Latent trait models and dichotomization of graded responses

This paper discusses thecompatibility of the polychotomous Rasch model with dichotomization of the response continuum. It is argued that in the case of graded responses, the response categories presented to the subject are essentially an arbitrary polychotomization of the response continuum, ranging for example from total rejection or disagreement to total acceptance or agreement of an item or statement. Because of this arbitrariness, the measurement outcome should be independent of the specific polychotomization applied, for example, presenting a specific multicategory response format should not affect the measurement outcome. When such is the case, the original polychotomous model is called compatible with dichotomization.A distinction is made between polychotomization or dichotomization before the fact, that is, in constructing the response format, and polycho- or dichotomization after the fact, for example in dichotomizing existing graded response data.It is shown that, at least in case of dichotomization after-the-fact, the polychotomous Rasch model is not compatible with dichotomization, unless a rather special condition of the model parameters is met. Insofar as it may be argued that dichotomization before the fact is not essentially different from dichotomization after the fact, the value of the unidimensional polychotomous Rasch model is consequently questionable. The impact of our conclusion on related models is also discussed.     

Conditions for rasch-dichotomizability of the unidimensional polytomous rasch model

Jansen and Roskam (1986) discussed the compatibility of the unidimensional polytomous Rasch model with dichotomization of the response continuum. They derived a rather strict condition in which dichotomization of multicategory data that fit the unidimensional polytomous Rasch model, results in dichotomous data which fit the dichotomous Research model with effectively the same subject parameter. In this paper a more general dichotomization condition is derived for the polytomous Rasch model, which appears less restrictive, but upholds that the intrinsic logic of the unidimensional polytomous Rasch model defies dichotomization in general. The robustness of dichotomous analysis investigated in a simulation study. It shows a close relation with the two-parameters (Birnbaum) model. Theoretical and methodological implications are discussed.The authors are indebted to H. Müller (personal communication, August 1986), for giving an example which pointed toward the core equation in this paper. The authors also acknowledge the critical comments of Th. Bezambinder and P. Wakker, and of Psychometrika's reviewers to an earlier version of this paper.     

An approximation for the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete

Lord developed an approximation for the bias function for the maximum likelihood estimate in the context of the three-parameter logistic model. Using Taylor's expansion of the likelihood equation, he obtained an equation that includes the conditional expectation, given true ability, of the discrepancy between the maximum likelihood estimate and true ability. All terms of orders higher thann ^?1 are ignored wheren indicates the number of items. Lord assumed that all item and individual parameters are bounded, all item parameters are known or well-estimated, and the number of items is reasonably large. In the present paper, an approximation for the bias function of the maximum likelihood estimate of the latent trait, or ability, will be developed using the same assumptions for the more general case where item responses are discrete. This will include the dichotomous response level, for which the three-parameter logistic model has been discussed, the graded response level and the nominal response level. Some observations will be made for both dichotomous and graded response levels.     

The bias function of the maximum likelihood estimate of ability for the dichotomous response level

Samejima has recently given an approximation for the bias function for the maximum likelihood estimate of the latent trait in the general case where item responses are discrete, generalizing Lord's bias function in the three-parameter logistic model for the dichotomous response level. In the present paper, observations are made about the behavior of this bias function for the dichotomous response level in general, and also with respect to several widely used mathematical models. Some empirical examples are given.     

Acceleration model in the heterogeneous case of the general graded response model

A new model, called acceleration model, is proposed in the framework of the heterogenous case of the graded response model, based on processing functions defined for a finite or enumerable number of steps. The model is expected to be useful in cognitive assessment, as well as in more traditional areas of application of latent trait models. Criteria for evaluating models are proposed, and soundness and robustness of the acceleration model are discussed. Graded response models based on individual choice behavior are also discussed, and criticisms on model selection in terms of fitnesses of models to the data are also given.This research was supported by the Office of Naval Research (N00014-90-J-1456).     

