nirt_to_Ab {multinomineq}

Nonparametric Item Response Theory (NIRT)

Description

Provides the inequality constraints on choice probabilities implied by nonparametric item response theory (NIRT; Karabatsos, 2001).

Usage

nirt_to_Ab(N, M, options = 2, axioms = c("W1", "W2"))

Arguments

N

number of persons / rows in item-response table

M

number of items / columns in item-response table

options

number of item categories/response options. If options=2, a dichotomous NIRT for product-binomial data is returned.

axioms

which axioms should be included in the polytope representation A*x <= b? See details.

Details

In contrast to parametric IRT models (e.g., the 1-parameter-logistic Rasch model), NIRT does not assume specific parametric shapes of the item-response and person-response functions. Instead, the necessary axioms for a unidimensional representation of the latent trait are tested directly.

The axioms are as follows:

"W1":

Weak row/subject independence: Persons can be ordered on an ordinal scale independent of items.

"W2":

Weak column/item independence: Items can be ordered on an ordinal scale independent of persons

"DC":

Double cancellation: A necessary condition for a joint ordering of (person,item) pairs and an additive representation (i.e., an interval scale).

Note that axioms W1 and W2 jointly define the ISOP model by Scheiblechner (1995; isotonic ordinal probabilistic model) and the double homogeneity model by Mokken (1971). If DC is added, we obtain the ADISOP model by Scheiblechner (1999; ).

References

Karabatsos, G. (2001). The Rasch model, additive conjoint measurement, and new models of probabilistic measurement theory. Journal of Applied Measurement, 2(4), 389–423.

Karabatsos, G., & Sheu, C.-F. (2004). Order-constrained Bayes inference for dichotomous models of unidimensional nonparametric IRT. Applied Psychological Measurement, 28(2), 110-125. doi:10.1177/0146621603260678

Mokken, R. J. (1971). A theory and procedure of scale analysis: With applications in political research (Vol. 1). Berlin: Walter de Gruyter.

Scheiblechner, H. (1995). Isotonic ordinal probabilistic models (ISOP). Psychometrika, 60(2), 281–304. doi:10.1007/BF02301417

Scheiblechner, H. (1999). Additive conjoint isotonic probabilistic models (ADISOP). Psychometrika, 64(3), 295–316. doi:10.1007/BF02294297

Examples

# 5 persons, 3 items
nirt_to_Ab(5, 3)