Marginal Maximum Research Articles

Psychometric validation of the French version of two scales measuring general (HLS19-Q12) and navigational (HLS19-NAV) health literacy using the Rasch model

BackgroundTo examine the psychometrics properties of the French version of two scales measuring general (HLS19-Q12) and navigational (HLS19-NAV) health literacy (HL) using validation methods based on modern psychometric test theories: a Rasch model analysis.MethodsThe data on representative samples of the French adult population came from the Health Literacy Survey (N = 2 003), conducted in France in two waves (2020 and 2021), and from the third wave of SLAVACO study (N = 2 022), conducted in December 2021. A Rasch analysis was performed using a partial credit model with marginal maximum likelihood estimation adapted to polytomous data. Category probability curves were used to examine whether participants found it consistently difficult to distinguish between response options. Differential item functioning (DIF) was also examined for each item in the two scales as a function of age, gender, and economic status. A Person Separation Index (PSI) of 0.7 has conventionally been considered to be the minimum acceptable PSI level. Chi-square fit statistics, expressed as infit and outfit mean square statistics, were calculated for each item in the two scales. An acceptable fit corresponded to average values between 0.7 and 1.3. The dimensionality of the scales was assessed using Principal Components Analysis (PCA) on the residual correlation matrix to identify whether items are locally independent. Additionally, Smith’s method was employed to detect potential multidimensionality.ResultsThe PSI values for both scales were greater than 0.90. No disordered categories were observed. No evidence of significant DIF was found when associated with gender and economic status. However, two items for both scales had a DIF which was dependent on age but did not have a significant impact on scale scores between younger and older people. Minor local dependence was noted on the HLS19-NAV scale. However, a PCA and Smith’s method demonstrated the unidimensionality of the French version of the scales measuring general and navigational HL.ConclusionsOur results, based on a rigorous statistical analysis, verified the psychometric parameters of the French version of the HLS19-Q12 and HLS19-NAV scales. Despite strong correlation between both scales, each measured a different latent trait. Moreover, despite the fact that presence of DIF was weak, attention must nonetheless be paid when comparing scale response scores between young and older respondents.

Latent Growth Model with Categorical Indicators: Comparison of the Different Estimators

Despite its potential usage in social and behavioral sciences, there has been limited research on Latent Growth Models (LGM) with categorical indicators. The categorical nature of binary or ordinal variables violates the multivariate normality assumption of the traditional maximum likelihood (ML) estimator. Alternative estimators can be employed, such as marginal maximum likelihood (MML), categorical weighted least squares (c-WLS), and categorical diagonally weighted least squares (c-DWLS). A Monte Carlo simulation study was conducted to compare the performance of the three estimators in the context of categorical LGM. The results revealed that the MML estimator performed the best for the correctly specified linear growth model, whereas the robust c-DWLS or c-WLS estimator excelled in correctly specified quadratic growth model. The robust c-DWLS and c-WLS estimators were more sensitive to model misspecifications. The findings of the present study provide important implications and practical recommendations for empirical researchers using categorical LGM.

Handling missing data in variational autoencoder based item response theory.

Recently Variational Autoencoders (VAEs) have been proposed as a method to estimate high dimensional Item Response Theory (IRT) models on large datasets. Although these improve the efficiency of estimation drastically compared to traditional methods, they have no natural way to deal with missing values. In this paper, we adapt three existing methods from the VAE literature to the IRT setting and propose one new method. We compare the performance of the different VAE-based methods to each other and to marginal maximum likelihood estimation for increasing levels of missing data in a simulation study for both three- and ten-dimensional IRT models. Additionally, we demonstrate the use of the VAE-based models on an existing algebra test dataset. Results confirm that VAE-based methods are a time-efficient alternative to marginal maximum likelihood, but that a larger number of importance-weighted samples are needed when the proportion of missing values is large.

