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.

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. Most link functions handle values up to 3 or 4. Some links can even handle values up to 9 but may suffer from catastrophic cancellation near the boundaries, be inefficient and slow because they are based on D.

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.

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.

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.

For positive parameters (i.e., greater than 0): loglink, negloglink, sqrtlink, powerlink.

For parameters greater than 1: logloglink, loglogloglink (greater than $e$).

For parameters between $-1$ and $1$: fisherzlink, rhobitlink.

For parameters between finite $A$ and $B$: extlogitlink, logofflink ($B = \infty$).

For unrestricted parameters (i.e., any value): identitylink, negidentitylink, reciprocallink, negreciprocallink.

References

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

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.

Author

T. W. Yee

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.