With models and research designs ever increasing in complexity, the foundational question of model identification is more important than ever. The determination of whether or not a model can be fit at all or fit to some particular data set is the essence of model identification. In this article, we pull from previously published work on data-independent model identification applicable to a broad set of structural equation models, and extend it further to include extremely flexible exogenous covariate effects and also to include data-dependent empirical model identification. For illustrative purposes, we apply this model identification solution to several small examples for which the answer is already known, including a real data example from the National Longitudinal Survey of Youth; however, the method applies similarly to models that are far from simple to comprehend. The solution is implemented in the open-source OpenMx package in R.

This article addresses problematic behaviors of Markov chain Monte Carlo (MCMC) methods for finite mixture models due to what we call degenerate nonidentifiability. We discuss the reasons for these behaviors, propose diagnostics to detect them, and show through simulations that using more informative priors than the vague defaults can mitigate the problems in growth mixture models (GMMs). Our motivating example is an application of GMMs to data from the National Longitudinal Survey of Youth (NLSY) to examine heterogeneity in the development of reading skills in children aged 6–14. We also suggest ways of describing and visualizing within-class heterogeneity in GMMs, provide a literature review of likelihood identification and Bayesian identification, propose a viable definition of Bayesian identification for latent variable models based on the marginal likelihood (integrated over the latent variables), and give a brief didactic description of Hamiltonian Monte Carlo (HMC) as implemented in Stan.

Establishing the effectiveness of treatments for psychopathology requires accurate models of its progression over time and the factors that impact it. Longitudinal data is however fraught with missingness, hindering accurate modeling. We re-analyse data on schizophrenia severity in a clinical trial using hidden Markov models (HMMs). We consider missing data in HMMs with a focus on situations where data is missing not at random (MNAR) and missingness depends on the latent states, allowing symptom severity to indirectly impact probability of missingness. In simulations, we show that including a submodel for state-dependent missingness reduces bias when data is MNAR and state-dependent, whilst not reducing accuracy when data is missing-at-random (MAR). When missingness depends on time, a model that allows missingness to be both state- and time-dependent is unbiased. Overall, these results show that modelling missingness as state-dependent and including relevant covariates is a useful strategy in applications of HMMs to time-series with missing data. Applying the model to data from a clinical trial, we find that drop-out is more likely for patients with less severe symptoms, which may lead to a biased assessment of treatment effectiveness.

Researchers interested in dyadic processes increasingly collect intensive longitudinal data (ILD), with the longitudinal actor–partner interdependence model (L-APIM) being a popular modeling approach. However, due to non-compliance and the use of conditional questions, ILD are almost always incomplete. These missing data issues become more prominent in dyadic studies, because partners often miss different measurement occasions or disagree about features that trigger conditional questions. Large amounts of missing data challenge the L-APIM’s estimation performance. Specifically, we found that non-convergence occurred when applying the L-APIM to pre-existing dyadic diary data with a lot of missing values. Using a simulation study, we systematically examined the performance of the L-APIM in dyadic ILD with missing values. Consistent with our illustrative data, we found that non-convergence often occurred in conditions with small sample sizes, while the fixed within-person actor and partner effects were well estimated when analyses did converge. Additionally, considering potential convergence failures with the L-APIM, we investigated 31 alternative models and evaluated their performance on simulated and empirical data, showing that multiple alternatives may alleviate the convergence problems. Overall, when the L-APIM fails to converge, we recommend fitting multiple alternative models to check the robustness of the results.

Despite the versatility of generalized linear mixed models in handling complex experimental designs, they often suffer from misspecification and convergence problems. This makes inference on the values of coefficients problematic. In addition, the researcher’s choice of random and fixed effects directly affects statistical inference correctness. To address these challenges, we propose a robust extension of the “two-stage summary statistics” approach using sign-flipping transformations of the score statistic in the second stage. Our approach efficiently handles within-variance structure and heteroscedasticity, ensuring accurate regression coefficient testing for 2-level hierarchical data structures. The approach is illustrated by analyzing the reduction of health issues over time for newly adopted children. The model is characterized by a binomial response with unbalanced frequencies and several categorical and continuous predictors. The proposed approach efficiently deals with critical problems related to longitudinal nonlinear models, surpassing common statistical approaches such as generalized estimating equations and generalized linear mixed models.

