Simulate Item Parameters for IRT Studies

sim_item_params() generates item parameters (difficulty \(\beta\) and discrimination \(\lambda\)) for Item Response Theory (IRT) simulation studies. It wraps the IRW irw_simu_diff() function for realistic difficulty distributions and provides multiple methods for generating correlated discriminations.

The function is designed with four key principles:

1. Realistic difficulties: Integration with Item Response Warehouse (IRW) for empirically-grounded difficulty distributions.

2. Correlated parameters: Support for the empirically observed negative correlation between difficulty and discrimination (Sweeney et al., 2022).

3. Marginal preservation: Copula method preserves exact marginal distributions while achieving target correlation.

4. Reliability targeting: Scale factor for subsequent calibration.

Usage

sim_item_params(
  n_items,
  model = c("rasch", "2pl"),
  source = c("irw", "parametric", "hierarchical", "custom"),
  method = c("copula", "conditional", "independent"),
  n_forms = 1L,
  difficulty_params = list(),
  discrimination_params = list(),
  hierarchical_params = list(),
  custom_params = list(),
  scale = 1,
  center_difficulties = TRUE,
  seed = NULL
)

Arguments

n_items

Integer. Number of items to generate per form.

model

Character. The data-generating model: "rasch" or "2pl".

source

Character. Source for generating difficulties:

"irw"

Use IRW difficulty pool (realistic, empirical)

"parametric"

Generate from parametric distribution

"hierarchical"

Joint MVN for both parameters (Glas & van der Linden)

"custom"

User-supplied parameters or function

method

Character. Method for generating discriminations (when model = "2pl"):

"copula"

Gaussian copula - preserves marginals exactly (RECOMMENDED)

"conditional"

Conditional normal regression on difficulty

"independent"

Independent generation (no correlation)

n_forms

Integer. Number of test forms to generate. Default is 1. When > 1, returns a data frame with form_id column.

difficulty_params

List. Parameters for difficulty generation:

For source = "irw":

pool - difficulty pool data frame

For source = "parametric":

mu, sigma, distribution

discrimination_params

List. Parameters for discrimination generation:

mu_log

Mean of log-discrimination (default: 0)

sigma_log

SD of log-discrimination (default: 0.3)

rho

Target correlation between \(\beta\) and \(\log(\lambda)\) (default: -0.3)

hierarchical_params

List. For source = "hierarchical":

mu

2-vector: means of \((\log\lambda, \beta)\)

tau

2-vector: SDs

rho

Correlation

custom_params

List. For source = "custom":

beta

Vector or function returning difficulties

lambda

Vector or function returning discriminations

scale

Numeric. Global discrimination scaling factor for reliability targeting. Final discriminations are \(\lambda_i^* = c \cdot \lambda_i\). Default is 1.

center_difficulties

Logical. If TRUE, center difficulties to sum to zero for identification. Default is TRUE.

seed

Integer. Random seed for reproducibility.

Value

An object of class "item_params" containing:

data

Data frame with columns: form_id, item_id, beta, lambda, lambda_unscaled

model

Model type used

source

Source used for generation

method

Method used for discrimination generation

n_items

Number of items per form

n_forms

Number of forms generated

scale

Scale factor applied

centered

Whether difficulties were centered

params

Parameters used for generation

achieved

Achieved statistics (correlations, moments)

Details

Why Copula Method is Recommended

When difficulties come from the IRW pool (which has realistic, often non-normal marginal distributions), the conditional normal method can distort the achieved correlation because it assumes linearity. The Gaussian copula method:

1. Transforms difficulties to uniform scale via empirical CDF

2. Generates correlated uniforms through Gaussian copula

3. Transforms back to desired marginals (log-normal for discrimination)

This guarantees:

• Exact preservation of difficulty marginal (whatever IRW provides)

• Exact log-normal marginal for discriminations

• Spearman correlation \(\approx \rho\) (rank-based, robust to non-normality

Connection to Reliability-Targeted Framework

The scale parameter implements "separation of structure and scale":

• Structure: Realistic item characteristics from IRW + correlation

• Scale: Global factor \(c\) calibrated for target reliability

References

Glas, C. A. W., & van der Linden, W. J. (2003). Computerized adaptive testing with item cloning. Applied Psychological Measurement, 27(4), 247-261.

Sweeney, S. M., et al. (2022). An investigation of the nature and consequence of the relationship between IRT difficulty and discrimination. EM:IP, 41(4), 50-67.

Zhang, L., et al. (2025). Realistic simulation of item difficulties. PsyArXiv.

Examples

# Example 1: Rasch with IRW difficulties
items1 <- sim_item_params(n_items = 25, model = "rasch", source = "irw")

# Example 2: 2PL with copula method (recommended)
items2 <- sim_item_params(
  n_items = 30, model = "2pl", source = "irw",
  method = "copula",
  discrimination_params = list(rho = -0.3)
)

# Example 3: Multiple forms
items3 <- sim_item_params(
  n_items = 20, model = "2pl", n_forms = 5,
  source = "irw", method = "copula"
)
#> Warning: collapsing to unique 'x' values

# Example 4: Hierarchical 2PL
items4 <- sim_item_params(
  n_items = 25, model = "2pl", source = "hierarchical",
  hierarchical_params = list(mu = c(0, 0), tau = c(0.25, 1), rho = -0.3)
)