Loglinear

Loglinear refers to a class of statistical models used for count data in contingency tables. A loglinear model describes the natural logarithm of the expected cell counts as a linear combination of parameters that correspond to main effects and interactions among categorical variables. If mu_{i j ...} denotes the expected count in a cell defined by levels of factors, a typical form is log(mu_{i j ...}) = lambda_0 + lambda_i + lambda_j + ... + lambda_{i j} + ..., where the lambda terms capture baseline levels, main effects, and interactions. The model is a member of the exponential family and can be estimated by maximum likelihood; in practice, Poisson regression with a log link yields equivalent estimates when applied to cell counts, possibly with an offset for exposure or population size.

Loglinear models are widely used to study associations among categorical variables in multi-way contingency tables. The independence model includes only main effects corresponding to each variable, implying no interaction; adding interaction terms tests for more complex associations. Model selection often relies on likelihood ratio tests and information criteria such as AIC or BIC, while the iterated proportional fitting (IPF) algorithm can compute expected counts under a specified model when margins are fixed.

Key concepts include the hierarchy principle (higher-order interactions are included only if their lower-order terms are present) and identifiability constraints (different parameterizations that yield the same likelihood). Interpretations of the parameters are on the log scale and, when exponentiated, relate to multiplicative effects on expected counts.

Applications span epidemiology, sociology, psychology, and any field analyzing categorical data. Limitations include sensitivity to small or zero cell counts, which can affect estimation and inference, and the need for careful model specification and interpretation in higher-dimensional tables.