clogloglink: Complementary Log-log Link Function

In VGAM: Vector Generalized Linear and Additive Models

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)

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.

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.

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.

Note

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.

Author(s)

Thomas W. Yee

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

## 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)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.