Logistic regression is a statistical regression model for Bernoulli-distributed dependent variables. It is a generalized linear model that uses the logit as its link function. Logistic regression is used extensively in the medical and social sciences. In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... In statistics, the generalized linear model (GLM) is a useful generalization of ordinary least squares regression. ... In mathematics, especially as applied in statistics, the logit (pronounced with a long o and a soft g, IPA ) of a number p between 0 and 1 is This function is used in logistic regression. ... In statistics, a generalized linear model is a model relating the expected value E(y) of a dependent variable y to one or more independent variables x1, ..., xn, with the relation stated as follows. ...

The logistic model takes the form

where there are n units with covariates X and

The logarithm of the odds (the probability divided by one minus the probability) of the outcome is modelled as a linear function of the explanatory variables, X_i. This can be written equivalently as In probability theory and statistics the odds in favor of an event or a proposition are the quantity p / (1 − p), where p is the probability of the event or proposition. ...

The interpretation of the β parameter estimates is as a multiplicative effect on the odds ratio. In the case of a dichotomous explanatory variable, for instance gender, e^β is the estimate of the odds-ratio of having the outcome for, say, males compared with females. The parameters are usually estimated by maximum likelihood. Note, the derivative is shown under single-layer, artificial neural network. The odds ratio is a statistical measure particularly important in Bayesian statistics and logistic regression. ... Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ... An artificial neural network (ANN) or commonly just neural network (NN) is an interconnected group of artificial neurons that uses a mathematical model or computational model for information processing based on a connectionist approach to computation. ...

Extensions of the model, such as polytomous regression, exist to cope with multi-category dependent variables and ordinal dependent variables. Multi-class classification by logistic regression is also known as multinomial logit modeling. A multinomial logit model is an econometric model which is an extension of the logit model for many cases (more than two). ...

Applications

One common application in the social sciences is the estimation of the probability of default from a historical data base on public corporations. The Jarrow Turnbull Model links the resulting logistic estimates of default probabilities to the value of credit risky securities. There are very few or no other articles that link to this one. ...

See also

• Artificial neural network
• Data mining
• Linear discriminant analysis
• Perceptron
• Probit model
• Variable rules analysis

An artificial neural network (ANN) or commonly just neural network (NN) is an interconnected group of artificial neurons that uses a mathematical model or computational model for information processing based on a connectionist approach to computation. ... Data mining (DM), also called Knowledge-Discovery in Databases (KDD) or Knowledge-Discovery and Data Mining, is the process of automatically searching large volumes of data for patterns using tools such as classification, association rule mining, clustering, etc. ... Linear discriminant analysis (LDA) and the related Fishers linear discriminant are used in statistics to find the linear combination of features which best separate two or more classes of object or event. ... The perceptron is a type of artificial neural network invented in 1957 at the Cornell Aeronautical Laboratory by Frank Rosenblatt. ... The probit model is a popular specification of a binary regression model, using a probit link function. ... In linguistics, variable rules analysis is a set of statistical analysis methods commonly used in sociolinguistics and historical linguistics to describe patterns of variation between alternative forms in language use. ...

External links

• Web-based logistic regression calculator

References

• Agresti, Alan, Categorical Data Analysis, 2nd ed., New York: Wiley-Interscience, 2002, ISBN 0-471-36093-7.
• Amemiya, T., Advanced Econometrics, Harvard University Press, 1985, ISBN 0-674-00560-0.
• Balakrishnan, N., Handbook of the Logistic Distribution, Marcel Dekker, Inc., 1991, ISBN-13: 978-0824785871.
• Green, William H., Econometric Analysis, fifth edition, Prentice Hall, 2003, ISBN 0-13-066189-9.
• Hosmer, David W. and Stanley Lemeshow, Applied Logistic Regression, 2nd ed., New York; Chichester, Wiley, 2000, ISBN 0-471-35632-8.