Quantitative psychology is concerned with the development and application of mathematical models in the behavioral sciences. Over time, models have become more complex, a consequence of the increasing complexity of research designs and experimental data, which is also a consequence of the utility of mathematical models in the science. As models have become more elaborate, the problems of estimating them have become increasingly challenging. This paper gives an introduction to a computing tool called automatic differentiation that is useful in calculating derivatives needed to estimate a model. As its name implies, automatic differentiation works in a routine way to produce derivatives accurately and quickly. Because so many features of model development require derivatives, the method has considerable potential in psychometric work. This paper reviews several examples to demonstrate how the methodology can be applied.

Two examples of behavioral measurement are explored—utility theory and a global psychophysical theory of intensity—that closely parallel the foundations of classical physical measurement in several ways. First, the qualitative attribute in question can be manipulated in two independent ways. Second, each method of manipulation is axiomatized and each leads to a measure of the attribute that, because they are order preserving, must be strictly monotonically related. Third, a law-like constraint, somewhat akin to the distribution property underlying, e.g., mass measurement, links the two types of manipulation. Fourth, given the numerical measures that result from each manipulation, the linking law between them can be recast as a functional equation that establishes the connection between the two measures of the same attribute. Fifth, a major difference from most physical measurement is that the resulting measures are themselves mathematical functions of underlying physical variables—of money and probability in the utility case and of physical intensity and numerical proportions in the psychophysical case. Axiomatizing these functions, although still problematic, appears to lead to interesting results and to limit the degrees of freedom in the representations.

Both the speed and accuracy of responding are important measures of performance. A well-known interpretive difficulty is that participants may differ in their strategy, trading speed for accuracy, with no change in underlying competence. Another difficulty arises when participants respond slowly and inaccurately (rather than quickly but inaccurately), e.g., due to a lapse of attention. We introduce an approach that combines response time and accuracy information and addresses both situations. The modeling framework assumes two latent competing processes. The first, the error-free process, always produces correct responses. The second, the guessing process, results in all observed errors and some of the correct responses (but does so via non-specific processes, e.g., guessing in compliance with instructions to respond on each trial). Inferential summaries of the speed of the error-free process provide a principled assessment of cognitive performance reducing the influences of both fast and slow guesses. Likelihood analysis is discussed for the basic model and extensions. The approach is applied to a data set on response times in a working memory test.

This paper focuses on model interpretation issues and employs a geometric approach to compare the potential value of using the Grade of Membership (GoM) model in representing population heterogeneity. We consider population heterogeneity manifolds generated by letting subject specific parameters vary over their natural range, while keeping other population parameters fixed, in the marginal space (based on marginal probabilities) and in the full parameter space (based on cell probabilities). The case of a 2 × 2 contingency table is discussed in detail, and a generalization to 2J tables with J ≥ 3 is sketched. Our approach highlights the main distinction between the GoM model and the probabilistic mixture of classes by demonstrating geometrically the difference between the concepts of partial and probabilistic memberships. By using the geometric approach we show that, in special cases, the GoM model can be thought of as being similar to an item response theory (IRT) model in representing population heterogeneity. Finally, we show that the GoM item parameters can provide quantities analogous to more general logistic IRT item parameters. As a latent structure model, the GoM model might be considered a useful alternative for a data analysis when both classes of extreme responses, and additional heterogeneity that cannot be captured by those latent classes, are expected in the population.

Glickman, Gray, and Morales (this issue) propose a statistical model for measuring the unobserved latency of stimulus-controlled processes. The model accounts for both speed and accuracy and does so by assuming that participants set an internal deadline. If a stimulus-controlled response is not produced by the deadline, the participant then guesses. The applicability of the model is discussed in this comment. The deadline model yields specific predictions for the case in which stimulus difficulty is manipulated in a within-block manner. In this case, it is reasonable to assume that stimulus difficulty does not affect the deadline. It is shown that in common perceptual and cognitive domains, extant data do not fully meet these predictions. Hence, practitioners need be aware of the possibility and consequences of model misspecification.

Rasch proposed an exact conditional inference approach to testing his model but never implemented it because it involves the calculation of a complicated probability. This paper furthers Rasch’s approach by (1) providing an efficient Monte Carlo methodology for accurately approximating the required probability and (2) illustrating the usefulness of Rasch’s approach for several important testing problems through simulation studies. Our Monte Carlo methodology is shown to compare favorably to other Monte Carlo methods proposed for this problem in two respects: it is considerably faster and it provides more reliable estimates of the Monte Carlo standard error.

