VGAM (version 1.0-4)

# 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 be replaced by `bmaxvalue` before computing the link function value. See `Links`.

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 `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.

## 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.

`Links`, `rhobit`, `atanh`, `logit`.

## Examples

Run this code
```# NOT RUN {
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)", col = "blue")
abline(v = (-1):1, h = 0, lty = 2, col = "gray")
# }
# 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, bminvalue = -1 + .Machine\$double.eps,
bmaxvalue =  1 - .Machine\$double.eps)  # Has no NAs
# }
```

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