VGAM (version 1.1-9)

logitlink: Logit Link Function

Description

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

Usage

logitlink(theta, bvalue = NULL, inverse = FALSE, deriv = 0,
    short = TRUE, tag = FALSE)
extlogitlink(theta, min = 0, max = 1, bminvalue = NULL,
    bmaxvalue = NULL, inverse = FALSE, deriv = 0,
    short = TRUE, tag = FALSE)

Value

For logitlink 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

eta / d

theta 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.

Arguments

theta

Numeric or character. See below for further details.

bvalue, bminvalue, bmaxvalue

See Links. These are boundary values. For extlogitlink, values of theta less than or equal to \(A\) or greater than or equal to \(B\) can be replaced by bminvalue and bmaxvalue.

min, max

For extlogitlink, min gives \(A\), max gives \(B\), and for out of range values, bminvalue and bmaxvalue.

inverse, deriv, short, tag

Details at Links.

Author

Thomas W. Yee

Details

The logit link function is very commonly used for parameters that lie in the unit interval. It is the inverse CDF of the logistic distribution. 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 extlogitlink 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.

References

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

See Also

Links, logitoffsetlink, probitlink, clogloglink, cauchitlink, logistic1, loglink, Logistic, multilogitlink.

Examples

p <- seq(0.01, 0.99, by = 0.01)
logitlink(p)
max(abs(logitlink(logitlink(p), inverse = TRUE) - p))  # 0?

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

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

if (FALSE)  par(mfrow = c(2,2), lwd = (mylwd <- 2))
y <- seq(-4, 4, length = 100)
p <- seq(0.01, 0.99, by = 0.01)
for (d in 0:1) {
  myinv <- (d > 0)
  matplot(p, cbind( logitlink(p, deriv = d, inv = myinv),
                   probitlink(p, deriv = d, inv = myinv)), las = 1,
          type = "n", col = "purple", ylab = "transformation",
          main = if (d ==  0) "Some probability link functions"
          else "1 / first derivative")
  lines(p,   logitlink(p, deriv = d, inverse = myinv), col = "limegreen")
  lines(p,  probitlink(p, deriv = d, inverse = myinv), col = "purple")
  lines(p, clogloglink(p, deriv = d, inverse = myinv), col = "chocolate")
  lines(p, cauchitlink(p, deriv = d, inverse = myinv), col = "tan")
  if (d ==  0) {
    abline(v = 0.5, h = 0, lty = "dashed")
    legend(0, 4.5, c("logitlink", "probitlink",
           "clogloglink", "cauchitlink"), col = c("limegreen", "purple",
           "chocolate", "tan"), lwd = mylwd)
  } else
    abline(v = 0.5, lty = "dashed")
}

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

p <- seq(0.21, 0.59, by = 0.01)
plot(p, extlogitlink(p, min = 0.2, max = 0.6), xlim = c(0, 1),
     type = "l", col = "black", ylab = "transformation",
     las = 1, main = "extlogitlink(p, min = 0.2, max = 0.6)")
par(lwd = 1)