Mixed-Effects Models

What It Is

Mixed-effects models (also called multilevel or hierarchical models) handle data where observations are grouped into clusters. Students nested in schools, patients nested in hospitals, repeated measures within participants – whenever observations within a group are more similar to each other than to observations in other groups, you have clustered data.

The “mixed” refers to two types of effects. Fixed effects are the predictors you care about (treatment, age, motivation) – the relationships you want to test. Random effects capture the shared variation within clusters. A random intercept means each cluster has its own baseline level; a random slope means the effect of a predictor varies across clusters.

Ignoring clustering inflates Type I error rates and produces misleadingly precise estimates. MCPower uses mixed-effects models to produce correct power estimates for clustered designs. Power is always tested for fixed effects (whether beta differs from zero), not random effects.

How It Works in MCPower

MCPower supports three random effect structures, configured via the formula and set_cluster():

Random intercepts – (1|school):

model = MCPower("satisfaction ~ treatment + motivation + (1|school)")
model.set_cluster("school", ICC=0.2, n_clusters=20)

Random slopes – (1 + x|school):

model = MCPower("y ~ x1 + (1 + x1|school)")
model.set_cluster("school", ICC=0.2, n_clusters=20,
                   random_slopes=["x1"], slope_variance=0.1, slope_intercept_corr=0.3)

Nested effects – (1|school/classroom):

model = MCPower("y ~ treatment + (1|school/classroom)")
model.set_cluster("school", ICC=0.15, n_clusters=10)
model.set_cluster("classroom", ICC=0.10, n_per_parent=3)

Guidelines

Statistical Models

Random intercepts:

\[y_{ij} = \mathbf{x}_{ij}'\boldsymbol{\beta} + b_j + e_{ij}, \quad b_j \sim \mathcal{N}(0, \tau^2), \quad e_{ij} \sim \mathcal{N}(0, \sigma^2)\]

Random slopes (adds slope variation per cluster):

\[y_{ij} = \mathbf{x}_{ij}'\boldsymbol{\beta} + b_{0j} + b_{1j} x_{ij} + e_{ij}\]
\[\begin{split}[b_{0j}, b_{1j}] \sim \mathcal{N}(\mathbf{0}, \mathbf{G}), \quad \mathbf{G} = \begin{bmatrix} \tau^2_{\text{int}} & \rho \cdot \tau_{\text{int}} \cdot \tau_{\text{slope}} \\ \rho \cdot \tau_{\text{int}} \cdot \tau_{\text{slope}} & \tau^2_{\text{slope}} \end{bmatrix}\end{split}\]

Nested (two levels of clustering):

\[y_{ijk} = \mathbf{x}_{ijk}'\boldsymbol{\beta} + b_{\text{school},j} + b_{\text{classroom},jk} + e_{ijk}\]

Design Recommendations

Recommendation

Rationale

10–20 clusters minimum

Below 10, random effect estimation becomes unstable

30+ clusters for random slopes

Slope variance estimation needs more cluster-level data

Minimum 5 obs/cluster

Enforced by MCPower; warning below 10

Slope variance has large impact

Even slope_var=0.05 can require 2–3x more observations

MCPower assigns treatment at the individual level within clusters. ICC and cluster allocation have minimal impact on fixed-effect power – power depends primarily on total N. See the Random Slopes tutorial for the factor that does matter: slope_variance.

Constraints

  • No crossed random effects (only nested and independent clusters)

  • No random slopes without intercept ((0 + x|group) not supported)

  • ICC valid range: 0 (no clustering) or 0.1 to 0.9. Values between 0 and 0.1 (exclusive) are rejected for numerical stability.

Common Patterns

Typical ICC Values by Research Domain

Domain

Grouping

Typical ICC

Education

Schools

0.10–0.25

Healthcare

Hospitals / clinics

0.05–0.20

Psychology

Therapists / sites

0.10–0.30

Organizational

Companies / teams

0.10–0.25

Which Random Effect Structure?

Scenario

Formula

Rationale

Observations clustered, effect same everywhere

(1|cluster)

Simplest; accounts for shared baseline

Effect of a predictor varies across clusters

(1 + x|cluster)

Captures slope heterogeneity

Two-level hierarchy (e.g., classrooms in schools)

(1|school/classroom)

Captures clustering at both levels

Learn More