R: Complementary Log-log Link Function

clogloglink {VGAM} | R Documentation

## Complementary Log-log Link Function

### Description

Computes the complementary log-log transformation, including its inverse and the first two derivatives.

### Usage

    clogloglink(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`.

### 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)

[Package VGAM version 1.1-3 Index]