The GLMM Tab

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.

For example, with student test scores collected from multiple schools, you can estimate fixed effects $x_i'\beta$ such as study hours while accounting for school-level differences through the random intercept $u_{j[i]}$ .

This model extends GLM by adding random effects, and is also known as a multilevel model. When test scores are correlated within schools and study hours vary across schools, ignoring the school differences with GLM leads to an underestimated standard error for the study-hours coefficient and a confidence interval that is narrower than it should be.

See GLMM Fundamentals for the mathematical background.

Basic Usage

Opening GLMM

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

Setting Up the Analysis

GLMM form configuration

Dataset selects the dataset to analyze. GLMM expects one row per observation, with a column identifying the group. If repeated measurements are spread across columns, convert them with Wide to Long in the Reshape tab.

Response Variable (Y) selects the response variable. Only numeric columns (interval/ratio scale) are available. For the Binomial family, use a column with 0/1 values. A boolean column is inferred as nominal scale by default and cannot be selected; change its scale to interval in the Data Table to make it selectable, and its values are treated as true=1, false=0.

Fixed Effects (X) selects predictor variables for fixed effects; select at least one. Only numeric columns are selectable. To use categorical variables, convert them with Dummy Coding first. The conversion creates a new derived dataset, so switch Dataset to that dataset afterwards.

Group Variable (Random Intercept) selects the grouping variable for random intercepts. Categorical (nominal/ordinal) or string columns are available. Columns that record groups as numbers, such as line numbers or school IDs, are inferred as interval scale and do not appear as candidates; change their scale to nominal in the Data Table to make them selectable.

Distribution Family selects the distribution family:

Family	Default Link	Available Links	Use Case
Gaussian (Normal)	Identity	Identity, Log	Continuous values
Binomial (Logistic)	Logit	Logit, Probit	Binary data
Poisson (Count)	Log	Log, Identity	Count data
Gamma	Inverse	Inverse, Log, Identity	Positive continuous

Link Function selects the link function. Defaults to the canonical link for the selected family. Available options depend on the selected family (see table above).

Link Function	Formula	Description
Identity	$\eta = \mu$	No transformation. Canonical link for Gaussian
Logit	$\eta = \log\!\bigl(\mu / (1 - \mu)\bigr)$	Log-odds transformation. Canonical link for Binomial
Log	$\eta = \log(\mu)$	Log transformation. Canonical link for Poisson. Ensures $\mu > 0$
Inverse	$\eta = 1/\mu$	Reciprocal transformation. Canonical link for Gamma
Probit	$\eta = \Phi^{-1}(\mu)$	Inverse CDF of the standard normal distribution. Corresponds to a latent normal variable model

See GLM Fundamentals: Choosing a Link Function for the mathematical properties of canonical links.

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

Confidence Level sets the confidence level for confidence intervals (default: 95%, range: 50--99.99%). This is reflected in the Lower N% / Upper N% columns of the Fixed Effects table. The Model Detail tab opened after saving has the same input pre-filled with the saved value, and you can change it there to recompute the CI without modifying the saved value.

Max Iterations sets the maximum number of optimization iterations (default: 100).

Convergence Tolerance sets the convergence threshold (default: 1e-6).

Running the Analysis

Click Run GLMM. The estimation algorithm differs by family (see details). While the analysis runs, a progress bar and the estimation stage appear below the form.

Changing the variable selection or the model settings after a run makes the displayed results no longer belong to the current settings. MIDAS removes those results from the view and shows a message asking you to run the analysis again. Restoring the previous settings brings the results back. Changing Dataset also resets the variable selection, so the results are discarded instead. Changing Confidence Level keeps the results, because that setting only recomputes the intervals from the estimated coefficients.

Understanding Results

GLMM analysis results (Random Effects, ICC, Fixed Effects, Model Fit)

Random Effects

Displays variance components for random effects.

Column	Description
Component	Name of the variance component. `Group (variable name)` for the group variable variance, `Residual` for residual variance
Variance	Variance estimate: $\sigma_u^2$ for the group variable, $\sigma_e^2$ for the residual
Std.Dev.	Square root of the variance

For Poisson and Binomial families, the Residual row is not shown because the dispersion parameter is fixed at $\phi = 1$ .

