Log-linear model explained

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 \left(c + \sum _{i} w _{i} f _{i(X)} \right) ,in which the are quantities that are functions of the variable, in general a vector of values, while and the 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, or more immediately, the transformed quantities 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

• Book: Gujarati, Damodar N. . Porter . Dawn C.. Dawn C. Porter . Basic Econometrics . New York . McGraw-Hill/Irwin . 2009 . 978-0-07-337577-9 . How to Measure Elasticity: The Log-Linear Model . 159–162 .