The inverse probability of treatment weighted (IPTW) estimator can be used to make causal inferences under two assumptions: (1) no unobserved confounders (ignorability) and (2) positive probability of treatment and of control at every level of the confounders (positivity), but is vulnerable to bias if by chance, the proportion of the sample assigned to treatment, or proportion of control, is zero at certain levels of the confounders. We propose to deal with this sampling zero problem, also known as practical violation of the positivity assumption, in a setting where the observed confounder is cluster identity, i.e., treatment assignment is ignorable within clusters. Specifically, based on a random coefficient model assumed for the potential outcome, we augment the IPTW estimating function with the estimated potential outcomes of treatment (or of control) for clusters that have no observation of treatment (or control). If the cluster-specific potential outcomes are estimated correctly, the augmented estimating function can be shown to converge in expectation to zero and therefore yield consistent causal estimates. The proposed method can be implemented in the existing software, and it performs well in simulated data as well as with real-world data from a teacher preparation evaluation study.

Parceling—using composites of observed variables as indicators for a common factor—strengthens loadings, but reduces the number of indicators. Factor indeterminacy is reduced when there are many observed variables per factor, and when loadings and factor correlations are strong. It is proven that parceling cannot reduce factor indeterminacy. In special cases where the ratio of loading to residual variance is the same for all items included in each parcel, factor indeterminacy is unaffected by parceling. Otherwise, parceling worsens factor indeterminacy. While factor indeterminacy does not affect the parameter estimates, standard errors, or fit indices associated with a factor model, it does create uncertainty, which endangers valid inference.

The $\theta$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta $$\end{document} metric in item response theory is often not the most useful metric for score reporting or interpretation. In this paper, I demonstrate that the filtered monotonic polynomial (FMP) item response model, a recently proposed nonparametric item response model (Liang & Browne in J Educ Behav Stat 40:5–34, 2015), can be used to specify item response models on metrics other than the $\theta$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta $$\end{document} metric. Specifically, I demonstrate that any item response function (IRF) defined within the FMP framework can be re-expressed as another FMP IRF by taking monotonic transformations of the latent trait. I derive the item parameter transformations that correspond to both linear and nonlinear transformations of the latent trait metric. These item parameter transformations can be used to define an item response model based on any monotonic transformation of the $\theta$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta $$\end{document} metric, so long as the metric transformation is approximated by a monotonic polynomial. I demonstrate this result by defining an item response model directly on the approximate true score metric and discuss the implications of metric transformations for applied testing situations.

To understand how SEM methods perform in practice where models always have misfit, simulation studies often involve incorrect models. To create a wrong model, traditionally one specifies a perfect model first and then removes some paths. This approach becomes difficult or even impossible to implement in moment structure analysis and fails to control the amounts of misfit separately and precisely for the mean and covariance parts. Most importantly, this approach assumes a perfect model exists and wrong models can eventually be made perfect, whereas in practice models are all implausible if taken literally and at best provide approximations of the real world. To improve the traditional approach, we propose a more realistic and flexible way to create model misfit for multiple group moment structure analysis. Given (a) the model $\mathbf{\mu }(·)$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varvec{{{\upmu }}} (\cdot ) $$\end{document} and $\mathbf{\Sigma }(·)$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varvec{{\Sigma }} (\cdot ) $$\end{document}, (b) population model parameters ${\mathbf{\theta }}_0$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document}, and (c) $F_1$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_1$$\end{document} and $F_2$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_2$$\end{document} specified by the researcher, our method creates ${\mathbf{\mu }}^{\ast }$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\mathbf{\Sigma }}^{\ast }$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} to simultaneously satisfy (a) ${\mathbf{\theta }}_0=argminF[{\mathbf{\mu }}^{\ast },{\mathbf{\Sigma }}^{\ast };\mathbf{\mu }(·),\mathbf{\Sigma }(·)]$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varvec{{{\uptheta }}} _0 = \arg \min F[\varvec{{{\upmu }}} ^*, \varvec{{\Sigma }} ^*; \varvec{{{\upmu }}} (\cdot ), \varvec{{\Sigma }} (\cdot )]$$\end{document}, (b) the mean structure’s misfit equals $F_1$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_1$$\end{document}, and (c) the covariance structure’s misfit equals $F_2$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_2$$\end{document}.

