## Description

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

## Usage

 ```1 2``` ```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.

## 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.

Thomas W. Yee

## References

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

`Links`, `rhobitlink`, `logitlink`.
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```theta <- seq(-0.99, 0.99, by = 0.01) y <- fisherzlink(theta) ## Not run: plot(theta, y, type = "l", las = 1, ylab = "", main = "fisherzlink(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)) fisherzlink(x) # Has NAs fisherzlink(x, bminvalue = -1 + .Machine\$double.eps, bmaxvalue = 1 - .Machine\$double.eps) # Has no NAs ```