Logistic regression
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In statistics, logistic regression (sometimes called the logistic model or logit model) is used for prediction of the probability of occurrence of an event by fitting data to a logistic curve. It is a generalized linear model used for binomial regression. Like many forms of regression analysis, it makes use of several predictor variables that may be either numerical or categorical. For example, the probability that a person has a heart attack within a specified time period might be predicted from knowledge of the person's age, sex and body mass index. Logistic regression is used extensively in the medical and social sciences as well as marketing applications such as prediction of a customer's propensity to purchase a product or cease a subscription.
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[edit] Lay explanation
An explanation of logistic regression begins with an explanation of the logistic function:
A graph of the function is shown in figure 1. The "input" is z and the "output" is f(z). The logistic function is useful because it can take as an input any value from negative infinity to positive infinity, whereas the output is confined to values between 0 and 1. The variable z represents the exposure to some set of risk factors, while f(z) represents the probability of a particular outcome, given that set of risk factors. The variable z is a measure of the total contribution of all the risk factors used in the model and is known as the logit.
The variable z is usually defined as
where β_{0} is called the "intercept" and β_{1}, β_{2}, β_{3}, and so on, are called the "regression coefficients" of x_{1}, x_{2}, x_{3} respectively. The intercept is the value of z when the value of all risk factors is zero (i.e., the value of z in someone with no risk factors). Each of the regression coefficients describes the size of the contribution of that risk factor. A positive regression coefficient means that that risk factor increases the probability of the outcome, while a negative regression coefficient means that risk factor decreases the probability of that outcome; a large regression coefficient means that the risk factor strongly influences the probability of that outcome; while a nearzero regression coefficient means that that risk factor has little influence on the probability of that outcome.
Logistic regression is a useful way of describing the relationship between one or more risk factors (e.g., age, sex, etc.) and an outcome such as death (which only takes two possible values: dead or not dead).
[edit] Example
The application of a logistic regression may be illustrated using a fictitious example of death from heart disease. This simplified model uses only three risk factors (age, sex, and blood cholesterol level) to predict the 10year risk of death from heart disease. This is the model that we fit:
 β_{0} = − 5.0 (the intercept)
 β_{1} = + 2.0
 β_{2} = − 1.0
 β_{3} = + 1.2
 x_{1} = age in decades, less 5.0
 x_{2} = sex, where 0 is male and 1 is female
 x_{3} = cholesterol level, in mmol/L less 5.0
Which means the model is
In this model, increasing age is associated with an increasing risk of death from heart disease (z goes up by 2.0 for every 10 years over the age of 50), female sex is associated with a decreased risk of death from heart disease (z goes down by 1.0 if the patient is female), and increasing cholesterol is associated with an increasing risk of death (z goes up by 1.2 for each 1 mmol/L increase in cholesterol above 5mmol/L).
We wish to use this model to predict Mr Petrelli's risk of death from heart disease: he is 50 years old and his cholesterol level is 7.0 mmol/L. Mr Petrelli's risk of death is therefore
This means that by this model, Mr Petrelli's risk of dying from heart disease in the next 10 years is 0.07 (or 7%).
[edit] Formal mathematical specification
Logistic regression analyzes binomially distributed data of the form
where the numbers of Bernoulli trials n_{i} are known and the probabilities of success p_{i} are unknown. An example of this distribution is the fraction of seeds (p_{i}) that germinate after n_{i} are planted.
The model proposes for each trial (value of i) there is a set of explanatory variables that might inform the final probability. These explanatory variables can be thought of as being in a k vector X_{i} and the model then takes the form
The logits of the unknown binomial probabilities (i.e., the logarithms of the odds) are modelled as a linear function of the X_{i}.
Note that a particular element of X_{i} can be set to 1 for all i to yield an intercept in the model. The unknown parameters β_{j} are usually estimated by maximum likelihood using a method common to all generalized linear models.
The interpretation of the β_{j} parameter estimates is as the additive effect on the log odds ratio for a unit change in the jth explanatory variable. 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 model has an equivalent formulation
This functional form is commonly called a singlelayer perceptron or singlelayer artificial neural network. A singlelayer neural network computes a continuous output instead of a step function. The derivative of p_{i} with respect to X = x_{1}...x_{k} is computed from the general form:
where f(X) is an analytic function in X. With this choice, the singlelayer network is identical to the logistic regression model. This function has a continuous derivative, which allows it to be used in backpropagation. This function is also preferred because its derivative is easily calculated:
[edit] Extensions
Extensions of the model cope with multicategory dependent variables and ordinal dependent variables, such as polytomous regression. Multiclass classification by logistic regression is known as multinomial logit modeling. An extension of the logistic model to sets of interdependent variables is the conditional random field.
[edit] See also
 Logistic function
 Sigmoid function
 Artificial neural network
 Data mining
 Linear discriminant analysis
 Perceptron
 Probit model
 Variable rules analysis
 JarrowTurnbull model
 Principle of maximum entropy
[edit] References
 Agresti, Alan. (2002). Categorical Data Analysis. New York: WileyInterscience. ISBN 0471360937.
 Amemiya, T. (1985). Advanced Econometrics. Harvard University Press. ISBN 0674005600.
 Balakrishnan, N. (1991). Handbook of the Logistic Distribution. Marcel Dekker, Inc.. ISBN 9780824785871.
 Greene, William H. (2003). Econometric Analysis, fifth edition. Prentice Hall. ISBN 0130661899.
 Hilbe, Joseph M. (2009). Logistic Regression Models. Chapman & Hall/CRC Press. ISBN 9781420075755.
 Hosmer, David W.; Stanley Lemeshow (2000). Applied Logistic Regression, 2nd ed.. New York; Chichester, Wiley. ISBN 0471356328.