On the existence and uniqueness of maximum-likelihood estimates in the Rasch model

Necessary and sufficient conditions for the existence and uniqueness of a solution of the so-called unconditional (UML) and the conditional (CML) maximum-likelihood estimation equations in the dichotomous Rasch model are given. The basic critical condition is essentially the same for UML and CML estimation. For complete data matricesA, it is formulated both as a structural property ofA and in terms of the sufficient marginal sums. In case of incomplete data, the condition is equivalent to complete connectedness of a certain directed graph. It is shown how to apply the results in practical uses of the Rasch model.Paper read at the European Meeting of the Psychometric Society, Groningen, June 19–21, 1980.Part of the research reported herein was done while the author was staying at the Pulmologisches Zentrum der Stadt Wien; he is indebted to Professor Dr. F. Muhar and Dr. R. Mutschlechner for providing excellent working conditions.     

A rasch model for continuous ratings

A unidimensional latent trait model for continuous ratings is developed. This model is an extension of Andrich's rating formulation which assumes that the response process at latent thresholds is governed by the dichotomous Rasch model. Item characteristic functions and information functions are used to illustrate that the model takes ceiling and floor effects into account. Both the dichotomous Rasch model and a linear model with normally distributed error can be derived as limiting cases. The separability of the structural and incidental parameters is demonstrated and a procedure for estimating the parameters is outlined.     

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.     

The essential process in a family of measurement models

Five members of the Rasch family of latent trait models which have appeared more or less independently in the literature are brought together and identified as one model. In addition to sharing the distinguishing characteristic of the dichotomous Rasch model—separable person and item parameters and hence sufficient statistics—all five models share a common algebraic form and have as their basic element the fundamental process defined by Rasch's simple logistic expression. In these models, the sufficient statistics for person and item parameters are counts of events constructed to be indicative of the variable being measured, and the measures they enable are ‘fundamental’.     

An extension of the rating scale model with an application to the measurement of change

The polytomous unidimensional Rasch model with equidistant scoring, also known as the rating scale model, is extended in such a way that the item parameters are linearly decomposed into certain basic parameters. The extended model is denoted as the linear rating scale model (LRSM). A conditional maximum likelihood estimation procedure and a likelihood-ratio test of hypotheses within the framework of the LRSM are presented. Since the LRSM is a generalization of both the dichotomous Rasch model and the rating scale model, the present algorithm is suited for conditional maximum likelihood estimation in these submodels as well. The practicality of the conditional method is demonstrated by means of a dichotomous Rasch example with 100 items, of a rating scale example with 30 items and 5 categories, and in the light of an empirical application to the measurement of treatment effects in a clinical study.Work supported in part by the Fonds zur Förderung der Wissenschaftlichen Forschung under Grant No. P6414.     

Some improved diagnostics for failure of the Rasch model

Although several goodness of fit tests have been developed for the Rasch model for dichotomous items, most of them are of a global, asymptotic, and confirmatory type. This paper, based on ideas from a recent thesis by Van den Wollenberg, offers some suggestions for local, small sample, and exploratory techniques: difficulty plots for person groups scoring right and wrong on a specific item, a slope test per item based on a binomial distribution per score group, and a unidimensionality check based on an extended hypergeometric distribution per score group. This paper owes much to the inspiring and pioneering work of Arnold Van den Wollenberg, of which only minor aspects are criticized. Thanks go to Charles Lewis for stimulating discussions and for solutions to some programming problems.     

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.     

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.     

Combining mixture distribution and multidimensional IRTree models for the measurement of extreme response styles

Personality constructs, attitudes and other non-cognitive variables are often measured using rating or Likert-type scales, which does not come without problems. Especially in low-stakes assessments, respondents may produce biased responses due to response styles (RS) that reduce the validity and comparability of the measurement. Detecting and correcting RS is not always straightforward because not all respondents show RS and the ones who do may not do so to the same extent or in the same direction. The present study proposes the combination of a multidimensional IRTree model with a mixture distribution item response theory model and illustrates the application of the approach using data from the Programme for the International Assessment of Adult Competencies (PIAAC). This joint approach allows for the differentiation between different latent classes of respondents who show different RS behaviours and respondents who show RS versus respondents who give (largely) unbiased responses. We illustrate the application of the approach by examining extreme RS and show how the resulting latent classes can be further examined using external variables and process data from computer-based assessments to develop a better understanding of response behaviour and RS.     

