Link functions for VGLM/VGAM/etc. families
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)theta |
Numeric or character.
This is usually theta (default) but can sometimes
be eta,
depending on the other arguments.
If |
someParameter |
Some parameter, e.g., an offset. |
bvalue |
Boundary value, positive if given.
If |
inverse |
Logical. If |
deriv |
Integer. Either 0, 1, or 2, specifying the order of the derivative. Some link functions handle values up to 3 or 4. |
short, tag |
Logical.
These are used for labelling the |
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 = Inf).
For unrestricted parameters (i.e., any value):
identitylink,
negidentitylink,
reciprocallink,
negreciprocallink.
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.
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!
VGAM link functions are generally not compatible with
other functions outside the package.
In particular, they won't work with
glm or any other package for fitting GAMs.
From October 2006 onwards,
all VGAM family functions will only
contain one default value for each link argument rather than giving a
vector of choices.
For example, rather than
binomialff(link = c("logitlink", "probitlink", "clogloglink",
"cauchitlink", "identitylink"), ...)
it is now
binomialff(link = "logitlink", ...).
No checking will be done to see if the user's choice is reasonable.
This means that the user can write his/her own VGAM link function
and use it within any VGAM family function.
Altogether this provides greater flexibility.
The downside is that the user must specify the full name of the
link function, by either assigning the link argument the full name as
a character string, or just the name itself. See the examples below.
From August 2012 onwards, a major change in link functions occurred.
Argument esigma (and the like such as earg)
used to be in VGAM prior to version 0.9-0 (released
during the 2nd half of 2012).
The major change is that arguments such as offset that used to
be passed in via those arguments can done directly through
the link function. For example,
gev(lshape = "logofflink", eshape = list(offset = 0.5))
is replaced by
gev(lshape = logofflink(offset = 0.5)).
The @misc slot no longer
has link and earg components, but two other
components replace these. Functions such as
dtheta.deta(),
d2theta.deta2(),
d3theta.deta3(),
eta2theta(),
theta2eta()
are modified.
From January 2019 onwards, all link function names ended
in "link". See above for details.
T. W. Yee
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.
logitlink("a")
logitlink("a", short = FALSE)
logitlink("a", short = FALSE, tag = TRUE)
logofflink(1:5, offset = 1) # Same as log(1:5 + 1)
powerlink(1:5, power = 2) # Same as (1:5)^2
## Not run: # This is old and no longer works:
logofflink(1:5, earg = list(offset = 1))
powerlink(1:5, earg = list(power = 2))
## End(Not run)
fit1 <- vgam(agaaus ~ altitude,
binomialff(link = "clogloglink"), hunua) # best
fit2 <- vgam(agaaus ~ altitude,
binomialff(link = clogloglink ), hunua) # okay
## Not run:
# This no longer works since "clog" is not a valid VGAM link function:
fit3 <- vgam(agaaus ~ altitude,
binomialff(link = "clog"), hunua) # not okay
# No matter what the link, the estimated var-cov matrix is the same
y <- rbeta(n = 1000, shape1 = exp(0), shape2 = exp(1))
fit1 <- vglm(y ~ 1, betaR(lshape1 = "identitylink",
lshape2 = "identitylink"),
trace = TRUE, crit = "coef")
fit2 <- vglm(y ~ 1, betaR(lshape1 = logofflink(offset = 1.1),
lshape2 = logofflink(offset = 1.1)), trace=TRUE)
vcov(fit1, untransform = TRUE)
vcov(fit1, untransform = TRUE) -
vcov(fit2, untransform = TRUE) # Should be all 0s
\dontrun{ # This is old:
fit1@misc$earg # Some 'special' parameters
fit2@misc$earg # Some 'special' parameters are here
}
par(mfrow = c(2, 2))
p <- seq(0.05, 0.95, len = 200) # A rather restricted range
x <- seq(-4, 4, len = 200)
plot(p, logitlink(p), type = "l", col = "blue")
plot(x, logitlink(x, inverse = TRUE), type = "l", col = "blue")
plot(p, logitlink(p, deriv=1), type="l", col="blue") # 1 / (p*(1-p))
plot(p, logitlink(p, deriv=2), type="l", col="blue") # (2*p-1)/(p*(1-p))^2
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.