This article is about the binary logit function. For other types of logit, see
discrete choice. For the basic regression technique that uses the logit function, see
logistic regression. For standard magnitudes combined by multiplication, see
logit (unit).
Because of this, the logit is also called the log-odds since it is equal to the
logarithm of the
odds where p is a probability. Thus, the logit is a type of function that maps probability values from to real numbers in ,[1] akin to the
probit function.
Definition
If p is a
probability, then p/(1 − p) is the corresponding
odds; the logit of the probability is the logarithm of the odds, i.e.:
The base of the
logarithm function used is of little importance in the present article, as long as it is greater than 1, but the
natural logarithm with base e is the one most often used. The choice of base corresponds to the choice of
logarithmic unit for the value: base 2 corresponds to a
shannon, base e to a “
nat”, and base 10 to a
hartley; these units are particularly used in information-theoretic interpretations. For each choice of base, the logit function takes values between negative and positive infinity.
The difference between the logits of two probabilities is the logarithm of the
odds ratio (R), thus providing a shorthand for writing the correct combination of odds ratios
only by adding and subtracting:
History
Several approaches have been explored to adapt linear regression methods to a domain where the output is a probability value , instead of any real number . In many cases, such efforts have focused on modeling this problem by mapping the range to and then running the linear regression on these transformed values.[2]
In 1934,
Chester Ittner Bliss used the cumulative normal distribution function to perform this mapping and called his model
probit, an abbreviation for "probability unit". This is, however, computationally more expensive.[2]
In 1944,
Joseph Berkson used log of odds and called this function logit, an abbreviation for "logistic unit", following the analogy for probit:
"I use this term [logit] for following Bliss, who called the analogous function which is linear on for the normal curve 'probit'."
Log odds was used extensively by
Charles Sanders Peirce (late 19th century).[4]G. A. Barnard in 1949 coined the commonly used term log-odds;[5][6] the log-odds of an event is the logit of the probability of the event.[7] Barnard also coined the term lods as an abstract form of "log-odds",[8] but suggested that "in practice the term 'odds' should normally be used, since this is more familiar in everyday life".[9]
The logit is also central to the probabilistic
Rasch model for
measurement, which has applications in psychological and educational assessment, among other areas.
The inverse-logit function (i.e., the
logistic function) is also sometimes referred to as the expit function.[10]
In plant disease epidemiology, the logistic, Gompertz, and monomolecular models are collectively known as the Richards family models.
The log-odds function of probabilities is often used in state estimation algorithms[11] because of its numerical advantages in the case of small probabilities. Instead of multiplying very small floating point numbers, log-odds probabilities can just be summed up to calculate the (log-odds) joint probability.[12][13]
As shown in the graph on the right, the logit and probit functions are extremely similar when the probit function is scaled, so that its slope at y = 0 matches the slope of the logit. As a result,
probit models are sometimes used in place of
logit models because for certain applications (e.g., in
item response theory) the implementation is easier.[14]
^Albert, James H. (2016). "Logit, Probit, and other Response Functions". Handbook of Item Response Theory. Vol. Two. Chapman and Hall. pp. 3–22.
doi:
10.1201/b19166-1.
Ashton, Winifred D. (1972). The Logit Transformation: with special reference to its uses in Bioassay. Griffin's Statistical Monographs & Courses. Vol. 32. Charles Griffin.
ISBN978-0-85264-212-2.