loglineaarne

Loglineaarne, or log-linear in English, is a statistical modeling approach used for analyzing counts in contingency tables. It describes how the logarithm of the expected cell frequencies can be written as a linear combination of parameters corresponding to main effects and interactions among categorical variables. The method is widely applied in fields such as sociology, epidemiology, marketing, and data science to study associations and dependencies among multiple categorical factors.

Mathematical form and interpretation: Consider a multi-way contingency table with cell counts N_i1...ik and expected counts mu_i1...ik. A log-linear model specifies log(mu_i1...ik) = lambda_0 + sum_r lambda_r f_r(i1...ik), where f_r are functions representing the presence of specific levels of factors and their interactions. Equivalently, mu_i1...ik = exp(linear predictor). The model can impose different restrictions to reflect assumptions about independence or interaction structure, such as mutual independence or joint independence of subsets of variables.

Estimation and uses: Parameters are typically estimated by maximum likelihood. The iterative proportional fitting (IPF) algorithm is a common method for fitting log-linear models to observed counts, and results can be compared using deviance or likelihood ratio tests. Log-linear models are also connected to Poisson regression, since modeling counts with a log link yields equivalent representations under appropriate exposure terms. This framework allows researchers to test hypotheses about associations between variables, assess goodness of fit, and explore higher-order interactions in multi-way tables.

Relation to related concepts: Log-linear models are related to, but distinct from, logistic regression. While log-linear models model counts in contingency tables, logistic regression models a binary or multinomial outcome as a function of predictors; the two approaches share underlying exponential-family structure and can be viewed as complementary tools for analyzing categorical data.