In statistics, the logit (/ˈloʊdʒɪt/ LOH-jit) 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 , so the logit is defined as
Because of this, the logit is also called the log-odds since it is equal to the logarithm of the odds where p is a probability. Thus, the logit is a type of function that maps probability values from to real numbers in , akin to the probit function.
If p is a probability, then p/(1 − p) is the corresponding odds; the logit of the probability is the logarithm of the odds, i.e.:
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 is given by the inverse-logit:
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:
There have been several efforts to adapt linear regression methods to a domain where the output is a probability value, , instead of any real number . In many cases, such efforts have focused on modeling this problem by mapping the range to 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";. However, this is computationally more expensive. In 1944, Joseph Berkson used log of odds and called this function logit, abbreviation for "logistic unit" following the analogy for probit:
I use this term [logit] for following Bliss, who called the analogous function which is linear on for the normal curve "probit."
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".
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 , where is the CDF of the standard normal distribution, as just mentioned:
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 Bayesian statistics) the implementation is easier.