Marginal Maximum Likelihood Estimation Research Articles

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.

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.

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.

The irtQ R package: a user-friendly tool for item response theorybased 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 (http://CRAN.R-project.org/), which simplifies IRTbased 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.

Ab initio uncertainty quantification in scattering analysis of microscopy.

Estimating parameters from data is a fundamental problem in physics, customarily done by minimizing a loss function between a model and observed statistics. In scattering-based analysis, it is common to work in the reciprocal space. Researchers often employ their domain expertise to select a specific range of wave vectors for analysis, a choice that can vary depending on the specific case. We introduce another paradigm that defines a probabilistic generative model from the beginning of data processing and propagates the uncertainty for parameter estimation, termed the ab initio uncertainty quantification (AIUQ). As an illustrative example, we demonstrate this approach with differential dynamic microscopy (DDM) that extracts dynamical information through minimizing a loss function for the squared differences of the Fourier-transformed intensities, at a selected range of wave vectors. We first show that the conventional way of estimation in DDM is equivalent to fitting a temporal variogram in the reciprocal space using a latent factor model as the generative model. Then we derive the maximum marginal likelihood estimator, which optimally weighs the information at all wave vectors, therefore eliminating the need to select the range of wave vectors. Furthermore, we substantially reduce the computational cost of computing the likelihood function without approximation, by utilizing the generalized Schur algorithm for Toeplitz covariances. Simulated studies of a wide range of dynamical systems validate that the AIUQ method improves estimation accuracy and enables model selection with automated analysis. The utility of AIUQ is also demonstrated by three distinct sets of experiments: first in an isotropic Newtonian fluid, pushing limits of optically dense systems compared to multiple particle tracking; next in a system undergoing a sol-gel transition, automating the determination of gelling points and critical exponent; and lastly, in discerning anisotropic diffusive behavior of colloids in a liquid crystal. These studies demonstrate that the new approach does not require manually selecting the wave vector range and enables automated analysis.

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.

The Rank-2PL IRT Models for Forced-Choice Questionnaires: Maximum Marginal Likelihood Estimation with an EM Algorithm

The rank two-parameter logistic (Rank-2PL) item response theory models refer to a set of models applying the 2PL model in a sequential ranking process that occurs in forced-choice questionnaires. The multi-unidimensional pairwise preference with 2PL model (MUPP-2PL) is a Rank-2PL model for items with two statements. Focusing on items with three statements, we develop a maximum marginal likelihood estimation with an expectation-maximization algorithm to estimate item parameters and their standard errors. A simulation study is conducted to check parameter recovery, and then the model is applied to a real dataset. Finally, the findings are summarized and discussed, and future research is suggested.

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.

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.

Estimation Method of Covariance Matrix in Atmospheric Inversion of CO2 Emissions

Atmospheric inversion of CO2 Emissions is based on the correction of prior carbon dioxide flux inventories using concentration monitoring data and atmospheric transport models to obtain posterior carbon dioxide flux. In atmospheric inversion studies, fixed covariance functions are commonly used to generate covariance matrices, and the hyperparameters in the covariance functions are empirically estimated. In this study, we design and implement an ideal experiment based on meteorological data from the central urban area of Zhengzhou, using WRF-STILT to generate sensitivity matrices and construct real carbon emission inventories and prior inventories. Based on the real carbon emission inventories and sensitivity matrices of monitoring stations, simulated observation concentration values are generated. Firstly, based on the observed concentration values, sensitivity matrices of monitoring stations, prior inventories, and constructed covariance matrices, the values of hyperparameters are determined based on maximum marginal likelihood estimation. Then, the influence of different prior covariance functions on the inversion results is tested, and it is found that the prior covariance matrix generated by the balgovind covariance function is most suitable for the experimental data.

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.

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.

Bayesian inverse problems with heterogeneous variance

AbstractWe consider inverse problems in Hilbert spaces under correlated Gaussian noise, and use a Bayesian approach to find their regularized solution. We focus on mildly ill‐posed inverse problems with fractional noise, using a novel wavelet‐based vaguelette–vaguelette approach. It allows us to apply sequence space methods without assuming that all operators are simultaneously diagonalizable. The results are proved for more general bases and covariance operators. Our primary aim is to study posterior contraction rate in such inverse problems over Sobolev classes and compare it to the derived minimax rate. Secondly, we study effect of plugging in a consistent estimator of variances in sequence space on the posterior contraction rate. This result is applied to the problem with error in forward operator. Thirdly, we show that empirical Bayes posterior distribution with a plugged‐in maximum marginal likelihood estimator of the prior scale contracts at the optimal rate, adaptively, in the minimax sense.

