VGAM (version 1.0-6)

cloglog: Complementary Log-log Link Function

Description

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

Usage

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

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.

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 \(e\).

Details

The complementary log-log link function is commonly used for parameters that lie in the unit interval. But unlike logit, probit and cauchit, 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, logit, probit, cauchit, pgumbel.

Examples

# NOT RUN {
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, bvalue = .Machine$double.eps)  # Has no NAs

# }
# NOT RUN {
p <- seq(0.01, 0.99, by = 0.01)
plot(p, logit(p), type = "l", col = "limegreen", lwd = 2, las = 1,
     main = "Some probability link functions", ylab = "transformation")
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)
# }
# NOT RUN {
# }
# NOT RUN {
# 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, 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 = "logit"),
            Rank = Rank)

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