grmtree: Fit a Graded Response Model Tree for Differential Item...

In grmtree: Recursive Partitioning for Graded Response Models

Fit a Graded Response Model Tree for Differential Item Functioning Detection

Description

This function implements a tree-based graded response model (GRM) using model-based recursive partitioning to detect and account for differential item functioning (DIF) in polytomous items. The GRMTree combines the statistical framework of item response theory with recursive partitioning to identify heterogeneous subgroups in the population where item parameters (discrimination and thresholds) vary systematically across covariates.

Usage

grmtree(
  formula,
  data,
  na.action = na.omit,
  control = grmtree.control(),
  mtry = NULL,
  ...
)

Arguments

formula	A formula specifying the model structure with the response matrix on the left and partitioning variables on the right (e.g., response_matrix ~ age + gender).
data	A data frame containing the variables in the model.
na.action	How to handle missing values (default: na.omit).
control	A list of control parameters created by grmtree.control().
mtry	Number of variables randomly sampled as candidates at each split. If NULL, all variables are considered.
...	Additional arguments passed to the fitting function.

Details

The algorithm works by first estimating a global GRM for the entire sample, then recursively testing for parameter instability with respect to available covariates. When significant DIF is detected, the sample is partitioned into homogeneous subgroups, each with their own set of item parameters. This approach allows for the identification of complex interaction effects and provides interpretable tree structures that visualize how item functioning varies across different patient subgroups.

GRMTree is particularly useful in health outcomes research where patient-reported outcome measures may function differently across diverse demographic, clinical, or socioeconomic subgroups. The resulting tree diagrams facilitate the development of personalized assessment strategies and can inform targeted interventions by identifying specific patient characteristics associated with differential item interpretation.

Conventional Graded Response Model (GRM)

Let Y_{im} denote the response of the i^{th} (i=1,\ldots,N) individual to the m^{th} (m = 1,2,\ldots,M) item. The graded response model is described as:

P(Y_{im} \geq j | \tau_{mj}, \lambda_m, \theta_i) = \frac{\exp(-(\tau_{mj} - \lambda_m \theta_i))}{1 + \exp(-(\tau_{mj} - \lambda_m \theta_i))}

where:

• P(Y_{im} \geq j | \tau_{mj}, \lambda_m, \theta_i) is the probability that individual i's response is in category j or higher on item m,

• \tau_{mj} is the threshold parameter between categories j-1 and j for item m,

• \lambda_m is the discrimination parameter for item m,

• \theta_i \sim N(0,1) is the latent trait score for individual i.

This parametrization is equivalent to the conventional IRT formulation where item discrimination is a_m = \lambda_m and item difficulty is b_{mj} = \tau_{mj} / \lambda_m.

Graded Response Model Tree (GRMTree) Implementation

The GRMTree is a hybrid model that integrates the GRM with model-based recursive partitioning to detect and account for differential item functioning (DIF) across subgroups defined by covariates. The algorithm proceeds through the following steps:

Step 1: Global Model Estimation

Estimate the GRM item parameters (\hat{\tau}_{mj}, \hat{\lambda}_m) jointly for all individuals in the study cohort at the root node via maximum likelihood estimation:

\hat{\beta}_{\text{global}} = \arg\max_{\beta} \sum_{i=1}^N \log L(\beta; \mathbf{y}_i)

where \beta = (a_1, \ldots, a_J, b_{11}, \ldots, b_{J,m-1}) contains all item parameters (discrimination and difficulty), providing a baseline model assuming parameter invariance.

Step 2: Parameter Stability Testing

For each available covariate X_p (p = 1, \ldots, P), assess the stability of the item parameters by conducting score-based structural change tests. This involves: 1. Calculating the score function contributions s(\hat{\beta}; y_i, x_i) for each individual, 2. Ordering these scores with respect to each covariate X_p, 3. Testing the null hypothesis H_0: \mathbb{E}[s(\hat{\beta}; y_i, x_i)] = 0 for all i against the alternative that scores fluctuate systematically with X_p, indicating parameter instability (DIF).

Step 3: Recursive Partitioning

If significant instability is detected (p < \alpha_{\text{adj}}):

• Covariate Selection: Identify the covariate X_p^* with the most significant instability (smallest adjusted p-value),

• Split Point Determination: Find the optimal cut-point c^* that maximizes the partitioned log-likelihood:

  \ell_{\text{left}}(\beta) + \ell_{\text{right}}(\beta) = \sum_{i: X_{pi} \leq c^*} \log L(\beta; y_i) + \sum_{i: X_{pi} > c^*} \log L(\beta; y_i)

  over all possible cut-points on X_p^*,

• Sample Splitting: Partition the sample into two child nodes based on the rule X_p^* \leq c^*.

Step 4: Recursive Application & Stopping Criteria

Repeat Steps 1-3 recursively within each resulting child node until one of the following stopping criteria is met:

1. No Significant Instability: No covariate shows significant parameter instability after multiple testing correction (\alpha_{\text{adj}} = \alpha/m, where multiple adjustment methods can be applied, including Bonferroni, Holm, Benjamini-Hochberg, etc.).

