Log-Linear Model

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

[math]\displaystyle{ \exp \left(c + \sum_{i} w_i f_i(X) \right) }[/math],

in which the f_i(X) are quantities that are functions of the variable X, in general a vector of values, while c and the w_i 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 f_i(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

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

Further reading

• 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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