A number of mathematical models for overcoming intransitive choice have been proposed and tested in the literature of decision theory. This article presents the development of a new stochastic choice model based on multidimensional scaling. This allows decision-makers to have multiple viewpoints, whereas current multidimensional scaling models are based on the assumption that a subject or group of subjects has only one viewpoint. The implication of our model is that subjects make an intransitive choice because they are able to shift their viewpoint. This paper also presents the maximum likelihood estimation of the proposed model, and reanalyzes Tversky’s gamble experiment data.

Two algorithms based on a latent class model are presented for discovering hierarchical relations that exist among a set of K dichotomous items. The two algorithms, stepwise forward selection and backward elimination, incorporate statistical criteria for selecting (or deleting) 0-1 response pattern vectors to form the subset of the total possible 2k vectors that uniquely describe the hierarchy. The performances of the algorithms are compared, using computer-constructed data, with those of three competing deterministic approaches based on ordering theory and the calculation of Phi/Phi-max coefficients. The discovery algorithms are also demonstrated on real data sets investigated in the literature.

Higher-order latent traits are proposed for specifying the joint distribution of binary attributes in models for cognitive diagnosis. This approach results in a parsimonious model for the joint distribution of a high-dimensional attribute vector that is natural in many situations when specific cognitive information is sought but a less informative item response model would be a reasonable alternative. This approach stems from viewing the attributes as the specific knowledge required for examination performance, and modeling these attributes as arising from a broadly-defined latent trait resembling the ϑ of item response models. In this way a relatively simple model for the joint distribution of the attributes results, which is based on a plausible model for the relationship between general aptitude and specific knowledge. Markov chain Monte Carlo algorithms for parameter estimation are given for selected response distributions, and simulation results are presented to examine the performance of the algorithm as well as the sensitivity of classification to model misspecification. An analysis of fraction subtraction data is provided as an example.

We illustrate a class of multidimensional item response theory models in which the items are allowed to have different discriminating power and the latent traits are represented through a vector having a discrete distribution. We also show how the hypothesis of unidimensionality may be tested against a specific bidimensional alternative by using a likelihood ratio statistic between two nested models in this class. For this aim, we also derive an asymptotically equivalent Wald test statistic which is faster to compute. Moreover, we propose a hierarchical clustering algorithm which can be used, when the dimensionality of the latent structure is completely unknown, for dividing items into groups referred to different latent traits. The approach is illustrated through a simulation study and an application to a dataset collected within the National Assessment of Educational Progress, 1996.

Computerized adaptive testing (CAT) is a mode of testing which enables more efficient and accurate recovery of one or more latent traits. Traditionally, CAT is built upon Item Response Theory (IRT) models that assume unidimensionality. However, the problem of how to build CAT upon latent class models (LCM) has not been investigated until recently, when Tatsuoka (J. R. Stat. Soc., Ser. C, Appl. Stat. 51:337–350, 2002) and Tatsuoka and Ferguson (J. R. Stat., Ser. B 65:143–157, 2003) established a general theorem on the asymptotically optimal sequential selection of experiments to classify finite, partially ordered sets. Xu, Chang, and Douglas (Paper presented at the annual meeting of National Council on Measurement in Education, Montreal, Canada, 2003) then tested two heuristics in a simulation study based on Tatsuoka’s theoretical work in the context of computerized adaptive testing. One of the heuristics was developed based on Kullback–Leibler information, and the other based on Shannon entropy. In this paper, we showcase the application of the optimal sequential selection methodology in item selection of CAT that is built upon cognitive diagnostic models. Two new heuristics are proposed, and are compared against the randomized item selection method and the two heuristics investigated in Xu et al. (Paper presented at the annual meeting of National Council on Measurement in Education, Montreal, Canada, 2003). Finally, we show the connection between the Kullback–Leibler-information-based approaches and the Shannon-entropy-based approach, as well as the connection between algorithms built upon LCM and those built upon IRT models.

