Solving the measurement invariance anchor item problem in item response theory

The efficacy of tests of differential item functioning (measurement invariance) has been well established. It is clear that when properly implemented, these tests can successfully identify differentially functioning (DF) items when they exist. However, an assumption of these analyses is that the metric for different groups is linked using anchor items that are invariant. In practice, however, it is impossible to be certain which items are DF and which are invariant. This problem of anchor items, or referent indicators, has long plagued invariance research, and a multitude of suggested approaches have been put forth. Unfortunately, the relative efficacy of these approaches has not been tested. This study compares 11 variations on 5 qualitatively different approaches from recent literature for selecting optimal anchor items. A large-scale simulation study indicates that for nearly all conditions, an easily implemented 2-stage procedure recently put forth by Lopez Rivas, Stark, and Chernyshenko (2009) provided optimal power while maintaining nominal Type I error. With this approach, appropriate anchor items can be easily and quickly located, resulting in more efficacious invariance tests. Recommendations for invariance testing are illustrated using a pedagogical example of employee responses to an organizational culture measure. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Improving measurement precision of test batteries using multidimensional item response models

A conventional way to analyze item responses in multiple tests is to apply unidimensional item response models separately, one test at a time. This unidimensional approach, which ignores the correlations between latent traits, yields imprecise measures when tests are short. To resolve this problem, one can use multidimensional item response models that use correlations between latent traits to improve measurement precision of individual latent traits. The improvements are demonstrated using 2 empirical examples. It appears that the multidimensional approach improves measurement precision substantially, especially when tests are short and the number of tests is large. To achieve the same measurement precision, the multidimensional approach needs less than half of the comparable items required for the unidimensional approach.     

Testing the conditional independence and monotonicity assumptions of item response theory

When item characteristic curves are nondecreasing functions of a latent variable, the conditional or local independence of item responses given the latent variable implies nonnegative conditional covariances between all monotone increasing functions of a set of item responses given any function of the remaining item responses. This general result provides a basis for testing the conditional independence assumption without first specifying a parametric form for the nondecreasing item characteristic curves. The proposed tests are simple, have known asymptotic null distributions, and possess certain optimal properties. In an example, the conditional independence hypothesis is rejected for all possible forms of monotone item characteristic curves.The author acknowledges Paul W. Holland for valuable conversations on the subject of this paper; Henry Braun and Fred Lord for comments at a presentation on this subject which led to improvements in the paper; Carl H. Haag for permission to use the data in §4; Bruce Kaplan for assistance with computing; and two referees for helpful suggestions. Requests for reprints should be sent to Paul R. Rosenbaum     

Summed-score linking using item response theory: application to depression measurement

An item response theory (IRT) approach to test linking based on summed scores is presented and demonstrated by calibrating a modified 23-item version of the Center for Epidemiologic Studies Depression Scale (CES-D) to the standard 20-item CES-D. Data are from the Depression Patient Outcomes Research Team, II, which used a modified CES-D to measure risk for depression. Responses (N = 1,120) to items on both the original and modified versions were calibrated simultaneously using F. Samejima's (1969, 1997) graded IRT model. The 2 scales were linked on the basis of derived summed-score-to-IRT-score translation tables. The established cut score of 16 on the standard CES-D corresponded most closely to a summed score of 20 on the modified version. The IRT summed-score approach to test linking is a straightforward, valid, and practical method that can be applied in a variety of situations.     

An item response theory analysis of the Problem Gambling Severity Index

Increases in the availability of gambling heighten the need for a short screening measure of problem gambling. The Problem Gambling Severity Index (PGSI) is a brief measure that allows for the assessment of characteristics of gambling behavior and severity and its consequences. The authors evaluate the psychometric properties of the PGSI using item response theory methods in a representative sample of the urban adult population in South Africa (N = 3,000). The PGSI items were evaluated for differential item functioning (DIF) due to language translation. DIF was not detected. The PGSI was found to be unidimensional, and use of the nominal categories model provided additional information at higher values of the underlying construct relative to a simpler binary model. This study contributes to the growing literature supporting the PGSI as the screen of choice for assessing gambling problems in the general population.     

