VGAM (version 1.1-2)

logloglink: Log-log and Log-log-log Link Functions

Description

Computes the two transformations, including their inverse and the first two derivatives.

Usage

logloglink(theta, bvalue = NULL, inverse = FALSE, deriv = 0,
           short = TRUE, tag = FALSE)
loglogloglink(theta, bvalue = NULL, inverse = FALSE, deriv = 0,
              short = TRUE, tag = FALSE)

Arguments

theta

Numeric or character. See below for further details.

bvalue

Values of theta which are less than or equal to 1 or \(e\) can be replaced by bvalue before computing the link function value. The component name bvalue stands for ``boundary value''. See Links for more information.

inverse, deriv, short, tag

Details at Links.

Value

For logloglink(): for deriv = 0, the log of log(theta), i.e., log(log(theta)) when inverse = FALSE, and if inverse = TRUE then exp(exp(theta)).

For loglogloglink(): for deriv = 0, the log of log(log(theta)), i.e., log(log(log(theta))) when inverse = FALSE, and if inverse = TRUE then exp(exp(exp(theta))).

For deriv = 1, then the function returns d theta / d eta 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 log-log link function is commonly used for parameters that are greater than unity. Similarly, the log-log-log link function is applicable for parameters that are greater than \(e\). Numerical values of theta close to 1 or \(e\) or out of range result in Inf, -Inf, NA or NaN. One possible application of loglogloglink() is to the \(k\) parameter (also called size) of negbinomial to Poisson-like data but with only a small amount of overdispersion; then \(k\) is a large number relative to munb. In such situations a loglink or loglog link may not be sufficient to draw the estimate toward the interior of the parameter space. Using a more stronger link function can help mitigate the Hauck-Donner effect hdeff.

References

McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London: Chapman & Hall.

See Also

Links, loglink, logofflink.

Examples

# NOT RUN {
x <- seq(0.8, 1.5, by = 0.1)
logloglink(x)  # Has NAs
logloglink(x, bvalue = 1.0 + .Machine$double.eps)  # Has no NAs

x <- seq(1.01, 10, len = 100)
logloglink(x)
max(abs(logloglink(logloglink(x), inverse = TRUE) - x))  # Should be 0
# }