# Logit

The logit ( LOH-jit) function is the inverse of the sigmoidal "logistic" function used in mathematics, especially in statistics.

Log-odds and logit are synonyms.[1]

## Definition

The logit of a number p between 0 and 1 is given by the formula:

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

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 "logistic" function of any number $\alpha$ is given by the inverse-logit:

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

If p is a probability, then p/(1 − p) is the corresponding odds; the logit of the probability is the logarithm of the odds. Similarly, 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:

$\operatorname{log}(R)=\log\left( \frac{{p_1}/(1-p_1)}{{p_2}/(1-p_2)} \right) =\log\left( \frac{p_1}{1-p_1} \right) - \log\left(\frac{p_2}{1-p_2}\right)=\operatorname{logit}(p_1)-\operatorname{logit}(p_2). \!\,$
Plot of logit(p) in the domain of 0 to 1, where the base of logarithm is e

## History

The logit model was introduced by Joseph Berkson in 1944, who coined the term. The term was borrowed by analogy from the very similar probit model developed by Chester Ittner Bliss in 1934.[2] G. A. Barnard in 1949 coined the commonly used term log-odds; the log-odds of an event is the logit of the probability of the event.[citation needed]

## Comparison with probit

Comparison of the logit function with a scaled probit (i.e. the inverse CDF of the normal distribution), comparing $\operatorname{logit}(x)$ vs. $\Phi^{-1}(x)/\sqrt{\frac{\pi}{8}}$, which makes the slopes the same at the y-origin.

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 $\Phi^{-1}(x)$, where $\Phi(x)$ is the CDF of the normal distribution, as just mentioned:

$\Phi(x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \operatorname{d}\!x$

As shown in the graph, the logit and probit functions are extremely similar, particularly 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 Bayesian statistics) implementation of them is easier.