Complementary Log-log Link Function
Computes the complementary log-log transformation, including its inverse and the first two derivatives.
clogloglink(theta, bvalue = NULL, inverse = FALSE, deriv = 0,
short = TRUE, tag = FALSE)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.
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.
Numerical instability may occur when theta is close to 1 or 0.
One way of overcoming this is to use bvalue.
Changing 1s to 0s and 0s to 1s in the response means that effectively
a loglog link is fitted. That is, tranform y by 1-y.
That's why only one of clogloglink
and logloglink is written.
With constrained ordination (e.g., cqo and
cao) used with binomialff, a complementary
log-log link function is preferred over the
default logitlink,
for a good reason. See the example below.
In terms of the threshold approach with cumulative probabilities for an ordinal response this link function corresponds to the extreme value distribution.
Thomas W. Yee
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.
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
## Not run:
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)
## End(Not run)
## Not run:
# 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))
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.