cauchitlink: Cauchit Link Function

In VGAM: Vector Generalized Linear and Additive Models

Cauchit Link Function

Description

Computes the cauchit (tangent) link transformation, including its inverse and the first two derivatives.

Usage

cauchitlink(theta, bvalue = .Machine$double.eps,
            inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)

Arguments

theta	Numeric or character. See below for further details.
bvalue	See Links.
inverse, deriv, short, tag	Details at Links.

Details

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.

Value

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.

Note

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).

Author(s)

Thomas W. Yee

References

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

See Also

logitlink, probitlink, clogloglink, loglink, cauchy, cauchy1, Cauchy.

Examples

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)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.