A Comparison of Limited Information Estimation Methods for the Two-Parameter Normal-Ogive Model with Locally Dependent Items

The two-parameter normal-ogive (2PNO) model is one of the most popular item response theory (IRT) models for analyzing dichotomous items. Consistent parameter estimation of the 2PNO model using marginal maximum likelihood estimation relies on the local independence assumption. However, the assumption of local independence might be violated in practice. Likelihood-based estimation of the local dependence structure is often computationally demanding. Moreover, many IRT models that model local dependence do not have a marginal interpretation of item parameters. In this article, limited information estimation methods are reviewed that allow the convenient and straightforward handling of local dependence in estimating the 2PNO model. In detail, pairwise likelihood, weighted least squares, and normal-ogive harmonic analysis robust method (NOHARM) estimation are compared with marginal maximum likelihood estimation that ignores local dependence. A simulation study revealed that item parameters can be consistently estimated with limited information methods. At the same time, marginal maximum likelihood estimation resulted in biased item parameter estimates in the presence of local dependence. From a practical perspective, there were only minor differences regarding the statistical quality of item parameter estimates of the different estimation methods. Differences between the estimation methods are also compared for two empirical datasets.

Interaction of black carbon surface mass with meteorological variables and spatial pattern across the 36 states of tropical Nigeria

Black carbon (BC) has been linked to cardio-pulmonary diseases and lung cancer. It is also the second largest contributor to climate change, after carbon dioxide (CO2), Effective mitigation of BC is of much benefits, but it requires detailed knowledge of its spatial distribution, as well as the influence of local meteorology on the spread of the pollutant. Many countries in sub-Sahara Africa including Nigeria, lack data on BC level. There is also limited information on meteorological impact on BC spatial distribution. This study therefore assessed the spatial and temporal distribution of BC across the 36 states of Nigeria and the influence of meteorology on the pollutant spread. Data on BC surface mass (BCMASS) and seven meteorological variables (air temperature, relative humidity, rainfall, wind speed, planetary boundary layer height (PBLH), atmospheric pressure and wind direction) were obtained from Modern Era Retrospective analysis for Research (MERRA-2). Spatial distribution of BCSMASS across 36 states of Nigeria, and interaction with meteorology was studied using correlation analyses, Bayesian ridge regression with marginal maximum likelihood estimation, and a Hybrid Single-Particle Lagrangian Integrated Trajectory (HYSPLIT) model. There was significant (p < 0.05) variation in BCSMASS across the states. The meteorological variables, except wind direction, significantly influenced BCSMASS and they accounted for 63% of the variation. Five-day backward trajectory modelled for each month of the year revealed that northern Nigeria is majorly under the influence of the dry and dusty tropical continental wind, and this could partially account for the lower BCSMASS concentration in the region, compared to southern Nigeria. Based on population-weighted exposure, states in southern Nigeria, especially the south-eastern part, should be prioritized in planning mitigation, for effective climate action.

Strengths Model for Youth (SM-Y) Case Management: Initial Findings on Youth Outcomes.

Case management is a widely utilized service in both youth and adult outpatient community mental health settings. Despite its widespread use, previous findings suggest that youth case management often lacks empirically tested models or frameworks. This article presents the results of a pilot study that involved adapting the Strengths Model, an adult case management model, for the child and adolescent outpatient community mental health population. The newly adapted model, known as the Strengths Model for Youth (SM-Y), was implemented in an urban community mental health center across five different youth case management teams. To assess changes over time in youth receiving SM-Y case management, marginal maximum likelihood multilevel modeling with adaptive Gaussian quadrature methods was applied. The study focused on three domains: socialization, education, and hospitalization. Utilizing the logit link function and Bernoulli conditional distribution due to the binary nature of the outcome data, three individual trajectories were drawn for socialization, education, and hospitalization. Positive findings indicated increases in socialization and educational performance among children and adolescents.

