VGAM (version 0.8-1)

cloglog: Complementary Log-log Link Function

Description

Computes the complementary log-log transformation, including its inverse and the first two derivatives.

Usage

cloglog(theta, earg = list(), inverse = FALSE, deriv = 0,
        short = TRUE, tag = FALSE)

Arguments

theta

Numeric or character. See below for further details.

earg

Optional 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

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 complimentary log-log of theta, i.e., log(-log(1 - theta)) when inverse = FALSE, and if inverse = TRUE then 1-exp(-exp(theta)),.

  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.

  Here, all logarithms are natural logarithms, i.e., to base $e$.

Details

The complementary log-log link function is commonly used for parameters that lie in the unit interval. 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

Links, logit, probit, cauchit.

Examples

p = seq(0.01, 0.99, by=0.01)
cloglog(p)
max(abs(cloglog(cloglog(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))
cloglog(p)  # Has NAs
cloglog(p, earg=list(bvalue= .Machine$double.eps))  # Has no NAs

plot(p, logit(p), type="l", col="limegreen", ylab="transformation",
     lwd=2, las=1, main="Some probability link functions")
lines(p, probit(p), col="purple", lwd=2)
lines(p, cloglog(p), col="chocolate", lwd=2)
lines(p, cauchit(p), col="tan", lwd=2)
abline(v=0.5, h=0, lty="dashed")
legend(0.1, 4, c("logit", "probit", "cloglog", "cauchit"),
       col=c("limegreen","purple","chocolate", "tan"), lwd=2)

# This example shows that a cloglog link is preferred over the logit
n = 500; p = 5; S = 3; Rank = 1  # Species packing model:
mydata = rcqo(n, p, S, EqualTol=TRUE, ESOpt=TRUE, EqualMax=TRUE,
              family="binomial", hiabundance=5, seed=123, Rank=Rank)
fitc = cqo(attr(mydata, "formula"), ITol=TRUE, data=mydata, 
           fam=binomialff(mv=TRUE, link="cloglog"), Rank=Rank)
fitl = cqo(attr(mydata, "formula"), ITol=TRUE, data=mydata, 
           fam=binomialff(mv=TRUE, link="logit"), Rank=Rank)

# Compare the fitted models (cols 1 and 3) with the truth (col 2)
cbind(ccoef(fitc), attr(mydata, "ccoefficients"), ccoef(fitl))