fisherz {VGAM} | R Documentation

## Fisher's Z Link Function

### Description

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

### Usage

    fisherz(theta, earg = list(), inverse = FALSE, deriv = 0,
            short = TRUE, tag = FALSE)

### Arguments

`theta` | Numeric or character. See below for further details.
`earg` | Optional list. Extra argument for passing in additional information. Values of `theta` which are less than or equal to -1 can be replaced by the `bminvalue` component of the list `earg` before computing the link function value. Values of `theta` which are greater than or equal to 1 can be replaced by the `bmaxvalue` component of the list `earg` before computing the link function value. See `Links` for general information about `earg`.
`inverse` | Logical. If `TRUE` the inverse function is computed.
`deriv` | Order of the derivative. Integer with value 0, 1 or 2.
`short` | Used for labelling the `blurb` slot of a `vglmff-class` object.
`tag` | Used for labelling the linear/additive predictor in the `initialize` slot of a `vglmff-class` object. Contains a little more information if `TRUE`.

### 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`. The arguments `short` and `tag` are used only if `theta` is character.

### Value

For `deriv = 0`, `0.5 * log((1+theta)/(1-theta))` when `inverse = FALSE`, and if `inverse = TRUE` then `(exp(2*theta)-1)/(exp(2*theta)+1)`.
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.

### Note

Numerical instability may occur when `theta` is close to -1 or 1. One way of overcoming this is to use `earg`.

The link function `rhobit` is very similar to `fisherz`, e.g., just twice the value of `fisherz`.

### 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`, `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)")
    abline(v=0, h=0, lty=2)
    ## 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, earg=list(bminvalue= -1 + .Machine$double.eps,
                        bmaxvalue=  1 - .Machine$double.eps))  # Has no NAs

[Package VGAM version 0.7-9 Index]