Marginal Maximum Likelihood Estimation Research Articles

Estimating a DIF decomposition model using a random-weights linear logistic test model approach.

A differential item functioning (DIF) decomposition model separates a testlet item DIF into two sources: item-specific differential functioning and testlet-specific differential functioning. This article provides an alternative model-building framework and estimation approach for a DIF decomposition model that was proposed by Beretvas and Walker (2012). Although their model is formulated under multilevel modeling with the restricted pseudolikelihood estimation method, our approach illustrates DIF decomposition modeling that is directly built upon the random-weights linear logistic test model framework with the marginal maximum likelihood estimation method. In addition to demonstrating our approach's performance, we provide detailed information on how to implement this new DIF decomposition model using an item response theory software program; using DIF decomposition may be challenging for practitioners, yet practical information on how to implement it has previously been unavailable in the measurement literature.

General partially linear additive transformation model with right-censored data

We propose a class of general partially linear additive transformation models (GPLATM) with right-censored survival data in this work. The class of models are flexible enough to cover many commonly used parametric and nonparametric survival analysis models as its special cases. Based on the B spline interpolation technique, we estimate the unknown regression parameters and functions by the maximum marginal likelihood estimation method. One important feature of the estimation procedure is that it does not need the baseline and censoring cumulative density distributions. Some numerical studies illustrate that this procedure can work very well for the moderate sample size.

Detecting Aberrant Responding on Unidimensional Pairwise Preference Tests

This study investigated the efficacy of the lz person fit statistic for detecting aberrant responding with unidimensional pairwise preference (UPP) measures, constructed and scored based on the Zinnes–Griggs item response theory (IRT) model, which has been used for a variety of recent noncognitive testing applications. Because UPP measures are used to collect both “self-” and “other” reports, the capability of lz to detect two of the most common and potentially detrimental response sets, namely fake good and random responding, was explored. The effectiveness of lz was studied using empirical and theoretical critical values for classification, along with test length, test information, the type of statement parameters, and the percentage of items answered aberrantly (20%, 50%, 100%). It was found that lz was ineffective in detecting fake good responding, with power approaching zero in the 100% aberrance conditions. However, lz was highly effective in detecting random responding, with power approaching 1.0 in long-test, high information conditions, and there was no diminution in efficacy when using marginal maximum likelihood estimates of statement parameters in place of the true values. Although using empirical critical values for classification provided slightly higher power and more accurate Type I error rates, theoretical critical values, corresponding to a standard normal distribution, provided nearly as good results.

IRT Modelling of Dichotomous Items with Linear Factor Analysis

Bearing on linear factor analytical techniques applied to matrices of tetrachoric and Pearson correlations of dichotomous variables, we describe heuristic methods of parameter estimation in the two-parameter normal ogive model and discuss some indices and methods of factor analysis and structural equation modeling which were never investigated systematically for the assessment of unidimensionality in this case. The results of a simulation study indicate that heuristic parameter estimation is surprisingly accurate when distributional assumptions are met, mostly being only slightly inferior to marginal maximum likelihood estimation. Moreover, recovery of unidimensionality appears feasible with residual-based indices and methods but also depends on type of correlation coefficient used. We discuss our findings with regard to their generalizability, their relevance to applied research, and the refinement of parameter estimation techniques in item response theory.

The Data-augmentation Techniques in Item Response Modeling: Current Approaches and New Developments

在心理与教育测量中，项目反应理论（Item Response Theory，IRT）模型的参数估计方法是理论研究与实践应用的基本工具。最近，由于IRT模型的不断扩展与EM（expectation-maximization）算法自身的固有问题，参数估计方法的改进与发展显得尤为重要。这里介绍了IRT模型中边际极大似然估计的发展，提出了它的阶段性特征，即联合极大似然估计阶段、确定性潜在心理特质“填补”阶段、随机潜在心理特质“填补”阶段，重点阐述了它的潜在心理特质“填补”（dataaugmentation）思想。EM算法与Metropolis-Hastings Robbins—Monro（MH-RM）算法作为不同的潜在心理特质“填补”方法，都是边际极大似然估计的思想跨越。目前，潜在心理特质“填补”的参数估计方法仍在不断发展与完善。