The factor analysis (FA) model does not permit unique estimation of the common and unique factor scores. This weakness is notorious as the factor indeterminacy in FA. Luckily, some part of the factor scores can be uniquely determined. Thus, as a whole, they can be viewed as a sum of determined and undetermined parts. The paper proposes to select the undetermined part, such that the resulting common factor scores have the following feature: the rows (i.e., individuals) of the common factor score matrix are as well classified as possible into few clusters. The clear benefit is that we can easily interpret the factor scores simply by focusing on the clusters. The procedure is called clustered common factor exploration (CCFE). An alternating least squares algorithm is developed for CCFE. It is illustrated with real data examples. The proposed approach can be viewed as a parallel to the rotation techniques in FA. They exploit another FA indeterminacy, the rotation indeterminacy, which is resolved by choosing the rotation that transforms the loading matrix into the ‘most’ interpretable one according to a pre-specified criterion. In contrast to the rotational indeterminacy, the factor indeterminacy is utilized to achieve well-clustered factor scores by CCFE. To the best of our knowledge, such an approach to the FA interpretation has not been studied yet.

Joint maximum likelihood (JML) estimation is one of the earliest approaches to fitting item response theory (IRT) models. This procedure treats both the item and person parameters as unknown but fixed model parameters and estimates them simultaneously by solving an optimization problem. However, the JML estimator is known to be asymptotically inconsistent for many IRT models, when the sample size goes to infinity and the number of items keeps fixed. Consequently, in the psychometrics literature, this estimator is less preferred to the marginal maximum likelihood (MML) estimator. In this paper, we re-investigate the JML estimator for high-dimensional exploratory item factor analysis, from both statistical and computational perspectives. In particular, we establish a notion of statistical consistency for a constrained JML estimator, under an asymptotic setting that both the numbers of items and people grow to infinity and that many responses may be missing. A parallel computing algorithm is proposed for this estimator that can scale to very large datasets. Via simulation studies, we show that when the dimensionality is high, the proposed estimator yields similar or even better results than those from the MML estimator, but can be obtained computationally much more efficiently. An illustrative real data example is provided based on the revised version of Eysenck’s Personality Questionnaire (EPQ-R).

Cognitive diagnosis models (CDMs) are useful statistical tools in cognitive diagnosis assessment. However, as many other latent variable models, the CDMs often suffer from the non-identifiability issue. This work gives the sufficient and necessary condition for identifiability of the basic DINA model, which not only addresses the open problem in Xu and Zhang (Psychometrika 81:625–649, 2016) on the minimal requirement for identifiability, but also sheds light on the study of more general CDMs, which often cover DINA as a submodel. Moreover, we show the identifiability condition ensures the consistent estimation of the model parameters. From a practical perspective, the identifiability condition only depends on the Q-matrix structure and is easy to verify, which would provide a guideline for designing statistically valid and estimable cognitive diagnosis tests.

In educational and psychological measurement, researchers and/or practitioners are often interested in examining whether the ability of an examinee is the same over two sets of items. Such problems can arise in measurement of change, detection of cheating on unproctored tests, erasure analysis, detection of item preknowledge, etc. Traditional frequentist approaches that are used in such problems include the Wald test, the likelihood ratio test, and the score test (e.g., Fischer, Appl Psychol Meas 27:3–26, 2003; Finkelman, Weiss, & Kim-Kang, Appl Psychol Meas 34:238–254, 2010; Glas & Dagohoy, Psychometrika 72:159–180, 2007; Guo & Drasgow, Int J Sel Assess 18:351–364, 2010; Klauer & Rettig, Br J Math Stat Psychol 43:193–206, 1990; Sinharay, J Educ Behav Stat 42:46–68, 2017). This paper shows that approaches based on higher-order asymptotics (e.g., Barndorff-Nielsen & Cox, Inference and asymptotics. Springer, London, 1994; Ghosh, Higher order asymptotics. Institute of Mathematical Statistics, Hayward, 1994) can also be used to test for the equality of the examinee ability over two sets of items. The modified signed likelihood ratio test (e.g., Barndorff-Nielsen, Biometrika 73:307–322, 1986) and the Lugannani–Rice approximation (Lugannani & Rice, Adv Appl Prob 12:475–490, 1980), both of which are based on higher-order asymptotics, are shown to provide some improvement over the traditional frequentist approaches in three simulations. Two real data examples are also provided.