Pairwise Likelihood Estimation of the 2PL Model with Locally Dependent Item Responses

The local independence assumption is crucial for the consistent estimation of item parameters in item response theory models. This article explores a pairwise likelihood estimation approach for the two-parameter logistic (2PL) model that treats the local dependence structure as a nuisance in the optimization function. Hence, item parameters can be consistently estimated without explicit modeling assumptions of the dependence structure. Two simulation studies demonstrate that the proposed pairwise likelihood estimation approach allows nearly unbiased and consistent item parameter estimation. Our proposed method performs similarly to the marginal maximum likelihood and pairwise likelihood estimation approaches, which also estimate the parameters for the local dependence structure.

The irtQ R package: a user-friendly tool for item response theory-based test data analysis and calibration.

Computerized adaptive testing (CAT) has become a widely adopted test design for high-stakes licensing and certification exams, particularly in the health professions in the United States, due to its ability to tailor test difficulty in real time, reducing testing time while providing precise ability estimates. A key component of CAT is item response theory (IRT), which facilitates the dynamic selection of items based on examinees' ability levels during a test. Accurate estimation of item and ability parameters is essential for successful CAT implementation, necessitating convenient and reliable software to ensure precise parameter estimation. This paper introduces the irtQ R package, which simplifies IRT-based analysis and item calibration under unidimensional IRT models. While it does not directly simulate CAT, it provides essential tools to support CAT development, including parameter estimation using marginal maximum likelihood estimation via the expectation-maximization algorithm, pretest item calibration through fixed item parameter calibration and fixed ability parameter calibration methods, and examinee ability estimation. The package also enables users to compute item and test characteristic curves and information functions necessary for evaluating the psychometric properties of a test. This paper illustrates the key features of the irtQ package through examples using simulated datasets, demonstrating its utility in IRT applications such as test data analysis and ability scoring. By providing a user-friendly environment for IRT analysis, irtQ significantly enhances the capacity for efficient adaptive testing research and operations. Finally, the paper highlights additional core functionalities of irtQ, emphasizing its broader applicability to the development and operation of IRT-based assessments.

Effects of the Quantity and Magnitude of Cross-Loading and Model Specification on MIRT Item Parameter Recovery.

In real-world situations, multidimensional data may appear on large-scale tests or psychological surveys. The purpose of this study was to investigate the effects of the quantity and magnitude of cross-loadings and model specification on item parameter recovery in multidimensional Item Response Theory (MIRT) models, especially when the model was misspecified as a simple structure, ignoring the quantity and magnitude of cross-loading. A simulation study that replicated this scenario was designed to manipulate the variables that could potentially influence the precision of item parameter estimation in the MIRT models. Item parameters were estimated using marginal maximum likelihood, utilizing the expectation-maximization algorithms. A compensatory two-parameter logistic-MIRT model with two dimensions and dichotomous item-responses was used to simulate and calibrate the data for each combination of conditions across 500 replications. The results of this study indicated that ignoring the quantity and magnitude of cross-loading and model specification resulted in inaccurate and biased item discrimination parameter estimates. As the quantity and magnitude of cross-loading increased, the root mean square of error and bias estimates of item discrimination worsened.

Using Auxiliary Item Information in the Item Parameter Estimation of a Graded Response Model for a Small to Medium Sample Size: Empirical Versus Hierarchical Bayes Estimation.

Marginal maximum likelihood estimation (MMLE) is commonly used for item response theory item parameter estimation. However, sufficiently large sample sizes are not always possible when studying rare populations. In this paper, empirical Bayes and hierarchical Bayes are presented as alternatives to MMLE in small sample sizes, using auxiliary item information to estimate the item parameters of a graded response model with higher accuracy. Empirical Bayes and hierarchical Bayes methods are compared with MMLE to determine under what conditions these Bayes methods can outperform MMLE, and to determine if hierarchical Bayes can act as an acceptable alternative to MMLE in conditions where MMLE is unable to converge. In addition, empirical Bayes and hierarchical Bayes methods are compared to show how hierarchical Bayes can result in estimates of posterior variance with greater accuracy than empirical Bayes by acknowledging the uncertainty of item parameter estimates. The proposed methods were evaluated via a simulation study. Simulation results showed that hierarchical Bayes methods can be acceptable alternatives to MMLE under various testing conditions, and we provide a guideline to indicate which methods would be recommended in different research situations. R functions are provided to implement these proposed methods.

