Log-linear model

Mathematical model

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A log-linear model is a mathematical model that takes the form of a function whose logarithm equals a linear combination of the parameters of the model, which makes it possible to apply (possibly multivariate) linear regression. That is, it has the general form

exp ⁡ ( c + ∑ i w i f i ( X ) ) {\displaystyle \exp \left(c+\sum _{i}w_{i}f_{i}(X)\right)} ,

in which the fi(X) are quantities that are functions of the variable X, in general a vector of values, while c and the wi stand for the model parameters.

The term may specifically be used for:

* A log-linear plot or graph, which is a type of semi-log plot.
* Poisson regression for contingency tables, a type of generalized linear model.

The specific applications of log-linear models are where the output quantity lies in the range 0 to ∞, for values of the independent variables X, or more immediately, the transformed quantities fi(X) in the range −∞ to +∞. This may be contrasted to logistic models, similar to the logistic function, for which the output quantity lies in the range 0 to 1. Thus the contexts where these models are useful or realistic often depends on the range of the values being modelled.

## See also

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* Log-linear analysis
* General linear model
* Generalized linear model
* Boltzmann distribution
* Elasticity

## Further reading

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* Gujarati, Damodar N.; Porter, Dawn C. (2009). "How to Measure Elasticity: The Log-Linear Model". Basic Econometrics. New York: McGraw-Hill/Irwin. pp. 159–162\. ISBN 978-0-07-337577-9.

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