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.

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.

Evaluation of the Psychometric Properties of the Arab Compulsive Internet Use Scale (CIUS) by Item Response Theory Modeling (IRT).

Introduction: The psychometric properties of the Arab translation of the Compulsive Internet Use Scale (CIUS) have been previously studied by confirmatory factor analysis (CFA) with AMOS software using the asymptotically distribution-free (ADF) estimator. Unidimensionality has been achieved at the cost of correlating several item variance errors. However, several reviews of SEM software packages and estimation methods indicate that the option of robust standard errors is not present in the AMOS package and that ADF estimation may yield biased parameter estimates. We therefore explored a second analysis through item response theory (IRT) using the parametric graded response model (GRM) and the marginal maximum likelihood (MML) estimation method embedded in the LTM package of R software. Differential item functioning (DIF) or item bias across subpopulations was also explored within IRT framework as different samples were investigated. The objective of the current study is to (1) analyze the Arab CIUS scale with IRT, (2) investigate DIF in three samples, and (3) contribute to the ongoing debate on Internet-use-related addictive behaviors using the CIUS items as a proxy. Methods: We assessed three samples of people, one in Algeria and two in Lebanon, with a total of 1520 participants. Results: Almost three out of every five items were highly related to the latent construct. However, the unidimensionality hypothesis was not supported. Furthermore, besides being locally dependent, the scale may be weakened by DIF across geographic regions. Some of the CIUS items related to increasing priority, impaired control, continued use despite harm, and functional impairment as well as withdrawal and coping showed good discriminative capabilities. Those items were endorsed more frequently than other CIUS items in people with higher levels of addictive Internet use. Conclusions: Contrary to earlier ADF estimation findings, unidimensionality of the CIUS scale was not supported by IRT parametric GRM in a large sample of Arab speaking participants. The results may be helpful for scale revision. By proxy, the study contributes to testing the validity of addiction criteria applied to Internet use related-addictive behaviors.

Power Analysis for the Wald, LR, Score, and Gradient Tests in a Marginal Maximum Likelihood Framework: Applications in IRT

The Wald, likelihood ratio, score, and the recently proposed gradient statistics can be used to assess a broad range of hypotheses in item response theory models, for instance, to check the overall model fit or to detect differential item functioning. We introduce new methods for power analysis and sample size planning that can be applied when marginal maximum likelihood estimation is used. This allows the application to a variety of IRT models, which are commonly used in practice, e.g., in large-scale educational assessments. An analytical method utilizes the asymptotic distributions of the statistics under alternative hypotheses. We also provide a sampling-based approach for applications where the analytical approach is computationally infeasible. This can be the case with 20 or more items, since the computational load increases exponentially with the number of items. We performed extensive simulation studies in three practically relevant settings, i.e., testing a Rasch model against a 2PL model, testing for differential item functioning, and testing a partial credit model against a generalized partial credit model. The observed distributions of the test statistics and the power of the tests agreed well with the predictions by the proposed methods in sufficiently large samples. We provide an openly accessible R package that implements the methods for user-supplied hypotheses.

Evaluating the effects of analytical decisions in large-scale assessments: analyzing PISA mathematics 2003-2012

Research questionThis paper examines the overarching question of to what extent different analytic choices may influence the inference about country-specific cross-sectional and trend estimates in international large-scale assessments. We take data from the assessment of PISA mathematics proficiency from the four rounds from 2003 to 2012 as a case study.MethodsIn particular, four key methodological factors are considered as analytical choices in the rescaling and analysis of the data: (1) The selection of country sub-samples for item calibration differing at three factor levels. (2) The item sample refering to two sets of mathematics items used within PISA. (3) The estimation method used for item calibration: marginal maximum likelihood estimation method as implemented in R package TAM or an pairwise row averaging approach as implemented in the R package pairwise. (4) The type of linking method: concurrent calibration or separate calibration with successive chain linking.FindingsIt turned out that analytical decisions for scaling did affect the PISA outcomes. The factors of choosing different calibration samples, estimation method and linking method tend to show only small effects on the country-specific cross-sectional and trend estimates. However, the selection of different link items seems to have a decisive influence on country ranking and development trends between and within countries.