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Empirical Quadrature Calibration (Algorithm 1: EQC/SQC)

Source: R/eqc_calibrate.R
eqc_calibrate.Rd

eqc_calibrate() implements Algorithm 1 (Empirical / Stochastic Quadrature Calibration, EQC/SQC) for reliability-targeted IRT simulation.

Given a target marginal reliability \(\rho^*\), a latent distribution generator sim_latentG() (for \(G\)) and an item parameter generator sim_item_params() (for \(H\)), the function searches for a global discrimination scale \(c^* > 0\) such that the population reliability \(\rho(c)\) of the Rasch/2PL model is approximately equal to \(\rho^*\).

The key idea is to:

  1. Draw a large fixed "quadrature" sample \(\{\theta_m\}_{m=1}^M \sim G\) and item parameters \(\{(\beta_i, \lambda_{i,0})\}_{i=1}^I \sim H\) once.

  2. For any scale \(c\), form \(\lambda_i(c) = c \cdot \lambda_{i,0}\) and compute the empirical approximation to population reliability \(\hat\rho_M(c)\) from the test information function.

  3. Solve the scalar equation \(\hat\rho_M(c^*) = \rho^*\) using deterministic root-finding (Brent's method via uniroot()).

Usage

eqc_calibrate(
  target_rho,
  n_items,
  model = c("rasch", "2pl"),
  latent_shape = "normal",
  item_source = "irw",
  latent_params = list(),
  item_params = list(),
  reliability_metric = c("msem", "info", "bar", "tilde"),
  M = 10000L,
  c_bounds = c(0.3, 3),
  tol = 1e-04,
  seed = NULL,
  verbose = FALSE
)

Arguments

target_rho

Numeric in (0, 1). Target marginal reliability \(\rho^*\).

n_items

Integer. Number of items in the test form.

model

Character. Measurement model: "rasch" or "2pl". For "rasch", all baseline discriminations are set to 1 before scaling.

latent_shape

Character. Shape argument passed to sim_latentG() (e.g. "normal", "bimodal", "heavy_tail", ...).

item_source

Character. Source argument passed to sim_item_params() (e.g. "irw", "parametric", "hierarchical", "custom").

latent_params

List. Additional arguments passed to sim_latentG().

item_params

List. Additional arguments passed to sim_item_params().

reliability_metric

Character. Reliability definition used inside EQC:

"msem"

MSEM-based marginal reliability (default, theoretically exact).

"info"

Average-information reliability (faster, more stable).

Synonyms: "bar" for "msem", "tilde" for "info".

M

Integer. Size of the empirical quadrature sample (default: 10000).

c_bounds

Numeric length-2 vector. Search bounds for \(c\). Default: c(0.3, 3).

tol

Numeric. Tolerance for uniroot(). Default: 1e-4.

seed

Optional integer for reproducibility.

verbose

Logical. If TRUE, print progress messages.

Value

An object of class "eqc_result" (a list) with elements:

c_star

Calibrated discrimination scale \(c^*\).

target_rho

Target reliability \(\rho^*\).

achieved_rho

Empirical quadrature estimate \(\hat\rho_M(c^*)\).

metric

Reliability metric used.

theta_quad

Length-M vector of quadrature abilities.

theta_var

Sample variance of theta_quad.

items_base

item_params object with scale = 1 (baseline).

items_calib

item_params object with discriminations scaled by c_star.

Details

Reliability Metrics

The function supports two reliability definitions:

  • MSEM-based ("msem"/"bar"): Uses the harmonic mean of test information, \(\bar{w}(c) = \sigma^2_\theta / (\sigma^2_\theta + E[1/\mathcal{J}(\theta;c)])\). This is theoretically exact but may have a lower ceiling for high reliability.

  • Average-information ("info"/"tilde"): Uses the arithmetic mean, \(\tilde{\rho}(c) = \sigma^2_\theta \bar{\mathcal{J}}(c) / (\sigma^2_\theta \bar{\mathcal{J}}(c) + 1)\). By Jensen's inequality, \(\tilde{\rho} \geq \bar{w}\), so this metric typically yields higher reliability values.

WLE vs EAP Reliability Interpretation

When validating with TAM, note that EAP reliability is systematically higher than WLE reliability. This is not a bug but a mathematical property of TAM's definitions. EAP reliability more directly corresponds to the MSEM-based population reliability targeted by EQC. For conservative inference, treat WLE as a lower bound and EAP as an upper bound for true measurement precision.

See also

spc_calibrate for the stochastic approximation alternative, compute_rho_bar and compute_rho_tilde for reliability computation utilities, compute_reliability_tam for TAM validation.

Examples

if (FALSE) { # \dontrun{
# Basic EQC calibration
eqc_result <- eqc_calibrate(
  target_rho = 0.80,
  n_items = 25,
  model = "rasch",
  latent_shape = "normal",
  item_source = "irw",
  seed = 42,
  verbose = TRUE
)
print(eqc_result)
} # }