Was There One Distractor Too Many?

Wainer, Howard; Wadkins, J. R. Jefferson; Rogers, Alfred N.

doi:10.1002/j.2330-8516.1983.tb00034.x

Cited by 1 publication

(2 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Not only are they guaranteed to be nondecreasing when all the IRFs are nondecreasing, but they will also be sensitive to some violations of the model in P 1 (θ ) if all items except X 1 are known to satisfy the UMIRT model. This sensitivity provides some justification for procedures such as those of Wainer (1983) and Wainer, Wadkins & Rogers (1984), who studied the probabilities of distractors of a defective item conditional on number correct for the remaining items.…”

Section: Dichotomous Itemsmentioning

confidence: 99%

“…The item-total regression has sometimes been replaced with the item-rest regression in studies of IRF shape and model fit. For example, Lord (1965) examined item-rest regressions to determine IRF shapes, and Wainer (1983) and Wainer, Wadkins, & Rogers (1984) used item-rest regressions to explore methods of identifying incorrectly keyed items.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Latent and Manifest Monotonicity in Item Response Models

Junker

Sijtsma

2000

Applied Psychological Measurement

104

View full text Add to dashboard Cite

The monotonicity of item response functions (IRF) is a central feature of most parametric and nonparametric item response models. Monotonicity allows items to be interpreted as measuring a trait, and it allows for a general theory of nonparametric inference for traits. This theory is based on monotone likelihood ratio and stochastic ordering properties. Thus, confirming the monotonicity assumption is essential to applications of nonparametric item response models. The results of two methods of evaluating monotonicity are presented: regressing individual item scores on the total test score and on the "rest" score, which is obtained by omitting the selected item from the total test score. It was found that the item-total regressions of some familiar dichotomous item response models with monotone IRFs exhibited nonmonotonicities that persist as the test length increased. However, item-rest regressions never exhibited nonmonotonicities under the nonparametric monotone unidimensional item response model. The implications of these results for exploratory analysis of dichotomous item response data and the application of these results to polytomous item response data are discussed. Index terms: elementary symmetric functions, essential unidimensionality, latent monotonicity, manifest monotonicity, monotone homogeneity, nonparametric item response models, strict unidimensionality.Most item response theory (IRT) models for dichotomous item scores (X 1 , X 2 , . . . , X J , taking values in {0,1}) assume that the probability of correctly responding to an item given a latent trait θ (P j (θ) = P [X j = 1|θ ]) is a monotonic, nondecreasing function of θ. Moreover, Hemker, Sijtsma, Molenaar, & Junker (1997) showed that for all graded response and partial-credit IRT models for polytomous items, the item step response functions (ISRFs) P * js (θ ) = P [X j > s|θ] are also nondecreasing in θ for each j and s, where s is an integer item score on a polytomous item.Monotonicity plays a central role in most nonparametric and parametric formulations of IRT because it captures the intuitive idea that the items measure θ; higher θs indicate a higher probability of answering an item correctly. The general nonparametric model discussed here has been studied before under many different names (e.g.van der Linden & Hambleton, 1997); here it is referred to as the nonparametric unidimensional monotone item response theory (UMIRT) model. It is defined aswhere x 1 , x 2 , . . . , x J are the observed values of the item response variables X 1 , X 2 , . . . , X J , P jx (θ) = P [X j = x|θ ] are the associated item category response functions, and dF(θ) is an arbitrary distribution function.The UMIRT model assumes unidimensionality (θ ∈ R), local independence (LI; P [X 1 = x 1 , x 2 , . . . ,x J = X J |θ ] = J j =1 P jx j (θ), and monotonicity. For the latter, the ISRFs P * js (θ ) = m j x=(s+1) P jx (θ ) are assumed to be nondecreasing in θ for each j and s. Throughout most of this paper X j ∈ {0, 1}, so that P * j 0 (θ) = P [X j = 1|θ ] = P j (θ)...

show abstract

Section: Dichotomous Itemsmentioning

confidence: 99%