Cauchit Link Function
Computes the cauchit (tangent) link transformation, including its inverse and the first two derivatives.
cauchitlink(theta, bvalue = .Machine$double.eps,
inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)This link function is an alternative link function for parameters that lie in the unit interval. This type of link bears the same relation to the Cauchy distribution as the probit link bears to the Gaussian. One characteristic of this link function is that the tail is heavier relative to the other links (see examples below).
Numerical values of theta close to 0 or 1 or out of range result
in Inf, -Inf, NA or NaN.
For deriv = 0, the tangent of theta, i.e.,
tan(pi * (theta-0.5)) when inverse = FALSE,
and if inverse = TRUE then
0.5 + atan(theta)/pi.
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.
Numerical instability may occur when theta is close to 1 or 0.
One way of overcoming this is to use bvalue.
As mentioned above,
in terms of the threshold approach with cumulative probabilities for
an ordinal response this link function corresponds to the
Cauchy distribution (see cauchy1).
Thomas W. Yee
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.
p <- seq(0.01, 0.99, by = 0.01)
cauchitlink(p)
max(abs(cauchitlink(cauchitlink(p), inverse = TRUE) - p)) # Should be 0
p <- c(seq(-0.02, 0.02, by=0.01), seq(0.97, 1.02, by = 0.01))
cauchitlink(p) # Has no NAs
## Not run:
par(mfrow = c(2, 2), lwd = (mylwd <- 2))
y <- seq(-4, 4, length = 100)
p <- seq(0.01, 0.99, by = 0.01)
for (d in 0:1) {
matplot(p, cbind(logitlink(p, deriv = d), probitlink(p, deriv = d)),
type = "n", col = "purple", ylab = "transformation",
las = 1, main = if (d == 0) "Some probability link functions"
else "First derivative")
lines(p, logitlink(p, deriv = d), col = "limegreen")
lines(p, probitlink(p, deriv = d), col = "purple")
lines(p, clogloglink(p, deriv = d), col = "chocolate")
lines(p, cauchitlink(p, deriv = d), col = "tan")
if (d == 0) {
abline(v = 0.5, h = 0, lty = "dashed")
legend(0, 4.5, c("logitlink", "probitlink", "clogloglink",
"cauchitlink"), lwd = mylwd,
col = c("limegreen", "purple", "chocolate", "tan"))
} else
abline(v = 0.5, lty = "dashed")
}
for (d in 0) {
matplot(y, cbind( logitlink(y, deriv = d, inverse = TRUE),
probitlink(y, deriv = d, inverse = TRUE)),
type = "n", col = "purple", xlab = "transformation", ylab = "p",
main = if (d == 0) "Some inverse probability link functions"
else "First derivative", las=1)
lines(y, logitlink(y, deriv = d, inverse = TRUE), col = "limegreen")
lines(y, probitlink(y, deriv = d, inverse = TRUE), col = "purple")
lines(y, clogloglink(y, deriv = d, inverse = TRUE), col = "chocolate")
lines(y, cauchitlink(y, deriv = d, inverse = TRUE), col = "tan")
if (d == 0) {
abline(h = 0.5, v = 0, lty = "dashed")
legend(-4, 1, c("logitlink", "probitlink", "clogloglink",
"cauchitlink"), lwd = mylwd,
col = c("limegreen", "purple", "chocolate", "tan"))
}
}
par(lwd = 1)
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.