A nonlinear mixed model framework for item response theory

The Mixed Effects Trend Vector Model

Maximum likelihood estimation of mixed effect baseline category logit models for multinomial longitudinal data can be prohibitive due to the integral dimension of the random effects distribution. We propose to use multidimensional unfolding methodology to reduce the dimensionality of the problem. As a by-product, readily interpretable graphical displays representing change are obtained. The methodology can be applied to both nominal and ordinal response variables. Relationships to standard statistical models for multinomial data are presented. Several empirical examples are given to show the merits of the proposed modeling framework.

Misuse of the linear mixed model when evaluating risk factors of cognitive decline

The linear mixed model (LMM), which is routinely used to describe change in outcomes over time and its association with risk factors, assumes that a unit change in any predictor is associated with a constant change in the outcome. When it is used on psychometric tests, this assumption may not hold. Indeed, psychometric tests usually suffer from ceiling and/or floor effects and curvilinearity (i.e., varying sensitivity to change). The authors aimed to determine the consequences of such misspecification when evaluating predictors of cognitive decline. As an alternative to the LMM, they considered 2 mixed models based on latent processes that handle discrete and bounded outcomes. Model differences are illustrated here using data on 4 psychometric tests from the Personnes Agées QUID (PAQUID) Study (1989-2004). The type I error of the Wald test for risk-factor regression parameters was formally assessed in a simulation study. It demonstrated that type I errors in the LMM could be dramatically inflated for some tests, such that spurious associations with risk factors were found. In particular, confusion between effects on mean level and effects on change over time was highlighted. The authors recommend use of the alternative mixed models when studying psychometric tests and more generally quantitative scales (quality of life, activities of daily living).

The king's foot of patient-reported outcomes: current practices and new developments for the measurement of change

PURPOSE

Assessing change remains a challenge in patient-reported outcomes. In June 2009, a group of psychometricians, biostatisticians, and behavioral researchers from other disciplines convened as a Longitudinal Analysis of Patient-Reported Outcomes Working group as part of the Statistical and Applied Mathematical Sciences Institute Summer Psychometric program to discuss the complex issues that arise when conceptualizing and operationalizing "change" in patient-reported outcome (PRO) measures and related constructs. This white paper summarizes these issues and provides recommendations and possible paths for dealing with the complexities of measuring change.

METHODS/RESULTS

This article presents and discusses issues associated with: (1) conceptualizing and operationalizing change in PRO measures; (2) modeling change using state-of-the-art statistical methods; (3) impediments to detecting true change; (4) new developments to deal with these challenges; and (5) important gaps that are fertile ground for future research.

CONCLUSIONS

There was a consensus that important research still needs to be performed in order develop and refine high-quality PRO measures and statistical methods to analyze and model change in PRO constructs.

Modellierung von Kompetenzen im Bereich „Sprache und Sprachgebrauch untersuchen” mit schwierigkeitsbestimmenden Aufgabenmerkmalen

Zusammenfassung. Für den Bereich „Sprache und Sprachgebrauch untersuchen” der KMK-Bildungsstandards wurden basierend auf Kategorien sprachbewusster Tätigkeiten neun schwierigkeitsbestimmende Aufgabenmerkmale abgeleitet. Auf Grundlage einer Stichprobe von 3.276 Schülerinnen und Schülern der dritten und vierten Klassenstufe wurde im ersten Analyseschritt der Zusammenhang der Aufgabenmerkmale mit den Schwierigkeiten von 107 Testaufgaben untersucht. Im zweiten Schritt wurde untersucht, inwieweit die schwierigkeitsgenerierenden Effekte der Aufgabenmerkmale von der sprachlichen Herkunft der Schülerinnen und Schüler unter Berücksichtigung des kulturellen Kapitals sowie der Klassenstufe moderiert werden, d.h. Differential Facet Functioning (DFF) vorliegt. Für Schritt 1 wurde das linear-logistische Testmodell (LLTM) in der Formulierung als generalisiertes lineares gemischtes Modell (GLMM) verwendet, welches für Schritt 2 um Schülermerkmale als Prädiktoren ergänzt wurde. Die Aufgabenmerkmale weisen einen sehr hohen Erklärungswert für die Aufgabenschwierigkeiten auf (R2 = .695), wobei die Beziehung zur Aufgabenschwierigkeit bei acht der neun Aufgabenmerkmale von mindestens einem der Schülermerkmale oder der Klassenstufe abhängt.