There are two well-known methods for obtaining a guaranteed globally optimal solution to the problem of least-squares unidimensional scaling of a symmetric dissimilarity matrix: (a) dynamic programming, and (b) branch-and-bound. Dynamic programming is generally more efficient than branch-and-bound, but the former is limited to matrices with approximately 26 or fewer objects because of computer memory limitations. We present some new branch-and-bound procedures that improve computational efficiency, and enable guaranteed globally optimal solutions to be obtained for matrices with up to 35 objects. Experimental tests were conducted to compare the relative performances of the new procedures, a previously published branch-and-bound algorithm, and a dynamic programming solution strategy. These experiments, which included both synthetic and empirical dissimilarity matrices, yielded the following findings: (a) the new branch-and-bound procedures were often drastically more efficient than the previously published branch-and-bound algorithm, (b) when computationally feasible, the dynamic programming approach was more efficient than each of the branch-and-bound procedures, and (c) the new branch-and-bound procedures require minimal computer memory and can provide optimal solutions for matrices that are too large for dynamic programming implementation.

Lazraq and Cléroux (Psychometrika, 2002, 411–419) proposed a test for identifying the number of significant components in redundancy analysis. This test, however, is ill-conceived. A major problem is that it regards each redundancy component as if it were a single observed predictor variable, which cannot be justified except for the rare situations in which there is only one predictor variable. Consequently, the proposed test leads to drastically biased results, particularly when the number of predictor variables is large, and it cannot be recommended for use. This is shown both theoretically and by Monte Carlo studies.

In this paper we propose two interpretations for the discrimination parameter in the two-parameter logistic model (2PLM). The interpretations are based on the relation between the 2PLM and two stochastic models. In the first interpretation, the 2PLM is linked to a diffusion model so that the probability of absorption equals the 2PLM. The discrimination parameter is the distance between the two absorbing boundaries and therefore the amount of information that has to be collected before a response to an item can be given. For the second interpretation, the 2PLM is connected to a specific type of race model. In the race model, the discrimination parameter is inversely related to the dependency of the information used in the decision process. Extended versions of both models with person-to-person variability in the difficulty parameter are considered. When fitted to a data set, it is shown that a generalization of the race model that allows for dependency between choices and response times (RTs) is the best-fitting model.

The assumptions underlying item response theory (IRT) models may be expressed as a set of equality and inequality constraints on the parameters of a latent class model. It is well known that the same assumptions imply that the parameters of the manifest distribution have to satisfy a more complicated set of inequality constraints which, however, are necessary but not sufficient. In this paper, we describe how the theory for likelihood-based inference under equality and inequality constraints may be used to test the underlying assumptions of IRT models. It turns out that the analysis based directly on the latent structure is simpler and more flexible. In particular, we indicate how several interesting extensions of the Rasch model may be obtained by partial relaxation of the basic constraints. An application to a data set provided by Educational Testing Service is used to illustrate the approach.

For the exploratory analysis of a matrix of proximities or (dis)similarities between objects, one often uses cluster analysis (CA) or multidimensional scaling (MDS). Solutions resulting from such analyses are sometimes interpreted using external information on the objects. Usually the procedures of CA, MDS and using external information are carried out independently and sequentially, although combinations of two of the three procedures (CA and MDS, or MDS and using external information) have been proposed in the literature. The present paper offers a procedure that combines all three procedures in one analysis, using a model that describes a partition of objects with cluster centroids represented in a low-dimensional space, which in turn is related to the information in the external variables. A simulation study is carried out to demonstrate that the method works satisfactorily for data with a known underlying structure. Also, to illustrate the method, it is applied to two empirical data sets.

In this paper, we reconsider the merits of unfolding solutions based on loss functions involving a normalization on the variance per subject. In the literature, solutions based on Stress-2 are often diagnosed to be degenerate in the majority of cases. Here, the focus lies on two frequently occurring types of degeneracies. The first type typically locates some subject points far away from a compact cluster of the other points. In the second type of solution, the object points lie on a circle. In this paper, we argue that these degenerate solutions are well fitting and informative. To reveal the information, we introduce mixtures of plots based on the ideal point model of unfolding, the vector model, and on the signed distance model. In addition to a different representation, we provide a new iterative majorization algorithm to optimize the average squared correlation between the distances in the configuration and the transformed data per individual. It is shown that this approach is equivalent to minimizing Kruskal’s Stress-2.

