We present a multidimensional data analysis framework for the analysis of ordinal response variables. Underlying the ordinal variables, we assume a continuous latent variable, leading to cumulative logit models. The framework includes unsupervised methods, when no predictor variables are available, and supervised methods, when predictor variables are available. We distinguish between dominance variables and proximity variables, where dominance variables are analyzed using inner product models, whereas the proximity variables are analyzed using distance models. An expectation–majorization–minimization algorithm is derived for estimation of the parameters of the models. We illustrate our methodology with three empirical data sets highlighting the advantages of the proposed framework. A simulation study is conducted to evaluate the performance of the algorithm.

In the last few decades, the study of ordinal data in which the variable of interest is not exactly observed but only known to be in a specific ordinal category has become important. To emphasize that the problem is not specific to a specific discipline we will use the neutral term coarsened observation. For single-equation models estimation of the latent linear model by Maximum Likelihood (ML) is routine. But, for higher-dimensional multivariate models it is computationally cumbersome as estimation requires the evaluation of multivariate normal distribution functions on a large scale. Our proposed alternative estimation method, based on the Generalized Method of Moments (GMM), circumvents this multivariate integration problem. It can be implemented by repeated application of standard techniques and provides a simpler and faster approach than the usual ML approach. It is applicable to multiple-equation models with $K$-dimensional error correlation matrices and ${J}_k$ response categories for the kth equation. It also yields a simple method to estimate polyserial and polychoric correlations. Comparison of our method with the outcomes of the Stata ML procedure cmp yields estimates that are not statistically different, while estimation by our method requires only a fraction of the computing time.

This article proposes a new statistical model to infer interpretable population-level preferences from ordinal comparison data. Such data is ubiquitous, e.g., ranked choice votes, top-10 movie lists, and pairwise sports outcomes. Traditional statistical inference on ordinal comparison data results in an overall ranking of objects, e.g., from best to worst, with each object having a unique rank. However, the ranks of some objects may not be statistically distinguishable. This could happen due to insufficient data or to the true underlying object qualities being equal. Because uncertainty communication in estimates of overall rankings is notoriously difficult, we take a different approach and allow groups of objects to have equal ranks or be rank-clustered in our model. Existing models related to rank-clustering are limited by their inability to handle a variety of ordinal data types, to quantify uncertainty, or by the need to pre-specify the number and size of potential rank-clusters. We solve these limitations through our proposed Bayesian Rank-Clustered Bradley–Terry–Luce (BTL) model. We accommodate rank-clustering via parameter fusion by imposing a novel spike-and-slab prior on object-specific worth parameters in the BTL family of distributions for ordinal comparisons. We demonstrate rank-clustering on simulated and real datasets in surveys, elections, and sports analytics.

We introduce a general framework for latent variable modeling, named Generalized Latent Variable Models for Location, Scale, and Shape parameters (GLVM-LSS). This framework extends the generalized linear latent variable model beyond the exponential family distributional assumption and enables the modeling of distributional parameters other than the mean (location parameter), such as scale and shape parameters, as functions of latent variables. Model parameters are estimated via maximum likelihood. We present two real-world applications on public opinion research and educational testing, and evaluate the model’s performance in terms of parameter recovery through extensive simulation studies. Our results suggest that the GLVM-LSS is a valuable tool in applications where modeling higher-order moments of the observed variables through latent variables is of substantive interest. The proposed model is implemented in the R package glvmlss, available online.

Two potential obstacles stand between the observation of a statistical correlation and the design (and deployment) of an effective intervention, omitted variable bias and reverse causality. Whereas the former has received ample attention, comparably scant focus has been devoted to the latter in the methodological literature. Many existing methods for reverse causality testing commence by postulating a structural model that may suffer from widely recognized issues such as the difficulty of properly setting temporal lags, which are critical to model validity. In this article, we draw upon advances in machine learning, specifically the recently established link between causal direction and the effectiveness of semi-supervised learning algorithms, to develop a novel method for reverse causality testing that circumvents many of the assumptions required by traditional methods. Mathematical analysis and simulation studies were carried out to demonstrate the effectiveness of our method. We also performed tests over a real-world dataset to show how our method may be used to identify causal relationships in practice.

Joint correspondence analysis (JCA) is a statistical method for obtaining a low-dimensional representation of multivariate categorical data. It was developed as an alternative to multiple correspondence analysis (MCA). Typically, the solution is visualized through a map that projects the data onto a reduced space. A joint map, which shows both object and category scores in the same space, helps users explore inter- and intra-relationships in objects and categories. However, unlike MCA, current JCA estimation methods do not allow the joint representation of objects and categories on the map, which limits the interpretability of JCA results. To overcome this limitation, we propose a simultaneous object and category score estimation method for JCA while addressing the underestimated variance problem that is inherent in MCA. In the proposed method, JCA parameters are estimated by minimizing the discrepancy between the observed categorical data and the JCA data model, rather than relying on the JCA covariance model used in existing estimation methods. Previous research has shown that JCA is comparable to exploratory factor analysis. We also address the factor-analytic interpretation of JCA solutions in addition to geometric interpretation. Two real data analysis examples are also presented to demonstrate the geometric and factor-analytic interpretations of the JCA solutions.