The Multidimensional Testlet-Effect Rasch Model

Testlet design has been widely adopted in educational and psychological assessment. A testlet is a cluster of items that share a common stimulus(e.g., a reading comprehension passage or a figure), and the possible local dependence among items within a testlet is called testlet-effect. Various models have been developed to take into account such testlet effect. Examples included the Rasch testlet model, two-parameter logistic Bayesian testlet model, and higher-order testlet model. However, these existing models all assume that an item is affected by only one single testlet effect. Therefore, they are essentially unidimensional testlet-effect models. In practice, multiple testlet effects may simultaneously affect item responses in a testlet. For example, in addition to common stimulus, items can be grouped according to their domains, knowledge units, or item format, such that multiple testlet effects are involved. In essence, an item measures multiple latent traits, in addition to the target latent trait(s) that the test was designed to measure. Existing unidimensional testlet-effect models become inapplicable when multiple testlet effects are involved. To account for multiple testlet effect, in this study we develop the so-called(within-item) multidimensional testlet-effect Rasch model. The parameters can be estimated with marginal maximum likelihood estimation methods or Bayesian methods with Markov chain Monte Carlo(MCMC) algorithms. In this study, a popular computer program for Rasch models, ConQuest, was used. A series of simulations were conducted to evaluate parameter recovery of the new model, consequences of model misspecification, and the effectiveness of model-data fit statistics. Results show that the parameters of the new model can be recovered fairly well; and ignoring the multiple testlet effects resulted in a biased estimation of item parameters, and an overestimation of test reliability. Additionally, it did little harm on parameter estimation to fit a more complicated model(i.e., the multidimensional testlet-effect Rasch model) to data with a simple structure. In conclusion, the new model is feasible and flexible.

Estimation of a Ramsay-Curve Item Response Theory Model by the Metropolis–Hastings Robbins–Monro Algorithm

In Ramsay curve item response theory (RC-IRT) modeling, the shape of the latent trait distribution is estimated simultaneously with the item parameters. In its original implementation, RC-IRT is estimated via Bock and Aitkin’s EM algorithm, which yields maximum marginal likelihood estimates. This method, however, does not produce the parameter covariance matrix as an automatic byproduct on convergence. In turn, researchers are limited in when they can employ RC-IRT, as the covariance matrix is needed for many statistical inference procedures. The present research remedies this problem by estimating the RC-IRT model parameters by the Metropolis–Hastings Robbins–Monro (MH-RM) algorithm. An attractive feature of MH-RM is that the structure of the algorithm makes estimation of the covariance matrix convenient. Additionally, MH-RM is ideally suited for multidimensional IRT, whereas EM is limited by the “curse of dimensionality.” Based on the current research, when RC-IRT or similar IRT models are eventually generalized to include multiple latent dimensions, MH-RM would appear to be the logical choice for estimation.

An accumulator model for responses and response times in tests based on the proportional hazards model

Latent trait models for responses and response times in tests often lack a substantial interpretation in terms of a cognitive process model. This is a drawback because process models are helpful in clarifying the meaning of the latent traits. In the present paper, a new model for responses and response times in tests is presented. The model is based on the proportional hazards model for competing risks. Two processes are assumed, one reflecting the increase in knowledge and the second the tendency to discontinue. The processes can be characterized by two proportional hazards models whose baseline hazard functions correspond to the temporary increase in knowledge and discouragement. The model can be calibrated with marginal maximum likelihood estimation and an application of the ECM algorithm. Two tests of model fit are proposed. The amenability of the proposed approaches to model calibration and model evaluation is demonstrated in a simulation study. Finally, the model is used for the analysis of two empirical data sets.

Partial marginal likelihood estimation for general transformation models

We consider a large class of transformation models introduced by Gu et al. (2005) [14]. They proposed an estimation procedure for calculating the maximum partial marginal likelihood estimator (MPMLE) of regression parameters. A big advantage of MPMLE is that it avoids estimating two infinitely dimensional nuisance parameters: baseline and censoring survival functions. And they showed the validity of MPMLE through extensive simulations. In this paper, we establish the asymptotic properties of MPMLE in the general transformation models for either right or left censored data. The difficulty in establishing these asymptotic results comes from the fact that the score function derived from the partial marginal likelihood does not have ordinary independence or martingale structure. We develop a novel discretization method to resolve the difficulty. The estimation procedure is further examined using simulation studies and the analysis of the ACTG019 data.

