Mixed Models – Interactive (LMM & GLMM)

Mixed Model = Fixed Effects (the same for all groups) + Random Effects (group-specific deviations that share information between groups)

🚀 How to get started Your own results in 3 steps.

1 Choose a likelihoodPick the distribution that fits your data.

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2 Compare poolingSwitch between Partial, No, and Complete Pooling.

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3 Read group differencesCompare effects below and spot shrinkage.

⚠ 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— distribution of the data

Pooling— group structure

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_j Unbalance (group sizes) 0.0

          0 = all groups the same size. Details → Help.
        

N = 60

Model Formula

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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!

RANDOM EFFECTS

Group Intercepts — No-Pooling vs. Partial-Pooling

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 GLMM Fixed Effect (γ₀₀, dashed).
            ·  ⚡ = Outlier Group
        

MODEL RESULTS

Estimated Parameters & Metrics

Learning Cards — Deeper Concepts

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.

But beware: such thresholds are only rough rules of thumb. Bliese (1998) showed that even noticeably smaller ICCs can already produce bias once a cluster structure is present — there is no reliable "safe" lower bound.

Practical tip: the trade-off usually favors the LMM. If you know a clustering variable exists, you're generally well advised to use an LMM instead of Complete Pooling right away — even if the ICC later turns out small, doing so costs you almost nothing.

Important: The ICC itself is not a parameter the LMM estimates — it's just a helper value computed afterward from τ₀ and σ, to roughly gauge how strongly clustered the data are. The model itself only "knows" τ₀ (the spread of the group intercepts) and accounts for exactly that τ₀ directly through the random effects — regardless of whether you compute the ICC at all. If τ₀ is large (and thus the ICC is high), the LMM pulls comparatively little toward the overall mean (weak shrinkage); if τ₀ is instead ≈ 0 (ICC ≈ 0), the LMM and Complete Pooling will typically give very similar results, because there's then hardly any real group difference left for partial pooling to account for.

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()

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).

Note on No Pooling: under "Random Intercept" it compares against a common slope there (y ~ x + group) — matching the RE structure (1|group). Group-specific slopes (y ~ x * group) only appear under "Intercept + Slope".

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.

Shrinkage

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

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

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 GLMM Fixed Effect

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.

R²_marginal / R²_conditional: R²_marginal explains variance via the fixed effects only, R²_conditional also via the group structure (random effects) — the difference shows its explanatory contribution. Both values always refer to the GLMM itself and do not change with the pooling strategy.

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

n_j Unbalance & Slope Comparison

The n_j Unbalance slider (Data Generation) lets group sizes vary linearly (0 = all groups the same size). This makes it visible that the GLMM Fixed Effect (Partial Pooling) and the No-Pooling Grand Mean are always identical for equally sized groups — only with unequal n_j do they diverge.

Grand Mean (No Pooling): a simple n_j-weighted average of the independent group estimates — larger groups automatically count more.
GLMM Fixed Effect (Partial Pooling): precision-weighted (weight ∝ n_j/(n_j·τ₀²+V₁)) — the actual shrinkage target. When between-group variance dominates (τ₀² large), every group counts almost equally, regardless of its size.

The Slope Comparison table (below Model Comparison) shows both values together with the Complete-Pooling slope — each row in the color of its line in the main plot.

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