Latent and Manifest Monotonicity in Item Response Models

Abstract

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)...

Desenvolvimento do subteste de escrita do Teste de Desempenho Escolar II

“…A versão A do subteste de escrita obteve o valor de 0,68, sendo, portanto, satisfató-rio. A monotonicidade dos itens foi avaliada, e todos os itens obtiveram valores acima de 0,46, o que é tido como adequado para a utilização de modelos paramétricos da TRI (Junker & Sijtsma, 2000). Dessa forma, o conjunto de itens da versão A é satisfatório para a realização das análises via TRI.…”

Some Exact Tests for Manifest Properties of Latent Trait Models

“…So when we observe 0 rejections out of 1,000 replications Given the above results, it seems that CSN and MM are rather general properties of multivariate binary data. In fact, by reviewing the theory underlying monotonicity, Junker and Sijtsma (2000) showed that MM holds for the 1PLM. For the 2PLM these authors construct three theoretical counterexamples in which MM fails.…”

Statistical Models for the Development of Psychological and Educational Tests

“…(16.14) Junker and Sijtsma (2000) showed that an MM result as in (] 6.14) is not obtained when R(_j) is rep]aced by X+. MM can be used to estimate the IRF by means of nonparametric regression.…”

“…Second, P[Xj~xjIR(_j)l has been shown not to be monotone in general, thus losing MM (Junker & Sijtsma, 2000). Specifically, a nondecreasing observable curve, P[Xj~xjIR(_j)J, is neither a necessary nor a sufficient condition for Assumption M, but much practical experience with simulated data suggests that such monotone curves tend to be supportive of Assumption M. Software for estimating these curves is available (Molenaar & Sijtsma, 2000;Ramsay, 2000).…”