Links: Link functions for VGLM/VGAM/etc. families
In VGAM: Vector Generalized Linear and Additive Models
Links R Documentation
Link functions for VGLM/VGAM/etc. families
Description
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.
Usage
TypicalVGAMlink(theta, someParameter = 0, bvalue = NULL, inverse = FALSE,
deriv = 0, short = TRUE, tag = FALSE)
Arguments
theta
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.
someParameter
Some parameter, e.g., an offset.
bvalue
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.
inverse
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.
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 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.
Details
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.
Value
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.
Warning
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.
Note
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.
Author(s)
T. W. Yee
References
McCullagh, P. and Nelder, J. A. (1989).
Generalized Linear Models, 2nd ed. London: Chapman & Hall.
See Also
TypicalVGAMfamilyFunction,
linkfun,
vglm,
vgam,
rrvglm.
cqo,
cao.
Examples
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)
VGAM documentation built on Sept. 19, 2023, 9:06 a.m.
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| Links | R Documentation |
Link functions for VGLM/VGAM/etc. families
Description
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.
Usage
TypicalVGAMlink(theta, someParameter = 0, bvalue = NULL, inverse = FALSE,
deriv = 0, short = TRUE, tag = FALSE)
Arguments
theta |
Numeric or character.
This is usually |
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 |
Details
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.
Value
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.
Warning
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.
Note
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.
Author(s)
T. W. Yee
References
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.
See Also
TypicalVGAMfamilyFunction,
linkfun,
vglm,
vgam,
rrvglm.
cqo,
cao.
Examples
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)
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