# Complementary Log-log Link Function Computes the complementary log-log transformation, including its inverse and the first two derivatives. The complementary log transformation is also computed. ```r 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. ## Returns 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 $e$. ## References McCullagh, P. and Nelder, J. A. (1989). **Generalized Linear Models**, 2nd ed. London: Chapman & Hall. ## Author(s) Thomas W. Yee ## 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. ## See Also `Links`, `logitoffsetlink`, `logitlink`, `probitlink`, `cauchitlink`, `pgumbel`. ## Examples ```r 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) ```

clogloglink function