logit {VGAM}R Documentation

Logit Link Function

Description

Computes the logit transformation, including its inverse and the first two derivatives.

Usage

logit(theta, earg = list(), inverse = FALSE, deriv = 0,
      short = TRUE, tag = FALSE)
elogit(theta, earg = list(min=0, max=1), inverse = FALSE, deriv = 0,
      short = TRUE, tag = FALSE)

Arguments

theta Numeric or character. See below for further details.
earg Optional list. Extra argument for passing in additional information. Values of theta which are less than or equal to 0 can be replaced by the bvalue component of the list earg before computing the link function value. Values of theta which are greater than or equal to 1 can be replaced by 1 minus the bvalue component of the list earg before computing the link function value. The component name bvalue stands for ``boundary value''. See Links for general information about earg. Similarly, for elogit, values of theta less than or equal to A or greater than or equal to B can be replaced by the bminvalue and bmaxvalue components of the list earg.
For elogit, earg should be a list with components min giving A, max giving B, and for out of range values, bminvalue and bmaxvalue. If earg is used, these component names should not be abbreviated.
inverse Logical. If TRUE the inverse function is computed. The inverse logit function is known as the expit function.
deriv Order of the derivative. Integer with value 0, 1 or 2.
short Used for labelling the blurb slot of a vglmff-class object.
tag Used for labelling the linear/additive predictor in the initialize slot of a vglmff-class object. Contains a little more information if TRUE.

Details

The logit link function is very commonly used for parameters that lie in the unit interval. Numerical values of theta close to 0 or 1 or out of range result in Inf, -Inf, NA or NaN.

The extended logit link function elogit should be used more generally for parameters that lie in the interval (A,B), say. The formula is

log((theta-A)/(B-theta))

and the default values for A and B correspond to the ordinary logit function. Numerical values of theta close to A or B or out of range result in Inf, -Inf, NA or NaN. However these can be replaced by values bminvalue and bmaxvalue first before computing the link function.

The arguments short and tag are used only if theta is character.

Value

For logit with deriv = 0, the logit of theta, i.e., log(theta/(1-theta)) when inverse = FALSE, and if inverse = TRUE then exp(theta)/(1+exp(theta)).
For deriv = 1, then the function returns d theta / d eta as a function of theta if inverse = FALSE, else if inverse = TRUE then it returns the reciprocal.
Here, all logarithms are natural logarithms, i.e., to base e.

Note

Numerical instability may occur when theta is close to 1 or 0 (for logit), or close to A or B for elogit. One way of overcoming this is to use earg.

In terms of the threshold approach with cumulative probabilities for an ordinal response this link function corresponds to the univariate logistic distribution (see logistic).

Author(s)

Thomas W. Yee

References

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

See Also

Links, probit, cloglog, cauchit, loge.

Examples

p = seq(0.01, 0.99, by=0.01)
logit(p)
max(abs(logit(logit(p), inverse=TRUE) - p)) # Should be 0

p = c(seq(-0.02, 0.02, by=0.01), seq(0.97, 1.02, by=0.01))
logit(p)  # Has NAs
logit(p, earg=list(bvalue= .Machine$double.eps))  # Has no NAs

p = seq(0.9, 2.2, by=0.1)
elogit(p, earg=list(min=1, max=2,
                    bminvalue = 1 + .Machine$double.eps,
                    bmaxvalue = 2 - .Machine$double.eps))  # Has no NAs

## Not run: 
par(mfrow=c(2,2))
y = seq(-4, 4, length=100)
for(d in 0:1) {
    matplot(p, cbind(logit(p, deriv=d), probit(p, deriv=d)),
            type="n", col="purple", ylab="transformation",
            lwd=2, las=1,
            main=if(d==0) "Some probability link functions"
            else "First derivative")
    lines(p, logit(p, deriv=d), col="limegreen", lwd=2)
    lines(p, probit(p, deriv=d), col="purple", lwd=2)
    lines(p, cloglog(p, deriv=d), col="chocolate", lwd=2)
    lines(p, cauchit(p, deriv=d), col="tan", lwd=2)
    if(d==0) {
        abline(v=0.5, h=0, lty="dashed")
        legend(0, 4.5, c("logit", "probit", "cloglog", "cauchit"),
               col=c("limegreen","purple","chocolate", "tan"), lwd=2)
    } else
        abline(v=0.5, lty="dashed")
}

for(d in 0) {
    matplot(y, cbind(logit(y, deriv=d, inverse=TRUE),
                     probit(y, deriv=d, inverse=TRUE)),
            type="n", col="purple", xlab="transformation", ylab="p",
            lwd=2, las=1,
            main=if(d==0) "Some inverse probability link functions"
            else "First derivative")
    lines(y, logit(y, deriv=d, inverse=TRUE), col="limegreen", lwd=2)
    lines(y, probit(y, deriv=d, inverse=TRUE), col="purple", lwd=2)
    lines(y, cloglog(y, deriv=d, inverse=TRUE), col="chocolate", lwd=2)
    lines(y, cauchit(y, deriv=d, inverse=TRUE), col="tan", lwd=2)
    if(d==0) {
        abline(h=0.5, v=0, lty="dashed")
        legend(-4, 1, c("logit", "probit", "cloglog", "cauchit"),
               col=c("limegreen","purple","chocolate", "tan"), lwd=2)
    }
}

p = seq(0.21, 0.59, by=0.01)
plot(p, elogit(p, earg=list(min=0.2, max=0.6)), lwd=2, 
     type="l", col="black", ylab="transformation", xlim=c(0,1),
     las=1, main="elogit(p, earg=list(min=0.2, max=0.6)")
## End(Not run)

[Package VGAM version 0.7-7 Index]