Marginal Maximum Likelihood Estimation Research Articles

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.

Bayesian Approaches for Detecting Differential Item Functioning Using the Generalized Graded Unfolding Model.

Differential item functioning (DIF) analysis is one of the most important applications of item response theory (IRT) in psychological assessment. This study examined the performance of two Bayesian DIF methods, Bayes factor (BF) and deviance information criterion (DIC), with the generalized graded unfolding model (GGUM). The Type I error and power were investigated in a Monte Carlo simulation that manipulated sample size, DIF source, DIF size, DIF location, subpopulation trait distribution, and type of baseline model. We also examined the performance of two likelihood-based methods, the likelihood ratio (LR) test and Akaike information criterion (AIC), using marginal maximum likelihood (MML) estimation for comparison with past DIF research. The results indicated that the proposed BF and DIC methods provided well-controlled Type I error and high power using a free-baseline model implementation, their performance was superior to LR and AIC in terms of Type I error rates when the reference and focal group trait distributions differed. The implications and recommendations for applied research are discussed.

Assessing Rasch measurement estimation methods across R packages with yes/no vocabulary test data

Instrument measurement conducted with Rasch analysis is a common process in language assessment research. A recent systematic review of 215 studies involving Rasch analysis in language testing and applied linguistics research reported that 23 different software packages had been utilized. However, none of the analyses were conducted with one of the numerous R-based Rasch analysis software packages, which generally employ one of the three estimation methods: conditional maximum likelihood estimation (CMLE), joint maximum likelihood estimation (JMLE), or marginal maximum likelihood estimation (MMLE). For this study, eRm, a CMLE-based R package, was utilized to conduct a dichotomous Rasch analysis of a Yes/No vocabulary test based on the academic word list. The resulting parameters and diagnostic statistics were compared with the equivalent results from four other R-based Rasch measurement software packages and Winsteps. Finally, all of the packages were utilized in the analysis of 1000 simulated datasets to investigate the extent to which results generated from the contrasting estimation methods converged or diverged. Overall, the differences between the results produced with the three estimation methods were negligible, and the discrepancies observed between datasets were attributable to the software choice as opposed to the estimation method.

[RETRACTED ARTICLE] irtplay: An R Package for Unidimensional Item Response Theory Modeling

Item response theory (IRT) is a general framework in which mathematical models are formulated to explain the relationship between an examinee's observable response on an item and the latent ability measured by a test. The application of IRT models and related statistical methods are commonly found in educational and psychological research. An important step in applying IRT models to test data is estimating the IRT model parameters. Accordingly, the successful application of IRT rests on the satisfactory statistical techniques and software for accurately estimating the model parameters. The irtplay R package was developed to provide users with a user-friendly experience and convenience when analyzing test data using unidimensional IRT models. The package can be used to fit the IRT models to a mixture of dichotomous and polytomous item data using marginal maximum likelihood estimation via the expectation-maximization, calibrate pretest items, and estimate examinees' latent ability parameters. In addition, the package provides practical tools that conveniently enable users to conduct many analyses related to IRT such as evaluating IRT model-data fit, analyzing differential item functioning, computing asymptotic variance-covariance matrices of item parameter estimates, calculating the conditional probability distribution of observed scores using the Lord and Wingersky (1984) formula, and importing item and ability parameter estimates from the output of popular IRT software. The main features of the irtplay package are illustrated using three data examples.

Modeling Not-Reached Items in Timed Tests: A Response Time Censoring Approach.

Time limits are imposed on many computer-based assessments, and it is common to observe examinees who run out of time, resulting in missingness due to not-reached items. The present study proposes an approach to account for the missing mechanisms of not-reached items via response time censoring. The censoring mechanism is directly incorporated into the observed likelihood of item responses and response times. A marginal maximum likelihood estimator is proposed, and its asymptotic properties are established. The proposed method was evaluated and compared to several alternative approaches that ignore the censoring through simulation studies. An empirical study based on the PISA 2018 Science Test was further conducted.

A Comprehensive Simulation Study of Estimation Methods for the Rasch Model

The Rasch model is one of the most prominent item response models. In this article, different item parameter estimation methods for the Rasch model are systematically compared through a comprehensive simulation study: Different alternatives of joint maximum likelihood (JML) estimation, different alternatives of marginal maximum likelihood (MML) estimation, conditional maximum likelihood (CML) estimation, and several limited information methods (LIM). The type of ability distribution (i.e., nonnormality), the number of items, sample size, and the distribution of item difficulties were systematically varied. Across different simulation conditions, MML methods with flexible distributional specifications can be at least as efficient as CML. Moreover, in many situations (i.e., for long tests), penalized JML and JML with ε adjustment resulted in very efficient estimates and might be considered alternatives to JML implementations currently used in statistical software. Moreover, minimum chi-square (MINCHI) estimation was the best-performing LIM method. These findings demonstrate that JML estimation and LIM can still prove helpful in applied research.

A Joint MLE Approach to Large-Scale Structured Latent Attribute Analysis

Structured latent attribute models (SLAMs) are a family of discrete latent variable models widely used in education, psychology, and epidemiology to model multivariate categorical data. A SLAM assumes that multiple discrete latent attributes explain the dependence of observed variables in a highly structured fashion. Usually, the maximum marginal likelihood estimation approach is adopted for SLAMs, treating the latent attributes as random effects. The increasing scope of modern assessment data involves large numbers of observed variables and high-dimensional latent attributes. This poses challenges to classical estimation methods and requires new methodology and understanding of latent variable modeling. Motivated by this, we consider the joint maximum likelihood estimation (MLE) approach to SLAMs, treating latent attributes as fixed unknown parameters. We investigate estimability, consistency, and computation in the regime where sample size, number of variables, and number of latent attributes all can diverge. We establish the statistical consistency of the joint MLE and propose efficient algorithms that scale well to large-scale data for several popular SLAMs. Simulation studies demonstrate the superior empirical performance of the proposed methods. An application to real data from an international educational assessment gives interpretable findings of cognitive diagnosis.

Bayesian Modal Estimation for the One-Parameter Logistic Ability-Based Guessing (1PL-AG) Model.

The calibration of the one-parameter logistic ability-based guessing (1PL-AG) model in item response theory (IRT) with a modest sample size remains a challenge for its implausible estimates and difficulty in obtaining standard errors of estimates. This article proposes an alternative Bayesian modal estimation (BME) method, the Bayesian Expectation-Maximization-Maximization (BEMM) method, which is developed by combining an augmented variable formulation of the 1PL-AG model and a mixture model conceptualization of the three-parameter logistic model (3PLM). By comparing with marginal maximum likelihood estimation (MMLE) and Markov Chain Monte Carlo (MCMC) in JAGS, the simulation shows that BEMM can produce stable and accurate estimates in the modest sample size. A real data example and the MATLAB codes of BEMM are also provided.