Extensions of the partial credit model

The partial credit model, developed by Masters (1982), is a unidimensional latent trait model for responses scored in two or more ordered categories. In the present paper some extensions of the model are presented. First, a marginal maximum likelihood estimation procedure is developed which allows for incomplete data and linear restrictions on both the item and the population parameters. Secondly, two statistical tests for evaluating model fit are presented: the former test has power against violation of the assumption about the ability distribution, the latter test offers the possibility of identifying specific items that do not fit the model.The authors are indepted to professor Wim van der Linden and Huub Verstralen for their helpful comments.     

On the existence and uniqueness of JML estimates for the partial credit model

A necessary and sufficient condition is given in this paper for the existence and uniqueness of the maximum likelihood (the so-called joint maximum likelihood) estimate of the parameters of the Partial Credit Model. This condition is stated in terms of a structural property of the pattern of the data matrix that can be easily verified on the basis of a simple iterative procedure. The result is proved by using an argument of Haberman (1977). The author wishes to thank the Editor and the anonymous reviewers for their comments that helped to substantially improve the final version of this paper. This research was supported in part by a MURST grant (ex 60%).     

Optimal designs for the generalized partial credit model

Analysing ordinal data is becoming increasingly important in psychology, especially in the context of item response theory. The generalized partial credit model (GPCM) is probably the most widely used ordinal model and has found application in many large-scale educational assessment studies such as PISA. In the present paper, optimal test designs are investigated for estimating persons’ abilities with the GPCM for calibrated tests when item parameters are known from previous studies. We find that local optimality may be achieved by assigning non-zero probability only to the first and last categories independently of a person's ability. That is, when using such a design, the GPCM reduces to the dichotomous two-parameter logistic (2PL) model. Since locally optimal designs require the true ability to be known, we consider alternative Bayesian design criteria using weight distributions over the ability parameter space. For symmetric weight distributions, we derive necessary conditions for the optimal one-point design of two response categories to be Bayes optimal. Furthermore, we discuss examples of common symmetric weight distributions and investigate under what circumstances the necessary conditions are also sufficient. Since the 2PL model is a special case of the GPCM, all of these results hold for the 2PL model as well.     

A comparison of three simple test theory models

In very simple test theory models such as the Rasch model, a single parameter is used to represent the ability of any examinee or the difficulty of any item. Simple models such as these provide very important points of departure for more detailed modeling when a substantial amount of data are available, and are themselves of real practical value for small or even medium samples. They can also serve a normative role in test design.As an alternative to the Rasch model, or the Rasch model with a correction for guessing, a simple model is introduced which characterizes strength of response in terms of the ratio of ability and difficulty parameters rather than their difference. This model provides a natural account of guessing, and has other useful things to contribute as well. It also offers an alternative to the Rasch model with the usual correction for guessing. The three models are compared in terms of statistical properties and fits to actual data. The goal of the paper is to widen the range of minimal models available to test analysts.This research was supported by grant AP320 from the Natural Sciences and Engineering Research Council of Canada. The author is grateful for discussions with M. Abrahamowicz, I. Molenaar, D. Thissen, and H. Wainer.     

Graphical regression models for polytomous variables

When modeling the relationship between two nominal categorical variables, it is often desirable to include covariates to understand how individuals differ in their response behavior. Typically, however, not all the relevant covariates are available, with the result that the measured variables cannot fully account for the associations between the nominal variables. Under the assumption that the observed and unobserved variables follow a homogeneous conditional Gaussian distribution, this paper proposesRC(M) regression models to decompose the residual associations between the polytomous variables. Based on Goodman's (1979, 1985)RC(M) association model, a distinctive feature ofRC(M) regression models is that they facilitate the joint estimation of effects due to manifest and omitted (continuous) variables without requiring numerical integration. TheRC(M) regression models are illustrated using data from the High School and Beyond study (Tatsuoka & Lohnes, 1988). This article was accepted for publication, when Willem J. Heiser was the Editor ofPsychometrika. This research was supported by grants from the National Science Foundation (#SBR96-17510 and #SBR94-09531) and the Bureau of Educational Research at the University of Illinois. We thank Jee-Seon Kim for comments and computational assistance.     