The link function does not affect the residual variance $\sigma_e^2$ , because $\sigma_e^2$ is a property of the distribution family (Gaussian: $\operatorname{Var}(Y \mid \mu) = \sigma_e^2$ ; Gamma: $\operatorname{Var}(Y \mid \mu) = \sigma_e^2 \mu^2$ ). What the link function does affect is the diagnostic residuals (deviance and Pearson), through the fitted values $\hat\mu = g^{-1}(\eta)$ computed from the linear predictor (see Saving and Diagnostics).

ICC (Intraclass Correlation Coefficient)

ICC represents the share of unexplained variance attributable to between-group differences ( $\text{ICC} = \sigma_u^2 / (\sigma_u^2 + \sigma_e^2)$ ). MIDAS computes ICC only for the following family+link combinations, where $\sigma_e^2$ has a theoretical basis:

Family	Link	$\sigma_e^2$
Gaussian	identity	REML estimate
Binomial	logit	$\pi^2/3$ (threshold model)
Binomial	probit	$1$ (threshold model)

For all other combinations — Poisson (all links), Gamma (all links), and Gaussian with the log link — no theoretically grounded residual variance exists, so N/A (ICC not defined) is shown instead of an ICC value (see GLMM Fundamentals for details). For these combinations, the magnitude of between-group variation can still be assessed from the Variance in the Random Effects table ( $\sigma_u^2$ on the link scale) and from BLUP.

Even for combinations where ICC is defined, N/A (degenerate fit) is shown when the variance component estimation breaks down and ICC cannot be computed.

The interpretation of ICC depends on the nature of the data and the research objective, and group size should also be considered (see When to Use GLMM vs GLM).

For Binomial models, ICC is computed on the latent (link) scale, not the probability scale. It does not equal the correlation of the observed binary responses.

Fixed Effects

Coefficient table for fixed effects.

Column	Description
Variable	Variable name
Estimate	Regression coefficient $\hat\beta$
Std. Error	Standard error. For Gaussian + identity, computed via $(X'V^{-1}X)^{-1}$ using the Woodbury formula. For all other combinations, an approximation based on the working weight matrix at PIRLS convergence
Lower N% / Upper N%	Wald-based confidence interval $\hat\beta \pm z_{1-\alpha/2} \times \text{SE}(\hat\beta)$ , where N is the selected confidence level. MIDAS always uses the standard normal distribution for GLMM fixed effects

When the link function is logit or log, the following columns are added:

Column	Description
OR / IRR / exp(Est.)	Exponentiated estimate $\exp(\hat\beta)$ . Displayed as odds ratio (OR) for logit link, incidence rate ratio (IRR) for log link with Poisson, and exp(Est.) for log link with other families
exp(Lower N%) / exp(Upper N%)	Exponentiated confidence interval bounds

Coefficients are interpreted on the link function scale, as in GLM. See GLM coefficient interpretation for details.

The confidence intervals are based on a normal approximation, which can be too narrow when the number of groups is small. See GLMM Fundamentals: Fixed Effect Inference for details.

The fixed effects coefficients table can be saved as a dataset using the Save as Dataset button for export to CSV. Using Add to Report also adds the BLUP table at the same time. Both require a saved model; if the model is not yet saved, the dialog asks for a model name and saves the model as part of the same action.

Model Fit

Metric	Description
REML Log-Likelihood / Log-Likelihood (Laplace)	Log-likelihood. REML for Gaussian + identity; the Laplace-approximated marginal log-likelihood for all other combinations
AIC	$-2\ell + 2k$ , where $k$ is the total number of fixed-effect parameters and variance components
BIC	$-2\ell + k\log n$ , where $n$ is the number of observations

Model comparison based on AIC and BIC has limitations (see GLMM Fundamentals: AIC/BIC Limitations). REML-based AIC (Gaussian + identity) can only compare models with identical fixed-effect structure. Models with different families or links cannot be compared by AIC, BIC, or log-likelihood, because the log-likelihood basis (REML vs. Laplace) and scale differ.

