Starting from an explicit scoring rule for time limit tasks incorporating both response time and accuracy, and a definite trade-off between speed and accuracy, a response model is derived. Since the scoring rule is interpreted as a sufficient statistic, the model belongs to the exponential family. The various marginal and conditional distributions for response accuracy and response time are derived, and it is shown how the model parameters can be estimated. The model for response accuracy is found to be the two-parameter logistic model. It is found that the time limit determines the item discrimination, and this effect is illustrated with the Amsterdam Chess Test II.

The paper proposes a composite likelihood estimation approach that uses bivariate instead of multivariate marginal probabilities for ordinal longitudinal responses using a latent variable model. The model considers time-dependent latent variables and item-specific random effects to be accountable for the interdependencies of the multivariate ordinal items. Time-dependent latent variables are linked with an autoregressive model. Simulation results have shown composite likelihood estimators to have a small amount of bias and mean square error and as such they are feasible alternatives to full maximum likelihood. Model selection criteria developed for composite likelihood estimation are used in the applications. Furthermore, lower-order residuals are used as measures-of-fit for the selected models.

Item replenishing is essential for item bank maintenance in cognitive diagnostic computerized adaptive testing (CD-CAT). In regular CAT, online calibration is commonly used to calibrate the new items continuously. However, until now no reference has publicly become available about online calibration for CD-CAT. Thus, this study investigates the possibility to extend some current strategies used in CAT to CD-CAT. Three representative online calibration methods were investigated: Method A (Stocking in Scale drift in on-line calibration. Research Rep. 88-28, 1988), marginal maximum likelihood estimate with one EM cycle (OEM) (Wainer & Mislevy In H. Wainer (ed.) Computerized adaptive testing: A primer, pp. 65–102, 1990) and marginal maximum likelihood estimate with multiple EM cycles (MEM) (Ban, Hanson, Wang, Yi, & Harris in J. Educ. Meas. 38:191–212, 2001). The objective of the current paper is to generalize these methods to the CD-CAT context under certain theoretical justifications, and the new methods are denoted as CD-Method A, CD-OEM and CD-MEM, respectively. Simulation studies are conducted to compare the performance of the three methods in terms of item-parameter recovery, and the results show that all three methods are able to recover item parameters accurately and CD-Method A performs best when the items have smaller slipping and guessing parameters. This research is a starting point of introducing online calibration in CD-CAT, and further studies are proposed for investigations such as different sample sizes, cognitive diagnostic models, and attribute-hierarchical structures.

Bi-factor analysis is a form of confirmatory factor analysis originally introduced by Holzinger and Swineford (Psychometrika 47:41–54, 1937). The bi-factor model has a general factor, a number of group factors, and an explicit bi-factor structure. Jennrich and Bentler (Psychometrika 76:537–549, 2011) introduced an exploratory form of bi-factor analysis that does not require one to provide an explicit bi-factor structure a priori. They use exploratory factor analysis and a bifactor rotation criterion designed to produce a rotated loading matrix that has an approximate bi-factor structure. Among other things this can be used as an aid in finding an explicit bi-factor structure for use in a confirmatory bi-factor analysis. They considered only orthogonal rotation. The purpose of this paper is to consider oblique rotation and to compare it to orthogonal rotation. Because there are many more oblique rotations of an initial loading matrix than orthogonal rotations, one expects the oblique results to approximate a bi-factor structure better than orthogonal rotations and this is indeed the case. A surprising result arises when oblique bi-factor rotation methods are applied to ideal data.

We consider the identification of a semiparametric multidimensional fixed effects item response model. Item response models are typically estimated under parametric assumptions about the shape of the item characteristic curves (ICCs), and existing results suggest difficulties in recovering the distribution of individual characteristics under nonparametric assumptions. We show that if the shape of the ICCs are unrestricted, but the shape is common across individuals and items, the individual characteristics are identified. If the shape of the ICCs are allowed to differ over items, the individual characteristics are identified in the multidimensional linear compensatory case but only identified up to a monotonic transformation in the unidimensional case. Our results suggest the development of two new semiparametric estimators for the item response model.

The infinitesimal jackknife, a nonparametric method for estimating standard errors, has been used to obtain standard error estimates in covariance structure analysis. In this article, we adapt it for obtaining standard errors for rotated factor loadings and factor correlations in exploratory factor analysis with sample correlation matrices. Both maximum likelihood estimation and ordinary least squares estimation are considered.

