We present an approach to quantifying errors in covariance structures in which adventitious error, identified as the process underlying the discrepancy between the population and the structured model, is explicitly modeled as a random effect with a distribution, and the dispersion parameter of this distribution to be estimated gives a measure of misspecification. Analytical properties of the resultant procedure are investigated and the measure of misspecification is found to be related to the root mean square error of approximation. An algorithm is developed for numerical implementation of the procedure. The consistency and asymptotic sampling distributions of the estimators are established under a new asymptotic paradigm and an assumption weaker than the standard Pitman drift assumption. Simulations validate the asymptotic sampling distributions and demonstrate the importance of accounting for the variations in the parameter estimates due to adventitious error. Two examples are also given as illustrations.

Wu and Browne (Psychometrika, 79, 2015) have proposed an innovative approach to modeling discrepancy between a covariance structure model and the population that the model is intended to represent. Their contribution is related to ongoing developments in the field of Uncertainty Quantification (UQ) on modeling and quantifying effects of model discrepancy. We provide an overview of basic principles of UQ and some relevant developments and we examine the Wu–Browne work in that context. We view the Wu–Browne contribution as a seminal development providing a foundation for further work on the critical problem of model discrepancy in statistical modeling in psychological research.

In this rejoinder we discuss the following aspects of our approach to model discrepancy: the interpretations of the two populations and adventitious error, the choice of inverse Wishart distribution, the perceived danger of justifying a model with bad fit, the relationship among our new approach, Chen’s (J R Stat Soc Ser B, 41:235–248, 1979) approach and the existing RMSEA-based approach, and the Pitman drift assumption.

Effort has been devoted to account for heteroscedasticity with respect to observed or latent moderator variables in item or test scores. For instance, in the multi-group generalized linear latent trait model, it could be tested whether the observed (polychoric) covariance matrix differs across the levels of an observed moderator variable. In the case that heteroscedasticity arises across the latent trait itself, existing models commonly distinguish between heteroscedastic residuals and a skewed trait distribution. These models have valuable applications in intelligence, personality and psychopathology research. However, existing approaches are only limited to continuous and polytomous data, while dichotomous data are common in intelligence and psychopathology research. Therefore, in present paper, a heteroscedastic latent trait model is presented for dichotomous data. The model is studied in a simulation study, and applied to data pertaining alcohol use and cognitive ability.

While estimation bias is a primary concern in psychological and educational measurement, the standard error is of equal importance in linking key aspects of the assessment structure, especially when the assessment goal concerns the classification of individuals into categories (e.g., master/non-mastery). In this paper, we show analytically how standard error of ability estimates affects expected classification accuracy and consistency when the decision is binary. When standard error decreases, the conditional classification accuracy and consistency increase. Given an examinee population and a cut score, smaller standard error over the entire latent trait continuum guarantees higher overall expected classification accuracy and consistency. We were also able to show the interrelationship between standard error, the expected classification consistency, and reliability. Utilizing the relationship between standard error and expected classification accuracy and consistency, we derive the upper bounds of the overall expected classification accuracy and consistency of a fixed-length computerized adaptive test. The lower bound of the expected classification accuracy and consistency is also derived given a number of stopping rules of variable-length computerized adaptive testing. Implications of these analytical results on operational tests are discussed.

Large sample theory states the asymptotic normality of the maximum likelihood estimator of the person parameter in the two parameter logistic (2PL) model. In short tests, however, the assumption of normality can be grossly wrong. As a consequence, intended coverage rates may be exceeded and confidence intervals are revealed to be overly conservative. Methods belonging to the higher-order-theory, more specifically saddlepoint approximations, are a convenient way to deal with small-sample problems. Confidence bounds obtained by these means hold the approximate confidence level for a broad range of the person parameter. Moreover, an approximation to the exact distribution permits to compute median unbiased estimates (MUE) that are as likely to overestimate as to underestimate the true person parameter. Additionally, in small samples, these MUE are less mean-biased than the often-used maximum likelihood estimator.

Posterior odds of cheating on achievement tests are presented as an alternative to $p$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p$$\end{document} values reported for statistical hypothesis testing for several of the probabilistic models in the literature on the detection of cheating. It is shown how to calculate their combinatorial expressions with the help of a reformulation of the simple recursive algorithm for the calculation of number-correct score distributions used throughout the testing industry. Using the odds avoids the arbitrary choice between statistical tests of answer copying that do and do not condition on the responses the test taker is suspected to have copied and allows the testing agency to account for existing circumstantial evidence of cheating through the specification of prior odds.

Missing data occur in many real world studies. Knowing the type of missing mechanisms is important for adopting appropriate statistical analysis procedure. Many statistical methods assume missing completely at random (MCAR) due to its simplicity. Therefore, it is necessary to test whether this assumption is satisfied before applying those procedures. In the literature, most of the procedures for testing MCAR were developed under normality assumption which is sometimes difficult to justify in practice. In this paper, we propose a nonparametric test of MCAR for incomplete multivariate data which does not require distributional assumptions. The proposed test is carried out by comparing the distributions of the observed data across different missing-pattern groups. We prove that the proposed test is consistent against any distributional differences in the observed data. Simulation shows that the proposed procedure has the Type I error well controlled at the nominal level for testing MCAR and also has good power against a variety of non-MCAR alternatives.