Departure from normal assumptions: A promise for future psychometrics with substantive mathematical modeling

Normal assumptions have been used in many psychometric methods, to the extent that most researchers do not even question their adequacy. With the rapid advancement of computer technologies in recent years, psychometrics has extended its territory to include intensive cognitive diagnosis, etcetera, and substantive mathematical modeling ha become essential. As a natural consequence, it is time to consider departure from normal assumptions seriously. As examples of models which are not based on normality or its approximation, the logistic positive exponent family of models is discussed. These models include the item task complexity as the third parameter, which determines the single principle of ordering individuals on the ability scale.     

A method of estimating item characteristic functions using the maximum likelihood estimate of ability

A method of estimating item characteristic functions is proposed, in which a set of test items, whose operating characteristics are known and which give a constant test information function for a substantially wide range of ability, are used. The method is based on the maximum likelihood estimates of ability for a group of several hundred examinees. Throughout the present study the Monte Carlo method is used.     

Some critical observations of the test information function as a measure of local accuracy in ability estimation

The test information function serves important roles in latent trait models and in their applications. Among others, it has been used as the measure of accuracy in ability estimation. A question arises, however, if the test information function is accurate enough for all meaningful levels of ability relative to the test, especially when the number of test items is relatively small (e.g., less than 50). In the present paper, using the constant information model and constant amounts of test information for a finite interval of ability, simulated data were produced for eight different levels of ability and for twenty different numbers of test items ranging between 10 and 200. Analyses of these data suggest that it is desirable to consider some modification of the test information function when it is used as the measure of accuracy in ability estimation.     

GT与IRT的比较: 北京奥运会男子10米跳台跳水分析

概化理论(GT)和项目反应理论(IRT)从两个不同的方向发展了经典测量理论, GT和IRT中的多面Rasch测量模型(MFRM)在主观评分中都可以用来估计评分中各变异来源对变异的贡献, 对测评的信度进行估计, 提出测评改进意见。12名运动员参加了2008北京奥运会男子10米跳台跳水决赛, 比赛共6个回合, 7名裁判独立对他们在各个回合的表现进行打分。GT和MFRM比较一致地认为运动员自身、回合、运动员与回合的交互效应是运动员得分的重要变异来源, 而裁判员对运动员得分差异的贡献不显著。MFRM同时还估计出难度系数是影响男子跳台跳水成绩的重要变异来源, 在评分等级6.5附近存在步校准错乱, 得出的运动员成绩排序与2008奥运实际排序有所不同。在GT中难度系数作为隐藏侧面, 其效应未能分离出来。GT和MFRM从两个不同的方面给测量提供改进意见: GT发现可以通过增加回合数来提高g系数, 而增加裁判数对其影响不大。MFRM给出各侧面的要素(如某裁判、运动员等)的估计值及其标准误, 它给出的诊断性拟合统计也有助于甄别异常得分或评分模式。     

Regularized Finite Mixture Models for Probability Trajectories

Finite mixture models are widely used in the analysis of growth trajectory data to discover subgroups of individuals exhibiting similar patterns of behavior over time. In practice, trajectories are usually modeled as polynomials, which may fail to capture important features of the longitudinal pattern. Focusing on dichotomous response measures, we propose a likelihood penalization approach for parameter estimation that is able to capture a variety of nonlinear class mean trajectory shapes with higher precision than maximum likelihood estimates. We show how parameter estimation and inference for whether trajectories are time-invariant, linear time-varying, or nonlinear time-varying can be carried out for such models. To illustrate the method, we use simulation studies and data from a long-term longitudinal study of children at high risk for substance abuse. This work was supported in part by NIAAA grants R37 AA07065 and R01 AA12217 to RAZ.