Comparison of CTT and Rasch-based approaches for the analysis of longitudinal Patient Reported Outcomes

Health sciences frequently deal with Patient Reported Outcomes (PRO) data for the evaluation of concepts, in particular health-related quality of life, which cannot be directly measured and are often called latent variables. Two approaches are commonly used for the analysis of such data: Classical Test Theory (CTT) and Item Response Theory (IRT). Longitudinal data are often collected to analyze the evolution of an outcome over time. The most adequate strategy to analyze longitudinal latent variables, which can be either based on CTT or IRT models, remains to be identified. This strategy must take into account the latent characteristic of what PROs are intended to measure as well as the specificity of longitudinal designs. A simple and widely used IRT model is the Rasch model. The purpose of our study was to compare CTT and Rasch-based approaches to analyze longitudinal PRO data regarding type I error, power, and time effect estimation bias. Four methods were compared: the Score and Mixed models (SM) method based on the CTT approach, the Rasch and Mixed models (RM), the Plausible Values (PV), and the Longitudinal Rasch model (LRM) methods all based on the Rasch model. All methods have shown comparable results in terms of type I error, all close to 5 per cent. LRM and SM methods presented comparable power and unbiased time effect estimations, whereas RM and PV methods showed low power and biased time effect estimations. This suggests that RM and PV methods should be avoided to analyze longitudinal latent variables.

Empirically indistinguishable multidimensional IRT and locally dependent unidimensional item response models

Multidimensionality is a core concept in the measurement and analysis of psychological data. In personality assessment, for example, constructs are mostly theoretically defined as unidimensional, yet responses collected from the real world are almost always determined by multiple factors. Significant research efforts have concentrated on the use of simulated studies to evaluate the robustness of unidimensional item response models when applied to multidimensional data with a dominant dimension. In contrast, in the present paper, I report the result from a theoretical investigation that a multidimensional item response model is empirically indistinguishable from a locally dependent unidimensional model, of which the single dimension represents the actual construct of interest. A practical implication of this result is that multidimensional response data do not automatically require the use of multidimensional models. Circumstances under which the alternative approach of locally dependent unidimensional models may be useful are discussed.

An introduction to recursive partitioning: rationale, application, and characteristics of classification and regression trees, bagging, and random forests

Recursive partitioning methods have become popular and widely used tools for nonparametric regression and classification in many scientific fields. Especially random forests, which can deal with large numbers of predictor variables even in the presence of complex interactions, have been applied successfully in genetics, clinical medicine, and bioinformatics within the past few years. High-dimensional problems are common not only in genetics, but also in some areas of psychological research, where only a few subjects can be measured because of time or cost constraints, yet a large amount of data is generated for each subject. Random forests have been shown to achieve a high prediction accuracy in such applications and to provide descriptive variable importance measures reflecting the impact of each variable in both main effects and interactions. The aim of this work is to introduce the principles of the standard recursive partitioning methods as well as recent methodological improvements, to illustrate their usage for low and high-dimensional data exploration, but also to point out limitations of the methods and potential pitfalls in their practical application. Application of the methods is illustrated with freely available implementations in the R system for statistical computing.

Investigating latent constructs with item response models: a MATLAB IRTm toolbox

Item response theory (IRT) models are the central tools in modern measurement and advanced psychometrics. We offer a MATLAB IRT modeling (IRTm) toolbox that is freely available and that follows an explicit design matrix approach, giving the end user control and flexibility in building a model that goes beyond standard models, such as the Rasch model (Rasch, 1960) and the two-parameter logistic model. As such, IRTm allows for a large variety of unidimensional IRT models for binary responses, the incorporation of additional person and item information, and deviations from common model assumptions. An exclusive key feature of the toolbox is the inclusion of copula IRT models to handle local item dependencies. Two appendixes for this report, containing example code and information on the general copula IRT in IRTm, may be downloaded from brm.psychonomic-journals.org/content/supplemental.

A multilevel item response theory model was investigated for longitudinal vision-related quality-of-life data

OBJECTIVE

To investigate how a multilevel item response theory (IRT) model for longitudinal dependent data could provide average and individual quality-of-life outcomes of low-vision rehabilitation.

STUDY DESIGN AND SETTING

In a nonrandomized longitudinal design, visually impaired older patients (n=296) were referred to multidisciplinary rehabilitation or to an optometric service. The five-dimensional Low Vision Quality of Life Questionnaire was administered at four time points. The IRT model was characterized by the graded response model for rating scales. Covariates were added to the model, mainly to correct for missing data. The invariance assumption across time points was investigated.

