Generalized Linear Mixed Model (GLMM)

The GLMM tab fits random intercept models $g(\mu_i) = x_i'\beta + u_{j[i]}$ , $u_j \sim N(0, \sigma_u^2)$ for data with group structure. It extends GLM by adding random effects. See GLMM Fundamentals for the mathematical background.

For example, when analyzing student test scores from multiple schools, GLMM can estimate the effect of study hours (fixed effect) while accounting for school-level differences (random intercept). Ignoring group structure with GLM leads to underestimated standard errors and inflated significance.

Basic Usage

Opening GLMM

Select Analysis > GLMM (Mixed Model)... from the menu bar.

Setting Up Variables

Dataset selects the dataset to analyze.

Dependent Variable (Y) selects the response variable. Only numeric columns are available.

Fixed Effects (X) selects predictor variables for fixed effects. Only numeric columns are selectable. To use categorical variables, convert them with Dummy Coding first.

Group Variable selects the grouping variable for random intercepts. Categorical (nominal/ordinal) or string columns are available.

Distribution Family selects the distribution family. The same options as GLM:

Family	Use Case
Gaussian (Normal)	Continuous values
Binomial (Logistic)	Binary data
Poisson (Count)	Count data
Gamma	Positive continuous

Link Function selects the link function. Defaults to the canonical link for the selected family.

Include intercept toggles the intercept term (default: on).

Advanced Options

Max Iterations: Maximum optimization iterations (default: 100)
Convergence Tolerance: Convergence threshold (default: 1e-6)

Running the Analysis

Click Run GLMM. The estimation algorithm differs by family (see details). A progress dialog shows the estimation stage.

Understanding Results

Random Effects

Displays variance components for random effects.

Item	Description
Group	Group variable variance $\sigma_u^2$ and standard deviation
Residual	Residual variance $\sigma_e^2$ and standard deviation (Gaussian only)

For non-Gaussian families, residual variance is not shown because the dispersion parameter is handled differently per family (Poisson/Binomial fix $\phi = 1$ ; Gamma estimates $\phi$ from profiled deviance).

ICC (Intraclass Correlation Coefficient)

ICC represents the proportion of total variance attributable to between-group differences ( $\text{ICC} = \sigma_u^2 / (\sigma_u^2 + \sigma_e^2)$ for Gaussian). See GLMM Fundamentals for non-Gaussian computation.

ICC Range	Interpretation
0 -- 0.05	Small group differences
0.05 -- 0.20	Small to moderate group effect
0.20 -- 0.50	Large group effect
0.50+	Group differences dominate

This table is a rough guide for the Gaussian family. For non-Gaussian families, ICC is computed on a latent scale and the interpretation differs (see GLMM Fundamentals). Group size should also be considered (see When to Use GLMM vs GLM).

Fixed Effects

Coefficient table for fixed effects.

Column	Description
Variable	Variable name
Estimate	Regression coefficient $\hat\beta$
Std. Error	Standard error (computed via $(X'V^{-1}X)^{-1}$ using the Woodbury formula)
z value	Wald statistic $z = \hat\beta / \text{SE}(\hat\beta)$
p-value	Two-sided p-value from the standard normal distribution
95% CI	Wald-based 95% confidence interval

Coefficient interpretation follows GLM conventions (on the link function scale). See GLM coefficient interpretation for details.

Model Fit

Metric	Description
Deviance	Conditional deviance
AIC	Akaike Information Criterion
BIC	Bayesian Information Criterion

For Gaussian, AIC and BIC are REML-based and should not be used to compare models with different fixed effects (details). For non-Gaussian families, AIC/BIC are based on the Laplace-approximated marginal log-likelihood.

BLUP (Random Effect Predictions)

Displays the predicted random intercept (BLUP) for each group. Smaller groups are shrunk more toward the overall mean (shrinkage details).

Saving and Diagnostics

Enter a model name in Model Name and click Save Model to save the model to the project. A diagnostic derived dataset is automatically created on save.

Column	Description
`fitted_values`	Predicted values (fixed + random effects)
`deviance_residuals`	Deviance residuals
`pearson_residuals`	Pearson residuals
`group_random_effect`	Group random intercept (BLUP)

After saving, Open Model Details and View Diagnostics buttons become available.

Notes

Current Limitations

The current GLMM implementation supports random intercept models ( $(1 \mid \text{Group})$ ) only. Random slopes ( $(x \mid \text{Group})$ ) and crossed random effects are not supported.

When to Use GLMM vs GLM

When ICC is small, ignoring group structure and using GLM produces nearly identical results. The impact depends not only on ICC but also on group size; the design effect $\text{DEFF} = 1 + (\bar n - 1) \times \text{ICC}$ provides a rough guide (see GLMM Fundamentals). When ICC is large, GLM violates the independence assumption between observations, leading to underestimated standard errors. GLMM explicitly models within-group correlation, enabling valid inference.

Automatic Exclusion of Missing Values

Rows containing missing values, non-numeric values, or infinity are automatically excluded. The count and reasons for excluded rows are displayed in the results.

Convergence Issues

If the model fails to converge:

Increase Max Iterations (100 → 500)
Relax Convergence Tolerance (1e-6 → 1e-4)
Very few groups (2-3) can make variance component estimation unstable
Large scale differences between predictors may cause the GLM initial-value estimation to fail (internal scaling is applied, but extreme cases may still have issues)