VGAM (version 1.1-12)

clogloglink: Complementary Log-log Link Function

Description

Computes the complementary log-log transformation, including its inverse and the first two derivatives. The complementary log transformation is also computed.

Usage

clogloglink(theta, bvalue = NULL, inverse = FALSE, deriv = 0,
            short = TRUE, tag = FALSE)
   cloglink(theta, bvalue = NULL, inverse = FALSE, deriv = 0,
            short = TRUE, tag = FALSE)

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

eta / d

theta

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 \(e\).

Arguments

theta

Numeric or character. See below for further details.

bvalue

See Links for general information about links.

inverse, deriv, short, tag

Details at Links.

Author

Thomas W. Yee

Details

The complementary log-log link function is commonly used for parameters that lie in the unit interval. But unlike logitlink, probitlink and cauchitlink, this link is not symmetric. It is the inverse CDF of the extreme value (or Gumbel or log-Weibull) distribution. Numerical values of theta close to 0 or 1 or out of range result in Inf, -Inf, NA or NaN.

The complementary log link function is the same as the complementary log-log but the outer log is omitted. This link is suitable for lrho in betabinomial because it handles probability-like parameters but also allows slight negative values in theory. In particular, cloglink safeguards against parameters exceeding unity.

References

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

See Also

Links, logitoffsetlink, logitlink, probitlink, cauchitlink, pgumbel.

Examples

p <- seq(0.01, 0.99, by = 0.01)
clogloglink(p)
max(abs(clogloglink(clogloglink(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))
clogloglink(p)  # Has NAs
clogloglink(p, bvalue = .Machine$double.eps)  # Has no NAs

if (FALSE) {
p <- seq(0.01, 0.99, by = 0.01)
plot(p, logitlink(p), type = "l", col = "limegreen", lwd = 2, las = 1,
     main = "Some probability link functions", ylab = "transformation")
lines(p, probitlink(p), col = "purple", lwd = 2)
lines(p, clogloglink(p), col = "chocolate", lwd = 2)
lines(p, cauchitlink(p), col = "tan", lwd = 2)
abline(v = 0.5, h = 0, lty = "dashed")
legend(0.1, 4, c("logitlink", "probitlink", "clogloglink", "cauchitlink"),
       col = c("limegreen", "purple", "chocolate", "tan"), lwd = 2)
}

if (FALSE) {
# This example shows that clogloglink is preferred over logitlink
n <- 500; p <- 5; S <- 3; Rank <- 1  # Species packing model:
mydata <- rcqo(n, p, S, eq.tol = TRUE, es.opt = TRUE, eq.max = TRUE,
               family = "binomial", hi.abundance = 5, seed = 123,
               Rank = Rank)
fitc <- cqo(attr(mydata, "formula"), I.tol = TRUE, data = mydata,
            fam = binomialff(multiple.responses = TRUE, link = "cloglog"),
            Rank = Rank)
fitl <- cqo(attr(mydata, "formula"), I.tol = TRUE, data = mydata,
            fam = binomialff(multiple.responses = TRUE, link = "logitlink"),
            Rank = Rank)

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