2. Minimum Node Size: The subsample size falls below a prespecified minimum (e.g., n < 10 \times the number of item parameters).

Formal GRMTree Structure

The final GRMTree provides a piecewise GRM where each terminal node represents a subgroup with homogeneous item parameters, explicitly modeling the detected DIF structure within the data. The resulting GRMTree model can be expressed as a mixture of subgroup-specific GRMs:

P(Y_{ij} = k | \theta_i, \mathbf{x}_i) = \sum_{b=1}^B I(\mathbf{x}_i \in \mathcal{X}_b) \cdot P_b(Y_{ij} = k | \theta_i)

where:

• B is the number of terminal nodes,

• \mathcal{X}_b is the covariate subspace defining terminal node b,

• P_b is the node-specific GRM with parameters \beta_b,

• I(\cdot) is the indicator function.

Each terminal node b contains a complete GRM with:

• Node-specific item parameters: \beta_b = (a_{1b}, \ldots, a_{Jb}, b_{11b}, \ldots, b_{J,m-1,b})

• Local ability distribution: \theta_i | \mathbf{x}_i \in \mathcal{X}_b \sim N(0, 1)

This approach allows differential item functioning (DIF) to be detected and modeled explicitly through the tree structure, where item parameters can vary across subgroups defined by covariates, while maintaining the conditional distribution of the latent trait within each subgroup.

Post-Hoc Multiple Comparison Adjustments

For holm, BH, BY, hochberg, and hommel methods, the algorithm employs a two-stage approach: (1) build a tree using initial_alpha as the splitting threshold, (2) apply global p-value adjustment across all splits, and (3) prune splits that do not meet the adjusted threshold alpha. This differs from Bonferroni correction, which is applied locally during tree construction via partykit::mob. The choice of initial_alpha represents a statistical trade-off. The default (min(3 * alpha, 0.20)) provides a balance between power and computational efficiency. Note that global post-hoc adjustments in hierarchical tree structures may be conservative compared to per-node adjustments, as they account for all tests performed during tree exploration. This global adjustment approach controls the tree-wide error rate and may be conservative (Type I error often <3%). This conservativeness ensures strong control of family-wise error rate or false discovery rate across all splits in the tree. Users requiring less conservative control may prefer the Bonferroni method, which applies per-node adjustment during tree construction.

Value

An object of class grmtree inheriting from modelparty containing the fitted tree structure.

Author(s)

Olayinka Imisioluwa Arimoro [e-mail], Lisa M. Lix, Tolulope T. Sajobi

References

Methodological Foundations

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement, 34, 100-114.

Strobl, C., Kopf, J., & Zeileis, A. (2015). Rasch trees: A new method for detecting differential item functioning in the Rasch model. Psychometrika, 80(2), 289-316.

Komboz, B., Strobl, C., & Zeileis, A. (2018). Tree-based global model tests for polytomous Rasch models. Educational and psychological measurement, 78(1), 128-166. https://doi.org/10.1177/0013164416664394

Arimoro, O. I., Lix, L. M., Patten, S. B., Sawatzky, R., Sebille, V., Liu, J., Wiebe, S., Josephson, C. B., & Sajobi, T. T. (2025). Tree-based latent variable model for assessing differential item functioning in patient-reported outcome measures: a simulation study. Quality of Life Research. https://doi.org/10.1007/s11136-025-04018-6

Applied Examples

Arimoro, O. I., Josephson, C. B., James, M. T., Patten, S. B., Wiebe, S., Lix, L. M., & Sajobi, T. T. (2024). Screening for depression in patients with epilepsy: same questions but different meaning to different patients. Quality of Life Research, 33(12), 3409-3419. https://doi.org/10.1007/s11136-024-03782-1

See Also

print.grmtree prints the detailed summary results of the grmtree object, grmtree.control creates a control object for grmtree, plot.grmtree creates plot for the grmtree object, grmforest for GRM Forests, varimp calculates the variable importance for GRM Forest, fscores_grmtree for computing factor scores, threshpar_grmtree for extracting threshold parameters, discrpar_grmtree for extracting discrimination parameters, itempar_grmtree for extracting item parameters

Examples

  library(grmtree)
  library(hlt)

  # Prepare the asti data (from the hlt package)
  data("asti", package = "hlt")
  asti$resp <- data.matrix(asti[, 1:4])

  # Fit GRM tree with gender and group as partitioning variables
  tree <- grmtree(resp ~ gender + group,
          data = asti,
          control = grmtree.control(minbucket = 30))

  ## Print and plot the tree
  print(tree)
  plot(tree)

  # Extract item parameters for specific subgroups
  discr_params <- discrpar_grmtree(tree)
  threshold_params <- threshpar_grmtree(tree)

grmtree documentation built on Sept. 2, 2026, 1:07 a.m.