A log-linear model is a
mathematical model
A mathematical model is an abstract and concrete, abstract description of a concrete system using mathematics, mathematical concepts and language of mathematics, language. The process of developing a mathematical model is termed ''mathematical m ...
that takes the form of a
function whose
logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
equals a
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
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 statistical model, model that estimates the relationship between a Scalar (mathematics), scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A mode ...
. That is, it has the general form
:
,
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
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 lo ...
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 curve) with the equation
f(x) = \frac
where
The logistic function has domain the real numbers, the limit as x \to -\infty is 0, and the limit as x \to +\infty is L.
...
, 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
The general linear model or general multivariate regression model is a compact way of simultaneously writing several multiple linear regression models. In that sense it is not a separate statistical linear model. The various multiple linear regre ...
*
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 ...
*
Boltzmann distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability tha ...
*
Elasticity
Further reading
*
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