# Getting Started with mstATA: A Complete Workflow

#### Hong

#### 2026-03-18

## Introduction

The `mstATA` package provides a comprehensive framework for automatic multistage test (MST) assembly using mixed-integer linear programming (MILP).

Specifically, `mstATA` supports

- flexible specification of constraints across multiple hierarchical levels, including module-, pathway-, panel-, and solution-level;

- partial selection of items associated with the same stimulus, allowing item-stimulus dependencies to be modeled without forcing all-in/all-out inclusion;

- flexible formulation of single or multiple objectives, including absolute and relative objectives;

- simultaneous assembly of multiple parallel MST panels with global control over exposure and content balancing;

- diagnostic feedback and reformulation tools for situations in which the optimization problem becomes infeasible or reaches time limit.

In addition to MST panel assembly, `mstATA` incorporates analytic methods for evaluating MST performance, such as measurement precision and classification accuracy. Collectively, these capabilities allow `mstATA` to unify a broad class of MST assembly and evaluation problems within a single modeling framework.

## Workflow Overview

Step | Description
--- | ---
Prepare item pool | Check attributes in the item pool.
Specify MST structure | Create mstATA_design object.
Identify hierarchical requirements | Understand test specifications.
Translate specifications to linear model | Create mstATA_constraint and mstATA_objective objects.
Execute assembly via solver | Create mstATA_model and mstATA_panel objects.
Diagnose infeasible models | Check the feasibility of individual and combined specifications.
Evaluate assembled panels | Produce tables, plots, and analytical results.

### Step 1: Prepare the Item Pool

Before assembly, the item pool must contain all required attributes (categorical, quantitative, logical) at item-, stimulus-, itemset- levels.

#### IRT Computation

- `compute_icc()`: item category probabilities
- `compute_iif()` : item information function
- `Pi_internal()`: helper function for `compute_icc()` and `compute_iif()`
- `plot_tif()`: item pool information preview

#### Attribute Check

- `get_attribute_val()`: extract categorical/quantitative attribute values
- `create_enemy_sets()`: create enemy item sets/enemy stimulus sets
- `concat_enemy_sets()`: combine enemy item sets/enemy stimulus sets from different sources (e.g., similarity, cluing, or other test security concern)
- `create_pivot_stimulus_map()`: create item-stimulus mapping

### Step 2: Specify the MST Structure

An MST design is defined using `mst_design()`. The `mstATA_design` object defines:

- MST design: number of stages, number of modules per stage
- Module/pathway lengths
- Routing structure: excluded pathways, routing decision points
- Decision variables for item-module selection in a panel

This step creates `mstATA_design` object, which is the required input for the subsequent process.

### Step 3: Identify Hierarchical Requirements

In addition to the type of specification—categorical, quantitative, or logical—each specification has two distinct dimensions: a defined level, which indicates the unit to which the requirement conceptually applies (item, stimulus, item set, module, pathway, panel, or solution), and an enforced level, which specifies the scope over which the requirement is operationally imposed (module, pathway, panel, or solution).

For requirements defined at the item, stimulus, or item-set levels, the enforcement level must be specified at a higher aggregation level (module, pathway, panel, or solution). The reason is that items, stimuli, and item sets are selection units, but they do not constitute independent test forms by themselves; they are selected in specific modules/pathways/a panel.

- Whether an item (or stimulus) “must be selected” is therefore meaningful only after specifying where it must be selected—e.g., in a particular module, somewhere within a pathway, anywhere in the panel, or across panels.
- Item-set exclusion requirements must be enforced at the pathway-level to ensure that no examinee can see enemy item pairs in a pathway.
- Item-set conditional inclusion requirements must be enforced at the module-level to ensure that items linked to a selected stimulus are administered in the same module.
- Item reusage requirements must be enforced at the panel-level (either strict item uniqueness within the entire panel or item reuse allowed within stage, but not within pathway) or the solution-level (the minimum or/and maximum number of times an item is selected across multiple panels.)

