get_posterior_expected_kl_information: Posterior expected Kullback-Leibler Information

In Karel-Kroeze/ShadowCAT: Multidimensional Computer Adaptive Testing with the Shadow Testing Procedure

Description

Kullback-Leibler divergence based on the EAP and ML estimates of ability under the posterior distribution of theta. Computes the numerical integral of the expectation of KL under the posterior distribution.

Usage

get_posterior_expected_kl_information(estimate, model, answers, administered,
  available, number_dimensions, estimator, alpha, beta, guessing, prior_form,
  prior_parameters, number_itemsteps_per_item,
  eap_estimation_procedure = "riemannsum")

Arguments

estimate	Vector containing current theta estimate, with covariance matrix as an attribute.
model	One of "3PLM", "GPCM", "SM" or "GRM", for the three-parameter logistic, generalized partial credit, sequential or graded response model, respectively.
answers	Vector with answers to administered items.
administered	vector with indices of administered items.
available	Vector with indices of available items.
number_dimensions	Number of dimensions of theta.
estimator	Type of estimator to be used, one of "maximum_likelihood", "maximum_aposteriori", or "expected_aposteriori"; see details.
alpha	Matrix of alpha parameters, one column per dimension, one row per item. Row names should contain the item keys. Note that so called within-dimensional models still use an alpha matrix, they simply have only one non-zero loading per item.
beta	Matrix of beta parameters, one column per item step, one row per item. Row names should contain the item keys. Note that shadowcat expects answer categories to be sequential, and without gaps. That is, the weight parameter in the GPCM model is assumed to be sequential, and equal to the position of the 'location' of the beta parameter in the beta matrix. The matrix should have a number of columns equal to the largest number of item steps over items, items with fewer answer categories should be right-padded with NA. NA values between answer categories are not allowed, and will lead to errors.
guessing	Matrix with one column of guessing parameters per item. Row names should contain the item keys. Optionally used in 3PLM model, ignored for all others.
prior_form	String indicating the form of the prior; one of "normal" or "uniform".
prior_parameters	List containing mu and Sigma of the normal prior: list(mu = ..., Sigma = ...), or the upper and lower bound of the uniform prior: list(lower_bound = ..., upper_bound = ...). The list element Sigma should always be in matrix form. List elements mu, lower_bound, and upper_bound should always be vectors. The length of mu, lower_bound, and upper_bound should be equal to the number of dimensions. For uniform prior in combination with expected aposteriori estimation, true theta should fall within lower_bound and upper_bound and be not too close to one of these bounds, in order to prevent errors.
number_itemsteps_per_item	Vector containing the number of non missing cells per row of the beta matrix.
eap_estimation_procedure	String indicating the estimation procedure if estimator is expected aposteriori and prior form is normal. One of "riemannsum" for integration via Riemannsum or "gauss_hermite_quad" for integration via Gaussian Hermite Quadrature. If prior form is uniform, estimation procedure should always be "riemannsum".

Details

Note that even with a simplified grid, the number of quadrature points which have to be calculated for each available item, at each step in the CAT is taken to the power Q. Use of KL information is likely to be slow in 3+ dimensional tests.

Value

Vector with PEKL information for each yet available item. Kullback Leibler Divergence for given items and pairs of thetas x posterior density. returns vector containing information for each yet available item

References

• Chang, H.-H, & Ying, Z. (1996). A Global Information Approach to Computerized Adaptive Testing. Applied Psychological Measurement, 20(3), 213 - 229. doi: 10.1177/014662169602000303.

• Mulder, J., & van der Linden, W. J. (2010). Multidimentional Adaptive testing with Kullback-Leibler information item selection. In W. J. van der Linden & C. A. W. Glas (Eds.), Elements of adaptive testing (pp. 77 - 101). New York: Springer.

• Wang, C., Chang, H.-H., & Boughton, K. A. (2010). Kullback-Leibler Information and Its Applications in Multi-Dimensional Adaptive Testing. Psychometrika, 76(1), 13 - 39. doi:10.1007/s11336-010-9186-0.

Karel-Kroeze/ShadowCAT documentation built on May 7, 2019, 12:28 p.m.