Comparing item characteristic curves

Test items are often evaluated and compared by contrasting the shapes of their item characteristics curves (ICC's) or surfaces. The current paper develops and applies three general (i.e., nonparametric) comparisons of the shapes of two item characteristic surfaces: (i) proportional latent odds, (ii) uniform relative difficulty, and (iii) item sensitivity. Two items may be compared in these ways while making no assumption about the shapes of item characteristic surfaces for other items, and no assumption about the dimensionality of the latent variable. Also studied is a method for comparing the relative shapes of two item characteristic curves in two examinee populations.The author is grateful to Paul Holland, Robert Mislevy, Tue Tjur, Rebecca Zwick, the editor and reviewers for valuable comments on the subject of this paper, to Mari A. Pearlman for advice on the pairing of items in the examples, and to Dorothy Thayer for assistance with computing.     

Items bundles

An item bundle is a small group of multiple choice items that share a common reading passage or graph, or a small group of matching items that share distractors. Item bundles are easily identified by paging through a copy of a test. Bundled items may violate the latent conditional independence assumption of unidimensional item response theory (IRT), but such a violation would not typically suggest the existence of a new fundamental human ability to read one specific reading passage or to interpret one specific graph. It is important, therefore, to have theoretical concepts and empirical checks that distinguish between, on the one hand, anticipated violations of latent conditional independence within item bundles, and, on the other hand, violations that cannot be attributed to idiosyncratic features of test format and instead suggest departures from unidimensionalty. To this end, two theorems on unidimensional IRT are extended to describe observable item response distributions when there is conditional independencebetween but not necessarilywithin item bundles.The author is grateful to Ivo Molenaar and the referees for many helpful suggestions, and to D. Thayer for assistance with computing.     

Does adaptive testing violate local independence?

Item response theory posits local independence, or conditional independence of item responses given item parameters and examinee proficiency parameters. The usual definition of local independence, however, addresses the context of fixed tests, and initially appears to yield incorrect response-pattern probabilities in the context of adaptive testing. The paradox is resolved by introducing additional notation to deal with the item selection mechanism.We are grateful to Charlie Lewis, Ming-Mei Wang, and Pao-Kuei Wu for discussions on this topic, and to the Editor, the reviewers, and Howard Wainer for helpful comments on an earlier version of the paper. The first author's work was supported in part by the National Center for Research on Evaluation, Standards, Student Testing (CRESST), Educational Research and Development Program, cooperative agreement number R117G10027 and CFDA catalog number 84.117G, as administered by the Office of Educational Research and Improvement, U.S. Department of Education.     

Psychometrics in Psychological Research: Role Model or Partner in Science?

This is a reaction to Borsboom's (2006) discussion paper on the issue that psychology takes so little notice of the modern developments in psychometrics, in particular, latent variable methods. Contrary to Borsboom, it is argued that latent variables are summaries of interesting data properties, that construct validation should involve studying nomological networks, that psychological research slowly but definitely will incorporate latent variable methods, and that the role of psychometrics in psychology is that of partner, not role model. Requests for reprints should be sent to Klaas Sijtsma, Department of Methodology and Statistics, FSW, Tilburg University, PO Box 90153, 5000 LE, Tilburg, The Netherlands.     

A general latent trait model for response processes

The purpose of the current paper is to propose a general multicomponent latent trait model (GLTM) for response processes. The proposed model combines the linear logistic latent trait (LLTM) with the multicomponent latent trait model (MLTM). As with both LLTM and MLTM, the general multicomponent latent trait model can be used to (1) test hypotheses about the theoretical variables that underlie response difficulty and (2) estimate parameters that describe test items by basic substantive properties. However, GLTM contains both component outcomes and complexity factors in a single model and may be applied to data that neither LLTM nor MLTM can handle. Joint maximum likelihood estimators are presented for the parameters of GLTM and an application to cognitive test items is described.This research was partially supported by the National Institute of Education grant number NIE-6-7-0156 to Susan Embretson (Whitely), principal investigator. However the optinions expressed herein do not necessarily reflect the position or policy of the National Institute of Education, and no official endorsement by the National Institute of Education should be inferred.     

An empirical look at the Defense Mechanism Test (DMT): reliability and construct validity

Although the Defense Mechanism Test (DMT) has been in use for almost half a century, there are still quite contradictory views about whether it is a reliable instrument, and if so, what it really measures. Thus, based on data from 39 female students, we first examined DMT inter-coder reliability by analyzing the agreement among trained judges in their coding of the same DMT protocols. Second, we constructed a "parallel" photographic picture that retained all structural characteristic of the original and analyzed DMT parallel-test reliability. Third, we examined the construct validity of the DMT by (a) employing three self-report defense-mechanism inventories and analyzing the intercorrelations between DMT defense scores and corresponding defenses in these instruments, (b) studying the relationships between DMT responses and scores on trait and state anxiety, and (c) relating DMT-defense scores to measures of self-esteem. The main results showed that the DMT can be coded with high reliability by trained coders, that the parallel-test reliability is unsatisfactory compared to traditional psychometric standards, that there is a certain generalizability in the number of perceptual distortions that people display from one picture to another, and that the construct validation provided meager empirical evidence for the conclusion that the DMT measures what it purports to measure, that is, psychological defense mechanisms.     

