MixedLM

MixedLM, short for mixed linear model, is a statistical framework that extends ordinary linear regression to handle data with hierarchical or grouped structure. It incorporates both fixed effects, which are constant across groups, and random effects, which vary by group, to account for correlations among observations within the same group.

In the standard formulation, the observed outcome y is modeled as y = Xβ + Zb + ε, where X is the fixed-effects design matrix, β the fixed-effect coefficients, Z is the random-effects design matrix, b the random effects, and ε the residual errors. The random effects are typically assumed to follow a multivariate normal distribution with mean zero and covariance G, while the residuals ε are assumed normal with covariance R. Random effects can represent random intercepts, random slopes, or more complex group-level deviations.

Estimation is usually performed by maximum likelihood (ML) or restricted maximum likelihood (REML). REML is commonly preferred for estimating variance components because it reduces bias. Software implementations, such as the MixedLM class in statsmodels, fit Gaussian outcomes with repeated measures or clustered data and allow specification of multiple random effects and covariance structures through the Z matrix and the random-effects covariance G. Outputs include fixed-effect estimates, variance components, standard errors, and fit statistics (log-likelihood, AIC, BIC). Inference can be conducted via Wald tests or likelihood ratio tests.

Assumptions include linearity, normality of random effects and residuals, and correct specification of the random-effects structure. MixedLM is most appropriate for continuous outcomes with hierarchical data; for non-Gaussian responses, generalized linear mixed models (GLMMs) or alternative approaches are used.

Common use cases involve longitudinal studies, educational or healthcare data with site or subject effects, and any scenario with grouped measurements requiring correlation modeling.