This paper studies three models for cognitive diagnosis, each illustrated with an application to fraction subtraction data. The objective of each of these models is to classify examinees according to their mastery of skills assumed to be required for fraction subtraction. We consider the DINA model, the NIDA model, and a new model that extends the DINA model to allow for multiple strategies of problem solving. For each of these models the joint distribution of the indicators of skill mastery is modeled using a single continuous higher-order latent trait, to explain the dependence in the mastery of distinct skills. This approach stems from viewing the skills as the specific states of knowledge required for exam performance, and viewing these skills as arising from a broadly defined latent trait resembling the θ of item response models. We discuss several techniques for comparing models and assessing goodness of fit. We then implement these methods using the fraction subtraction data with the aim of selecting the best of the three models for this application. We employ Markov chain Monte Carlo algorithms to fit the models, and we present simulation results to examine the performance of these algorithms.

The present work explores the connections between cognitive diagnostic models (CDM) and knowledge space theory (KST) and shows that these two quite distinct approaches overlap. It is proved that in fact the Multiple Strategy DINA (Deterministic Input Noisy AND-gate) model and the CBLIM, a competence-based extension of the basic local independence model (BLIM), are equivalent. To demonstrate the benefits that arise from integrating the two theoretical perspectives, it is shown that a fairly complete picture on the identifiability of these models emerges by combining results from both camps. The impact of the results is illustrated by an empirical example, and topics for further research are pointed out.

A general model is presented for homogeneous, dichotomous items when the answer key is not known a priori. The model is structurally related to the two-class latent structure model with the roles of respondents and items interchanged. For very small sets of respondents, iterative maximum likelihood estimates of the parameters can be obtained by existing methods. For other situations, new estimation methods are developed and assessed with Monte Carlo data. The answer key can be accurately reconstructed with relatively small sets of respondents. The model is useful when a researcher wants to study objectively the knowledge possessed by members of a culturally coherent group that the researcher is not a member of.

A multilevel latent transition analysis (LTA) with a mixture IRT measurement model (MixIRTM) is described for investigating the effectiveness of an intervention. The addition of a MixIRTM to the multilevel LTA permits consideration of both potential heterogeneity in students’ response to instructional intervention as well as a methodology for assessing stage sequential change over time at both student and teacher levels. Results from an LTA–MixIRTM and multilevel LTA–MixIRTM were compared in the context of an educational intervention study. Both models were able to describe homogeneities in problem solving and transition patterns. However, ignoring a multilevel structure in LTA–MixIRTM led to different results in group membership assignment in empirical results. Results for the multilevel LTA–MixIRTM indicated that there were distinct individual differences in the different transition patterns. The students receiving the intervention treatment outscored their business as usual (i.e., control group) counterparts on the curriculum-based Fractions Computation test. In addition, 27.4 % of the students in the sample moved from the low ability student-level latent class to the high ability student-level latent class. Students were characterized differently depending on the teacher-level latent class.

The specification of the $\mathit{Q}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\varvec{Q}}$$\end{document} matrix in cognitive diagnosis models is important for correct classification of attribute profiles. Researchers have proposed many methods for estimation and validation of the data-driven $\mathit{Q}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\varvec{Q}}$$\end{document} matrices. However, inference of the number of attributes in the general restricted latent class model remains an open question. We propose a Bayesian framework for general restricted latent class models and use the spike-and-slab prior to avoid the computation issues caused by the varying dimensions of model parameters associated with the number of attributes, K. We develop an efficient Metropolis-within-Gibbs algorithm to estimate K and the corresponding $\mathit{Q}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\varvec{Q}}$$\end{document} matrix simultaneously. The proposed algorithm uses the stick-breaking construction to mimic an Indian buffet process and employs a novel Metropolis–Hastings transition step to encourage exploring the sample space associated with different values of K. We evaluate the performance of the proposed method through a simulation study under different model specifications and apply the method to a real data set related to a fluid intelligence matrix reasoning test.

