dich_response_deriv: Derivatives and Information for the Dichotomous Response...

In mlthom/CogIRT: Cognitive Testing Using Item Response Theory

Derivatives and Information for the Dichotomous Response Model

Description

This function calculates the matrix of first partial derivatives, the matrix of second partial derivatives, and information matrix for the posterior distribution with respect to omega The formulas are based on Segall (1996; 2009).

Usage

dich_response_deriv(
  y,
  nu,
  lambda,
  kappa = NULL,
  gamma,
  omega,
  zeta,
  omega_mu,
  omega_sigma2,
  zeta_mu,
  zeta_sigma2,
  link = "probit"
)

Arguments

y	Matrix of item responses (K by IJ).
nu	Matrix of item intercept parameters (K by IJ).
lambda	Matrix of item structure parameters (IJ by JM).
kappa	Matrix of item guessing parameters (K by IJ).
gamma	Matrix of experimental structure parameters (JM by MN).
omega	Examinee-level effects of the experimental manipulation (K by MN).
zeta	Condition-level effects of the experimental manipulation (K by JM).
omega_mu	Vector of means prior for omega (1 by MN).
zeta_mu	Vector of means prior for zeta (1 by JM).
link	Choose between logit or probit link functions.
omega_sigma@	Covariance matrix prior for omega (MN by MN).
zeta_sigma@	Covariance matrix prior for zeta (JM by JM).

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.

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

Examples

dich_response_deriv(y = sdirt$y, nu = sdirt$nu, lambda = sdirt$lambda,
                gamma = sdirt$gamma, omega = sdirt$omega, zeta = sdirt$zeta,
                omega_mu = sdirt$omega_mu, omega_sigma2 = sdirt$omega_sigma2,
                zeta_mu = sdirt$zeta_mu, zeta_sigma2 = sdirt$zeta_sigma2,
                link  = "probit")

mlthom/CogIRT documentation built on June 13, 2022, 7:45 a.m.