Fisher's Z Link Function

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

Usage

fisherzlink(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 be replaced by bmaxvalue before computing the link function value. See Links.

inverse, deriv, short, tag

Details at Links.

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.

References

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

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, e.g., bminvalue.

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

Examples

theta <- seq(-0.99, 0.99, by = 0.01)
y <- fisherzlink(theta)
if (FALSE)  plot(theta, y, type = "l", las = 1, ylab = "",
   main = "fisherzlink(theta)", col = "blue")
abline(v = (-1):1, h = 0, lty = 2, col = "gray")  # \dontrun{}
#> Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet

x <- c(seq(-1.02, -0.98, by = 0.01), seq(0.97, 1.02, by = 0.01))
fisherzlink(x)  # Has NAs
#>  [1]       NaN       NaN      -Inf -2.646652 -2.297560  2.092296  2.297560
#>  [8]  2.646652       Inf       NaN       NaN
fisherzlink(x, bminvalue = -1 + .Machine$double.eps,
               bmaxvalue =  1 - .Machine$double.eps)  # Has no NAs
#>  [1] -18.368400 -18.368400 -18.368400  -2.646652  -2.297560   2.092296
#>  [7]   2.297560   2.646652  18.368400  18.368400  18.368400