Generalized IRT Models for Extreme Response Style

Extreme response style (ERS) is a systematic tendency for a person to endorse extreme options (e.g., strongly disagree, strongly agree) on Likert-type or rating-scale items. In this study, we develop a new class of item response theory (IRT) models to account for ERS so that the target latent trait is free from the response style and the tendency of ERS is quantified. Parameters of these new models can be estimated with marginal maximum likelihood estimation methods or Bayesian methods. In this study, we use the freeware program WinBUGS, which implements Bayesian methods. In a series of simulations, we find that the parameters are recovered fairly well; ignoring ERS by fitting standard IRT models resulted in biased estimates, and fitting the new models to data without ERS did little harm. Two empirical examples are provided to illustrate the implications and applications of the new models.

Response Time Modeling Based on the Proportional Hazards Model

Response time data are regularly analyzed in psychology. When several response times are assessed per participant, it is common practice to use latent trait models in order to account for the dependency of the response times from the same participant. One such model is the proportional hazards model with random effects. Despite its popularity in survival analysis, this model is rarely used in psychology because of the difficulty of model estimation when latent variables are present. In this article, a new estimation method is proposed. This method is based on the rank correlation matrix containing Kendall's Tau coefficients and unweighted least squares estimation (Kendall, 1938). Compared with marginal maximum likelihood estimation, the new estimation approach is simple, not computationally intensive, and almost as efficient. Additionally, the approach allows the implementation of a test for model fit. Feasibility of the estimation method and validity of the fit test is demonstrated with a simulation study. An application of the model to a real data set is provided.

Flexible marginalized models for bivariate longitudinal ordinal data

Random effects models are commonly used to analyze longitudinal categorical data. Marginalized random effects models are a class of models that permit direct estimation of marginal mean parameters and characterize serial correlation for longitudinal categorical data via random effects (Heagerty, 1999). Marginally specified logistic-normal models for longitudinal binary data. Biometrics 55, 688-698; Lee and Daniels, 2008. Marginalized models for longitudinal ordinal data with application to quality of life studies. Statistics in Medicine 27, 4359-4380). In this paper, we propose a Kronecker product (KP) covariance structure to capture the correlation between processes at a given time and the correlation within a process over time (serial correlation) for bivariate longitudinal ordinal data. For the latter, we consider a more general class of models than standard (first-order) autoregressive correlation models, by re-parameterizing the correlation matrix using partial autocorrelations (Daniels and Pourahmadi, 2009). Modeling covariance matrices via partial autocorrelations. Journal of Multivariate Analysis 100, 2352-2363). We assess the reasonableness of the KP structure with a score test. A maximum marginal likelihood estimation method is proposed utilizing a quasi-Newton algorithm with quasi-Monte Carlo integration of the random effects. We examine the effects of demographic factors on metabolic syndrome and C-reactive protein using the proposed models.

A multicomponent latent trait model for diagnosis.

This paper presents a noncompensatory latent trait model, the multicomponent latent trait model for diagnosis (MLTM-D), for cognitive diagnosis. In MLTM-D, a hierarchical relationship between components and attributes is specified to be applicable to permit diagnosis at two levels. MLTM-D is a generalization of the multicomponent latent trait model (MLTM; Whitely in Psychometrika, 45:479-494, 1980; Embretson in Psychometrika, 49:175-186, 1984) to be applicable to measures of broad traits, such as achievement tests, in which component structure varies between items. Conditions for model identification are described and marginal maximum likelihood estimators are presented, along with simulation data to demonstrate parameter recovery. To illustrate how MLTM-D can be used for diagnosis, an application to a large-scale test of mathematics achievement is presented. An advantage of MLTM-D for diagnosis is that it may be more applicable to large-scale assessments with more heterogeneous items than are latent class models.

How should we assess the fit of Rasch-type models? Approximating the power of goodness-of-fit statistics in categorical data analysis.

