Logit transformation

logit() maps probabilities from (0, 1) to the real line. Inputs must lie in [0, 1]; values outside this probability interval are rejected. Valid probabilities below eps and above 1 - eps are clipped before the transformation, because the mathematical logit is infinite at the boundary.

Usage

logit(p, eps = .Machine$double.eps)

Arguments

p

Numeric vector of probabilities in [0, 1].

eps

Positive clipping constant in (0, 0.5) used before applying the logit.

Value

A numeric vector on the logit scale with the same length as p.

Details

For a probability \(p \in (0, 1)\), the logit is

$$\operatorname{logit}(p) = \log\left(\frac{p}{1 - p}\right).$$

The transformation is monotone increasing and maps probabilities below \(0.5\) to negative values, \(0.5\) to zero, and probabilities above \(0.5\) to positive values. Because the expression is not finite at \(p = 0\) or \(p = 1\), the implementation first computes

$$p^* = \min\{\max(p, \epsilon), 1 - \epsilon\},$$

where \(\epsilon\) is eps, and then returns \(\operatorname{logit}(p^*)\). The returned vector has the same length as p.

Examples

probabilities <- data.frame(p = c(0.05, 0.25, 0.5, 0.75, 0.95)) |>
  dplyr::mutate(
    logit_p = logit(p),
    recovered = inv_logit(logit_p)
  )

probabilities
#>      p   logit_p recovered
#> 1 0.05 -2.944439      0.05
#> 2 0.25 -1.098612      0.25
#> 3 0.50  0.000000      0.50
#> 4 0.75  1.098612      0.75
#> 5 0.95  2.944439      0.95