Item Parameter Recovery: Sensitivity to Prior Distribution.

Marginal maximum likelihood, a common estimation method for item response theory models, is not inherently a Bayesian procedure. However, due to estimation difficulties, Bayesian priors are often applied to the likelihood when estimating 3PL models, especially with small samples. Little focus has been placed on choosing the priors for marginal maximum estimation. In this study, using sample sizes of 1,000 or smaller, not using priors often led to extreme, implausible parameter estimates. Applying prior distributions to the c-parameters alleviated the estimation problems with samples of 500 or more; for the samples of 100, priors on both the a-parameters and c-parameters were needed. Estimates were biased when the mode of the prior did not match the true parameter value, but the degree of the bias did not depend on the strength of the prior unless it was extremely informative. The root mean squared error (RMSE) of the a-parameters and b-parameters did not depend greatly on either the mode or the strength of the prior unless it was extremely informative. The RMSE of the c-parameters, like the bias, depended on the mode of the prior for c.

Parameter Estimation of KST-IRT Model under Local Dependence

A mantra often repeated in the introductory material to psychometrics and Item Response Theory (IRT) is that a Rasch model is a probabilistic version of a Guttman scale. The idea comes from the observation that a sigmoidal item response function provides a probabilistic version of the characteristic function that models an item response in the Guttman scale. It appears, however, more difficult to reconcile the assumption of local independence, which traditionally accompanies the Rasch model, with the item dependence existing in a Guttman scale. In recent work, an alternative probabilistic version of a Guttman scale was proposed, combining Knowledge Space Theory (KST) with IRT modeling, here referred to as KST-IRT. The present work has, therefore, a two-fold aim. Firstly, the estimation of the parameters involved in KST-IRT models is discussed. More in detail, two estimation methods based on the Expectation Maximization (EM) procedure are suggested, i.e., Marginal Maximum Likelihood (MML) and Gibbs sampling, and are compared on the basis of simulation studies. Secondly, for a Guttman scale, the estimates of the KST-IRT models are compared with those of the traditional combination of the Rasch model plus local independence under the interchange of the data generation processes. Results show that the KST-IRT approach might be more effective in capturing local dependence as it appears to be more robust under misspecification of the data generation process, but it comes with the price of an increased number of parameters.

Random Effects Multinomial Processing Tree Models: A Maximum Likelihood Approach

The present article proposes and evaluates marginal maximum likelihood (ML) estimation methods for hierarchical multinomial processing tree (MPT) models with random and fixed effects. We assume that an identifiable MPT model with S parameters holds for each participant. Of these S parameters, R parameters are assumed to vary randomly between participants, and the remaining S-R parameters are assumed to be fixed. We also propose an extended version of the model that includes effects of covariates on MPT model parameters. Because the likelihood functions of both versions of the model are too complex to be tractable, we propose three numerical methods to approximate the integrals that occur in the likelihood function, namely, the Laplace approximation (LA), adaptive Gauss–Hermite quadrature (AGHQ), and Quasi Monte Carlo (QMC) integration. We compare these three methods in a simulation study and show that AGHQ performs well in terms of both bias and coverage rate. QMC also performs well but the number of responses per participant must be sufficiently large. In contrast, LA fails quite often due to undefined standard errors. We also suggest ML-based methods to test the goodness of fit and to compare models taking model complexity into account. The article closes with an illustrative empirical application and an outlook on possible extensions and future applications of the proposed ML approach.

