Mixed Models – Interactive (LMM & GLMM)

⚠ Illustration tool: Estimates via IRLS + Method-of-Moments — no Laplace approximation (lme4/glmmTMB). Direction and order of magnitude are correct, exact values may differ.

LIKELIHOOD:

POOLING:

Fixed Effects

γ₀₀ · Grand Intercept2.00

γ₁₀ · Fixed Slope0.60

Random Effects

τ₀ · SD Random Intercept1.00

τ₁ · SD Random Slope0.30

ρ · Correlation u₀↔u₁0.00

          ρ>0: Groups with high intercept have steep slope (fan-out)

          ρ<0: Groups with high intercept have flat slope (fan-in)

σ · Residual SD0.60

Data Generation

Groups J5

Obs. per group n_j12

x-Range±3.0

x-Group offset (Simpson) 0.0

N = 60

Model Formula

–

MAIN PLOT — DATA & REGRESSION LINES

Partial Pooling: Group lines with shrinkage

⚠Simpson risk: Complete pooling line shows a different direction than the within-group effects — do not ignore the group structure!

SHRINKAGE DIAGRAM

No Pooling vs. Partial Pooling Intercepts

          How to read? Each row = one group, sorted by no-pooling intercept (top = largest). ○ = No-Pooling  ·  ● = Partial-Pooling. Arrow shows direction and strength of shrinkage toward the grand mean (γ̂₀₀, dashed).
            ·  ⚡ = Outlier Group
        

MODEL RESULTS

Estimated Parameters & Metrics

What is a Mixed Model?

A Mixed Model combines Fixed Effects (γ — the same for all groups) with Random Effects (u₀ⱼ — group-specific deviations). The basic model is:

y_ij = γ₀₀ + γ₁₀·x_ij + u₀ⱼ + ε_ij

The u₀ⱼ are not estimated directly but modeled as random variables: u₀ⱼ ~ Normal(0, τ₀²). This enables information sharing between groups — each group benefits from the data of the others.

Complete · No · Partial Pooling

Complete Pooling: One line for all — group structure is ignored. Efficient, but biased when groups truly differ.

No Pooling: Independent lines per group. No information sharing — overfitting with small n_j.

Partial Pooling (Mixed Model): The golden middle ground. Group-specific estimates are pulled toward the grand mean (Shrinkage). Groups with little data are pulled more strongly — the model "trusts" them less.

ICC — Intraclass Correlation

ICC = τ₀² / (τ₀² + σ²)

The proportion of total variance that lies between groups. Rule of thumb: ICC > 0.05 → LMM is necessary, since observations within a group are not independent.

Shrinkage factor λ = τ₀² / (τ₀² + σ²/n_j)

Determines how strongly the group-specific estimate is pulled toward the grand mean. With small n_j or small τ₀ → strong shrinkage. Visible in the shrinkage diagram as arrow length.

Simpson's Paradox

When x correlates with group membership (x-group offset > 0), the marginal trend (complete pooling over all data) can be opposite to the within-group trend.

Example: Within each group y increases with x — but groups with high x also have lower intercepts. The complete pooling line shows a negative overall trend, even though the true effect is positive.

Solution: Partial pooling separates the within-group effect (γ₁₀) from the between-group effect.

Random Intercept vs. Intercept + Slope

Random Intercept (τ₀ > 0): Groups differ in their baseline level — parallel lines with different y-intercepts. Formula: y ~ x + (1|group)

Random Intercept + Slope (τ₀, τ₁ > 0): Groups also have different slopes — no fan-out or fan-in. Formula: y ~ x + (1+x|group)

The parameter ρ controls the correlation: ρ > 0 → groups with high intercept also have steep slope (fan-out). ρ < 0 → lines cross each other (fan-in).

GLMM — When to use which likelihood?

The mixed model principle applies to all distribution families:

Gaussian (LMM): Continuous, symmetric DV. y ~ Normal(μ, σ)

Poisson (GLMM): Count data (0, 1, 2, …). Log link: log(λ) = η — λ stays positive.

Gamma (GLMM): Positive, right-skewed data (reaction times, costs). Log link.

Logistic (GLMM): Binary DV (0/1). Logit link: log(p/(1−p)) = η — p stays in (0,1).

In brms: family = poisson(), family = Gamma(link="log"), family = bernoulli()

ℹ (G)LMM Interactive — Help

What do I learn here?

When data have a group structure (persons, schools, clinics), simple regressions are problematic — observations within a group are not independent. Mixed Models solve this through Random Effects: each group gets its own intercept u₀ⱼ, drawn from a common distribution N(0, τ₀).

Recommended exploration

Start with Partial Pooling — this is the LMM/GLMM
Compare No Pooling vs. Complete Pooling — where do the lines diverge?
Vary τ₀ (random effect SD) — when does shrinkage become visible?
Toggle ⚡ Outlier Group — how does each pooling strategy react?
Vary number of groups J and group sizes n_j — how does shrinkage change?
Compare AIC/BIC in the model results — when does Partial Pooling outperform No/Complete Pooling?
Load the ⚠ Simpson preset and observe Complete Pooling — why is Partial Pooling indispensable here?

The most important panels

Group lines: Fixed Effect (γ₀₀ + γ₁₀·x) + group-specific deviation u₀ⱼ — Partial Pooling shrinks all lines toward the grand mean

Shrinkage diagram: No-pooling estimate → Partial-pooling estimate; arrows show the pull. Groups with small n are pulled more strongly.

ICC bar: τ₀²/(τ₀²+σ²) — proportion of variance between groups. ICC > 0.05 indicates that a mixed model is needed.

Simpson warning: appears when Complete Pooling reverses the direction of the slope — the classic Simpson's Paradox.

GLM families

Gaussian: continuous data — classic LMM
Poisson: count data (0,1,2,…) — log link
Gamma: positive continuous data — log link
Logistic: 0/1 data — logit link