A log-linear model is a

mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...

that takes the form of a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

whose

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...

equals a linear combination of the

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

s of the model, which makes it possible to apply (possibly multivariate)

linear regression In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is ...

. That is, it has the general form :

\exp \left(c + \sum_ 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 In science and engineering, a semi-log plot/graph or semi-logarithmic plot/graph has one axis on a logarithmic scale, the other on a linear scale. It is useful for data with exponential relationships, where one variable covers a large range ...

. *

Poisson regression In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable ''Y'' has a Poisson distribution, and assumes the loga ...

for contingency tables, a type of

generalized linear model In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a ''link function'' and by ...

. 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 A logistic function or logistic curve is a common S-shaped curve (sigmoid function, sigmoid curve) with equation f(x) = \frac, where For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is ...

, 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