Exploring Approaches for Estimating Parameters in Cognitive Diagnosis Models with Small Sample Sizes

Cognitive diagnostic models (CDMs) are increasingly being used in various assessment contexts to identify cognitive processes and provide tailored feedback. However, the most commonly used estimation method for CDMs, marginal maximum likelihood estimation with Expectation–Maximization (MMLE-EM), can present difficulties when sample sizes are small. This study compares the results of different estimation methods for CDMs under varying sample sizes using simulated and empirical data. The methods compared include MMLE-EM, Bayes modal, Markov chain Monte Carlo, a non-parametric method, and a parsimonious parametric model such as Restricted DINA. We varied the sample size, and assessed the bias in the estimation of item parameters, the precision in attribute classification, the bias in the reliability estimate, and computational cost. The findings suggest that alternative estimation methods are preferred over MMLE-EM under low sample-size conditions, whereas comparable results are obtained under large sample-size conditions. Practitioners should consider using alternative estimation methods when working with small samples to obtain more accurate estimates of CDM parameters. This study aims to maximize the potential of CDMs by providing guidance on the estimation of the parameters.

Fast estimation of multiple group generalized linear latent variable models for categorical observed variables

A computationally efficient method for marginal maximum likelihood estimation of multiple group generalized linear latent variable models for categorical data is introduced. The approach utilizes second-order Laplace approximations of the integrals in the likelihood function. It is demonstrated how second-order Laplace approximations can be utilized highly efficiently for generalized linear latent variable models by considering symmetries that exist for many types of model structures. In a simulation with binary observed variables and four correlated latent variables in four groups, the method has similar bias and mean squared error compared to adaptive Gauss-Hermite quadrature with five quadrature points while substantially improving computational efficiency. An empirical example from a large-scale educational assessment illustrates the accuracy and computational efficiency of the method when compared against adaptive Gauss-Hermite quadrature with three, five, and 13 quadrature points.

Margin-Based Modal Adaptive Learning for Visible-Infrared Person Re-Identification.

Visible-infrared person re-identification (VIPR) has great potential for intelligent transportation systems for constructing smart cities, but it is challenging to utilize due to the huge modal discrepancy between visible and infrared images. Although visible and infrared data can appear to be two domains, VIPR is not identical to domain adaptation as it can massively eliminate modal discrepancies. Because VIPR has complete identity information on both visible and infrared modalities, once the domain adaption is overemphasized, the discriminative appearance information on the visible and infrared domains would drain. For that, we propose a novel margin-based modal adaptive learning (MMAL) method for VIPR in this paper. On each domain, we apply triplet and label smoothing cross-entropy functions to learn appearance-discriminative features. Between the two domains, we design a simple yet effective marginal maximum mean discrepancy (M3D) loss function to avoid an excessive suppression of modal discrepancies to protect the features' discriminative ability on each domain. As a result, our MMAL method could learn modal-invariant yet appearance-discriminative features for improving VIPR. The experimental results show that our MMAL method acquires state-of-the-art VIPR performance, e.g., on the RegDB dataset in the visible-to-infrared retrieval mode, the rank-1 accuracy is 93.24% and the mean average precision is 83.77%.

Accelerating item factor analysis on GPU with Python package xifa.

Item parameter estimation is a crucial step when conducting item factor analysis (IFA). From the view of frequentist estimation, marginal maximum likelihood (MML) seems to be the gold standard. However, fitting a high-dimensional IFA model by MML is still a challenging task. The current study demonstrates that with the help of a GPU (graphics processing unit) and carefully designed vectorization, the computational time of MML could be largely reduced for large-scale IFA applications. In particular, a Python package called xifa (accelerated item factor analysis) is developed, which implements a vectorized Metropolis-Hastings Robbins-Monro (VMHRM) algorithm. Our numerical experiments show that the VMHRM on a GPU may run 33 times faster than its CPU version. When the number of factors is at least five, VMHRM (on GPU) is much faster than the Bock-Aitkin expectation maximization, MHRM implemented by mirt (on CPU), and the importance-weighted autoencoder (on GPU). The GPU-implemented VMHRM is most appropriate for high-dimensional IFA with large data sets. We believe that GPU computing will play a central role in large-scale psychometric modeling in the near future.

