loglinær

Loglinær, in a statistical context, refers to log-linear models used for analyzing count data in contingency tables. The approach models the expected cell counts on a logarithmic scale as a linear combination of effects for the categorical variables and their interactions. This framework is especially suitable for multi-way tables where researchers want to understand the relationships among several categorical factors.

In a typical log-linear model, let n be the observed cell counts in a table defined by categorical variables such as A, B, and C. The model assumes n follows a Poisson distribution with mean μ, and the log of μ is expressed as a linear predictor: log μ = λ + λA(i) + λB(j) + λC(k) + λAB(ij) + λAC(ik) + λBC(jk) + λABC(ijk), with constraints to ensure identifiability. The parameters correspond to main effects and interactions among the variables. Estimation is usually performed via maximum likelihood, and the model can be fitted as a Poisson generalized linear model with a log link. Software implementations include standard GLM tools and dedicated functions in statistical packages.

Log-linear models are used to assess independence and interaction patterns in categorical data, test hypotheses about associations among variables, and compare nested models using likelihood ratio tests, AIC, or BIC. They contrast with logistic regression, which is typically applied to binary outcomes rather than cell counts in a contingency table. Practical considerations include handling sparse data, choosing a hierarchical structure of effects, and ensuring interpretability of interaction terms. While powerful for exploratory analysis of relationships among categorical factors, conclusions about causality require careful study design and additional methods.