An IRT model with a parameter-driven process for change is proposed. Quantitative differences between persons are taken into account by a continuous latent variable, as in common IRT models. In addition, qualitative interindividual differences and autodependencies are accounted for by assuming within-subject variability with respect to the parameters of the IRT model. In particular, the parameters of the IRT model are governed by an unobserved or “hidden'” homogeneous Markov process. The model includes the mixture linear logistic test model (Mislevy & Verhelst, 1990), the mixture Rasch model (Rost, 1990), and the Saltus model (Wilson, 1989) as specific instances. The model is applied to a longitudinal experiment on discontinuity in conservation acquisition (van der Maas, 1993).

The sum score is often used to order respondents on the latent trait measured by the test. Therefore, it is desirable that under the chosen model the sum score stochastically orders the latent trait. It is known that unlike dichotomous item response theory (IRT) models, most polytomous IRT models do not imply stochastic ordering. It is unknown, however, (1) whether stochastic ordering is often or rarely violated and (2) whether violations yield a serious problem for practical data analysis. These are the central issues of this paper. First, some unanswered questions that pertain to polytomous IRT models implying stochastic ordering were investigated. Second, simulation studies were conducted to evaluate stochastic ordering in practical situations. It was found that for most polytomous IRT models that do not imply stochastic ordering, the sum score can be used safely to order respondents on the latent trait.

Two new tests for a model for the response times on pure speed tests by Rasch (1960) are proposed. The model is based on the assumption that the test response times are approximately gamma distributed, with known index parameters and unknown rate parameters. The rate parameters are decomposed in a subject ability parameter and a test difficulty parameter. By treating the ability as a gamma distributed random variable, maximum marginal likelihood (MML) estimators for the test difficulty parameters and the parameters of the ability distribution are easily derived. Also the model tests proposed here pertain to the framework of MML. Two tests or modification indices are proposed. The first one is focused on the assumption of local stochastic independence, the second one on the assumption of the test characteristic functions. The tests are based on Lagrange multiplier statistics, and can therefore be computed using the parameter estimates under the null model. Therefore, model violations for all items and pairs of items can be assessed as a by-product of one single estimation run. Power studies and applications to real data are included as numerical examples.

Complete response vectors of all answer options in multiple-choice items can be used to estimate ability. The rising selection ratios criterion is necessary for scoring individuals because it implies that estimated ability always increases when the correct alternative is selected. This paper introduces the generalized DLT model, which assumes rising selection ratios and uses three parameters to describe each incorrect alternative. It is shown that the nominal categories and the Thissen and Steinberg model do not meet the criterion. A simulation study on goodness of recovery and an example with real data are also included.

Multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by approximately equal distances between points from different sets. A definition of an absolutely degenerate solution is proposed, which makes clear that these solutions only occur when an intercept is present in the transformation function. Many solutions for the degeneracy problem have been proposed and tested, but with little success so far. In this paper, we offer a substantial modification of an approach initiated bythat introduced a normalization factor based on thevariance in the usual least squares loss function. Heiser unpublishedthesis, (1981) and showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies. The factor proposed in the present paper, based on the coefficient of variation, discourages or penalizes nonmetric transformations of the proximities with small variation, so that the procedure steers away from solutions with small variation in the interpoint distances. An algorithm is described for minimizing the re-adjusted loss function, based on iterative majorization. The results of a simulation study are discussed, in which the optimal range of the penalty parameters is determined. Two empirical data sets are analyzed by our method, clearly showing the benefits of the proposed loss function.

Several hierarchical classes models can be considered for the modeling of three-way three-mode binary data, including the INDCLAS model (Leenen, Van Mechelen, De Boeck, and Rosenberg, 1999), the Tucker3-HICLAS model (Ceulemans, Van Mechelen, and Leenen, 2003), the Tucker2-HICLAS model (Ceulemans and Van Mechelen, 2004), and the Tucker1-HICLAS model that is introduced in this paper. Two questions then may be raised: (1) how are these models interrelated, and (2) given a specific data set, which of these models should be selected, and in which rank? In the present paper, we deal with these questions by (1) showing that the distinct hierarchical classes models for three-way three-mode binary data can be organized into a partially ordered hierarchy, and (2) by presenting model selection strategies based on extensions of the well-known scree test and on the Akaike information criterion. The latter strategies are evaluated by means of an extensive simulation study and are illustrated with an application to interpersonal emotion data. Finally, the presented hierarchy and model selection strategies are related to corresponding work by Kiers (1991) for principal component models for three-way three-mode real-valued data.