Log-log and Log-log-log Link Functions
Computes the two transformations, including their inverse and the first two derivatives.
logloglink(theta, bvalue = NULL, inverse = FALSE, deriv = 0,
short = TRUE, tag = FALSE)
loglogloglink(theta, bvalue = NULL, inverse = FALSE, deriv = 0,
short = TRUE, tag = FALSE)theta |
Numeric or character. See below for further details. |
bvalue |
Values of |
inverse, deriv, short, tag |
Details at |
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.
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.
Numerical instability may occur when theta is
close to 1 or e
unless bvalue is used.
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.
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
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.