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The logit function is an important part of logistic regression
Logistic regression

In statistics, logistic regression is a model used for prediction of the probability of occurrence of an event by fitting data to a logistic curve....
: for more information, please see that article.


The logit function is the inverse of the "sigmoid", or "logistic" function
Logistic function

A logistic function or logistic curve is the most common sigmoid curve. It modelsthe S-curve of growth of some set P, where P might...
 used in mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, especially in statistics
Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
. The logit of a number p between 0 and 1 is given by the formula:

Logit is with a long "o" and a soft "g".

The base of the logarithm
Logarithm

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
 function used is of little importance in the present article, as long as it is greater than 1, but the natural logarithm
Natural logarithm

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e , where e is an irrational number constant approximately equal to 2.718281828....
 with base e
E (mathematical constant)

The mathematical constant e is the unique real number such that the function ex has the same value as the derivative, for all values of x....
 is the one most often used.

If p is a probability
Probability

Probability, or wikt:chance, is a way of expressing knowledge or belief that an Event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about t...
 then p/(1 − p) is the corresponding odds
Odds

In probability theory and statistics the odds in favour of an event or a proposition are the quantity , where p is the probability of the event or proposition....
, 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
Odds ratio

The odds ratio is a measure of effect size, describing the strength of association or non-independence between two binary data values. It is used as a descriptive statistics, and plays an important role in logistic regression....
 (R), thus providing a shorthand for writing the correct combination of odds-ratios only by adding and subtracting
Additive function

Different definitions exist depending on the specific field of application. Traditionally, an additive function is a function that preserves the addition operation:for any two elements x and y in the domain....
:


logit model was introduced by Joseph Berkson
Joseph Berkson

Joseph Berkson was initially trained as a physicist. Later in his career he became primarily concerned with studying statistics. In 1950, while working at the Division of Biometry and Medical Statistics, Mayo Clinic, Rochester, Minnesota, , Berkson wrote a key paper entitled Are there two regressions?....
 in 1944, who coined the term.






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Encyclopedia


The logit function is an important part of logistic regression
Logistic regression

In statistics, logistic regression is a model used for prediction of the probability of occurrence of an event by fitting data to a logistic curve....
: for more information, please see that article.


The logit function is the inverse of the "sigmoid", or "logistic" function
Logistic function

A logistic function or logistic curve is the most common sigmoid curve. It modelsthe S-curve of growth of some set P, where P might...
 used in mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, especially in statistics
Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
. The logit of a number p between 0 and 1 is given by the formula:

Logit is with a long "o" and a soft "g".

The base of the logarithm
Logarithm

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
 function used is of little importance in the present article, as long as it is greater than 1, but the natural logarithm
Natural logarithm

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e , where e is an irrational number constant approximately equal to 2.718281828....
 with base e
E (mathematical constant)

The mathematical constant e is the unique real number such that the function ex has the same value as the derivative, for all values of x....
 is the one most often used.

If p is a probability
Probability

Probability, or wikt:chance, is a way of expressing knowledge or belief that an Event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about t...
 then p/(1 − p) is the corresponding odds
Odds

In probability theory and statistics the odds in favour of an event or a proposition are the quantity , where p is the probability of the event or proposition....
, 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
Odds ratio

The odds ratio is a measure of effect size, describing the strength of association or non-independence between two binary data values. It is used as a descriptive statistics, and plays an important role in logistic regression....
 (R), thus providing a shorthand for writing the correct combination of odds-ratios only by adding and subtracting
Additive function

Different definitions exist depending on the specific field of application. Traditionally, an additive function is a function that preserves the addition operation:for any two elements x and y in the domain....
:

Logit

History

The logit model was introduced by Joseph Berkson
Joseph Berkson

Joseph Berkson was initially trained as a physicist. Later in his career he became primarily concerned with studying statistics. In 1950, while working at the Division of Biometry and Medical Statistics, Mayo Clinic, Rochester, Minnesota, , Berkson wrote a key paper entitled Are there two regressions?....
 in 1944, who coined the term. The term was borrowed by analogy from the very similar probit
Probit

In probability theory and statistics, the probit function is the inverse function cumulative distribution function , or quantile function associated with the standard normal distribution....
 model developed by Chester Ittner Bliss
Chester Ittner Bliss

Chester Ittner Bliss was primarily a biologist, who is best-known for his contributions to statistics. He was born in Springfield, Ohio in 1899 and died in 1979....
 in 1934. .

Uses and properties


  • The logit in logistic regression is a special case of a link function in a generalized linear model
    Generalized linear model

    In statistics, the generalized linear model is a flexible generalization of ordinary linear regression. It relates the random distribution of the measured variable of the experiment to the systematic portion of the experiment through a function called the link function....
    : it is the canonical link function for the binomial distribution
    Binomial distribution

    In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n statistical independence yes/no experiments, each of which yields success with probability p....
    .
  • The logit function is the negative of the derivative
    Derivative

    In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
     of the binary entropy function
    Binary entropy function

    In information theory, the binary entropy function, denoted or , is defined as the information entropy of a Bernoulli trial with probability of success p....
    .
  • The logit is also central to the probabilistic Rasch model
    Rasch model

    Rasch models are used for analysing data from assessments to measure things such as abilities, attitudes, and personality traits. For example, they may be used to estimate a student's reading ability from answers to questions on a reading assessment, or the extremity of a person's attitude to capital punishment from responses on a questionnai...
     for measurement
    Measurement

    Measurement is the process of assigning a number to an attribute according to a rule or set of rules. The term can also be used to refer to the result obtained after performing the process....
    , which has applications in psychological and educational assessment, among other areas.


See also

  • Daniel McFadden
    Daniel McFadden

    Daniel Little McFadden is an econometrics who won the 2000 Nobel Memorial Prize in Economic Sciences; McFadden's share of the prize was "for his development of theory and methods for analyzing Discrete Choice Modelling"....
    , a Nobel Prize winner for development of a particular logit model used in economics
  • Logit analysis in marketing
  • Perceptron
    Perceptron

    The perceptron is a type of artificial neural network invented in 1957 at the Cornell Aeronautical Laboratory by Frank Rosenblatt. It can be seen as the simplest kind of feedforward neural network: a linear classifier....
  • Probit
    Probit

    In probability theory and statistics, the probit function is the inverse function cumulative distribution function , or quantile function associated with the standard normal distribution....
  • Logistic regression
    Logistic regression

    In statistics, logistic regression is a model used for prediction of the probability of occurrence of an event by fitting data to a logistic curve....
  • Logistic function
    Logistic function

    A logistic function or logistic curve is the most common sigmoid curve. It modelsthe S-curve of growth of some set P, where P might...