VGAM (version 0.7-5)

cauchit: Cauchit Link Function

Description

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

Usage

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

Arguments

theta

Numeric or character. See below for further details.

earg

List. Extra argument for passing in additional information. Values of theta which are less than or equal to 0 can be replaced by the bvalue component of the list earg before computing the link function value.

inverse

Logical. If TRUE the inverse function is computed.

deriv

Order of the derivative. Integer with value 0, 1 or 2.

short

Used for labelling the blurb slot of a vglmff-class object.

tag

Used for labelling the linear/additive predictor in the initialize slot of a vglmff-class object. Contains a little more information if TRUE.

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 theta / d eta 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. The arguments short and tag are used only if theta is character.

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))
y = seq(-4, 4, length=100)

for(d in 0:1) {
    matplot(p, cbind(logit(p, deriv=d), probit(p, deriv=d)),
            type="n", col="purple", ylab="transformation",
            lwd=2, las=1, main=if(d==0) "Some probability link functions"
            else "First derivative")
    lines(p, logit(p, deriv=d), col="limegreen", lwd=2)
    lines(p, probit(p, deriv=d), col="purple", lwd=2)
    lines(p, cloglog(p, deriv=d), col="chocolate", lwd=2)
    lines(p, cauchit(p, deriv=d), col="tan", lwd=2)
    if(d==0) {
        abline(v=0.5, h=0, lty="dashed")
        legend(0, 4.5, c("logit", "probit", "cloglog", "cauchit"),
               col=c("limegreen","purple","chocolate", "tan"), lwd=2)
    } 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", lwd=2, las=1)
    lines(y, logit(y, deriv=d, inverse=TRUE), col="limegreen", lwd=2)
    lines(y, probit(y, deriv=d, inverse=TRUE), col="purple", lwd=2)
    lines(y, cloglog(y, deriv=d, inverse=TRUE), col="chocolate", lwd=2)
    lines(y, cauchit(y, deriv=d, inverse=TRUE), col="tan", lwd=2)
    if(d==0) {
        abline(h=0.5, v=0, lty="dashed")
        legend(-4, 1, c("logit", "probit", "cloglog", "cauchit"),
               col=c("limegreen","purple","chocolate", "tan"), lwd=2)
    }
}