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Compute Marginal Reliability from Simulated Test Information

Source: R/reliability_utils.R
compute_rho_bar.Rd

Low-level utilities for mapping a discrimination scale \(c\) and simulated person/item parameters to marginal reliability.

These functions implement the same reliability definitions used inside both eqc_calibrate() and spc_calibrate():

  • compute_rho_bar(): MSEM-based marginal reliability \(\bar{w}(c) = \sigma_\theta^2 / (\sigma_\theta^2 + E[1/\mathcal{J}(\theta;c)])\)

  • compute_rho_tilde(): Average-information reliability \(\tilde{\rho}(c) = \sigma_\theta^2 \bar{\mathcal{J}}(c) / (\sigma_\theta^2 \bar{\mathcal{J}}(c) + 1)\)

Usage

compute_rho_bar(c, theta_vec, beta_vec, lambda_base, theta_var = NULL)

compute_rho_tilde(c, theta_vec, beta_vec, lambda_base, theta_var = NULL)

Arguments

c

Numeric scalar. Global discrimination scaling factor.

theta_vec

Numeric vector of abilities \(\theta_m\).

beta_vec

Numeric vector of item difficulties \(\beta_i\).

lambda_base

Numeric vector of baseline discriminations \(\lambda_{i,0}\) (before scaling by c).

theta_var

Optional numeric. Pre-calculated variance of theta. If NULL, computed from theta_vec.

Value

Numeric scalar reliability value.

Details

The computation proceeds as:

  1. Form scaled discriminations \(\lambda_i(c) = c \cdot \lambda_{i,0}\).

  2. Compute item response probabilities \(p_{mi} = \text{logit}^{-1}\{\lambda_i(c)(\theta_m - \beta_i)\}\).

  3. Item information: \(\mathcal{J}_{mi} = \lambda_i(c)^2 p_{mi}(1-p_{mi})\).

  4. Test information at each \(\theta_m\): \(\mathcal{J}_m = \sum_i \mathcal{J}_{mi}\).

  5. Reliability:

    • compute_rho_bar(): harmonic-mean-based MSEM \(\text{MSEM} = E[1/\mathcal{J}_m]\), \(\bar{w}(c) = \sigma_\theta^2 / (\sigma_\theta^2 + \text{MSEM})\).

    • compute_rho_tilde(): arithmetic-mean-based information \(\bar{\mathcal{J}} = E[\mathcal{J}_m]\), \(\tilde{\rho}(c) = \sigma_\theta^2 \bar{\mathcal{J}} / (\sigma_\theta^2 \bar{\mathcal{J}} + 1)\).

A small floor (1e-10) is applied to test information to avoid numerical problems when taking reciprocals.

Examples

# Simple toy example
set.seed(1)
theta <- rnorm(1000)
beta  <- rnorm(20)
lambda0 <- rep(1, 20)

compute_rho_bar(1, theta, beta, lambda0)
#> [1] 0.7795393
compute_rho_tilde(1, theta, beta, lambda0)
#> [1] 0.7870079

# With pre-calculated theta variance (recommended for SPC)
theta_var_fixed <- var(rnorm(10000))  # Pre-calculate from large sample
compute_rho_bar(1, theta, beta, lambda0, theta_var = theta_var_fixed)
#> [1] 0.7684109