VGAM (version 0.9-4)

fisherz: 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

theta

Numeric or character. See below for further details.

bminvalue, bmaxvalue

Optional boundary values. Values of theta which are less than or equal to $-1$ can be replaced by bminvalue before computing the link function value. Values of theta which are greater than or equal to $1$ can

inverse, deriv, short, tag

Details at Links.

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 theta / d eta 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.

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.

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)
plot(theta, y, type = "l", las = 1, ylab = "",
   main = "fisherz(theta)", col = "blue")
abline(v = (-1):1, h = 0, lty = 2, col = "gray")

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