RESULTS

Average and individual rehabilitation effects were estimated. For multidisciplinary rehabilitation, significant average deterioration was seen on three dimensions after 4.4 years. Many individuals in the optometric service group significantly improved on the "reading small print" dimension (18.5%); in both groups, many individuals significantly deteriorated on "visual (motor) skills" (22.2-30.0%). Invariance across time points could be assumed for all dimensions, except for "adjustment." Gender, education, visual acuity, and health status were significantly associated with the outcome.

CONCLUSION

We present how a multilevel IRT model can be applied to describe longitudinal dependent vision-related quality-of-life data, while focusing on average and individual effects.

Psychometric approaches for developing commensurate measures across independent studies: traditional and new models

When conducting an integrative analysis of data obtained from multiple independent studies, a fundamental problem is to establish commensurate measures for the constructs of interest. Fortunately, procedures for evaluating and establishing measurement equivalence across samples are well developed for the linear factor model and commonly used item response theory models. A newly proposed moderated nonlinear factor analysis model generalizes these models and procedures, allowing for items of different scale types (continuous or discrete) and differential item functioning across levels of categorical and/or continuous variables. The potential of this new model to resolve the problem of measurement in integrative data analysis is shown via an empirical example examining changes in alcohol involvement from ages 10 to 22 years across 2 longitudinal studies.

Modeling life-span growth curves of cognition using longitudinal data with multiple samples and changing scales of measurement

The authors use multiple-sample longitudinal data from different test batteries to examine propositions about changes in constructs over the life span. The data come from 3 classic studies on intellectual abilities in which, in combination, 441 persons were repeatedly measured as many as 16 times over 70 years. They measured cognitive constructs of vocabulary and memory using 8 age-appropriate intelligence test batteries and explore possible linkage of these scales using item response theory (IRT). They simultaneously estimated the parameters of both IRT and latent curve models based on a joint model likelihood approach (i.e., NLMIXED and WINBUGS). They included group differences in the model to examine potential interindividual differences in levels and change. The resulting longitudinal invariant Rasch test analyses lead to a few new methodological suggestions for dealing with repeated constructs based on changing measurements in developmental studies.

Solution Strategies and Achievement in Dutch Complex Arithmetic: Latent Variable Modeling of Change

In the Netherlands, national assessments at the end of primary school (Grade 6) show a decline of achievement on problems of complex or written arithmetic over the last two decades. The present study aims at contributing to an explanation of the large achievement decrease on complex division, by investigating the strategies students used in solving the division problems in the two most recent assessments carried out in 1997 and in 2004. The students' strategies were classified into four categories. A data set resulted with two types of repeated observations within students: the nominal strategies and the dichotomous achievement scores (correct/incorrect) on the items administered.It is argued that latent variable modeling methodology is appropriate to analyze these data. First, latent class analyses with year of assessment as a covariate were carried out on the multivariate nominal strategy variables. Results showed a shift from application of the traditional long division algorithm in 1997, to the less accurate strategy of stating an answer without writing down any notes or calculations in 2004, especially for boys. Second, explanatory IRT analyses showed that the three main strategies were significantly less accurate in 2004 than they were in 1997.

Replenishing a computerized adaptive test of patient-reported daily activity functioning

PURPOSE

Computerized adaptive testing (CAT) item banks may need to be updated, but before new items can be added, they must be linked to the previous CAT. The purpose of this study was to evaluate 41 pretest items prior to including them into an operational CAT.

METHODS

We recruited 6,882 patients with spine, lower extremity, upper extremity, and nonorthopedic impairments who received outpatient rehabilitation in one of 147 clinics across 13 states of the USA. Forty-one new Daily Activity (DA) items were administered along with the Activity Measure for Post-Acute Care Daily Activity CAT (DA-CAT-1) in five separate waves. We compared the scoring consistency with the full item bank, test information function (TIF), person standard errors (SEs), and content range of the DA-CAT-1 to the new CAT (DA-CAT-2) with the pretest items by real data simulations.

RESULTS

We retained 29 of the 41 pretest items. Scores from the DA-CAT-2 were more consistent (ICC = 0.90 versus 0.96) than DA-CAT-1 when compared with the full item bank. TIF and person SEs were improved for persons with higher levels of DA functioning, and ceiling effects were reduced from 16.1% to 6.1%.

CONCLUSIONS

Item response theory and online calibration methods were valuable in improving the DA-CAT.