I address two issues that were inspired by my work on the Dutch Committee on Tests and Testing (COTAN). The first issue is the understanding of problems test constructors and researchers using tests have of psychometric knowledge. I argue that this understanding is important for a field, like psychometrics, for which the dissemination of psychometric knowledge among test constructors and researchers in general is highly important. The second issue concerns the identification of psychometric research topics that are relevant for test constructors and test users but in my view do not receive enough attention in psychometrics. I discuss the influence of test length on decision quality in personnel selection and quality of difference scores in therapy assessment, and theory development in test construction and validity research. I also briefly mention the issue of whether particular attributes are continuous or discrete.

The Graded Response Model (GRM; Samejima, Estimation of ability using a response pattern of graded scores, Psychometric Monograph No. 17, Richmond, VA: The Psychometric Society, 1969) can be derived by assuming a linear regression of a continuous variable, Z, on the trait, θ, to underlie the ordinal item scores (Takane & de Leeuw in Psychometrika, 52:393–408, 1987). Traditionally, a normal distribution is specified for Z implying homoscedastic error variances and a normally distributed θ. In this paper, we present the Heteroscedastic GRM with Skewed Latent Trait, which extends the traditional GRM by incorporation of heteroscedastic error variances and a skew-normal latent trait. An appealing property of the extended GRM is that it includes the traditional GRM as a special case. This enables specific tests on the normality assumption of Z. We show how violations of normality in Z can lead to asymmetrical category response functions. The ability to test this normality assumption is beneficial from both a statistical and substantive perspective. In a simulation study, we show the viability of the model and investigate the specificity of the effects. We apply the model to a dataset on affect and a dataset on alexithymia.

The likelihood for generalized linear models with covariate measurement error cannot in general be expressed in closed form, which makes maximum likelihood estimation taxing. A popular alternative is regression calibration which is computationally efficient at the cost of inconsistent estimation. We propose an improved regression calibration approach, a general pseudo maximum likelihood estimation method based on a conveniently decomposed form of the likelihood. It is both consistent and computationally efficient, and produces point estimates and estimated standard errors which are practically identical to those obtained by maximum likelihood. Simulations suggest that improved regression calibration, which is easy to implement in standard software, works well in a range of situations.

Latent class regression models relate covariates and latent constructs such as psychiatric disorders. Though full maximum likelihood estimation is available, estimation is often in three steps: (i) a latent class model is fitted without covariates; (ii) latent class scores are predicted; and (iii) the scores are regressed on covariates. We propose a new method for predicting class scores that, in contrast to posterior probability-based methods, yields consistent estimators of the parameters in the third step. Additionally, in simulation studies the new methodology exhibited only a minor loss of efficiency. Finally, the new and the posterior probability-based methods are compared in an analysis of mobility/exercise.

The issue of compensation in multidimensional response modeling is addressed. We show that multidimensional response models are compensatory in their ability parameters if and only if they are monotone. In addition, a minimal set of assumptions is presented under which the MLEs of the ability parameters are also compensatory. In a recent series of articles, beginning with Hooker, Finkelman, and Schwartzman (2009) in this journal, the second type of compensation was presented as a paradoxical result for certain multidimensional response models, leading to occasional unfairness in maximum-likelihood test scoring. First, it is indicated that the compensation is not unique and holds generally for any multiparameter likelihood with monotone score functions. Second, we analyze why, in spite of its generality, the compensation may give the impression of a paradox or unfairness.

This paper proposes a semiparametric Bayesian framework for the analysis of associations among multivariate longitudinal categorical variables in high-dimensional data settings. This type of data is frequent, especially in the social and behavioral sciences. A semiparametric hierarchical factor analysis model is developed in which the distributions of the factors are modeled nonparametrically through a dynamic hierarchical Dirichlet process prior. A Markov chain Monte Carlo algorithm is developed for fitting the model, and the methodology is exemplified through a study of the dynamics of public attitudes toward science and technology in the United States over the period 1992–2001.

In practice, the sum of the item scores is often used as a basis for comparing subjects. For items that have more than two ordered score categories, only the partial credit model (PCM) and special cases of this model imply that the subjects are stochastically ordered on the common latent variable. However, the PCM is very restrictive with respect to the constraints that it imposes on the data. In this paper, sufficient conditions for the stochastic ordering of subjects by their sum score are obtained. These conditions define the isotonic (nonparametric) PCM model. The isotonic PCM is more flexible than the PCM, which makes it useful for a wider variety of tests. Also, observable properties of the isotonic PCM are derived in the form of inequality constraints. It is shown how to obtain estimates of the score distribution under these constraints by using the Gibbs sampling algorithm. A small simulation study shows that the Bayesian p-values based on the log-likelihood ratio statistic can be used to assess the fit of the isotonic PCM to the data, where model-data fit can be taken as a justification of the use of the sum score to order subjects.