Plausible values (PVs) are a standard multiple imputation tool for analysis of large education survey data, which measures latent proficiency variables. When latent proficiency is the dependent variable, we reconsider the standard institutionally generated PV methodology and find it applies with greater generality than shown previously. When latent proficiency is an independent variable, we show that the standard institutional PV methodology produces biased inference because the institutional conditioning model places restrictions on the form of the secondary analysts’ model. We offer an alternative approach that avoids these biases based on the mixed effects structural equations model of Schofield (Modeling measurement error when using cognitive test scores in social science research. Doctoral dissertation. Department of Statistics and Heinz College of Public Policy. Pittsburgh, PA: Carnegie Mellon University, 2008).

Many large-scale standardized tests are intended to measure skills related to ability rather than the rate at which examinees can work. Time limits imposed on these tests make it difficult to distinguish between the effect of low proficiency and the effect of lack of time. This paper proposes a mixture cure-rate model approach to address this issue. Maximum likelihood estimation is proposed for parameter and variance estimation for three cases: when examinee parameters are to be estimated given precalibrated item parameters, when item parameters are to be calibrated given known examinee parameters, and when item parameters are to be estimated without assuming known examinee parameters. Large-sample properties are established for the cases under suitable regularity conditions. Simulation studies suggest that the proposed approach is appropriate for inferences concerning model parameters. In addition, not distinguishing between the effect of low proficiency and the effect of lack of time is shown to have considerable consequences for parameter estimation. A real data example is presented to demonstrate the new model. Choice of survival models for the latent power times is also discussed.

The component loadings are interpreted by considering their magnitudes, which indicates how strongly each of the original variables relates to the corresponding principal component. The usual ad hoc practice in the interpretation process is to ignore the variables with small absolute loadings or set to zero loadings smaller than some threshold value. This, in fact, makes the component loadings sparse in an artificial and a subjective way. We propose a new alternative approach, which produces sparse loadings in an optimal way. The introduced approach is illustrated on two well-known data sets and compared to the existing rotation methods.

Latent trait models for responses and response times in tests are often pure statistical models without a close connection to features of the assumed response process. In the present paper, a new model is presented that is more closely related to assumptions about the response process. The model is based on two increasing stochastic processes. Each stochastic process represents the accumulation of knowledge with respect to one of two response options, the correct and incorrect response. Both accumulators compete and the accumulator that first exceeds a critical level determines the response. General assumptions about the accumulators result in a race between two response times that follow a bivariate Birnbaum Saunders distribution. The model can be calibrated with marginal maximum likelihood estimation. Feasibility of the estimation approach is demonstrated in a simulation study. Additionally, a test of model fit is proposed. Finally, the model will be used for the analysis of an empirical data set.

Classification and Regression Trees (CART), and their successors—bagging and random forests, are statistical learning tools that are receiving increasing attention. However, due to characteristics of censored data collection, standard CART algorithms are not immediately transferable to the context of survival analysis. Questions about the occurrence and timing of events arise throughout psychological and behavioral sciences, especially in longitudinal studies. The prediction power and other key features of tree-based methods are promising in studies where an event occurrence is the outcome of interest. This article reviews existing tree algorithms designed specifically for censored responses as well as recently developed survival ensemble methods, and introduces available computer software. Through simulations and a practical example, merits and limitations of these methods are discussed. Suggestions are provided for practical use.

The present paper proposes a hierarchical, multi-unidimensional two-parameter logistic item response theory (2PL-MUIRT) model extended for a large number of groups. The proposed model was motivated by a large-scale integrative data analysis (IDA) study which combined data (N = 24,336) from 24 independent alcohol intervention studies. IDA projects face unique challenges that are different from those encountered in individual studies, such as the need to establish a common scoring metric across studies and to handle missingness in the pooled data. To address these challenges, we developed a Markov chain Monte Carlo (MCMC) algorithm for a hierarchical 2PL-MUIRT model for multiple groups in which not only were the item parameters and latent traits estimated, but the means and covariance structures for multiple dimensions were also estimated across different groups. Compared to a few existing MCMC algorithms for multidimensional IRT models that constrain the item parameters to facilitate estimation of the covariance matrix, we adapted an MCMC algorithm so that we could directly estimate the correlation matrix for the anchor group without any constraints on the item parameters. The feasibility of the MCMC algorithm and the validity of the basic calibration procedure were examined using a simulation study. Results showed that model parameters could be adequately recovered, and estimated latent trait scores closely approximated true latent trait scores. The algorithm was then applied to analyze real data (69 items across 20 studies for 22,608 participants). The posterior predictive model check showed that the model fit all items well, and the correlations between the MCMC scores and original scores were overall quite high. An additional simulation study demonstrated robustness of the MCMC procedures in the context of the high proportion of missingness in data. The Bayesian hierarchical IRT model using the MCMC algorithms developed in the current study has the potential to be widely implemented for IDA studies or multi-site studies, and can be further refined to meet more complicated needs in applied research.