fisherzlink.RdComputes the Fisher Z transformation, including its inverse and the first two derivatives.
fisherzlink(theta, bminvalue = NULL, bmaxvalue = NULL,
inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)Numeric or character. See below for further details.
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.
Details at Links.
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.
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.
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.
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.
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