Function in statistics
In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in data transformations.
Mathematically, the logit is the inverse of the standard logistic function formula_1, so the logit is defined as
 formula_2
Because of this, the logit is also called the log-odds since it is equal to the logarithm of the odds formula_3 where p is a probability. Thus, the logit is a type of function that maps probability values from formula_4 to real numbers in formula_5, akin to the probit function.
Definition.
If p is a probability, then is the corresponding odds; the logit of the probability is the logarithm of the odds, i.e.:
 formula_6
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 formula_7 is given by the inverse-logit:
 formula_8
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:
 formula_9
History.
Several approaches have been explored to adapt linear regression methods to a domain where the output is a probability value formula_4, instead of any real number formula_5. In many cases, such efforts have focused on modeling this problem by mapping the range formula_4 to formula_5 and then running the linear regression on these transformed values.
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.
In 1944, Joseph Berkson used log of odds and called this function "logit", an abbreviation for "logistic unit", following the analogy for probit:
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Log odds was used extensively by Charles Sanders Peirce (late 19th century). 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. Barnard also coined the term "lods" as an abstract form of "log-odds", but suggested that "in practice the term 'odds' should normally be used, since this is more familiar in everyday life".
Comparison with probit.
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 formula_14, where formula_15 is the CDF of the standard normal distribution, as just mentioned:
 formula_16
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.
References.
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