On the relationship between item response theory and factor analysis of discretized variables

Equivalence of marginal likelihood of the two-parameter normal ogive model in item response theory (IRT) and factor analysis of dichotomized variables (FA) was formally proved. The basic result on the dichotomous variables was extended to multicategory cases, both ordered and unordered categorical data. Pair comparison data arising from multiple-judgment sampling were discussed as a special case of the unordered categorical data. A taxonomy of data for the IRT and FA models was also attempted.The work reported in this paper has been supported by Grant A6394 to the first author from the Natural Sciences and Engineering Research Council of Canada.     

Bayesian modeling of measurement error in predictor variables using item response theory

It is shown that measurement error in predictor variables can be modeled using item response theory (IRT). The predictor variables, that may be defined at any level of an hierarchical regression model, are treated as latent variables. The normal ogive model is used to describe the relation between the latent variables and dichotomous observed variables, which may be responses to tests or questionnaires. It will be shown that the multilevel model with measurement error in the observed predictor variables can be estimated in a Bayesian framework using Gibbs sampling. In this article, handling measurement error via the normal ogive model is compared with alternative approaches using the classical true score model. Examples using real data are given.This paper is part of the dissertation by Fox (2001) that won the 2002 Psychometric Society Dissertation Award.     

Structure and measurement of depression in youths: applying item response theory to clinical data

Our goals in this article were to use item response theory (IRT) to assess the relation of depressive symptoms to the underlying dimension of depression and to demonstrate how IRT-based measurement strategies can yield more reliable data about depression severity than conventional symptom counts. Participants were 3,403 children and adolescents from 12 contributing clinical and nonclinical samples; all participants had received the Kiddie Schedule of Affective Disorders and Schizophrenia for School-Aged Children. Results revealed that some symptoms reflected higher levels of depression and were more discriminating than others. Furthermore, use of IRT-based information about symptom severity and discriminability in the measurement of depression severity was shown to reduce measurement error and increase measurement fidelity.     

A note on monotonicity of item response functions for ordered polytomous item response theory models

A monotone relationship between a true score (τ) and a latent trait level (θ) has been a key assumption for many psychometric applications. The monotonicity property in dichotomous response models is evident as a result of a transformation via a test characteristic curve. Monotonicity in polytomous models, in contrast, is not immediately obvious because item response functions are determined by a set of response category curves, which are conceivably non-monotonic in θ. The purpose of the present note is to demonstrate strict monotonicity in ordered polytomous item response models. Five models that are widely used in operational assessments are considered for proof: the generalized partial credit model (Muraki, 1992, Applied Psychological Measurement, 16, 159), the nominal model (Bock, 1972, Psychometrika, 37, 29), the partial credit model (Masters, 1982, Psychometrika, 47, 147), the rating scale model (Andrich, 1978, Psychometrika, 43, 561), and the graded response model (Samejima, 1972, A general model for free-response data (Psychometric Monograph no. 18). Psychometric Society, Richmond). The study asserts that the item response functions in these models strictly increase in θ and thus there exists strict monotonicity between τ and θ under certain specified conditions. This conclusion validates the practice of customarily using τ in place of θ in applied settings and provides theoretical grounds for one-to-one transformations between the two scales.     

Increasing item complexity: A possible cause of scale shrinkage for unidimensional item response theory

When the three-parameter logistic model is applied to tests covering a broad range of difficulty, there frequently is an increase in mean item discrimination and a decrease in variance of item difficulties and traits as the tests become more difficult. To examine the hypothesis that this unexpected scale shrinkage effect occurs because the items increase in complexity as they increase in difficulty, an approximate relationship is derived between the unidimensional model used in data analysis and a multidimensional model hypothesized to be generating the item responses. Scale shrinkage is successfully predicted for several sets of simulated data.The author is grateful to Robert Mislevy for kindly providing a copy of his computer program, RESOLVE.     

