Optimal item difficulty for the three-parameter normal ogive response model

In tailored testing, it is important to determine the optimal difficulty of the next item to present to the examinee. This paper shows that the difference that maximizes information for the three-parameter normal ogive response model is approximately 1.7 times the optimal difference –b for the three-parameter logistic model. Under the normal model, calculation of the optimal difficulty for minimizing the Bayes risk is equivalent to maximizing an associated information function.The views expressed herein, are those of the author and do not necessarily reflect those of the Department of the Navy.     

Comparing score tests and other local dependence diagnostics for the graded response model

Score tests for identifying locally dependent item pairs have been proposed for binary item response models. In this article, both the bifactor and the threshold shift score tests are generalized to the graded response model. For the bifactor test, the generalization is straightforward; it adds one secondary dimension associated only with one pair of items. For the threshold shift test, however, multiple generalizations are possible: in particular, conditional, uniform, and linear shift tests are discussed in this article. Simulation studies show that all of the score tests have accurate Type I error rates given large enough samples, although their small‐sample behaviour is not as good as that of Pearson's Χ² and M₂ as proposed in other studies for the purpose of local dependence (LD) detection. All score tests have the highest power to detect the LD which is consistent with their parametric form, and in this case they are uniformly more powerful than Χ² and M₂; even wrongly specified score tests are more powerful than Χ² and M₂ in most conditions. An example using empirical data is provided for illustration.     

Conditional covariance structure of generalized compensatory multidimensional items

Some nonparametric dimensionality assessment procedures, such as DIMTEST and DETECT, use nonparametric estimates of item pair conditional covariances given an appropriately chosen subtest score as their basic building blocks. Such conditional covariances given some subtest score can be regarded as an approximation to the conditional covariances given an appropriately chosen unidimensional latent composite, where the composite is oriented in the multidimensional test space direction in which the subtest score measures best. In this paper, the structure and properties of such item pair conditional covariances given a unidimensional latent composite are thoroughly investigated, assuming a semiparametric IRT modeling framework called a generalized compensatory model. It is shown that such conditional covariances are highly informative about the multidimensionality structure of a test. The theory developed here is very useful in establishing properties of dimensionality assessment procedures, current and yet to be developed, that are based upon estimating such conditional covariances.In particular, the new theory is used to justify the DIMTEST procedure. Because of the importance of conditional covariance estimation, a new bias reducing approach is presented. A byproduct of likely independent importance beyond the study of conditional covariances is a rigorous score information based definition of an item's and a score's direction of best measurement in the multidimensional test space.This paper is based on a chapter of the first author's doctoral dissertation, written at the University of Illinois and supervised by the second author. Part of this research has been presented at the annual meeting of the National Council on Measurement in Education, San Francisco, April 1995.The authors would like to thank Jeff Douglas, Xuming He and Ming-mei Wang for their comments and suggestions. The research of the first author was partially supported by an ETS/GREB Psychometric Fellowship, and by Educational Testing Service Research Allocation Project 884-01. The research of the second author was partially supported by NSF grant DMS 97-04474.     

使用题组反应模型缓解局部题目依赖性对多阶段测验的危害

尽管多阶段测验(MST)在保持自适应测验优点的同时允许测验编制者按照一定的约束条件去建构每一个模块和题板,但建构测验时若因忽视某些潜在的因素而导致题目之间出现局部题目依赖性(LID)时,也会对MST测验结果带来一定的危害。为探究"LID对MST的危害"这一问题,本研究首先介绍了MST和LID等相关概念;然后通过模拟研究比较探讨该问题,结果表明LID的存在会影响被试能力估计的精度但仍为估计偏差较小,且该危害不限于某一特定的路由规则;之后为消除该危害,使用了题组反应模型作为MST施测过程中的分析模型,结果表明尽管该方法能够消除部分危害但效果有限。这一方面表明LID对MST中被试能力估计精度所带来的危害确实值得关注,另一方面也表明在今后关于如何消除MST中由LID造成危害的方法仍值得进一步探究的。     

Essential independence and likelihood-based ability estimation for polytomous items

