deriv_lambda: Derivatives and Information for Lambda

In cogirt: Cognitive Testing Using Item Response Theory

Derivatives and Information for Lambda

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

Usage

deriv_lambda(
  y = NULL,
  omega = NULL,
  gamma = NULL,
  lambda = NULL,
  zeta = NULL,
  kappa = NULL,
  nu = NULL,
  lambda_mu = NULL,
  lambda_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).
kappa	Item guessing matrix (IJ by 1). Defaults to 0.
nu	Item intercept matrix (IJ by 1).
lambda_mu	Mean prior for lambda (1 by JM)
lambda_sigma2	Covariance prior for lambda (JM by JM)
link	Choose between "logit" or "probit" link functions.

Value

List with elements fpd (1 by JM vector of first partial derivatives for alpha), spd (JM by JM matrix of second partial derivatives for alpha), post_info (JM by JM posterior information matrix for alpha), and fisher_info (JM by JM Fisher information matrix for alpha). 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.