Posterior contraction and testing for multivariate isotonic regression

We consider the nonparametric regression problem with multiple predictors and an additive error, where the regression function is assumed to be coordinatewise nondecreasing. We propose a Bayesian approach to make inferences on the multivariate monotone regression function, obtain the posterior contraction rate, and construct a universally consistent Bayesian testing procedure for multivariate monotonicity. To facilitate posterior analysis, we temporarily set aside the shape restrictions, and endow a prior on blockwise constant regression functions with independently normally distributed heights. The unknown variance of the error term is either estimated by the marginal maximum likelihood estimate, or is equipped with an inverse-gamma prior. Then the unrestricted block-heights are a posteriori also independently normally distributed given the error variance, by conjugacy. To comply with the shape restrictions, we project samples from the unrestricted posterior onto the class of multivariate monotone functions, inducing the “projection-posterior distribution”, to be used for making an inference. Under an L1-metric, we show that the projection-posterior based on n independent samples contracts around the true monotone regression function at the optimal rate n−1∕(2+d). Then we construct a Bayesian test for multivariate monotonicity based on the posterior probability of a shrinking neighborhood of the class of multivariate monotone functions. We show that the test is universally consistent, that is, the level of the Bayesian test goes to zero, and the power at any fixed alternative goes to one. Moreover, we show that for a smooth alternative function, power goes to one as long as its distance to the class of multivariate monotone functions is at least of the order of the estimation error for a smooth function. To the best of our knowledge, no other test for multivariate monotonicity is available in the Bayesian or the frequentist literature.

Interpolating log-determinant and trace of the powers of matrix textbf{A} + ttextbf{B}

We develop heuristic interpolation methods for the functions tmapsto log det left( textbf{A} + ttextbf{B} right) and tmapsto {{,textrm{trace},}}left( (textbf{A} + ttextbf{B})^{p} right) where the matrices textbf{A} and textbf{B} are Hermitian and positive (semi) definite and p and t are real variables. These functions are featured in many applications in statistics, machine learning, and computational physics. The presented interpolation functions are based on the modification of sharp bounds for these functions. We demonstrate the accuracy and performance of the proposed method with numerical examples, namely, the marginal maximum likelihood estimation for Gaussian process regression and the estimation of the regularization parameter of ridge regression with the generalized cross-validation method.

Robustness of the performance of the optimized hierarchical two-parameter logistic IRT model for small-sample item calibration

Hierarchical Bayesian modeling is beneficial when complex models with many parameters of the same type, such as item response theory (IRT) models, are to be estimated with sparse data. Recently, Koenig et al. (Applied Psychological Measurement, 44, 311–326, 2020) illustrated in an optimized hierarchical Bayesian two-parameter logistic model (OH2PL) how to avoid bias due to unintended shrinkage or degeneracies of the posterior, and how to benefit from this approach in small samples. The generalizability of their findings, however, is limited because they investigated only a single specification of the hyperprior structure. Consequently, in a comprehensive simulation study, we investigated the robustness of the performance of the novel OH2PL in several specifications of their hyperpriors under a broad range of data conditions. We show that the novel OH2PL in the half-Cauchy or Exponential configuration yields unbiased (in terms of bias) model parameter estimates in small samples of N = 50. Moreover, it outperforms (especially in terms of the RMSE of the item discrimination parameters) marginal maximum likelihood (MML) estimation and its nonhierarchical counterpart. This further corroborates the possibility that hierarchical Bayesian IRT models behave differently than general hierarchical Bayesian models. We discuss these results regarding the applicability of complex IRT models in small-scale situations typical in psychological research, and illustrate the extended applicability of the 2PL IRT model with an empirical example.