VGAM (version 1.0-4)

# 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. 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.

`Links`, `logitoffsetlink`, `logit`, `probit`, `cauchit`.

## Examples

Run this code
```# 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))
# }
```

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