Applying multilevel item response theory to vision-related quality of life in Dutch visually impaired elderly

PURPOSE

Instead of applying the usual longitudinal methods to assess the outcome of low-vision rehabilitation services in terms of vision-related quality of life, a three-level Item Response Theory (IRT) method was proposed.

METHODS

The translated Vision-Related Quality of Life Core Measure (VCM1) and Low Vision Quality Of Life (LVQOL) questionnaires were used in a nonrandomized follow-up study among elderly patients (n = 296) referred to two different low-vision rehabilitation services in the Netherlands. Factor analysis was performed on the matrix of polychoric correlations to investigate (uni-)dimensionality and to prepare both questionnaires for the multilevel IRT analyses. A statistical model, which was characterized by a graded response model for rating scales, was developed. Threshold and item difficulty parameters and group by time-specific mean fixed effects were estimated. Random individual effects were predicted. Measurement invariance across occasions was tested.

RESULTS

The VCM1 and the LVQOL "reading and fine work" dimension showed item parameter drift. In the multidisciplinary rehabilitation center patients, deterioration was found on the "mobility" dimension after 1 year and improvement was found on "adjustment" and "visual (motor) skills" after 5 months (p < 0.05). Patients in both low-vision services showed improvement on the "reading small print" subscale at both follow-up time points (p < 0.05).

CONCLUSIONS

Except for improvement in "reading small print," low-vision rehabilitation services did not seem to contribute substantially to any other dimensions of vision-related quality of life. The results showed a change in only a limited number of individual patients. However, with regard to the field of low-vision rehabilitation, the proposed IRT method seemed to be successful in the follow-up of individuals. IRT specific software was unnecessary. The data did not have to be complete and the use of cumulative logits made the proposed IRT method an economical and efficient approach. Because of item parameter drift, the VCM1 was difficult to interpret. The use of multilevel IRT models with longitudinal data and dependent observations is recommended.

Thurstonian-Based Analyses: Past, Present, and Future Utilities

Current psychometric models of choice behavior are strongly influenced by Thurstone's (1927, 1931) experimental and statistical work on measuring and scaling preferences. Aided by advances in computational techniques, choice models can now accommodate a wide range of different data types and sources of preference variability among respondents induced by such diverse factors as person-specific choice sets or different functional forms for the underlying utility representations. At the same time, these models are increasingly challenged by behavioral work demonstrating the prevalence of choice behavior that is not consistent with the underlying assumptions of these models. I discuss new modeling avenues that can account for such seemingly inconsistent choice behavior and conclude by emphasizing the interdisciplinary frontiers in the study of choice behavior and the resulting challenges for psychometricians.

Statistical inference in generalized linear mixed models: a review

We present a review of statistical inference in generalized linear mixed models (GLMMs). GLMMs are an extension of generalized linear models and are suitable for the analysis of non-normal data with a clustered structure. A GLMM contains parameters common to all clusters (fixed regression effects and variance components) and cluster-specific parameters. The latter parameters are assumed to be randomly drawn from a population distribution. The parameters of this population distribution (the variance components) have to be estimated together with the fixed effects. We focus on the case in which the cluster-specific parameters are normally distributed. The cluster-specific effects are integrated out of the likelihood so that the fixed effects and variance components can be estimated. Unfortunately, the integral over the cluster-specific effects is intractable for most GLMMs with a normal mixing distribution. Within a classical statistical framework, we distinguish between two broad classes of methods to handle this intractable integral: methods that rely on a numerical approximation to the integral and methods that use an analytical approximation to the integrand. Finally, we present an overview of available methods for testing hypotheses about the parameters of GLMMs.

Validating, Improving Reliability, and Estimating Correlation of the Four Subscales in the WHOQOL-BREF using Multidimensional Rasch Analysis

OBJECTIVE

This study examined the construct validity, and improved the test reliability and the estimation accuracy for the correlation between domains of the WHOQOL-BREF using multidimensional Rasch analysis.

METHOD

A total of 13,083 adults were administered the 28-item WHOQOL-BREF Taiwan version, which consists of 4 subscales (domains). The multidimensional form of the partial credit model was used to examine the fit of the 4 subscales. For comparison, each subscale individually was also fitted to the unidimensional partial credit model. Standard item fit statistics and analysis of differential item functioning (DIF) were used to check model-data fit.

