RandomEffectsModells

RandomEffectsModells, or random effects models, are statistical models designed for data that are grouped or hierarchical. They extend fixed-effects models by including random effects that capture unobserved heterogeneity across groups. The typical linear mixed effects model for a continuous outcome is y_ij = X_ij beta + Z_ij b_j + epsilon_ij, where i indexes observations within group j, b_j ~ N(0, D) is a vector of group-specific random effects, and epsilon_ij ~ N(0, sigma^2) is the residual error. A common special case is the random intercept model: y_ij = beta0 + beta'X_ij + u_j + e_ij, with u_j ~ N(0, sigma_u^2) and e_ij ~ N(0, sigma_e^2). Random effects induce correlation among observations within the same group and can be extended to include random slopes and multiple random effects.

Assumptions typically include that random effects are uncorrelated with the fixed effects, random effects and residuals are normally distributed, and groups are independent. For non-Gaussian outcomes, generalized linear mixed models (GLMMs) are used, incorporating non-normal link functions and distributions.

Estimation and inference are usually performed via maximum likelihood (ML) or restricted maximum likelihood (REML). REML provides less biased estimates of variance components. Inference often relies on likelihood ratio tests, Wald tests, or Bayesian methods. The Hausman test is commonly employed to decide between random effects and fixed effects models when there is concern about correlation between group effects and covariates.

Applications include longitudinal and panel data, repeated measures, multi-site studies, and hierarchical data in fields such as psychology, ecology, and econometrics. Extensions cover random slopes, cross-classified and nested designs, and GLMMs for non-normal outcomes. Software implementations are available in R (lme4, nlme), SAS PROC MIXED, Stata mixed, and Python (statsmodels).