When analyzing scaling conditions in latent variable structural equation models (SEMs) with continuous observed variables, analysts scaling a latent variable typically set the factor loading of one indicator to one and either set its intercept to zero or the mean of its latent variable to zero. When binary and ordinal observed variables are part of SEMs, the identification and scaling choices are more varied and multifaceted. Longitudinal data further complicate this. In SEM software, such as lavaan and Mplus, fixing the underlying variables’ variances or the error variances to one are two primary scaling conventions. As demonstrated in this paper, choosing between these constraints can significantly impact longitudinal analysis, affecting model fit, degrees of freedom, and assumptions about the dynamic process and error structure. We explore alternative parameterizations and conditions of model equivalence with categorical repeated measures.

Using data from the National Longitudinal Survey of Youth 1997, we empirically explore how different parameterizations lead to varying conclusions in longitudinal categorical analysis. More specifically, we provide insights into the specifications of the autoregressive latent trajectory model and its special cases—the linear growth curve and first-order autoregressive models—for categorical repeated measures. These findings have broader implications for a wide range of longitudinal models.

Structured latent curve models (SLCMs) for continuous repeated measures data have been the subject of considerable recent research activity. In this article, we develop a first-order SLCM for repeated measures count data where the underlying change process is theorized to develop in distinct phases. Parameters of the multiphase or piecewise growth model, including changepoints, are allowed to vary across individuals. Exposure is allowed to vary across both individuals and time. We demonstrate our modeling approach on empirical expressive language data (grammatical morpheme counts) drawn from multiple distinct corpora available in the Child Language Data Exchange System (CHILDES), where the acquisition of grammatical morphology is understood to occur in distinct phases in typically developing children. A multiphase SLCM is fit to summarize individuals’ data as well as the average developmental pattern. Change in time-varying dispersion (unexplained variability in morpheme counts) over the course of early childhood is modeled concurrently to provide additional insights into acquisition. Unique characteristics of count data create modeling, identification, estimation, and diagnostic challenges that are exacerbated by incorporating growth models with nonlinear random effects. These are discussed at length. We provide annotated software code for each of models used in the empirical example.

This paper presents a model specification for group comparisons regarding a functional trend over time within a trial and learning across a series of trials in intensive binary longitudinal eye-tracking data. The functional trend and learning effects are modeled using by-variable smooth functions. This model specification is formulated as a generalized additive mixed model, which allowed for the use of the freely available mgcv package (Wood in Package ‘mgcv.’ https://cran.r-project.org/web/packages/mgcv/mgcv.pdf, 2023) in R. The model specification was applied to intensive binary longitudinal eye-tracking data, where the questions of interest concern differences between individuals with and without brain injury in their real-time language comprehension and how this affects their learning over time. The results of the simulation study show that the model parameters are recovered well and the by-variable smooth functions are adequately predicted in the same condition as those found in the application.

To assess country-level progress toward these educational goals it is important to monitor trends in educational outcomes over time. The purpose of this article is to demonstrate how optimally predictive growth models can be constructed to monitor the pace of progress at which countries are moving toward (or way from) the education sustainable development goals as specified by the United Nations. A number of growth curve models can be specified to estimate the pace of progress, however, choosing one model and using it for predictive purposes assumes that the chosen model is the one that generated the data, and this choice runs the risk of “over-confident inferences and decisions that are more risky than one thinks they are” (Hoeting et al., 1999). To mitigate this problem, we adapt and apply Bayesian stacking to form mixtures of predictive distributions from an ensemble of individual models specified to predict country-level pace of progress. We demonstrate Bayesian stacking using country-level data from the Program on International Student Assessment. Our results show that Bayesian stacking yields better predictive accuracy than any single model as measured by the Kullback–Leibler divergence. Issues of Bayesian model identification and estimation for growth models are also discussed.

For classroom teaching and learning, classifying students’ skills into more than two categories (e.g., no, basic, and advanced masteries) is more instructionally relevant. Such classifications require polytomous attributes, for which most existing cognitive diagnosis models (CDMs) are inapplicable. This paper proposes the saturated polytomous cognitive diagnosis model (sp-CDM), a general model that subsumes existing CDMs for polytomous attributes as special cases. The generalization is shown by mathematically illustrating the relationships between the proposed and existing CDMs. Moreover, algorithms to estimate the proposed model is proposed. A simulation study is conducted to evaluate the parameter recovery of the sp-CDM using the proposed estimation algorithms, as well as to illustrate the consequences of improperly fitting constrained or unnecessarily complex polytomous-attribute CDMs. A real-data example involving polytomous attributes is presented to demonstrate the practical utility of the proposed model.

