deriv_omega: Derivatives and Information for Omega

In cogirt: Cognitive Testing Using Item Response Theory

Derivatives and Information for Omega

Description

This function calculates the matrix of first partial derivatives, the matrix of second partial derivatives, and matrix of posterior and Fisher information for the posterior distribution with respect to omega (ability) based on the slope-intercept form of the 1-, 2-, or 3-parameter item response theory model.

Usage

deriv_omega(
  y = NULL,
  omega = NULL,
  gamma = NULL,
  lambda = NULL,
  zeta = NULL,
  nu = NULL,
  kappa = NULL,
  omega_mu = NULL,
  omega_sigma2 = NULL,
  zeta_mu = NULL,
  zeta_sigma2 = NULL,
  est_zeta = TRUE,
  link = NULL
)

Arguments

y	Item response matrix (K by IJ).
omega	Contrast effects matrix (K by MN).
gamma	Contrast codes matrix (JM by MN).
lambda	Item slope matrix (IJ by JM).
zeta	Specific effects matrix (K by JM).
nu	Item intercept matrix (IJ by 1).
kappa	Item guessing matrix (IJ by 1). Defaults to 0.
omega_mu	Mean prior for omega (1 by MN).
omega_sigma2	Covariance prior for omega (MN by MN).
zeta_mu	Mean prior for zeta (1 by JM).
zeta_sigma2	Covariance prior for zeta (JM by JM).
est_zeta	Logical indicating whether or not to estimate zeta derivatives
link	Choose between "logit" or "probit" link functions.

Value

List with elements fpd (1 by MN vector of first partial derivatives for omega), spd (MN by MN matrix of second partial derivatives for omega), post_info (MN by MN posterior information matrix for omega), and fisher_info (MN by MN Fisher information matrix for omega). Within each of these elements, there are sub-elements for all K examinees.

Dimensions

I = Number of items per condition; J = Number of conditions; K = Number of examinees; M Number of ability (or trait) dimensions; N Number of contrasts (should include intercept).

A Note About Model Notation

The function converts GLLVM notation to the more typical IRT notation used by Segall (1996) for ease of referencing formulas (with the exception of using the slope-intercept form of the item response model).

References

Segall, D. O. (1996). Multidimensional adaptive testing. Psychometrika, 61(2), 331-354. https://doi.org/10.1007/BF02294343

Segall, D. O. (2009). Principles of Multidimensional Adaptive Testing. In W. J. van der Linden & C. A. W. Glas (Eds.), Elements of Adaptive Testing (pp. 57-75). https://doi.org/10.1007/978-0-387-85461-8_3

cogirt documentation built on April 3, 2025, 8:14 p.m.