This paper provides results on a form of adaptive testing that is used frequently in intelligence testing. In these tests, items are presented in order of increasing difficulty. The presentation of items is adaptive in the sense that a session is discontinued once a test taker produces a certain number of incorrect responses in sequence, with subsequent (not observed) responses commonly scored as wrong. The Stanford-Binet Intelligence Scales (SB5; Riverside Publishing Company, 2003) and the Kaufman Assessment Battery for Children (KABC-II; Kaufman and Kaufman, 2004), the Kaufman Adolescent and Adult Intelligence Test (Kaufman and Kaufman 2014) and the Universal Nonverbal Intelligence Test (2nd ed.) (Bracken and McCallum 2015) are some of the many examples using this rule. He and Wolfe (Educ Psychol Meas 72(5):808–826, 2012. https://doi.org/10.1177/0013164412441937) compared different ability estimation methods in a simulation study for this discontinue rule adaptation of test length. However, there has been no study, to our knowledge, of the underlying distributional properties based on analytic arguments drawing on probability theory, of what these authors call stochastic censoring of responses. The study results obtained by He and Wolfe (Educ Psychol Meas 72(5):808–826, 2012. https://doi.org/10.1177/0013164412441937) agree with results presented by DeAyala et al. (J Educ Meas 38:213–234, 2001) as well as Rose et al. (Modeling non-ignorable missing data with item response theory (IRT; ETS RR-10-11), Educational Testing Service, Princeton, 2010) and Rose et al. (Psychometrika 82:795–819, 2017. https://doi.org/10.1007/s11336-016-9544-7) in that ability estimates are biased most when scoring the not observed responses as wrong. This scoring is used operationally, so more research is needed in order to improve practice in this field. The paper extends existing research on adaptivity by discontinue rules in intelligence tests in multiple ways: First, an analytical study of the distributional properties of discontinue rule scored items is presented. Second, a simulation is presented that includes additional scoring rules and uses ability estimators that may be suitable to reduce bias for discontinue rule scored intelligence tests.

Stochastic actor-oriented models (SAOMs) can be used to analyse dynamic network data, collected by observing a network and a behaviour in a panel design. The parameters of SAOMs are usually estimated by the method of moments (MoM) implemented by a stochastic approximation algorithm, where statistics defining the moment conditions correspond in a natural way to the parameters. Here, we propose to apply the generalized method of moments (GMoM), using more statistics than parameters. We concentrate on statistics depending jointly on the network and the behaviour, because of the importance of their interdependence, and propose to add contemporaneous statistics to the usual cross-lagged statistics. We describe the stochastic algorithm developed to approximate the GMoM solution. A small simulation study supports the greater statistical efficiency of the GMoM estimator compared to the MoM.

Typical Bayesian methods for models with latent variables (or random effects) involve directly sampling the latent variables along with the model parameters. In high-level software code for model definitions (using, e.g., BUGS, JAGS, Stan), the likelihood is therefore specified as conditional on the latent variables. This can lead researchers to perform model comparisons via conditional likelihoods, where the latent variables are considered model parameters. In other settings, however, typical model comparisons involve marginal likelihoods where the latent variables are integrated out. This distinction is often overlooked despite the fact that it can have a large impact on the comparisons of interest. In this paper, we clarify and illustrate these issues, focusing on the comparison of conditional and marginal Deviance Information Criteria (DICs) and Watanabe–Akaike Information Criteria (WAICs) in psychometric modeling. The conditional/marginal distinction corresponds to whether the model should be predictive for the clusters that are in the data or for new clusters (where “clusters” typically correspond to higher-level units like people or schools). Correspondingly, we show that marginal WAIC corresponds to leave-one-cluster out cross-validation, whereas conditional WAIC corresponds to leave-one-unit out. These results lead to recommendations on the general application of the criteria to models with latent variables.

Using the well-known strategy in which parameters are linked to the sampling distribution via an identification analysis, we offer an interpretation of the item parameters in the one-parameter logistic with guessing model (1PL-G) and the nested Rasch model. The interpretations are based on measures of informativeness that are defined in terms of odds of correctly answering the items. It is shown that the interpretation of what is called the difficulty parameter in the random-effects 1PL-G model differs from that of the item parameter in a random-effects Rasch model. It is also shown that the traditional interpretation of the guessing parameter in the 1PL-G model changes, depending on whether fixed-effects or random-effects versions of both models are considered.

