Linear mixed models for a continuous outcome with grouped or clustered data — repeated measures, students within classrooms, sites in a trial. The grouping soaks up correlation that plain OLS would ignore.

Examples

• Repeated measures: random intercept per subject systolic_bp ~ phase + (1|patient) — does a continuous outcome change over repeated visits, each subject as its own control.
• Treatment x time interaction (random intercept) pain_score ~ treatment * week + (1|patient) — does the trajectory over time differ between two arms (difference-in-differences).
• Growth curve: random intercept and slope of time seedling_height ~ week + (week | seedling) — the average time slope judged against scattering individual growth rates.
• Conditional growth: treatment moderating individual slopes seedling_height ~ fertilizer * week + (1 + week | seedling) — whether two arms diverge in growth rate, with random intercepts and slopes.
• Two-level cluster-randomized trial (random intercept) cholesterol ~ treatment + (1|clinic) — a trial randomising whole clusters, one continuous outcome per member.
• Cluster RCT with a baseline covariate (adjusted) blood_pressure ~ treatment + baseline_bp + (1|clinic) — a cluster trial adjusted for a baseline covariate.
• Multisite trial: treatment effect varying across sites recovery_days ~ treatment + (treatment | site) — a treatment effect allowed to vary across sites (random slope).
• Within-subjects one-way design: repeated condition factor enzyme_activity ~ condition + (1|sample) — a repeated 3-level condition factor compared within each subject.