BLUP (Random Effect Predictions)

BLUP table showing random effect predictions by group

Displays the BLUP (Best Linear Unbiased Predictor) of the random intercept for each group.

Column	Description
Group	Group variable value
Conditional Mode	Predicted value of the random effect $u_j$ . Smaller groups are shrunk more toward the overall mean (0) (shrinkage details)
Std. Error	Standard error of the prediction. The square root of the conditional variance; larger for smaller groups
Rank	Rank of Conditional Mode in descending order

The defining property of the BLUP — the smallest mean squared prediction error among linear unbiased predictors — holds only for Gaussian + identity. For all other combinations, the predicted value is the mode (the point of highest density) of the distribution of $u_j$ conditional on the observed data — the conditional mode — which is the quantity shown in the Conditional Mode column (see GLMM Fundamentals: Estimation and Prediction).

For combinations other than Gaussian + identity, the Std. Error is based on a normal approximation around the conditional mode, so an interval of the form (conditional mode) ± z × SE is not guaranteed to be an accurate prediction interval.

The BLUP table can be saved as a dataset using the Save as Dataset button for export to CSV. Saving the BLUP dataset requires a saved model; if the model is not yet saved, the dialog also asks for a model name and saves the model together with the dataset. To add it to a report, use the Add to Report button in the Fixed Effects section.

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)

For Gaussian + identity, the deviance and Pearson residuals both equal the raw residual $y_i - \hat y_i$ , so deviance_residuals and pearson_residuals hold the same values.

After saving, View Model Details and View Diagnostics buttons become available. Model Detail displays the fixed effects coefficient table and a BLUP table (per-group random intercept estimates). The BLUP table shows up to 50 rows by default; when there are more groups, click Show all N rows (N is the total number of groups) to expand the full list. Use the Add to Report button to add both the coefficients table and the BLUP table to a report.

Notes

Current Limitations

The current GLMM implementation supports random intercept models only. Random slopes (models where a predictor's effect varies by group) and crossed random effects (models where observations belong to multiple group variables) are not supported (see GLMM Fundamentals).

The GLM tab offers the Negative Binomial family, but GLMM does not.

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 and distorts the standard errors of the coefficients. The size of the distortion depends on the predictor: the more a predictor's variation lies between groups (such as a school-level tuition fee, which takes the same value within each group), the more strongly its standard error is underestimated, while predictors that vary mainly within groups are less affected. GLMM avoids this problem by explicitly modeling within-group correlation.

When you want to compare a small number of specific groups and estimate their effects directly, another option is to convert the groups to dummy variables with Dummy Coding and treat them as fixed effects in GLM (see GLMM Fundamentals).

Automatic Exclusion of Missing Values

Rows containing missing values, non-numeric values, or infinity are automatically excluded. The Observations value in the results is the number of observations after exclusion. This is listwise deletion. See Missing Data Mechanisms for conditions under which it yields valid estimates.

Convergence Issues

If the model fails to converge, try the following:

Increase Max Iterations (for example, from 100 to 500)
Relax Convergence Tolerance (for example, from 1e-6 to 1e-4)
Standardize predictors whose scales differ greatly

Predictors are scaled internally, but extreme scale differences can make the GLM estimation that provides the initial values fail. Convergence problems are also more likely with very few groups (2 or 3), where variance component estimation becomes unstable.

Singular Fit

A "Singular fit" warning appears when the random effect variance becomes negligibly small and the estimate reaches the boundary of the parameter space. The check tests whether the scale-invariant relative covariance parameter $\theta$ used in variance component optimization (see GLMM Fundamentals) is close enough to the zero boundary, so it does not depend on the units of the response.

Singular fit occurs when the group variable explains very little variation in the response, or when the sample size or number of groups is too small to separate group-level variation from residual variation.

When singular fit occurs, ICC and variance component estimates should be interpreted with caution. A fixed-effects-only model (GLM) may be more appropriate.