Logit

From Academic Kids

The logit (pronounced with a long "o" and a soft "g", IPA ) of a number p between 0 and 1 is

<math>{\rm logit}(p)=\log\left( \frac{p}{1-p} \right) =\log(p)-\log(1-p).<math>
Plot of logit in the range 0 to 1, base is e
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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.

Logits are used for various purposes by statisticians. In particular there is the "logit model" of which the simplest sort is

<math>{\rm logit}(p_i)=a+bx_i<math>

where xi is some quantity on which success or failure in the i-th in a sequence of Bernoulli trials may depend, and pi 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 (another instance of the reason why the words "success" and "failure" in speaking of Bernoulli trials should be taken with large doses of salt). 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.

A logistic regression model is identical to a neural network with no hidden units. For a neural network with hidden units, each hidden unit computes a logistic regression (different for each hidden unit), and the output is therefore a weighted sum of logistic regression outputs.

The logit in logistic regression is a special case of a link function in generalized linear models. Another example is the probit model, which is more concerned with the tails of the response curve.

The logit model was introduced by Joseph Berkson in 1944, who coined the term. 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.

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