Bi-factor analysis is a form of confirmatory factor analysis widely used in psychological and educational measurement. The use of a bi-factor model requires specifying an explicit bi-factor structure on the relationship between the observed variables and the group factors. In practice, the bi-factor structure is sometimes unknown, in which case, an exploratory form of bi-factor analysis is needed. Unfortunately, there are few methods for exploratory bi-factor analysis, with the exception of a rotation-based method proposed in Jennrich and Bentler ([2011, Psychometrika 76, pp. 537–549], [2012, Psychometrika 77, pp. 442–454]). However, the rotation method does not yield an exact bi-factor loading structure, even after hard thresholding. In this article, we propose a constraint-based optimization method that learns an exact bi-factor loading structure from data, overcoming the issue with the rotation-based method. The key to the proposed method is a mathematical characterization of the bi-factor loading structure as a set of equality constraints, which allows us to formulate the exploratory bi-factor analysis problem as a constrained optimization problem in a continuous domain and solve the optimization problem with an augmented Lagrangian method. The power of the proposed method is shown via simulation studies and a real data example.

Factor rotation is a crucial step in interpreting the results of exploratory factor analysis. Several rotation methods have been developed for simple structure solutions, but their extensions to bi-factor analysis are often not well established. In this article, we propose a mathematical framework that incorporates customized factor structure as a regularization to produce the optimal orthogonal or oblique rotation. We demonstrate the utility of the framework using examples of simple structure rotation and bi-factor rotation. Through detailed simulations, we show that the new method is accurate and robust in recovering the factor structures and latent correlations when bi-factor analysis is applied. The new method is applied to a test data and a Quality of Life survey data. Results show that our method can reveal bi-factor structures that are consistent with the theories.

Fitting propensity (FP) analysis quantifies model complexity but has been impeded in item response theory (IRT) due to the computational infeasibility of uniformly and randomly sampling multinomial item response patterns under a full-information approach. We adopt a limited-information (LI) approach, wherein we generate data only up to the lower-order margins of the complete item response patterns. We present an algorithm that builds upon classical work on sampling contingency tables with fixed margins by implementing a Sequential Importance Sampling algorithm to Quickly and Uniformly Obtain Contingency tables (SISQUOC). Theoretical justification and comprehensive validation demonstrate the effectiveness of the SISQUOC algorithm for IRT and offer insights into sampling from the complete data space defined by the lower-order margins. We highlight the efficiency and simplicity of the LI approach for generating large and uniformly random datasets of dichotomous and polytomous items. We further present an iterative proportional fitting procedure to reconstruct joint multinomial probabilities after LI-based data generation, facilitating FP evaluation using traditional estimation strategies. We illustrate the proposed approach by examining the FP of the graded response model and generalized partial credit model, with results suggesting that their functional forms express similar degrees of configural complexity.

This article introduces item response models for rating relational data. The relational data are obtained via ratings of senders and receivers in a directed network. The proposed models allow comparisons of senders and receivers on a one-dimensional latent scale while accounting for unobserved homophilic relationships. We show that the approach effectively captures reciprocity and clustering phenomena in the relational data. We estimate model parameters using a Bayesian specification and utilize Markov Chain Monte Carlo methods to approximate the full conditional posterior distributions. Simulation studies demonstrate that model parameters can be recovered satisfactorily even when the dimensionality of the network is small. We also present an extensive empirical application to illustrate the usefulness of the proposed models for complete and incomplete networks.

Factor score indeterminacy is a characteristic property of factor analysis (FA) models. This research introduces a novel procedure, regression-based factor score exploration (RFE), which uniquely determines factor scores and simultaneously estimates other parameters of the FA model. RFE uniquely determines factor scores by minimizing a loss function that balances FA and multivariate regression, regulated by a tuning parameter. Theoretical aspects of RFE, including the uniqueness of factor scores, the relationship between observed and latent variables, and rotational indeterminacy, are examined. Additionally, clustering-based factor exploration (CFE) is presented as a variant of RFE, derived by generalizing the penalty term to enable the clustering of factor scores. It is demonstrated that CFE creates cluster structures more accurately than the existing method. A simulation study shows that the proposed procedures accurately recover true parameter matrices even in the presence of error-contaminated data, with lower computational demand compared to existing methods. Real data examples illustrate that the proposed procedures provide interpretable results, demonstrating high relevance to the factor scores obtained by existing methods.

