VGAM (version 1.0-1)

cauchit: Cauchit Link Function

Description

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

Usage

cauchit(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.

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.

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.

References

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

See Also

logit, probit, cloglog, loge, cauchy, cauchy1.

Examples

p <- seq(0.01, 0.99, by=0.01)
cauchit(p)
max(abs(cauchit(cauchit(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))
cauchit(p)  # Has no NAs

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(logit(p, deriv = d), probit(p, deriv = d)),
          type = "n", col = "purple", ylab = "transformation",
          las = 1, main = if (d == 0) "Some probability link functions"
          else "First derivative")
  lines(p,   logit(p, deriv = d), col = "limegreen")
  lines(p,  probit(p, deriv = d), col = "purple")
  lines(p, cloglog(p, deriv = d), col = "chocolate")
  lines(p, cauchit(p, deriv = d), col = "tan")
  if (d == 0) {
    abline(v = 0.5, h = 0, lty = "dashed")
    legend(0, 4.5, c("logit", "probit", "cloglog", "cauchit"), lwd = mylwd,
           col = c("limegreen","purple","chocolate", "tan"))
  } else
    abline(v = 0.5, lty = "dashed")
}

for (d in 0) {
  matplot(y, cbind( logit(y, deriv = d, inverse = TRUE),
                   probit(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,   logit(y, deriv = d, inverse = TRUE), col = "limegreen")
  lines(y,  probit(y, deriv = d, inverse = TRUE), col = "purple")
  lines(y, cloglog(y, deriv = d, inverse = TRUE), col = "chocolate")
  lines(y, cauchit(y, deriv = d, inverse = TRUE), col = "tan")
  if (d == 0) {
      abline(h = 0.5, v = 0, lty = "dashed")
      legend(-4, 1, c("logit", "probit", "cloglog", "cauchit"), lwd = mylwd,
             col = c("limegreen", "purple", "chocolate", "tan"))
  }
}
par(lwd = 1)