Logit


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, 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 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.

Definition

If is a probability, then is the corresponding odds; the 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 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 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-:
The difference between the s of two probabilities is the logarithm of the odds ratio, thus providing a shorthand for writing the correct combination of odds ratios only by adding and subtracting:
The Taylor series for the logit function is given by:

History

Several approaches have been explored 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". 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:
Log odds was used extensively by Charles Sanders Peirce. 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".

Uses and properties

Closely related to the function are the probit function and probit model. The and 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 of a probability distribution. In fact, the is the quantile function of the logistic distribution, while the is the quantile function of the normal distribution. The function is denoted, where is the CDF of the standard normal distribution, as just mentioned:
As shown in the graph on the right, the and functions are extremely similar when the function is scaled, so that its slope at matches the slope of the. As a result, probit models are sometimes used in place of logit models because for certain applications the implementation is easier.