A variety of joint space multidimensional scaling (MDS) methods have been utilized for the spatial analysis of two- or three-way dominance data involving subjects’ preferences, choices, considerations, intentions, etc. so as to provide a parsimonious spatial depiction of the underlying relevant dimensions, attributes, stimuli, and/or subjects’ utility structures in the same joint space representation. We demonstrate that care must be taken with respect to a key assumption in existent joint space MDS models such that all estimated dimensions are utilized by each and every subject in the sample, as this assumption can lead to serious distortions with respect to the derived joint spaces. We develop a new Bayesian dimension selection methodology for the multidimensional unfolding model which accommodates heterogeneity with respect to such dimensional utilization at the individual subject level for the analysis of two or three-way dominance data. A consumer psychology application regarding the preference for Over-the-Counter (OTC) analgesics is provided. We conclude by discussing the practical implications of the results, as well as directions for future research.

In an addendum to his seminal 1969 article Jöreskog stated two sets of conditions for rotational identification of the oblique factor solution under utilization of fixed zero elements in the factor loadings matrix (Jöreskog in Advances in factor analysis and structural equation models, pp. 40–43, 1979). These condition sets, formulated under factor correlation and factor covariance metrics, respectively, were claimed to be equivalent and to lead to global rotational uniqueness of the factor solution. It is shown here that the conditions for the oblique factor correlation structure need to be amended for global rotational uniqueness, and, hence, that the condition sets are not equivalent in terms of unicity of the solution.

Latent trait models for response times in tests have become popular recently. One challenge for response time modeling is the fact that the distribution of response times can differ considerably even in similar tests. In order to reduce the need for tailor-made models, a model is proposed that unifies two popular approaches to response time modeling: Proportional hazard models and the accelerated failure time model with log–normally distributed response times. This is accomplished by resorting to discrete time. The categorization of response time allows the formulation of a response time model within the framework of generalized linear models by using a flexible link function. Item parameters of the proposed model can be estimated with marginal maximum likelihood estimation. Applicability of the proposed approach is demonstrated with a simulation study and an empirical application. Additionally, means for the evaluation of model fit are suggested.

A multinormal partial credit model for factor analysis of polytomously scored items with ordered response categories is derived using an extension of the Dutch Identity (Holland in Psychometrika 55:5–18, 1990). In the model, latent variables are assumed to have a multivariate normal distribution conditional on unweighted sums of item scores, which are sufficient statistics. Attention is paid to maximum likelihood estimation of item parameters, multivariate moments of latent variables, and person parameters. It is shown that the maximum likelihood estimates can be found without the use of numerical integration techniques. More general models are discussed which can be used for testing the model, and it is shown how models with different numbers of latent variables can be tested against each other. In addition, multi-group extensions are proposed, which can be used for testing both measurement invariance and latent population differences. Models and procedures discussed are demonstrated in an empirical data example.

Multidimensional computer adaptive testing (MCAT) can provide higher precision and reliability or reduce test length when compared with unidimensional CAT or with the paper-and-pencil test. This study compared five item selection procedures in the MCAT framework for both domain scores and overall scores through simulation by varying the structure of item pools, the population distribution of the simulees, the number of items selected, and the content area. The existing procedures such as Volume (Segall in Psychometrika, 61:331–354, 1996), Kullback–Leibler information (Veldkamp & van der Linden in Psychometrika 67:575–588, 2002), Minimize the error variance of the linear combination (van der Linden in J. Educ. Behav. Stat. 24:398–412, 1999), and Minimum Angle (Reckase in Multidimensional item response theory, Springer, New York, 2009) are compared to a new procedure, Minimize the error variance of the composite score with the optimized weight, proposed for the first time in this study. The intent is to find an item selection procedure that yields higher precisions for both the domain and composite abilities and a higher percentage of selected items from the item pool. The comparison is performed by examining the absolute bias, correlation, test reliability, time used, and item usage. Three sets of item pools are used with the item parameters estimated from real live CAT data. Results show that Volume and Minimum Angle performed similarly, balancing information for all content areas, while the other three procedures performed similarly, with a high precision for both domain and overall scores when selecting items with the required number of items for each domain. The new item selection procedure has the highest percentage of item usage. Moreover, for the overall score, it produces similar or even better results compared to those from the method that selects items favoring the general dimension using the general model (Segall in Psychometrika 66:79–97, 2001); the general dimension method has low precision for the domain scores. In addition to the simulation study, the mathematical theories for certain procedures are derived. The theories are confirmed by the simulation applications.