RESULTS

After excluding 2 overall items and deleting 7 DIF items, the remaining items of each subscale in the WHOQOL-BREF constituted a single construct. The test reliabilities and correlations between domains obtained from the multidimensional approach, (0.82-0.86) and (0.79-0.89), respectively, were much higher than those obtained from the unidimensional approach, (0.67-0.75) and (0.53-0.65), respectively.

CONCLUSION

The 19-item WHOQOL-BREF measures more succinct latent traits than the original design. The multidimensional approach yields not only more accurate estimates for the correlation between domains but also substantially higher reliabilities, than the standard unidimensional approach.

Using SAS PROC NLMIXED to fit item response theory models

Researchers routinely construct tests or questionnaires containing a set of items that measure personality traits, cognitive abilities, political attitudes, and so forth. Typically, responses to these items are scored in discrete categories, such as points on a Likert scale or a choice out of several mutually exclusive alternatives. Item response theory (IRT) explains observed responses to items on a test (questionnaire) by a person's unobserved trait, ability, or attitude. Although applications of IRT modeling have increased considerably because of its utility in developing and assessing measuring instruments, IRT modeling has not been fully integrated into the curriculum of colleges and universities, mainly because existing general purpose statistical packages do not provide built-in routines with which to perform IRT modeling. Recent advances in statistical theory and the incorporation of those advances into general purpose statistical software such as the Statistical Analysis System (SAS) allow researchers to analyze measurement data by using a class of models known as generalized linear mixed effects models (McCulloch & Searle, 2001), which include IRT models as special cases. The purpose of this article is to demonstrate the generality and flexibility of using SAS to estimate IRT model parameters. With real data examples, we illustrate the implementations of a variety of IRT models for dichotomous, polytomous, and nominal responses. Since SAS is widely available in educational institutions, it is hoped that this article will contribute to the spread of IRT modeling in quantitative courses.

Validity of an abbreviated quality of life enjoyment and satisfaction questionnaire (Q-LES-Q-18) for schizophrenia, schizoaffective, and mood disorder patients

We sought to identify a core subset of Quality of Life Enjoyment and Satisfaction Questionnaire (Q-LES-Q) items that maintains the validity and psychometric properties of the basic version. A parsimonious subset of items from the Q-LES-Q that can accurately predict the basic Q-LES-Q domain mean scores was sought and evaluated in 339 inpatients meeting DSM-IV criteria for schizophrenia, schizoaffective, and mood disorders. Three additional data sets were used for validation. Assessments included Q-LES-Q, Quality of Life Scale, Lancashire Quality of Life Profile, rating scales for psychopathology, medication side effects, and self-reported emotional distress, self-esteem, self-efficacy, and social support. We found that 18-items predicted basic Q-LES-Q domains (physical health, subjective feelings, leisure time activities, social relationships) and general index scores with high accuracy. Q-LES-Q-18 showed high reliability, validity, and stability of test-retest ratings. Thus, Q-LES-Q-18, a brief, self-administered questionnaire may aid in monitoring quality of life outcomes of schizophrenia, schizoaffective, and mood disorder patients.

People are variables too: multilevel structural equations modeling

The article uses confirmatory factor analysis (CFA) as a template to explain didactically multilevel structural equation models (ML-SEM) and to demonstrate the equivalence of general mixed-effects models and ML-SEM. An intuitively appealing graphical representation of complex ML-SEMs is introduced that succinctly describes the underlying model and its assumptions. The use of definition variables (i.e., observed variables used to fix model parameters to individual specific data values) is extended to the case of ML-SEMs for clustered data with random slopes. Empirical examples of multilevel CFA and ML-SEM with random slopes are provided along with scripts for fitting such models in SAS Proc Mixed, Mplus, and Mx. Methodological issues regarding estimation of complex ML-SEMs and the evaluation of model fit are discussed. Further potential applications of ML-SEMs are explored.

Persönlichkeitspsychologie: Stand und Perspektiven

Zusammenfassung. Es wird gezeigt, dass die Persönlichkeitspsychologie eine zur Zeit sehr dynamische sowie interdisziplinär stark vernetzte Teildisziplin der Psychologie ist. Dies geschieht exemplarisch anhand von sechs Forschungsfeldern: (a) Entwicklung statistischer und mathematischer Methoden (z.B. Strukturgleichungsmodelle, Mischverteilungsmodelle), welche an individuellen Unterschieden ansetzen und deren Analyse optimieren; (b) Implizite Assoziationstests als Ergänzung zu Selbstberichten im Rahmen der Persönlichkeitsmessung; (c) Forschungen zu den kognitiven und neuronalen Quellen individueller Unterschiede in der Intelligenz; (d) Forschungen zu biologischen Einflussfaktoren auf Temperamentsmerkmale; (e) quantitative und molekulare Verhaltensgenetik unter besonderer Berücksichtigung der Beziehung zwischen normaler Variation und psychischen Störungen; und (f) Vorhersage von Ausbildungs- und Berufserfolg.

