Logit

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In mathematics, especially as applied in statistics, the logit (pronounced with a long "o" and a soft "g", IPA /loʊdʒɪt/) of a number p between 0 and 1 is

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

Plot of logit in the range 0 to 1, base is e

(The base of the logarithm function used here is of little importance in the present article, as long as it is greater than 1.) The logit function is the inverse of the "sigmoid", or "logistic" function. If p is a probability then p/(1 − p) is the corresponding odds, and 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, thus providing an additive mechanism for combining odds-ratios.

1 History
2 Use
- 2.1 Logistic regression
- 2.2 Other uses and properties
3 See also
4 External links

[edit] 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. 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.

[edit] Use

Logits are used for various purposes by statisticians, most notably for logistic regression.

[edit] Logistic regression

In particular there is the "logit model" of which the simplest sort is

$\operatorname{logit}(p_i)=a+bx_i\,\!$

where x_i is some quantity on which success or failure in the i-th in a sequence of Bernoulli trials may depend, and p_i is the probability of success in the i-th case. For example, x may be the age of a patient admitted to a hospital with a heart attack, and "success" may be the event that the patient dies before leaving the hospital. Having observed the values of x in a sequence of cases and whether there was a "success" or a "failure" in each such case, a statistician will often estimate the values of the coefficients a and b by the method of maximum likelihood. The result can then be used to assess the probability of "success" in a subsequent case in which the value of x is known. Estimation and prediction by this method are called logistic regression.

[edit] Other uses and properties

The logit function is the negative of the derivative of the binary entropy function.
The logit in logistic regression is a special case of a link function in a generalized linear model.
The concept of a logit is also central to the probabilistic Rasch model for measurement, which has applications in psychological and educational assessment, among other areas.