In other words, low-level requirements require an explicit enforcement scope to determine which subset of decision variables is constrained. In contrast, for requirements defined at the module, pathway, panel, or solution levels, the enforcement level is typically the same as the defined level.

### Step 4: Translate specifications into an optimization model

This step converts specifications into a MILP model.

#### Objectives

Objective terms are created using `objective_term()` and compiled into a `compiled_objective` object.

Available objective strategies:

- `single_obj`: minimize or maximize a single objective.
- `weighted_obj()`: minimize or maximize the weighted sum of multiple objectives.
- `maximin_obj()`: maximize a common minimum value across multiple objectives,, optionally applying a fixed penalty to any amount that exceeds this bound.
- `capped_maximin_obj()`: maximize a common minimum value across multiple objectives, penalized by the tolerance for any overflow.
- `minimax_obj()`: minimize a common maximum deviation from multiple goals.
- `goal_programming_obj()`: minimize the (weighted or unweighted) sum of deviations from multiple goals.

#### Constraints

Constraints are created as `mstATA_constraint` objects.

Structural constraints

- `mst_structure_con()`: a higher-level wrapper that jointly constructs module/pathway length constraints `test_itemcount_con()` and routing decision point constraints `test_rdp_con()`.

- `dvlink_item_solution()`: an internal function (not intended for direct user calls) invoked by `multipanel_spec()` to define solution-level item indicator variables and generate constraints for item exposure control across multiple panels.

Item-level constraints:

- `itemcat_con()`: constrain an item must or must not be selected.
- `itemquant_con()`: constrain quantitative attribute for an item to be selected.

Stimulus-level constraints:

- `stimcat_con()`: constrain a stimulus must or must not be selected.
- `stimquant_con()`: constrain quantitative attribute for a stimulus to be selected.

Itemset-level constraints:

- `enemyitem_exclu_con()`: exclude the enemy item pair appearing in the same pathway.
- `enemystim_exclu_con()`: exclude the enemy stimulus pair appearing in the same pathway.
- `stim_itemcount_con()`: constrain the min/exact/max number of items selected conditional on the selection of a stimulus.
- `stim_itemcat_con()`: constrain the min/exact/max number of items selected from category c conditional on the selection of a stimulus.
- `stim_itemquant_con()`: constrain the min/exact/max values for the sum of item quantitative attribute values within a selected stimulus.

Module-/Pathway-level constraints:

- `test_itemcat_con()`, `test_itemcat_range_con()`: constrain the min/equal/max/range number of items from specific categories in a module or pathway.
- `test_itemquant_con()`, `test_itemquant_range_con()`: constrain the min/equal/max/range for the sum of the item quantitative attribute in a module or pathway.
- `test_stimcount_con()`: constrain the min/equal/max number of stimuli in a module or a pathway.
- `test_stimcat_con()`: constrain the min/equal/max number of stimuli from specific categories in a module or pathway.
- `test_stimquant_con()`: constrain the min/equal/max for the sum of the stimulus quantitative attribute in a module or pathway.

Panel-level constraints:

- `panel_itemreuse_con()`: constrain item exposure within a panel.
- `panel_itemcat_con()`: constrain the min/equal/max number of items from specific categories within a panel.
- `panel_stimcat_con()`: constrain the min/equal/max number of stimuli from specific categories within a panel.

Solution-level constraints:

- `solution_itemcount_con()`: constraint the min/equal/max number of unique items across multiple panels.
- `solution_itemcat_con()`: constrain the min/equal/max number of unique items from specific categories across multiple panels.
- `solution_stimcount_con()`: constraint the min/equal/max number of unique stimuli across multiple panels.
- `solution_stimcat_con()`: constrain the min/equal/max number of unique stimuli from specific categories across multiple panels.

### Step 5: Execute Assembly via a Solver

After objectives and constraints are compiled, create `mstATA_model` via `onepanel_spec()` or `multipanel_spec()`.