A Mixed-Effects Regression Model for Longitudinal Multivariate Ordinal Data

A mixed-effects item response theory model that allows for three-level multivariate ordinal outcomes and accommodates multiple random subject effects is proposed for analysis of multivariate ordinal outcomes in longitudinal studies. This model allows for the estimation of different item factor loadings (item discrimination parameters) for the multiple outcomes. The covariates in the model do not have to follow the proportional odds assumption and can be at any level. Assuming either a probit or logistic response function, maximum marginal likelihood estimation is proposed utilizing multidimensional Gauss-Hermite quadrature for integration of the random effects. An iterative Fisher scoring solution, which provides standard errors for all model parameters, is used. An analysis of a longitudinal substance use data set, where four items of substance use behavior (cigarette use, alcohol use, marijuana use, and getting drunk or high) are repeatedly measured over time, is used to illustrate application of the proposed model.

Condensed version of the Quality of Life Scale for schizophrenia for use in outcome studies

The Quality of Life Scale (QLS(21)) is widely used in clinical trials involving schizophrenia patients. This study aimed to identify a core subset of QLS(21) items that maintains the validity and psychometric properties of the complete version. A parsimonious subset of items from the QLS(21) that can accurately predict the total scale score was sought and evaluated in 133 schizophrenia patients, using the heuristic algorithm for a regression model. Two additional data sets were used for model validation: a subset of 124 patients who participated in the model construction and who completed the QLS(21) 1 year later as well as a new sample of 40 inpatients. Patients were examined with the Positive and Negative Syndrome Scale (PANSS), the Quality of Life Enjoyment and Satisfaction Questionnaire (Q-LES-Q), the Talbieh Brief Distress Inventory (TBDI), the Extrapyramidal Symptom Rating Scale (ESRS) and the Global Assessment of Functioning Scale (GAF). Using only five QLS items (social initiatives, adequacy, acquaintances, motivation, and time utilization; QLS(5)) as predictors, the correlation was 0.9805 between the predicted and true QLS totals. Two validation samples confirmed this finding. Additional analyses indicate that the QLS(5) exhibited similar performance to the QLS(21) regarding construct validity, test-retest reliability and responsiveness to changes over time. Thus, the five-item condensed Quality of Life Scale for schizophrenia maintains the validity of the full QLS, and has the advantage of shorter administration time. Utilization of the revised QLS(5) in routine care and clinical trials may potentially facilitate evaluation of treatment outcomes in schizophrenia.

A Conceptual and Psychometric Framework for Distinguishing Categories and Dimensions

An important, sometimes controversial feature of all psychological phenomena is whether they are categorical or dimensional. A conceptual and psychometric framework is described for distinguishing whether the latent structure behind manifest categories (e.g., psychiatric diagnoses, attitude groups, or stages of development) is category-like or dimension-like. Being dimension-like requires (a) within-category heterogeneity and (b) between-category quantitative differences. Being category-like requires (a) within-category homogeneity and (b) between-category qualitative differences. The relation between this classification and abrupt versus smooth differences is discussed. Hybrid structures are possible. Being category-like is itself a matter of degree; the authors offer a formalized framework to determine this degree. Empirical applications to personality disorders, attitudes toward capital punishment, and stages of cognitive development illustrate the approach.

Estimation of the MIRID: A program and a SAS-based approach

The MIRID CML program is a program for the estimation of the parameter values of two different componential IRT models: the Rasch-MIRID and the OPLM-MIRID (Butter, 1994; Butter, De Boeck, & Verhelst, 1998). To estimate the parameters of both models, the program uses a CML approach. The model parameters can also be estimated with a MML approach that can be implemented in PROC NLMIXED of SAS Version 8. Both the MIRID CML program and the MML SAS approach are explained and compared in a simulation study. The results showed that they did about equally well in estimating the values of the item parameters but that there were some differences in the estimation of the person parameters, as could be expected from the differential assumptions regarding the distribution of the persons. The SAS MML approach is much slower than the MIRID CML program, but it is more flexible.