Multidimensional scaling models for reaction times and same-different judgments

A method for joint analysis of reaction times and same-different judgments is discussed. A set of stimuli is assumed to have some parametric representation which uniquely defines dissimilarities between the stimuli. Those dissimilarities are then related to the observed reaction times and same-different judgments through a model of psychological processes. Three representation models of dissimilarities are considered, the Minkowski power distance model, the linear model, and Tversky's feature matching model. Maximum likelihood estimation procedures are developed and implemented in the form of a FORTRAN program. An example is given to illustrate the kind of analyses that can be performed by the proposed method.The work reported in this paper is supported by Grant A6394 to the first author from the Natural Sciences and Engineering Research Council of Canada. Portions of this study have been presented at the Psychometric Society meeting in Chapel Hill, N.C., in May, 1981. We thank Tony Marley, Jim Ramsay and anonymous reviewers for their helpful comments. MAXRT, a computer program which performs the computations described in this paper may be obtained by writing to the first author.     

The unique correspondence of the item response function and item category response functions in polytomously scored item response models

The item response function (IRF) for a polytomously scored item is defined as a weighted sum of the item category response functions (ICRF, the probability of getting a particular score for a randomly sampled examinee of ability ). This paper establishes the correspondence between an IRF and a unique set of ICRFs for two of the most commonly used polytomous IRT models (the partial credit models and the graded response model). Specifically, a proof of the following assertion is provided for these models: If two items have the same IRF, then they must have the same number of categories; moreover, they must consist of the same ICRFs. As a corollary, for the Rasch dichotomous model, if two tests have the same test characteristic function (TCF), then they must have the same number of items. Moreover, for each item in one of the tests, an item in the other test with an identical IRF must exist. Theoretical as well as practical implications of these results are discussed.This research was supported by Educational Testing Service Allocation Projects No. 79409 and No. 79413. The authors wish to thank John Donoghue, Ming-Mei Wang, Rebecca Zwick, and Zhiliang Ying for their useful comments and discussions. The authors also wish to thank three anonymous reviewers for their comments.     

On equivalence between a partial credit item and a set of independent Rasch binary items

Given a Masters partial credit item withn known step difficulties, conditions are stated for the existence of a set of (locally) independent Rasch binary items such that their raw score and the partial credit raw score have identical probability density functions. The conditions are those for the existence ofn positive values with predetermined elementary symmetric functions and include the requirement that then step difficulties form an increasing sequence.     

Semi‐parametric proportional hazards models with crossed random effects for psychometric response times

The semi‐parametric proportional hazards model with crossed random effects has two important characteristics: it avoids explicit specification of the response time distribution by using semi‐parametric models, and it captures heterogeneity that is due to subjects and items. The proposed model has a proportionality parameter for the speed of each test taker, for the time intensity of each item, and for subject or item characteristics of interest. It is shown how all these parameters can be estimated by Markov chain Monte Carlo methods (Gibbs sampling). The performance of the estimation procedure is assessed with simulations and the model is further illustrated with the analysis of response times from a visual recognition task.     

Decomposition of a Rasch partial credit item into independent binary and indecomposable trinary items

For each Rasch (Masters) partial credit item, there exists a set of independent Rasch binary and indecomposable trinary items for which the sum of the scores and the partial credit score have identical probability density functions. If each indecomposable trinary item is further expressed as the sum of two binary items, then the binary items are positively dependent and cannot be both of the Rasch type. This paper was written while the author was working with Steve Ferrara and Hillary Michaels on some technical aspects of the Maryland School Performance Assessment Program. The author had been puzzled by the fact that most MSPAP assessment items have three or less score categories. With a psychometric justification now being apparent, this paper is dedicated to both of them.     

Empirical underidentification in estimating random utility models: The role of choice sets and standardizations

A standard approach to distinguishing people’s risk preferences is to estimate a random utility model using a power utility function to characterize the preferences and a logit function to capture choice consistency. We demonstrate that with often-used choice situations, this model suffers from empirical underidentification, meaning that parameters cannot be estimated precisely. With simulations of estimation accuracy and Kullback–Leibler divergence measures we examined factors that potentially mitigate this problem. First, using a choice set that guarantees a switch in the utility order between two risky gambles in the range of plausible values leads to higher estimation accuracy than randomly created choice sets or the purpose-built choice sets common in the literature. Second, parameter estimates are regularly correlated, which contributes to empirical underidentification. Examining standardizations of the utility scale, we show that they mitigate this correlation and additionally improve the estimation accuracy for choice consistency. Yet, they can have detrimental effects on the estimation accuracy of risk preference. Finally, we also show how repeated versus distinct choice sets and an increase in observations affect estimation accuracy. Together, these results should help researchers make informed design choices to estimate parameters in the random utility model more precisely.