The VGAM package provides a number of (parameter) link functions which are described in general here. Collectively, they offer the user considerable choice and flexibility for modelling data.
TypicalVGAMlink(theta, someParameter = 0, bvalue = NULL, inverse = FALSE,
deriv = 0, short = TRUE, tag = FALSE)
Returns one of: the link function value or its first or second derivative, the inverse link or its first or second derivative, or a character description of the link.
Here are the general details.
If inverse = FALSE and deriv = 0 (default) then the
ordinary link
function \(\eta = g(\theta)\) is returned.
If inverse = TRUE and deriv = 0 then the inverse
link function value is returned, hence theta is really
\(\eta\) (the only occasion this happens).
If inverse = FALSE and deriv = 1 then it is
\(d\eta / d\theta\)
as a function of
\(\theta\).
If inverse = FALSE and deriv = 2 then it is
\(d^2\eta / d\theta^2\)
as a function of
\(\theta\).
If inverse = TRUE and deriv = 1 then it is
\(d\theta / d\eta\)
as a function of
\(\theta\).
If inverse = TRUE and deriv = 2 then it is
\(d^2\theta / d\eta^2\)
as a function of
\(\theta\).
It is only when deriv = 1 that
linkfun(theta, deriv = 1, inverse = TRUE)
and
linkfun(theta, deriv = 1, inverse = FALSE)
are reciprocals of each other. In particular,
linkfun(theta, deriv = 2, inverse = TRUE)
and
linkfun(theta, deriv = 2, inverse = FALSE)
are not reciprocals of each other in general.
Numeric or character.
This is usually \(\theta\) (default) but can sometimes
be \(\eta\),
depending on the other arguments.
If theta is character then inverse and
deriv are ignored.
The name theta should always be the name of the first argument.
Some parameter, e.g., an offset.
Boundary value, positive if given.
If 0 < theta then
values of theta which are less than or equal to 0 can be
replaced by bvalue
before computing the link function value.
Values of theta which are greater than or equal to 1 can be
replaced by 1 minus bvalue
before computing the link function value.
The value bvalue = .Machine$double.eps is sometimes a reasonable
value, or something slightly higher.
Logical. If TRUE and deriv = 0 then
the inverse link value
\(\theta\) is returned, hence the argument
theta is really \(\eta\).
In all other cases, the argument theta is
really \(\theta\).
Integer. Either 0, 1, or 2, specifying the order of the derivative. Some link functions handle values up to 3 or 4.
Logical.
These are used for labelling the blurb slot of a
vglmff-class object.
These arguments are used only if theta is character,
and gives the formula for the link in character form.
If tag = TRUE then the result is preceeded by a little
more information.
The output of link functions changed at VGAM 0.9-9
(date was around 2015-07).
Formerly, linkfun(theta, deriv = 1) is now
linkfun(theta, deriv = 1, inverse = TRUE), or equivalently,
1 / linkfun(theta, deriv = 1, inverse = TRUE).
Also, formerly, linkfun(theta, deriv = 2) was
1 / linkfun(theta, deriv = 2, inverse = TRUE).
This was a bug.
Altogether, these are big changes and the user should beware!
In VGAM 1.0-7 (January 2019)
all link function names were made to
end in the characters "link",
e.g.,
loglink replaces loge,
logitlink replaces logit.
For this most of them were renamed.
Upward compatability holds for older link function names,
however, users should adopt the new names immediately.
T. W. Yee
Almost all VGAM link functions have something similar to the argument list as given above. In this help file we have \(\eta = g(\theta)\) where \(g\) is the link function, \(\theta\) is the parameter and \(\eta\) is the linear/additive predictor. The link \(g\) must be strictly monotonic and twice-differentiable in its range.
The following is a brief enumeration of all VGAM link functions.
For parameters lying between 0 and 1 (e.g., probabilities):
logitlink,
probitlink,
clogloglink,
cauchitlink,
foldsqrtlink,
logclink,
gordlink,
pordlink,
nbordlink.
For positive parameters (i.e., greater than 0):
loglink,
negloglink,
powerlink.
For parameters greater than 1:
logloglink,
loglogloglink (greater than \(e\)).
For parameters between \(-1\) and \(1\):
fisherzlink,
rhobitlink.
For parameters between \(A\) and \(B\):
extlogitlink,
logofflink (\(B = \infty\)).
For unrestricted parameters (i.e., any value):
identitylink,
negidentitylink,
reciprocallink,
negreciprocallink.
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.
TypicalVGAMfamilyFunction,
linkfun,
vglm,
vgam,
rrvglm.
cqo,
cao.