 # Loglinear models

Loglinear models:

Loglinear models are models that postulate a linear relationship between the independent variables and the logarithm of the dependent variable, for example:

 log(y) = a0 + a1 x1 + a2 x2 … + aN xN

where y is the dependent variable; xi, i=1,…,N are independent variables, and {ai, i=0,…,N} are parameters (coefficients) of the model.

Loglinear models, for example, are widely used to analyze categorical data represented as a contingency table . In this case, the main reason to transform frequencies (counts) or probabilities to their log-values is that, provided the independent variables are not correlated with each other, the relationship between the new transformed dependent variable and the independent variables is a linear (additive) one. For example, a simple bivariate independence model for two categorical variables X and Y

 pij = P(X=i) P(Y=j); i=1,…,M; j=1,…,N

transforms to:

 log(pij) = liX + ljY;

where

 liX = logP(X=i);
 ljY = logP(Y=j).

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