intermediate
9 min read
Life sciences · Topic
Linear Mixed Models
Linear mixed models (LMMs) extend OLS to handle correlated data — repeated measurements, clustered subjects, and multi-site studies. They partition variance into fixed effects (population-level) and random effects (subject- or group-specific).
The model
\[\mathbf{y} = X\boldsymbol{\beta} + Z\mathbf{b} + \boldsymbol{\varepsilon}\]where:
- $X\boldsymbol{\beta}$ — fixed effects (population means)
- $Z\mathbf{b}$ — random effects, $\mathbf{b} \sim \mathcal{N}(\mathbf{0}, G)$
- $\boldsymbol{\varepsilon}$ — residuals, $\boldsymbol{\varepsilon} \sim \mathcal{N}(\mathbf{0}, R)$
The marginal distribution is $\mathbf{y} \sim \mathcal{N}(X\boldsymbol{\beta},\; ZGZ^\top + R)$.
REML estimation
Restricted Maximum Likelihood (REML) estimates variance components with less bias than full ML by marginalising out fixed effects first. It maximises:
\[\ell_{REML}(G, R) = \log \int \mathcal{L}(\boldsymbol{\beta}, G, R;\, \mathbf{y})\, d\boldsymbol{\beta}\]Clinical trial applications
- Repeated measures: each patient measured at multiple time points — the correlation within patients is modelled by random intercepts and slopes.
- Multi-site trials: random site effects absorb between-centre variability.
- Missing data: LMMs are valid under Missing At Random (MAR) without imputation.