mixedmodels

Mixed models, or mixed-effects models, are statistical models that include both fixed effects and random effects to handle correlation and non-independence in data arising from clustering, repeated measurements, or hierarchical structure. Fixed effects describe systematic, population-level influences, while random effects capture unit-specific deviations and variance components associated with grouping factors such as subjects, sites, or schools. The standard linear mixed model uses a normal response and can be written as y = Xβ + Zb + ε, where Xβ represents fixed effects, Zb represents random effects with b ~ N(0, D), and ε ~ N(0, σ^2 I) is residual error. Generalized linear mixed models extend this framework to non-normal responses through a link function: g(E[y|b]) = Xβ + Zb.

Estimation typically proceeds by maximum likelihood or restricted maximum likelihood (REML); Bayesian approaches using MCMC are also common. Linear mixed models (LMMs) assume a continuous, approximately normal response; generalized linear mixed models (GLMMs) handle binary, count, or other data types via appropriate distributions (e.g., binomial, Poisson) and link functions. Nonlinear and multilevel variants further extend the framework.

Random effects can include random intercepts, random slopes, or more complex structures; designs may be nested (e.g., students within classrooms) or crossed (e.g., students observed across multiple teachers). Data can be unbalanced or contain missing values, which mixed models handle more gracefully than simple repeated-measures models.

Applications span psychology, ecology, education, biostatistics, and econometrics, especially with longitudinal and clustered data. Model selection and diagnostics involve likelihood ratio tests, information criteria (AIC/BIC), and measures of explained variance such as ICC or conditional/marginal R^2. Software implementations include R (lme4, nlme), Python (statsmodels, PyMC3 for Bayesian), SAS PROC MIXED, STATA, and Julia (MixedModels.jl).