Logistic regression

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Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]

Overview

In statistics, logistic regression is a regression model for binomially distributed response/dependent variables. It is useful for modeling the probability of an event occurring as a function of other factors. 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. Other names for logistic regression used in various other application areas include logistic model, logit model, and maximum-entropy classifier.

Logistic regression analyzes binomially distributed data of the form

<math>Y_i \ \sim B(p_i,n_i),\text{ for }i = 1, \dots , m,</math>

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 is then that for each trial (value of i) there is a set of explanatory/independent 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

<math>p_i = \operatorname{E}\left(\left.\frac{Y_i}{n_{i}}\right|X_i \right). \,\!</math>

The logits of the unknown binomial probabilities (i.e., the logarithms of the odds) are modelled as a linear function of the X_i.

<math>\operatorname{logit}(p_i)=\ln\left(\frac{p_i}{1-p_i}\right) = \beta_0 + \beta_1 x_{1,i} + \cdots + \beta_k x_{k,i}.</math>

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.

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, <math>e^\beta</math> is the estimate of the odds ratio of having the outcome for, say, males compared with females.

The model has an equivalent formulation as

<math>p_i = \frac{1}{1+e^{-(\beta_0 + \beta_1 x_{1,i} + \cdots + \beta_k x_{k,i})}}. \,\!</math>

This functional form is commonly identified as a single-layer "perceptron" or single-layer artificial neural network. A single-layer neural network computes a continuous output instead of a step function. The derivative of p_i with respect to X = x₁...x_k is computed from the general form:

<math>y = \frac{1}{1+e^{-f(X)}}</math>

where f(X) is an analytic function in X. With this choice, the single-layer 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:

<math>y' = y(1-y)\frac{\mathrm{d}f}{\mathrm{d}X}\,\!</math>

Extensions

Extensions of the model exist to cope with multi-category dependent variables and ordinal dependent variables, such as polytomous regression. Multi-class 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.

Example

Let p(x) be the probability of success when the value of the predictor variable is x. Then let

<math>p(x) = \frac{1}{1+e^{-(B_0+B_1x)}} = \frac{e^{B_0 + B_1x}}{1+e^{B_0+B_1x}}.</math>

Algebraic manipulation shows that

<math>\frac{p(x)}{1-p(x)} = e^{B_0+B_1x},</math>

where <math>\frac{p(x)}{1-p(x)}</math> is the odds in favor of success.

If we take a simple example, say p(50) = 2/3, then

<math>\frac{p(50)}{1-p(50)} = \frac{\frac{2}{3}}{1-\frac{2}{3}} = 2.</math>

So when x = 50, a success is twice as likely as a failure. Or, it can be simply said that the odds are 2 to 1.

See also

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

External links

• Web-based logistic regression calculator
• A highly optimized Maximum Entropy modeling package
• MALLET Java library, includes a trainer for logistic models

References

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