherz(x, bminvalue = -1 + .Machine$double.eps,
           bmaxvalue =  1 - .Machine$double.eps)  # Has no NAs