deriv_nu: Derivatives and Information for Nu

In cogirt: Cognitive Testing Using Item Response Theory

Derivatives and Information for Nu

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 nu (easiness) based on the slope-intercept form of the 1-, 2-, or 3-P item response theory model.

Usage

deriv_nu(
  y = NULL,
  omega = NULL,
  gamma = NULL,
  lambda = NULL,
  zeta = NULL,
  nu = NULL,
  kappa = NULL,
  nu_mu = NULL,
  nu_sigma2 = NULL,
  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.
nu_mu	Mean prior for nu (1 by 1)
nu_sigma2	Covariance prior for nu (1 by 1)
link	Choose between "logit" or "probit" link functions.

Value

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

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

Carlson, J. E. (1987). Multidimensional Item Response Theory Estimation: A computer program (Reprot No. ONR87-2). The American College Testing Program. https://apps.dtic.mil/sti/pdfs/ADA197160.pdf

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.