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.:

${\displaystyle \operatorname {logit} (p)=\ln \left({\frac {p}{1-p}}\right)=\ln(p)-\ln(1-p)=-\ln \left({\frac {1}{p}}-1\right)=2\operatorname {atanh} (2p-1).}$ 

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 “logistic” function of any number ${\displaystyle \alpha }$  is given by the inverse-logit:

${\displaystyle \operatorname {logit} ^{-1}(\alpha )=\operatorname {logistic} (\alpha )={\frac {1}{1+\exp(-\alpha )}}={\frac {\exp(\alpha )}{\exp(\alpha )+1}}={\frac {\tanh({\frac {\alpha }{2}})+1}{2}}}$ 

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:

${\displaystyle \ln(R)=\ln \left({\frac {p_{1}/(1-p_{1})}{p_{2}/(1-p_{2})}}\right)=\ln \left({\frac {p_{1}}{1-p_{1}}}\right)-\ln \left({\frac {p_{2}}{1-p_{2}}}\right)=\operatorname {logit} (p_{1})-\operatorname {logit} (p_{2})\,.}$ 

Several approaches have been explored to adapt linear regression methods to a domain where the output is a probability value ${\displaystyle (0,1)}$ , instead of any real number ${\displaystyle (-\infty ,+\infty )}$ . In many cases, such efforts have focused on modeling this problem by mapping the range ${\displaystyle (0,1)}$  to ${\displaystyle (-\infty ,+\infty )}$  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 ${\displaystyle \ln p/q}$  following Bliss, who called the analogous function which is linear on ⁠${\displaystyle x}$ ⁠ for the normal curve 'probit'."

— Joseph Berkson (1944)^[3]

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 in logistic regression is a special case of a link function in a generalized linear model: it is the canonical link function for the Bernoulli distribution.
• More abstractly, the logit is the natural parameter for the binomial distribution; see Exponential family § Binomial distribution.
• The logit function is the negative of the derivative of the binary entropy function.
• 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]

Closely related to the logit function (and logit model) are the probit function and probit model. The logit and probit are both sigmoid functions with a domain between 0 and 1, which makes them both quantile functions – i.e., inverses of the cumulative distribution function (CDF) of a probability distribution. In fact, the logit is the quantile function of the logistic distribution, while the probit is the quantile function of the normal distribution. The probit function is denoted ${\displaystyle \Phi ^{-1}(x)}$ , where ${\displaystyle \Phi (x)}$  is the CDF of the standard normal distribution, as just mentioned:

${\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-y^{2}/2}dy.}$ 

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]

• Sigmoid function, inverse of the logit function
• Discrete choice on binary logit, multinomial logit, conditional logit, nested logit, mixed logit, exploded logit, and ordered logit
• Limited dependent variable
• Logit analysis in marketing
• Multinomial logit
• Ogee, curve with similar shape
• Perceptron
• Probit, another function with the same domain and range as the logit
• Ridit scoring
• Data transformation (statistics)
• Arcsin (transformation)
• Rasch model

• Which Link Function — Logit, Probit, or Cloglog? 12.04.2023

• Ashton, Winifred D. (1972). The Logit Transformation: with special reference to its uses in Bioassay. Griffin's Statistical Monographs & Courses. Vol. 32. Charles Griffin. ISBN 978-0-85264-212-2.