VGAM (version 1.1-8)

clogloglink: Complementary Log-log Link Function

Description

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

Usage

clogloglink(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.

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))
}