Analytic Pre-Calibration (APC) Initialization — compute_apc

Computes an initial value for the scaling factor using the closed-form approximation under Gaussian Rasch assumptions.

Usage

compute_apc_init(target_rho, n_items, sigma_beta = 1)

Arguments

target_rho: Numeric. Target reliability.
n_items: Integer. Number of items.
sigma_beta: Numeric. SD of item difficulties (default: 1.0).

Value

Numeric. Initial scaling factor c_init.

Details

Under the Gaussian Rasch setting with $\theta \sim N(0,1)$ and $\beta \sim N(0, \sigma_\beta^2)$, the expected item information involves the logistic-normal convolution: $$\kappa(\sigma^2) = \int \frac{e^z}{(1+e^z)^2} \phi(z; 0, \sigma^2) dz$$

Approximating $\kappa \approx 0.25 / \sqrt{1 + \sigma^2 \pi^2/3}$, the closed-form pre-calibration is: $$c_{init} = \sqrt{\frac{\rho^*}{I \cdot \kappa \cdot (1 - \rho^*)}}$$

Examples

# Compute initial c for target reliability of 0.80 with 25 items
compute_apc_init(target_rho = 0.80, n_items = 25)
#> [1] 1.327405

# With different difficulty spread
compute_apc_init(target_rho = 0.75, n_items = 20, sigma_beta = 1.5)
#> [1] 1.432348