logit: 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

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.

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.

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 $\log ((θ - A) / (B - θ))$ 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.

References

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

Examples

Run this code

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

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)")

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