Mixed Effects Models Reference Cheat Sheet

Key Facts

• A basic random-intercept mixed model is $y_{ij} = \beta_0 + \beta_1 x_{ij} + u_{0j} + \varepsilon_{ij}$, where $u_{0j}$ is the group-specific random intercept.
• Random effects are commonly modeled as $u_{0j} \sim N(0, \tau_{00})$ and residual errors as $\varepsilon_{ij} \sim N(0, \sigma^2)$.
• The intraclass correlation for a random-intercept model is $ICC = \frac{\tau_{00}}{\tau_{00} + \sigma^2}$.
• A random-slope model can be written as $y_{ij} = \beta_0 + \beta_1 x_{ij} + u_{0j} + u_{1j}x_{ij} + \varepsilon_{ij}$.
• In lme4 syntax, a random intercept is written as $\text{y} \sim \text{x} + (1\mid \text{group})$.
• In lme4 syntax, a correlated random intercept and slope are written as $\text{y} \sim \text{x} + (1 + \text{x}\mid \text{group})$.
• In lme4 syntax, uncorrelated random intercepts and slopes are written as $\text{y} \sim \text{x} + (1 + \text{x}\Vert \text{group})$.
• Use maximum likelihood to compare models with different fixed effects, and use restricted maximum likelihood for final estimation of variance components.

Vocabulary

Fixed effect

A coefficient such as $\beta_1$ that estimates the average relationship between a predictor and the response across the population.

Random effect

A group-specific deviation such as $u_{0j}$ or $u_{1j}$ that allows intercepts or slopes to vary across clusters.

Cluster

A grouping unit, such as a school, patient, classroom, or site, within which observations may be correlated.

Variance component

A parameter such as $\tau_{00}$ or $\sigma^2$ that measures variability from random effects or residual error.

Intraclass correlation

The proportion of total variance attributable to between-cluster differences, computed as $ICC = \frac{\tau_{00}}{\tau_{00} + \sigma^2}$.

Random slope

A group-specific change in a predictor effect, represented by $u_{1j}$ in a model such as $y_{ij} = \beta_0 + \beta_1 x_{ij} + u_{1j}x_{ij} + \varepsilon_{ij}$.