get_fisher_information: Fisher Information

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

Description

Fisher Information (expected information) per item.

Usage

get_fisher_information(estimate, model, number_dimensions, alpha, beta,
  guessing, number_itemsteps_per_item)

Arguments

estimate	Vector with current estimate of latent trait theta. Length should be equal to the number of dimensions.
model	One of "3PLM", "GPCM", "SM" or "GRM", for the three-parameter logistic, generalized partial credit, sequential or graded response model, respectively.
number_dimensions	Number of dimensions of theta.
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.
number_itemsteps_per_item	Vector containing the number of non missing cells per row of the beta matrix.

Details

Fisher Information is given as:

\mathcal{I}(θ) = - {E} ≤ft[≤ft. \frac{\partial^2}{\partialθ^2} \log f(X;θ)\right|θ \right]

(minus expectation of second derivative of the Log-Likelihood of f(theta)) and is calculated as the weighted sum of second derivatives for all response categories. Information for multiple items is simply the sum of the individual information matrices.

Note: get_fisher_information always returns the 'raw' information; information given by prior distributions is added by the calling functions.

Value

Three dimensional array of information matrices, where dimensions one and two run along the Q dimensions of the model, and three runs along items.

References

• Muraki, E. (1992). A Generized Partial Credit Model: Application of an EM Algorithm. Applied Psychological Measurement, 16(2), 159 - 176. Doi:10.1177/014662169201600206.

• Samejima, F. (1970). Estimation of latent trait ability using a response pattern of graded scores. Psychometrika, 35(1), 139 - 139. Doi: 10.1007/BF02290599.

• Segall, D. O. (2000). Principles of multidimensional adaptive testing. In W. J. van der Linden & en C. A. W. Glas (Eds.), Computerized adaptive testing: Theory and practice (pp. 53 - 74). Dordrecht: Kluwer Academic Publishers.

• Tutz, G. (1986). Bradley-Terry-Luce model with an ordered response. Journal of Mathematical Psychology, 30(1), 306 - 316. doi: 10.1016/0022-2496(86)90034-9.

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