VGAM (version 1.0-1)

loglog: Log-log Link Function

Description

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

Usage

loglog(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 can be replaced by bvalue before computing the link function value. The component name bvalue stands for ``boundary value''. See

inverse, deriv, short, tag

Details at Links.

Value

• For deriv = 0, the log of theta, i.e., log(log(theta)) when inverse = FALSE, and if inverse = TRUE then 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. Numerical values of theta close to 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.

See Also

Links, loge, logoff.

Examples

x <- seq(0.8, 1.5, by = 0.1)
loglog(x)  # Has NAs
loglog(x, bvalue = 1.0 + .Machine$double.eps)  # Has no NAs

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