VGAM (version 0.8-1)

Links: 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

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

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

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.

Details

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 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 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 ($B=\infty$).

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

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

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

# 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!