This paper uses log-linear models with latent variables (Hagenaars, in Loglinear Models with Latent Variables, 1993) to define a family of cognitive diagnosis models. In doing so, the relationship between many common models is explicitly defined and discussed. In addition, because the log-linear model with latent variables is a general model for cognitive diagnosis, new alternatives to modeling the functional relationship between attribute mastery and the probability of a correct response are discussed.

A Bayesian nonparametric model is introduced for score equating. It is applicable to all major equating designs, and has advantages over previous equating models. Unlike the previous models, the Bayesian model accounts for positive dependence between distributions of scores from two tests. The Bayesian model and the previous equating models are compared through the analysis of data sets famous in the equating literature. Also, the classical percentile-rank, linear, and mean equating models are each proven to be a special case of a Bayesian model under a highly-informative choice of prior distribution.

When data are not missing at random (NMAR), maximum likelihood (ML) procedure will not generate consistent parameter estimates unless the missing data mechanism is correctly modeled. Understanding NMAR mechanism in a data set would allow one to better use the ML methodology. A survey or questionnaire may contain many items; certain items may be responsible for NMAR values in other items. The paper develops statistical procedures to identify the responsible items. By comparing ML estimates (MLE), statistics are developed to test whether the MLEs are changed when excluding items. The items that cause a significant change of the MLEs are responsible for the NMAR mechanism. Normal distribution is used for obtaining the MLEs; a sandwich-type covariance matrix is used to account for distribution violations. The class of nonnormal distributions within which the procedure is valid is provided. Both saturated and structural models are considered. Effect sizes are also defined and studied. The results indicate that more missing data in a sample does not necessarily imply more significant test statistics due to smaller effect sizes. Knowing the true population means and covariances or the parameter values in structural equation models may not make things easier either.

The generalized logit–linear item response model (GLLIRM) is a linearly constrained nominal categories model (NCM) that computes the scale and intercept parameters for categories as a weighted sum of basic parameters. This paper addresses the problems of the identifiability of the basic parameters and the equivalence between different GLLIRM models. It is shown that the identifiability of the basic parameters depends on the size and rank of the coefficient matrix of the linear functions. Moreover, two models are observationally equivalent if the product of the respective coefficient matrices has full column rank. Finally, the paper also explores the relations between the parameters of nested models.

Several criteria from the optimal design literature are examined for use with item selection in multidimensional adaptive testing. In particular, it is examined what criteria are appropriate for adaptive testing in which all abilities are intentional, some should be considered as a nuisance, or the interest is in the testing of a composite of the abilities. Both the theoretical analyses and the studies of simulated data in this paper suggest that the criteria of A-optimality and D-optimality lead to the most accurate estimates when all abilities are intentional, with the former slightly outperforming the latter. The criterion of E-optimality showed occasional erratic behavior for this case of adaptive testing, and its use is not recommended. If some of the abilities are nuisances, application of the criterion of As-optimality (or Ds-optimality), which focuses on the subset of intentional abilities is recommended. For the measurement of a linear combination of abilities, the criterion of c-optimality yielded the best results. The preferences of each of these criteria for items with specific patterns of parameter values was also assessed. It was found that the criteria differed mainly in their preferences of items with different patterns of values for their discrimination parameters.

In this paper, we propose a cluster-MDS model for two-way one-mode continuous rating dissimilarity data. The model aims at partitioning the objects into classes and simultaneously representing the cluster centers in a low-dimensional space. Under the normal distribution assumption, a latent class model is developed in terms of the set of dissimilarities in a maximum likelihood framework. In each iteration, the probability that a dissimilarity belongs to each of the blocks conforming to a partition of the original dissimilarity matrix, and the rest of parameters, are estimated in a simulated annealing based algorithm. A model selection strategy is used to test the number of latent classes and the dimensionality of the problem. Both simulated and classical dissimilarity data are analyzed to illustrate the model.

Ideal point discriminant analysis is a classification tool which uses highly intuitive multidimensional scaling procedures. However, in the last paper, Takane wrote about it. He concludes that the interpretation is rather intricate and calls that a weakness of the model. We summarize the conditions that provide an easy interpretation and show that in maximum dimensionality they can be obtained without any loss. For reduced dimensionality, it is conjectured that loss is minor which is examined using several data sets.

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.

This article discusses large-scale assessment of change in student achievement and takes the study by Hickendorff, Heiser, Van Putten, and Verhelst (2009) as an example. This study compared the achievement of students in the Netherlands in 1997 and 2004 on written division problems. Based on this comparison, they claim that there is a performance decline in this subdomain of mathematics, and that there is a move from applying the digit-based long division algorithm to a less accurate way of working without writing down anything. In our discussion of this study, we address methodological challenges that come in when investigating long-term trends in student achievements, such as the need for adequate operationalizations, the influence of the time of measurement and the necessity of the comparability of assessments, the effect of the assessment format, and the importance of inclusion relevant covariates in item response models. All these issues matter when assessing change in student achievement.

In this rejoinder, we discuss substantive and methodological validity issues of large-scale assessments of trends in student achievement, commenting on the discussion paper by Van den Heuvel-Panhuizen, Robitzsch, Treffers, and Köller (2009). We focus on methodological challenges in deciding what to measure, how to measure it, and how to foster stability. Next, we discuss what to do with trends that are found. Finally, we reflect on how the research findings were received.