Generalised linear mixed models: a binary (logistic or probit) or count (Poisson) outcome with grouped or clustered data, including crossed and nested grouping structures. The mixed-model counterpart of GLM.

Examples

Binary (logistic)

• Cluster-randomised binary trial (random intercept per cluster) infection ~ treatment + (1|hospital) — one between-cluster treatment effect on a yes/no outcome, clusters randomised whole.
• Longitudinal binary outcome over time (random intercept per subject) symptom_present ~ month + treatment + (1|patient) — a yes/no outcome tracked over time within subjects across two arms.
• Difference-in-differences on a binary outcome (group x time GLMM) employed ~ policy_group * period + (1|individual) — the group-by-time interaction (DiD) on a clustered binary outcome.
• Logistic GLMM with a continuous predictor and random slope species_present ~ temperature + (1 + temperature|site) — a continuous predictor whose slope varies across groups.

Binary (probit)

• Dose-response trial on an adverse response (longitudinal probit) adverse_response ~ dose + (1|subject) — repeated dose-level measurements within subject, binary outcome via the probit link.

Count (Poisson)

• Cluster-randomised trial on adverse-event counts (clustered Poisson) adverse_events ~ treatment + (1|clinic) — a count outcome, whole clinics randomised to control vs treatment.
• Two arms followed over months on a clinic-visit count visits ~ month + treatment + (1|patient) — a longitudinal count outcome tracked over time within subjects across two arms.
• Pollutant-exposure effect on species counts with site-varying slopes count ~ exposure + (1 + exposure|site) — a continuous predictor on a count outcome whose slope varies across groups.

Advanced structure

• Survey agreement across respondents and items: crossed random effects (logistic) agree ~ condition + (1|respondent) + (1|item) — two independent, fully-crossed grouping factors on a binary outcome.
• Nested cluster trial on a binary pass/fail outcome: classrooms inside schools passed ~ treatment + (1|school/classroom) — a nested 3-level grouping structure on a binary outcome.