- `solve_model()`: supported solvers include `gurobi`, `HiGHS`, `Symphony`, `GLPK` and `lpsolve`.
- `assembled_panel()`: organizes the solver results by panel, module, and pathway.

### Step 6: Diagnosing Infeasibility

- `check_singblock_feasibility()`: tests the feasibility of an individual specification by solving a reduced model that contains the core constraints plus exactly one additional constraint block.
- `check_comblock_feasibility()`: tests the feasibility of multiple specifications by solving a reduced model that contains the core constraints plus user-specified combination of constraint blocks.

Recover feasibility for an infeasible `mstATA_model`

- `solve_with_slack()`: attempt to recover feasibility for an infeasible `mstATA_model`.

### Step 7: Evaluate assembled panels

After assembly, panels can be evaluated from multiple perspectives.

Content and structural reports:

- `report_test_itemcat()`: summarizes the number of selected items belonging to specified categorical attribute levels in specific modules/pathways.
- `report_test_itemquant()`: summarizes quantitative item attributes (e.g., test information, time, difficulty) in specific modules/pathways.
- `report_test_tcc()`: computes and reports test characteristic curves (TCC) aggregated at the module or pathway level for assembled MST panels.
- `report_test_tif()`: computes and reports test information functions (TIF) aggregated at the module or pathway level for assembled MST panels.

Psychometric evaluation:

- `plot_panel_tcc()`: plots test characteristic curves aggregated at the module or pathway level, supports both single-panel and multi-panel visualization.
- `plot_panel_tif()`: plots test information functions aggregated at the module or pathway level, supports both single-panel and multi-panel visualization.

Precision and classification:

- `analytic_mst_precision()`: computes conditional bias and conditional standard error of measurement for MST panels using a recursion-based evaluation method (Lim, 2000).
- `analytic_mst_classification()`: computes classification evaluation criterion using Rudner’s method (2000, 2005).

Supporting utilities:

- `expected_score()`: computes the expected total score implied by item category probability curves.
- `inverse_tcc()`: computes ITCC estimates from test characteristic curve.
- `module_score_dist()`: computes the conditional distribution of total module scores given a/multiple ability values.
- `joint_module_score_dist()`: computes the joint (cumulative) score distribution after administering a next-stage module, conditional on routing into a branch defined by a set of reachable previous scores.

## Summary

`mstATA` provides a unified framework for hierarchical constraint modeling, multi-strategy objective optimization, multi-panel MST assembly, and psychometric evaluation.

- Each specification corresponds to a clearly named function

- Each function produces explicit linear constraints

- All constraints can be inspected prior to solving

- Infeasibility diagnostics are available

- Bonus: An item–module eligibility set can be specified to filter the active binary decision variables. In addition, explicitly providing objective values can further improve the computational efficiency of the mixed-integer linear programming formulation.

References

van der Linden, W. J. (2005). Linear models for optimal test design. Springer. https://doi.org/10.1007/0-387-29054-0

van der Linden, W. J. (2000). Optimal assembly of tests with item sets. Applied Psychological Measurement, 24(3), 225–240. https://doi.org/10.1177/01466210022031697

van der Linden, W. J., & Boekkooi-Timminga, E. (1989). A maximin model for test design with practical constraints. Psychometrika, 54(2), 237–247. https://doi.org/10.1007/BF02294518

Lim, H., Davey, T., & Wells, C. S. (2020).
A recursion-based analytical approach to evaluate the performance of MST.
Journal of Educational Measurement, 58(2), 154–178.
https://doi.org/10.1111/jedm.12276

Rudner, L. M. (2000). Computing the expected proportions of misclassified examinees. Practical Assessment, Research, and Evaluation, 7(1). https://doi.org/10.7275/an9m-2035

Rudner, L. M. (2005). Expected classification accuracy. Practical Assessment, Research, and Evaluation, 10(1). https://doi.org/10.7275/56a5-6b14