Logit

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This article discusses the binary logit function only. See discrete choice for a discussion of multinomial logit, conditional logit, nested logit, mixed logit, exploded logit, and ordered logit. For the basic regression technique that uses the logit function, see logistic regression.
Plot of logit(p) in the domain of 0 to 1, where the base of logarithm is e

The logit (/ˈlɪt/ LOH-jit) function is the inverse of the sigmoidal "logistic" function or logistic transform used in mathematics, especially in statistics. When the function's parameter represents a probability p, the logit function gives the log-odds, or the logarithm of the odds p/(1-p).[1]

Definition[edit]

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

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). \!\,

History[edit]

Log odds was used extensively by Charles Sanders Peirce (late 19th century).[2] 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.[3] 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]

Uses and properties[edit]

Comparison with probit[edit]

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) the implementation is easier.

See also[edit]

References[edit]

  1. ^ http://itl.nist.gov/div898/software/dataplot/refman2/auxillar/logoddra.htm
  2. ^ Stigler, Stephen M. (1986). The history of statistics : the measurement of uncertainty before 1900. Cambridge, Mass: Belknap Press of Harvard University Press. ISBN 0-674-40340-1.  edit
  3. ^ a b J. S. Cramer (2003). "The origins and development of the logit model". Cambridge UP. 
  4. ^ http://www.stat.ucl.ac.be/ISdidactique/Rhelp/library/msm/html/expit.html

Further reading[edit]

  • Ashton, Winifred D. (1972). The Logit Transformation: with special reference to its uses in Bioassay. Griffin's Statistical Monographs & Courses 32. Charles Griffin. ISBN 0-85264-212-1.