The basic local independence model (BLIM) is a probabilistic model developed in knowledge space theory (KST). Recently, Stefanutti, de Chiusole, et al. (2020, Psychometrika 85, 684–715) proposed the polytomous local independence model (PoLIM), which is an extension of the BLIM to items with more than two response alternatives (polytomous items). In a Commentary to this paper, Chiu et al. (2023, Psychometrika 88, 656–671) claimed that (i) the BLIM is just a deterministic input noisy AND-gate (DINA) model where every item has a single skill and, as a consequence of this, (ii) the “PoLIM is simply a paraphrase of a DINA model in cognitive diagnosis (CD) for polytomous items” (p. 656). This rejoinder shows that such statements are invalid and totally misleading. Its aim is to clarify the nature of the relationship between the BLIM and the DINA, as well as that between the PoLIM and the Polytomous DINA. It builds upon formal results by Heller, et al. (2015, Psychometrika 80(4), 995–1019) on the intimate relation between KST and CD notions, and shows that the BLIM/PoLIM may be conceived as marginal models for whole classes of CD models.

Mediation analysis is one of the most popularly used methods in social sciences and related areas. To estimate the indirect effect, the least-squares regression is routinely applied, which is also the most efficient when the errors are normally distributed. In practice, however, real data sets are often non-normally distributed, either heavy-tailed or skewed, so that the least-squares estimators may behave very badly. To overcome this problem, we propose a robust M-estimation for the indirect effect via a general loss function, with a main focus on the Huber loss which is more slowly varying at large values than the squared loss. We further propose a data-driven procedure to select the optimal tuning constant by minimizing the asymptotic variance of the Huber estimator, which is more robust than the least-squares estimator facing outliers and non-normal data, and more efficient than the least-absolute-deviation estimator. Simulation studies compare the finite sample performance of the Huber loss with the existing competitors in terms of the mean square error, the type I error rate, and the statistical power. Finally, the usefulness of the proposed method is also illustrated using two real data examples.

The family of polychoric models (PM) categories ordinal data with latent multivariate normal variables. This modeling framework is commonly used to study the association between ordinal variables, often leading to a polychoric correlation model (PCM). Moreover, PM subsumes several well-known psychometric models, such as the structural equation modeling (SEM) with ordinal data. That said, the identifiability of PM has not been addressed in the literature. Meanwhile, in recent years researchers have suggested that the latent variables underlying PM could be generalized to the family of elliptical distributions, such as the multivariate logistic and t distributions. This article concerns the identifiability of PM and PCM with latent elliptical distributions, for which we show that PM is not identifiable and PCM is identifiable. In particular, we prove the identifiability of the polychoric t correlation model based on the copula representation. We then move on to find the set of identifiability constraints of PM through an “equivalence-classes approach of identifiability,” and demonstrate its use in two applications: one concerns the identifiability of PM on Likert scales and on comparative judgment, and the other concerns the identifiability of ordinal SEM and item factor analysis. Possible implications induced by these identifiability constraints are discussed.

This article introduces a novel methodology to model the hierarchical dependence structure of manifest variables (MVs). This is done by reconstructing their correlation matrix considering a hierarchy of latent factors which forms an ultrametric correlation matrix. The proposed ultrametric factor analysis model will be shown to obtain reliable, unidimensional, and unique hierarchical factors. This approach addresses the limitations of traditional factor analysis methods that often oversimplify the multidimensional and complex relationships among MVs. The article provides an in-depth mathematical description of the proposed model, as well as an algorithm for parameter estimation. An extensive simulation study with $3,000$ generated samples validates the proposal across twelve different scenarios. Finally, we demonstrate the potential of the proposed model using a real data set that is a benchmark in psychological research.

Conditional dependence (CD) reflects potential interactions between persons and items in measurement, offering valuable information for deriving personalized diagnoses, evaluations, and feedback. The recent integration of psychometric models with latent space provides an effective way to visualize and quantify person–item interactions unexplained by latent variables and item parameters. In such applications, it is important to recognize the relative nature of CD, as models with different structures and complexities (e.g., due to factor dimensionality and item parameters) produce varying systematic explanations of person and item effects, leading to differing residual variations in both quantitative and qualitative sense. To demonstrate this relativity, we extend the previously developed unidimensional Rasch-based latent space item response model by incorporating between-item multidimensionality and item discrimination parameters. The resulting model can be reduced to simpler models with appropriate constraints, allowing us to explore the relativity in CD by comparing them. Simulation studies demonstrate that (1) the most complex proposed model properly recovers its parameters, (2) it outperforms the traditional IRT models by accounting for CD, and (3) the models in comparison exhibit distinctive extents of CD. The study continues with empirical examples that further illustrate relative changes in the extent and configurations of CD with practical implications.