R: Fisher's Z Link Function

fisherzlink {VGAM} | R Documentation

## Fisher's Z Link Function

### Description

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.

### 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.

### Author(s)

Thomas W. Yee

### References

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

### See Also

`Links`, `rhobitlink`, `logitlink`.

### Examples

    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

[Package VGAM version 1.1-3 Index]