A definition ofessential independence is proposed for sequences of polytomous items. For items satisfying the reasonable assumption that the expected amount of credit awarded increases with examinee ability, we develop a theory ofessential unidimensionality which closely parallels that of Stout. Essentially unidimensional item sequences can be shown to have a unique (up to change-of-scale) dominant underlying trait, which can be consistently estimated by a monotone transformation of the sum of the item scores. In more general polytomous-response latent trait models (with or without ordered responses), anM-estimator based upon maximum likelihood may be shown to be consistent for under essentially unidimensional violations of local independence and a variety of monotonicity/identifiability conditions. A rigorous proof of this fact is given, and the standard error of the estimator is explored. These results suggest that ability estimation methods that rely on the summation form of the log likelihood under local independence should generally be robust under essential independence, but standard errors may vary greatly from what is usually expected, depending on the degree of departure from local independence. An index of departure from local independence is also proposed.This work was supported in part by Office of Naval Research Grant N00014-87-K-0277 and National Science Foundation Grant NSF-DMS-88-02556. The author is grateful to William F. Stout for many helpful comments, and to an anonymous reviewer for raising the questions addressed in section 2. A preliminary version of section 6 appeared in the author's Ph.D. thesis.     

The equivalence of two methods of parameter estimation for the rasch model

Two methods of estimating parameters in the Rasch model are compared. It is shown that estimates for a certain loglinear model for the score × item × response table are equivalent to the unconditional maximum likelihood estimates for the Rasch model.     

On the explaining‐away phenomenon in multivariate latent variable models

Many probabilistic models for psychological and educational measurements contain latent variables. Well‐known examples are factor analysis, item response theory, and latent class model families. We discuss what is referred to as the ‘explaining‐away’ phenomenon in the context of such latent variable models. This phenomenon can occur when multiple latent variables are related to the same observed variable, and can elicit seemingly counterintuitive conditional dependencies between latent variables given observed variables. We illustrate the implications of explaining away for a number of well‐known latent variable models by using both theoretical and real data examples.     

A nonparametric approach for assessing latent trait unidimensionality

Assuming a nonparametric family of item response theory models, a theory-based procedure for testing the hypothesis of unidimensionality of the latent space is proposed. The asymptotic distribution of the test statistic is derived assuming unidimensionality, thereby establishing an asymptotically valid statistical test of the unidimensionality of the latent trait. Based upon a new notion of dimensionality, the test is shown to have asymptotic power 1. A 6300 trial Monte Carlo study using published item parameter estimates of widely used standardized tests indicates conservative adherence to the nominal level of significance and statistical power averaging 81 out of 100 rejections for examinee sample sizes and psychological test lengths often incurred in practice.The referees' comments were remarkably detailed and greatly enhanced the writeup and sensitized the author to certain pertinent issues. Discussions with Fritz Drasgow, Lloyd Humphreys, Dennis Jennings, Brian Junker, Robert Linn, Ratna Nandakumar, and Robin Shealy were also very useful.This research was supported by the Office of Naval Research under grant N00014-84-K-0186; NR 150-533, and by the National Science Foundation under grant DMS 85-03321.     

Optimal sequential designs for on-line item estimation

Replenishing item pools for on-line ability testing requires innovative and efficient data collection designs. By generating localD-optimal designs for selecting individual examinees, and consistently estimating item parameters in the presence of error in the design points, sequential procedures are efficient for on-line item calibration. The estimating error in the on-line ability values is accounted for with an item parameter estimate studied by Stefanski and Carroll. LocallyD-optimaln-point designs are derived using the branch-and-bound algorithm of Welch. In simulations, the overall sequential designs appear to be considerably more efficient than random seeding of items.This report was prepared under the Navy Manpower, Personnel, and Training R&D Program of the Office of the Chief of Naval Research under Contract N00014-87-0696. The authors wish to acknowledge the valuable advice and consultation given by Ronald Armstrong, Charles Davis, Bradford Sympson, Zhaobo Wang, Ing-Long Wu and three anonymous reviewers.