VGAM (version 1.0-4)

Links: 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)



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.

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.


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!

One day in the future, all VGAM link functions may be renamed to end in the characters "link".


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): logit, probit, cloglog, cauchit, foldsqrt, logc, golf, polf, nbolf.

For positive parameters (i.e., greater than 0): loge, negloge, powerlink.

For parameters greater than 1: loglog.

For parameters between \(-1\) and \(1\): fisherz, rhobit.

For parameters between \(A\) and \(B\): extlogit, logoff (\(B = \infty\)).

For unrestricted parameters (i.e., any value): identity, negidentity, reciprocal, negreciprocal.


McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London: Chapman & Hall.

See Also

TypicalVGAMfamilyFunction, linkfun, vglm, vgam, rrvglm. cqo, cao.


Run this code
logit("a", short = FALSE)
logit("a", short = FALSE, tag = TRUE)

logoff(1:5, offset = 1)  # Same as log(1:5 + 1)
powerlink(1:5, power = 2)  # Same as (1:5)^2

# }
 # This is old and no longer works:
logoff(1:5, earg = list(offset = 1))
powerlink(1:5, earg = list(power = 2))
# }
fit1 <- vgam(agaaus ~ altitude, binomialff(link = "cloglog"), hunua)  # best
fit2 <- vgam(agaaus ~ altitude, binomialff(link =  cloglog ), hunua)  # okay

# }
# 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 = logoff(offset = 1.1),
                          lshape2 = logoff(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, logit(p), type = "l", col = "blue")
plot(x, logit(x, inverse = TRUE), type = "l", col = "blue")
plot(p, logit(p, deriv = 1), type = "l", col = "blue")  # 1 / (p*(1-p))
plot(p, logit(p, deriv = 2), type = "l", col = "blue")  # (2*p-1)/(p*(1-p))^2
# }

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