A. Formulas Used

Linear Mixed Model (general form):
y_ij = β₀ + β₁x_ij + u_j + ε_ij
u_j ~ N(0, σ²_u), ε_ij ~ N(0, σ²_ε)

Nested random effects:
y_ijk = β₀ + u_k + v_j(k) + ε_ijk

Crossed random effects:
y_ijk = β₀ + u_j + w_k + ε_ijk (j and k vary independently)

Variance components (ANOVA-based estimator used here):
σ²_between = (MS_between − MS_within) / n̄
σ²_within = MS_within

Intraclass correlation (ICC):
ICC = σ²_between / (σ²_between + σ²_within)

Where:
y_ij = outcome for observation i in cluster j
β₀ = fixed intercept (population grand mean)
u_j = random intercept for cluster j
ε_ij = residual
σ²_u = between-cluster variance
σ²_ε = within-cluster (residual) variance
n̄ = harmonic mean of cluster sizes
MS = mean square (between or within)

B. Technical Notes

• Estimator: This tool uses the method-of-moments / ANOVA-based variance component estimator. For balanced and approximately balanced designs, it agrees closely with REML estimates from lme4::lmer.
• Fixed effect: The grand mean is treated as the only fixed effect; predictors can be tested by comparing cluster means.
• Likelihood-ratio test (approximate): An F-test on between-cluster vs within-cluster mean squares serves as an LRT analogue for the random intercept.
• Crossed vs nested: If nested is selected, the second factor's variance is partitioned within the first. If crossed, the two factors contribute independently.
• Assumption check: Residuals are inspected via Q-Q plot; visible departures suggest a non-Gaussian family (logit/log link).
• Convergence: Variance components clipped at zero; negative ANOVA estimates set to 0 (consistent with the boundary of the parameter space).

Use a generalized linear mixed model with crossed or nested random effects whenever your observations are not independent — they fall inside groups, and the groups themselves come from a wider population you want to generalize to.

Decision Checklist

• ✅ Observations are clustered (students in classes, plots in sites, repeated measures on subjects)
• ✅ The grouping factor has many levels and you want the result to generalize to more levels
• ✅ You can identify whether grouping factors are nested or crossed
• ✅ Sample size: ≥ 5–6 levels per random factor; ideally ≥ 30 observations per group
• ❌ Do not use if the grouping factor has only 2–3 levels — treat it as a fixed effect instead
• ❌ Do not use if observations are independent (use OLS regression)
• ❌ Do not use a single-level GLMM if the data are nested across multiple levels (use multilevel GLMM)

Real-World Examples

1. Education research: Test scores from students nested within classrooms nested within schools — three random levels.
2. Psycholinguistics: Reaction times where subjects and items are crossed — every subject sees every item.
3. Wildlife ecology: Camera trap detection counts at plots within sites within reserves — nested spatial design.
4. Clinical trials: Recovery time of patients nested within hospitals, with hospital-level random intercepts.
5. Sensory science: Tasters and food samples are crossed — every taster rates every sample.

Sample Size Guidance

• ≥ 5 levels per random factor for stable variance components
• ≥ 30 total observations per group; thumb-rule: 30/30 (groups × within-group n)
• Power analysis via simr in R is recommended for confirmatory studies

Related Methods — Decision Tree

Independent observations? ─── Yes ── Linear regression / GLM
                          │
                          └── No ── Clustered? ─── Yes ── Random factor with many levels?
                                                                ├── Yes ── GLMM (this tool)
                                                                │           ├── 1 grouping → simple random intercept
                                                                │           ├── 2+ grouping, nested → nested GLMM
                                                                │           └── 2+ grouping, crossed → crossed GLMM
                                                                └── No  ── Treat as fixed effect (ANOVA / regression)

1. Enter Your Data — three input options. Paste comma-separated values into each cluster textarea, upload a CSV/Excel file with one cluster per column, or enter values manually. Default placeholder shows the format: 52, 48, 55, 61, 47, ...
2. Choose a Sample Dataset — five built-in datasets cover education (students in classrooms), psycholinguistics (subjects × items), camera traps (sites within reserves), clinical trials (patients within hospitals), and vegetation surveys (plots within transects).
3. Configure Settings — set the design (nested vs crossed) and the alpha level (0.01, 0.05, 0.10). Alpha controls the confidence interval and the significance threshold.
4. Run the Analysis — click Run GLMM Analysis. The tool computes variance components, ICC, fixed-effect grand mean, F-statistic, p-value, AIC, and BIC.
5. Read the Summary Cards — top metrics at a glance: ICC, total variance, p-value, and number of clusters. Green = significant random-effect variance; amber = trivial.
6. Read the Full Results Table — includes grand mean, SE, t-value, p-value, between-cluster variance, within-cluster variance, ICC, AIC, BIC, and 95% CI.
7. Examine the Variance Components Table — partitioned variance for each random factor, plus their proportions.
8. View the Visualizations — four plots: cluster means with error bars, variance partition pie/bar, residuals vs fitted, and Q-Q plot of residuals.
9. Read the Plain-Language Interpretation — dynamic, paragraph-level explanation of every key statistic.
10. Export Your Results — Download Report (.txt) for plain-text archives; Download PDF for sharing or attaching to manuscripts.