VGAM (version 0.9-3)

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 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.

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, bvalue = .Machine$double.eps)  # Has no NAs

p <- seq(0.01, 0.99, by = 0.01)
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, eq.tol = TRUE, es.opt = TRUE, eq.max = TRUE,
               family = "binomial", hi.abundance = 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(concoef(fitc), attr(mydata, "ccoefficients"), concoef(fitl))