# 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

- ,

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*stand for the model parameters.

_{i}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

## 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.