It is well known that the classical exploratory factor analysis (EFA) of data with more observations than variables has several types of indeterminacy. We study the factor indeterminacy and show some new aspects of this problem by considering EFA as a specific data matrix decomposition. We adopt a new approach to the EFA estimation and achieve a new characterization of the factor indeterminacy problem. A new alternative model is proposed, which gives determinate factors and can be seen as a semi-sparse principal component analysis (PCA). An alternating algorithm is developed, where in each step a Procrustes problem is solved. It is demonstrated that the new model/algorithm can act as a specific sparse PCA and as a low-rank-plus-sparse matrix decomposition. Numerical examples with several large data sets illustrate the versatility of the new model, and the performance and behaviour of its algorithmic implementation.

Parametric likelihood estimation is the prevailing method for fitting cognitive diagnosis models—also called diagnostic classification models (DCMs). Nonparametric concepts and methods that do not rely on a parametric statistical model have been proposed for cognitive diagnosis. These methods are particularly useful when sample sizes are small. The general nonparametric classification (GNPC) method for assigning examinees to proficiency classes can accommodate assessment data conforming to any diagnostic classification model that describes the probability of a correct item response as an increasing function of the number of required attributes mastered by an examinee (known as the “monotonicity assumption”). Hence, the GNPC method can be used with any model that can be represented as a general DCM. However, the statistical properties of the estimator of examinees’ proficiency class are currently unknown. In this article, the consistency theory of the GNPC proficiency-class estimator is developed and its statistical consistency is proven.

In best linear prediction (BLP), a true test score is predicted by observed item scores and by ancillary test data. If the use of BLP rather than a more direct estimate of a true score has disparate impact for different demographic groups, then a fairness issue arises. To improve population invariance but to preserve much of the efficiency of BLP, a modified approach, penalized best linear prediction, is proposed that weights both mean square error of prediction and a quadratic measure of subgroup biases. The proposed methodology is applied to three high-stakes writing assessments.

In item response theory (IRT), it is often necessary to perform restricted recalibration (RR) of the model: A set of (focal) parameters is estimated holding a set of (nuisance) parameters fixed. Typical applications of RR include expanding an existing item bank, linking multiple test forms, and associating constructs measured by separately calibrated tests. In the current work, we provide full statistical theory for RR of IRT models under the framework of pseudo-maximum likelihood estimation. We describe the standard error calculation for the focal parameters, the assessment of overall goodness-of-fit (GOF) of the model, and the identification of misfitting items. We report a simulation study to evaluate the performance of these methods in the scenario of adding a new item to an existing test. Parameter recovery for the focal parameters as well as Type I error and power of the proposed tests are examined. An empirical example is also included, in which we validate the pediatric fatigue short-form scale in the Patient-Reported Outcome Measurement Information System (PROMIS), compute global and local GOF statistics, and update parameters for the misfitting items.

The assumption of latent monotonicity is made by all common parametric and nonparametric polytomous item response theory models and is crucial for establishing an ordinal level of measurement of the item score. Three forms of latent monotonicity can be distinguished: monotonicity of the cumulative probabilities, of the continuation ratios, and of the adjacent-category ratios. Observable consequences of these different forms of latent monotonicity are derived, and Bayes factor methods for testing these consequences are proposed. These methods allow for the quantification of the evidence both in favor and against the tested property. Both item-level and category-level Bayes factors are considered, and their performance is evaluated using a simulation study. The methods are applied to an empirical example consisting of a 10-item Likert scale to investigate whether a polytomous item scoring rule results in item scores that are of ordinal level measurement.

Item calibration is a technique to estimate characteristics of questions (called items) for achievement tests. In computerized tests, item calibration is an important tool for maintaining, updating and developing new items for an item bank. To efficiently sample examinees with specific ability levels for this calibration, we use optimal design theory assuming that the probability to answer correctly follows an item response model. Locally optimal unrestricted designs have usually a few design points for ability. In practice, it is hard to sample examinees from a population with these specific ability levels due to unavailability or limited availability of examinees. To counter this problem, we use the concept of optimal restricted designs and show that this concept naturally fits to item calibration. We prove an equivalence theorem needed to verify optimality of a design. Locally optimal restricted designs provide intervals of ability levels for optimal calibration of an item. When assuming a two-parameter logistic model, several scenarios with D-optimal restricted designs are presented for calibration of a single item and simultaneous calibration of several items. These scenarios show that the naive way to sample examinees around unrestricted design points is not optimal.