R: Logit Link Function

logit {VGAM} | R Documentation

## Logit Link Function

### Description

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

### Usage

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

### Arguments

`theta` | Numeric or character. See below for further details.
`bvalue, bminvalue, bmaxvalue` | See `Links`. These are boundary values. For `extlogit`, 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 `extlogit`, `min` gives A, `max` gives B, and for out of range values, `bminvalue` and `bmaxvalue`.
`inverse, deriv, short, tag` | Details at `Links`.

### 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 `extlogit` 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.

### 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 `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.

### Note

Numerical instability may occur when `theta` is close to 1 or 0 (for `logit`), or close to A or B for `extlogit`. One way of overcoming this is to use, e.g., `bvalue`.

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`, `logitoffsetlink`, `probit`, `cloglog`, `cauchit`, `logistic1`, `loge`, `plogis`, `multilogit`.

### 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, bvalue = .Machine$double.eps)  # Has no NAs

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

    ## Not run:  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( logit(p, deriv = d, inverse = myinv),
                       probit(p, deriv = d, inverse = myinv)),
              type = "n", col = "purple", ylab = "transformation", las = 1,
              main = if (d ==  0) "Some probability link functions"
              else "1 / first derivative")
      lines(p,   logit(p, deriv = d, inverse = myinv), col = "limegreen")
      lines(p,  probit(p, deriv = d, inverse = myinv), col = "purple")
      lines(p, cloglog(p, deriv = d, inverse = myinv), col = "chocolate")
      lines(p, cauchit(p, deriv = d, inverse = myinv), col = "tan")
      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 = mylwd)
      } 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)), las = 1,
              type = "n", col = "purple", xlab = "transformation", ylab = "p",
              main = if (d ==  0) "Some inverse probability link functions"
              else "First derivative")
      lines(y,   logit(y, deriv = d, inverse = TRUE), col = "limegreen")
      lines(y,  probit(y, deriv = d, inverse = TRUE), col = "purple")
      lines(y, cloglog(y, deriv = d, inverse = TRUE), col = "chocolate")
      lines(y, cauchit(y, deriv = d, inverse = TRUE), col = "tan")
      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 = mylwd)
      }
    }

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

    ## End(Not run)

[Package VGAM version 1.0-3 Index]