In this article, we propose a series of latent trait models for the responses and the response times on low stakes tests where some test takers respond preliminary without making full effort to solve the items. The models consider individual differences in capability and persistence. Core of the models is a race between the solution process and a process of disengagement that interrupts the solution process. The different processes are modeled with the linear ballistic accumulator model. Within this general framework, we develop different model variants that differ in the number of accumulators and the way the response is generated when the solution process is interrupted. We distinguish no guessing, random guessing and informed guessing where the guessing probability depends on the status of the solution process. We conduct simulation studies on parameter recovery and on trait estimation. The simulation study suggests that parameter values and traits can be recovered well under certain conditions. Finally, we apply the model variants to empirical data.

Cognitive diagnosis models (CDMs) have been popularly used in fields such as education, psychology, and social sciences. While parametric likelihood estimation is a prevailing method for fitting CDMs, nonparametric methodologies are attracting increasing attention due to their ease of implementation and robustness, particularly when sample sizes are relatively small. However, existing consistency results of the nonparametric estimation methods often rely on certain restrictive conditions, which may not be easily satisfied in practice. In this article, the consistency theory for the general nonparametric classification method is reestablished under weaker and more practical conditions.

Peer grading is an educational system in which students assess each other’s work. It is commonly applied under Massive Open Online Course (MOOC) and offline classroom settings. With this system, instructors receive a reduced grading workload, and students enhance their understanding of course materials by grading others’ work. Peer grading data have a complex dependence structure, for which all the peer grades may be dependent. This complex dependence structure is due to a network structure of peer grading, where each student can be viewed as a vertex of the network, and each peer grade serves as an edge connecting one student as a grader to another student as an examinee. This article introduces a latent variable model framework for analyzing peer grading data and develops a fully Bayesian procedure for its statistical inference. This framework has several advantages. First, when aggregating multiple peer grades, the average score and other simple summary statistics fail to account for grader effects and, thus, can be biased. The proposed approach produces more accurate model parameter estimates and, therefore, more accurate aggregated grades by modeling the heterogeneous grading behavior with latent variables. Second, the proposed method provides a way to assess each student’s performance as a grader, which may be used to identify a pool of reliable graders or generate feedback to help students improve their grading. Third, our model may further provide insights into the peer grading system by answering questions such as whether a student who performs better in coursework also tends to be a more reliable grader. Finally, thanks to the Bayesian approach, uncertainty quantification is straightforward when inferring the student-specific latent variables as well as the structural parameters of the model. The proposed method is applied to two real-world datasets.

Intensive longitudinal data (ILD) collected in mobile health (mHealth) studies contain rich information on the dynamics of multiple outcomes measured frequently over time. Motivated by an mHealth study in which participants self-report the intensity of many emotions multiple times per day, we describe a dynamic factor model that summarizes ILD as a low-dimensional, interpretable latent process. This model consists of (i) a measurement submodel—a factor model—that summarizes the multivariate longitudinal outcome as lower-dimensional latent variables and (ii) a structural submodel—an Ornstein–Uhlenbeck (OU) stochastic process—that captures the dynamics of the multivariate latent process in continuous time. We derive a closed-form likelihood for the marginal distribution of the outcome and the computationally-simpler sparse precision matrix for the OU process. We propose a block coordinate descent algorithm for estimation and use simulation studies to show that it has good statistical properties with ILD. Then, we use our method to analyze data from the mHealth study. We summarize the dynamics of 18 emotions using models with one, two, and three time-varying latent factors, which correspond to different behavioral science theories of emotions. We demonstrate how results can be interpreted to help improve behavioral science theories of momentary emotions, latent psychological states, and their dynamics.

This study incorporates a random forest (RF) approach to probe complex interactions and nonlinearity among predictors into an item response model with the goal of using a hybrid approach to outperform either an RF or explanatory item response model (EIRM) only in explaining item responses. In the specified model, called EIRM-RF, predicted values using RF are added as a predictor in EIRM to model the nonlinear and interaction effects of person- and item-level predictors in person-by-item response data, while accounting for random effects over persons and items. The results of the EIRM-RF are probed with interpretable machine learning (ML) methods, including feature importance measures, partial dependence plots, accumulated local effect plots, and the H-statistic. The EIRM-RF and the interpretable methods are illustrated using an empirical data set to explain differences in reading comprehension in digital versus paper mediums, and the results of EIRM-RF are compared with those of EIRM and RF to show empirical differences in modeling the effects of predictors and random effects among EIRM, RF, and EIRM-RF. In addition, simulation studies are conducted to compare model accuracy among the three models and to evaluate the performance of interpretable ML methods.