Log-linear model

{{Short description|Mathematical model}}

{{More citations needed|date=July 2012}}

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

:$\backslash exp\; \backslash left(c\; +\; \backslash sum\_\{i\}\; w\_i\; f\_i(X)\; \backslash right)$,

in which the {{math|f_i(X)}} are quantities that are functions of the variable {{math|X}}, in general a vector of values, while {{math|c}} and the {{math|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 {{math|X}}, or more immediately, the transformed quantities {{math|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.