On the sampling theory roundations of item response theory models

Item response theory (IT) models are now in common use for the analysis of dichotomous item responses. This paper examines the sampling theory foundations for statistical inference in these models. The discussion includes: some history on the stochastic subject versus the random sampling interpretations of the probability in IRT models; the relationship between three versions of maximum likelihood estimation for IRT models; estimating versus estimating -predictors; IRT models and loglinear models; the identifiability of IRT models; and the role of robustness and Bayesian statistics from the sampling theory perspective.A presidential address can serve many different functions. This one is a report of investigations I started at least ten years ago to understand what IRT was all about. It is a decidedly one-sided view, but I hope it stimulates controversy and further research. I have profited from discussions of this material with many people including: Brian Junker, Charles Lewis, Nicholas Longford, Robert Mislevy, Ivo Molenaar, Donald Rock, Donald Rubin, Lynne Steinberg, Martha Stocking, William Stout, Dorothy Thayer, David Thissen, Wim van der Linden, Howard Wainer, and Marilyn Wingersky. Of course, none of them is responsible for any errors or misstatements in this paper. The research was supported in part by the Cognitive Science Program, Office of Naval Research under Contract No. Nooo14-87-K-0730 and by the Program Statistics Research Project of Educational Testing Service.     

Sequential sampling designs for the two-parameter item response theory model

In optimal design research, designs are optimized with respect to some statistical criterion under a certain model for the data. The ideas from optimal design research have spread into various fields of research, and recently have been adopted in test theory and applied to item response theory (IRT) models. In this paper a generalized variance criterion is used for sequential sampling in the two-parameter IRT model. Some general principles are offered to enable a researcher to select the best sampling design for the efficient estimation of item parameters.     

A modular approach for item response theory modeling with the R package flirt

The new R package flirt is introduced for flexible item response theory (IRT) modeling of psychological, educational, and behavior assessment data. flirt integrates a generalized linear and nonlinear mixed modeling framework with graphical model theory. The graphical model framework allows for efficient maximum likelihood estimation. The key feature of flirt is its modular approach to facilitate convenient and flexible model specifications. Researchers can construct customized IRT models by simply selecting various modeling modules, such as parametric forms, number of dimensions, item and person covariates, person groups, link functions, etc. In this paper, we describe major features of flirt and provide examples to illustrate how flirt works in practice.     

Optimum examinee samples for item parameter estimation in item response theory: A multi-objective programming approach

This paper proposes a multi-objective programming method for determining samples of examinees needed for estimating the parameters of a group of items. In the numerical experiments, optimum samples are compared to uniformly and normally distributed samples. The results show that the samples usually recommended in the literature are well suited for estimating the difficulty parameters. Furthermore, they are also adequate for estimating the discrimination parameters in the three-parameter model, butnot for the guessing parameters.     

Detecting differential item functioning with confirmatory factor analysis and item response theory: toward a unified strategy

In this article, the authors developed a common strategy for identifying differential item functioning (DIF) items that can be implemented in both the mean and covariance structures method (MACS) and item response theory (IRT). They proposed examining the loadings (discrimination) and the intercept (location) parameters simultaneously using the likelihood ratio test with a free-baseline model and Bonferroni corrected critical p values. They compared the relative efficacy of this approach with alternative implementations for various types and amounts of DIF, sample sizes, numbers of response categories, and amounts of impact (latent mean differences). Results indicated that the proposed strategy was considerably more effective than an alternative approach involving a constrained-baseline model. Both MACS and IRT performed similarly well in the majority of experimental conditions. As expected, MACS performed slightly worse in dichotomous conditions but better than IRT in polytomous cases where sample sizes were small. Also, contrary to popular belief, MACS performed well in conditions where DIF was simulated on item thresholds (item means), and its accuracy was not affected by impact.     

Detection and validation of unscalable item score patterns using item response theory: an illustration with Harter's Self-Perception Profile for Children

We illustrate the usefulness of person-fit methodology for personality assessment. For this purpose, we use person-fit methods from item response theory. First, we give a nontechnical introduction to existing person-fit statistics. Second, we analyze data from Harter's (1985) Self-Perception Profile for Children (Harter, 1985) in a sample of children ranging from 8 to 12 years of age (N = 611) and argue that for some children, the scale scores should be interpreted with care and caution. Combined information from person-fit indexes and from observation, interviews, and self-concept theory showed that similar score profiles may have a different interpretation. For some children in the sample, item scores did not adequately reflect their trait level. Based on teacher interviews, this was found to be due most likely to a less developed self-concept and/or problems understanding the meaning of the questions. We recommend investigating the scalability of score patterns when using self-report inventories to help the researcher interpret respondents' behavior correctly.