The single intradermal comparative cervical tuberculin (SICCT) test and post-mortem examination are the main diagnostic tools for bovine tuberculosis (bTB) in cattle in the British Isles. Latent class modelling is often used to estimate the bTB test characteristics due to the absence of a gold standard. However, the reported sensitivity of especially the SICCT test has shown a lot of variation. We applied both the Hui–Walter latent class model under the Bayesian framework and the Bayesian model specified at the animal level, including various risk factors as predictors, to estimate the SICCT test and post-mortem test characteristics. Data were collected from all cattle slaughtered in abattoirs in Northern Ireland in 2015. Both models showed comparable posterior median estimation for the sensitivity of the SICCT test (88.61% and 90.56%, respectively) using standard interpretation and for post-mortem examination (53.65% and 53.79%, respectively). Both models showed almost identical posterior median estimates for the specificity (99.99% vs. 99.80% for SICCT test at standard interpretation and 99.66% vs. 99.86% for post-mortem examination). The animal-level model showed slightly narrower posterior 95% credible intervals. Notably, this study was carried out in slaughtered cattle which may not be representative for the general cattle population.

In the absence of perfect reference standard, classical techniques result in biased diagnostic accuracy and prevalence estimates. By statistically defining the true disease status, latent class models (LCM) constitute a promising alternative. However, LCM is a complex method which relies on parametric assumptions, including usually a conditional independence between tests and might suffer from data sparseness. We carefully applied LCMs to assess new campylobacter infection detection tests for which bacteriological culture is an imperfect reference standard. Five diagnostic tests (culture, polymerase chain reaction and three immunoenzymatic tests) of campylobacter infection were collected in 623 patients from Bordeaux and Lyon Hospitals, France. Their diagnostic accuracy were estimated with standard and extended LCMs with a thorough examination of models goodness-of-fit. The model including a residual dependence specific to the immunoenzymatic tests best complied with LCM assumptions. Asymptotic results of goodness-of-fit statistics were substantially impaired by data sparseness and empirical distributions were preferred. Results confirmed moderate sensitivity of the culture and high performances of immunoenzymatic tests. LCMs can be used to estimate diagnostic tests accuracy in the absence of perfect reference standard. However, their implementation and assessment require specific attention due to data sparseness and limitations of existing software.

A latent tuberculosis infection (LTBI) prevalence survey was conducted using tuberculin skin test (TST) and Quantiferon test (QFT) in 1218 healthcare workers (HCWs) in Medellín, Colombia. In order to improve the prevalence estimates, a latent class model was built using a Bayesian approach with informative priors on the sensitivity and specificity of the TST. The proportion of concordant results (TST+,QFT+) was 41% and the discordant results contributed 27%. The marginal estimate of the prevalence P(LTBI+) was 62·1% [95% credible interval (CrI) 53·0–68·2]. The probability of LTBI+ given positive results for both tests was 99·6% (95% CrI 98·1–99·9). Sensitivity was 88·5 for TST and 74·3 for QFT, and specificity was 87·8 for TST and 97·6 for QFT. A high LTBI prevalence was found in HCWs with time-accumulated exposure in hospitals that lack control plans. In a context of intermediate tuberculosis (TB) incidence it is recommended to use only one test (either QFT or TST) in prevalence surveys or as pre-employment tests. Results will be useful to help implement TB infection control plans in hospitals where HCWs may be repeatedly exposed to unnoticed TB patients, and to inform the design of TB control policies.

Agricultural production estimates have often differentiated and estimated different technologies within a sample of farms. The common approach is to use observable farm characteristics to split the sample into groups and subsequently estimate different functions for each group. Alternatively, unique technologies can be determined by econometric procedures such as latent class models. This paper compares the results of a latent class model with the use of a priori information to split the sample using dairy farm data. Latent class separation appears to be a superior method of separating heterogeneous technologies and suggests that technology differences are multifaceted.

We investigate heterogeneous consumer preferences and willingness to pay (WTP) for various sweet cherry attributes using choice experiments. A mixed logit model and a latent-class logit model are used to estimate consumer WTP for the attributes and identify groups of consumers based on those preferences. We find that consumers of sweet cherries will pay the greatest premium for sweetness and the smallest premium for fruit size. Three groups of consumers are identified—flavor sensitive, price sensitive, and storage sensitive. The results are useful for suppliers of sweet cherries when adopting targeted marketing strategies.