Links {VGAM}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 flexibility for modelling data.

Usage

TypicalVGAMlinkFunction(theta, earg=list(), inverse=FALSE,
                        deriv=0, short=TRUE, tag=FALSE)

Arguments

Almost all VGAM link functions have something similar to the argument list as given above. That is, there is a matching earg for each link argument. In the following we have eta=g(theta) where g is the link function, theta is the parameter and eta is the linear/additive predictor.

theta Numeric or character. Actually this can be theta (default) or eta, depending on the other arguments. If theta is character then inverse and deriv are ignored.
earg List. Extra argument allowing for additional information, specific to the link function. For example, for logoff, this will contain the offset value. The argument earg is always a list with named components. See each specific link function to find the component names for the list.
Almost all VGAM family functions with a single link function have an argument (often called earg) which will allow parameters to be inputted for that link function. For VGAM family functions with more than one link function there usually will be an earg-type argument for each link. For example, if there are two links called lshape and lscale then the earg-type arguments for these might be called eshape and escale, say.
inverse Logical. If TRUE the inverse link value theta is returned, hence the argument theta is really eta.
deriv Integer. Either 0, 1, or 2 specifying the order of the derivative.
short, tag Logical. Used for labelling the blurb slot of a vglmff-class object. Used only if theta is character, and gives the formula for the link in character form. If tag=TRUE then the result contains a little more information.

Details

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, loglog, fsqrt, logc, golf, polf, nbolf.

For positive parameters (i.e., greater than 0): loge, nloge, powl.

For parameters greater than 1: loglog.

For parameters between -1 and 1: fisherz, rhobit.

For parameters between A and B: elogit, logoff (B=Inf).

For unrestricted parameters (i.e., any value): identity, nidentity, reciprocal, nreciprocal.

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=FALSE and deriv=1 then it is d theta / d eta as a function of theta. If inverse=FALSE and deriv=2 then it is d^2 theta / d eta^2 as a function of theta.
If inverse=TRUE and deriv=0 then the inverse link function is returned, hence theta is really eta. If inverse=TRUE and deriv is positive then the reciprocal of the same link function with (theta=theta, earg=earg, inverse=TRUE, deriv=deriv) is returned.

Note

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("logit", "probit", "cloglog", "cauchit", "identity"), ...) it is now binomialff(link="logit", ...) 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.

Author(s)

T. W. Yee

References

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

See Also

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

Examples

logit("a")
logit("a", short=FALSE)
logit("a", short=FALSE, tag=TRUE)

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

data(hunua)
fit1 = vgam(agaaus ~ altitude, binomialff(link=cloglog), hunua)    # ok
fit2 = vgam(agaaus ~ altitude, binomialff(link="cloglog"), hunua)  # ok

## Not run: 
# This no longer works since "clog" is not a valid VGAM link function:
fit3 = vgam(agaaus ~ altitude, binomialff(link="clog"), hunua) # not ok

# 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, beta.ab(lshape1="identity", lshape2="identity"),
            trace = TRUE, crit="c")
fit2 = vglm(y ~ 1, beta.ab(lshape1=logoff, eshape1=list(offset=1.1),
                           lshape2=logoff, eshape2=list(offset=1.1)),
            trace = TRUE, crit="c")
vcov(fit1, untran=TRUE)
vcov(fit1, untran=TRUE)-vcov(fit2, untran=TRUE)  # Should be all 0s
fit1@misc$earg   # No 'special' parameters
fit2@misc$earg   # Some 'special' parameters are here

par(mfrow=c(2,2))
p = seq(0.01, 0.99, len=200)
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") # reciprocal!
plot(p, logit(p, deriv=2), type="l", col="blue") # reciprocal!
## End(Not run)

[Package VGAM version 0.7-7 Index]