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.     

Distractor selection ratios

A latent variable representation for multiple-choice item and option characteristic curves is presented. Under standard assumptions of conditional independence of item responses and monotonicity of item characteristic curves, a criterion for distractors is proposed based on distractor selection ratios. A connection is made between the proposed criterion and the theory of individual choice behavior, providing new insight. The main results allow for the testing of the criterion from observable data without first specifying a parametric form for the characteristic curves. A series of examples apply the method.     

Tail-measurability in monotone latent variable models

We consider latent variable models for an infinite sequence (or universe) of manifest (observable) variables that may be discrete, continuous or some combination of these. The main theorem is a general characterization by empirical conditions of when it is possible to construct latent variable models that satisfy unidimensionality, monotonicity, conditional independence, andtail-measurability. Tail-measurability means that the latent variable can be estimated consistently from the sequence of manifest variables even though an arbitrary finite subsequence has been removed. The characterizing,necessary and sufficient, conditions that the manifest variables must satisfy for these models are conditional association and vanishing conditional dependence (as one conditions upon successively more other manifest variables). Our main theorem considerably generalizes and sharpens earlier results of Ellis and van den Wollenberg (1993), Holland and Rosenbaum (1986), and Junker (1993). It is also related to the work of Stout (1990).The main theorem is preceded by many results for latent variable modelsin general—not necessarily unidimensional and monotone. They pertain to the uniqueness of latent variables and are connected with the conditional independence theorem of Suppes and Zanotti (1981). We discuss new definitions of the concepts of true-score and subpopulation, which generalize these notions from the stochastic subject, random sampling, and domain sampling formulations of latent variable models (e.g., Holland, 1990; Lord & Novick, 1968). These definitions do not require the a priori specification of a latent variable model.The authors made equivalent contributions to the results of this article. Ellis' research was supported by the Dutch Interuniversitary Graduate School of Psychometrics and Sociometrics. Junker's research was supported by ONR Grant N00014-87-K-0277, NIMH Grant MH15758, and a Carnegie Mellon University Faculty Development Grant. In addition Junker would like to acknowledge the hospitality of the Nijmegen Institute for Cognition and Information during his visit to the University of Nijmegen in August 5–10, 1993.     

Logistic positive exponent family of models: Virtue of asymmetric item characteristic curves

The paper addresses and discusses whether the tradition of accepting point-symmetric item characteristic curves is justified by uncovering the inconsistent relationship between the difficulties of items and the order of maximum likelihood estimates of ability. This inconsistency is intrinsic in models that provide point-symmetric item characteristic curves, and in this paper focus is put on the normal ogive model for observation. It is also questioned if in the logistic model the sufficient statistic has forfeited the rationale that is appropriate to the psychological reality. It is observed that the logistic model can be interpreted as the case in which the inconsistency in ordering the maximum likelihood estimates is degenerated.The paper proposes a family of models, called the logistic positive exponent family, which provides asymmetric item chacteristic curves. A model in this family has a consistent principle in ordering the maximum likelihood estimates of ability. The family is divided into two subsets each of which has its own principle, and includes the logistic model as a transition from one principle to the other. Rationale and some illustrative examples are given.     

Testing the latent factor structure and construct validity of the Ten-Item Personality Inventory

Recent efforts have aimed to develop relatively short measures of the Five-Factor Model (FFM) of personality, particularly for when time and/or space is limited. We evaluate the Ten-Item Personality Inventory (TIPI), a non-proprietary FFM measure with two items per dimension. We use a latent variable methodology to examine the TIPI’s factor structure and convergent validity with the 50-item International Personality Item Pool (IPIP) FFM measure. We provide correlations between the scale scores and latent factors, and compare each measure’s pattern of correlations with measures of other individual difference constructs. Results were favorable in terms of the factor structure and convergent validity of the TIPI, particularly regarding the correlations between the respective latent factors of the TIPI and the IPIP–FFM measures.     

等级反应模型下项目特征曲线等值法在大型考试中的应用

在中国最大的资格考试之一的经济专业资格考试中,为保证不同年度间考试的可比性、进行题库建设和为计算机自适应考试做准备,应用项目反应理论中等级反应模型下的项目特征曲线等值法,采用铆测验等值设计,实现了4个年度考试资料的项目参数和能力参数的等值,并成功地组建了经济专业题库。在此基础上,利用等值技术对不同年份试卷的划界分数进行了比较,为经济考试的合格标准制定、确保考试的公平性提供了实证依据。     

A taxonomy of item response models

A number of models for categorical item response data have been proposed in recent years. The models appear to be quite different. However, they may usefully be organized as members of only three distinct classes, within which the models are distinguished only by assumptions and constraints on their parameters. “Difference models” are appropriate for ordered responses, “divide-by-total” models may be used for either ordered or nominal responses, and “left-side added” models are used for multiple-choice responses with guessing. The details of the taxonomy and the models are described in this paper. The present study was supported in part by two postdoctoral fellowships awarded to Lynne Steinberg: an Educational Testing Service Postdoctoral Fellowship at ETS, Princeton, NJ and an NIMH Individual National Research Service Award at Stanford University, Stanford, CA. Helpful comments by the editor and three anonymous reviewers are gratefully acknowledged.