We investigate the performance of three statistics, R1, R2 (Glas in Psychometrika 53:525-546, 1988), and M2 (Maydeu-Olivares & Joe in J. Am. Stat. Assoc. 100:1009-1020, 2005, Psychometrika 71:713-732, 2006) to assess the overall fit of a one-parameter logistic model (1PL) estimated by (marginal) maximum likelihood (ML). R1 and R2 were specifically designed to target specific assumptions of Rasch models, whereas M2 is a general purpose test statistic. We report asymptotic power rates under some interesting violations of model assumptions (different item discrimination, presence of guessing, and multidimensionality) as well as empirical rejection rates for correctly specified models and some misspecified models. All three statistics were found to be more powerful than Pearson's X(2) against two- and three-parameter logistic alternatives (2PL and 3PL), and against multidimensional 1PL models. The results suggest that there is no clear advantage in using goodness-of-fit statistics specifically designed for Rasch-type models to test these models when marginal ML estimation is used.

Maximum Marginal Likelihood Estimation for Nonnegative Dictionary Learning in the Gamma-Poisson Model

In this paper we describe an alternative to standard nonnegative matrix factorization (NMF) for nonnegative dictionary learning, i.e., the task of learning a dictionary with nonnegative values from nonnegative data, under the assumption of nonnegative expansion coefficients. A popular cost function used for NMF is the Kullback-Leibler divergence, which underlies a Poisson observation model. NMF can thus be considered as maximization of the joint likelihood of the dictionary and the expansion coefficients. This approach lacks optimality because the number of parameters (which include the expansion coefficients) grows with the number of observations. In this paper we describe variational Bayes and Monte-Carlo EM algorithms for optimization of the marginal likelihood, i.e., the likelihood of the dictionary where the expansion coefficients have been integrated out (given a Gamma prior). We compare the output of both maximum joint likelihood estimation (i.e., standard NMF) and maximum marginal likelihood estimation (MMLE) on real and synthetical datasets. In particular we present face reconstruction results on CBCL dataset and text retrieval results over the musiXmatch dataset, a collection of word counts in song lyrics. The MMLE approach is shown to prevent overfitting by automatically pruning out irrelevant dictionary columns, i.e., embedding automatic model order selection.

General partially linear varying-coefficient transformation model with right censored data

In this paper, a unified maximum marginal likelihood estimation procedure is proposed for the analysis of right censored data using general partially linear varying-coefficient transformation models (GPLVCTM), which are flexible enough to include many survival models as its special cases. Unknown functional coefficients in the models are approximated by cubic B-spline polynomial. We estimate B-spline coefficients and regression parameters by maximizing marginal likelihood function. One advantage of this procedure is that it is free of both baseline and censoring distribution. Through simulation studies and a real data application (VA data from the Veteran's Administration Lung Cancer Study Clinical Trial), we illustrate that the proposed estimation procedure is accurate, stable and practical.

A comparison of methods for estimating quadratic effects in nonlinear structural equation models.

Two Monte Carlo simulations were performed to compare methods for estimating and testing hypotheses of quadratic effects in latent variable regression models. The methods considered in the current study were (a) a 2-stage moderated regression approach using latent variable scores, (b) an unconstrained product indicator approach, (c) a latent moderated structural equation method, (d) a fully Bayesian approach, and (e) marginal maximum likelihood estimation. Of the 5 estimation methods, it was found that overall the methods based on maximum likelihood estimation and the Bayesian approach performed best in terms of bias, root-mean-square error, standard error ratios, power, and Type I error control, although key differences were observed. Similarities as well as disparities among methods are highlight and general recommendations articulated. As a point of comparison, all 5 approaches were fit to a reparameterized version of the latent quadratic model to educational reading data.

Detecting Specific Genotype by Environment Interactions Using Marginal Maximum Likelihood Estimation in the Classical Twin Design

Considerable effort has been devoted to the analysis of genotype by environment (G × E) interactions in various phenotypic domains, such as cognitive abilities and personality. In many studies, environmental variables were observed (measured) variables. In case of an unmeasured environment, van der Sluis et al. (2006) proposed to study heteroscedasticity in the factor model using only MZ twin data. This method is closely related to the Jinks and Fulker (1970) test for G × E, but slightly more powerful. In this paper, we identify four challenges to the investigation of G × E in general, and specifically to the heteroscedasticity approaches of Jinks and Fulker and van der Sluis et al. We propose extensions of these approaches purported to solve these problems. These extensions comprise: (1) including DZ twin data, (2) modeling both A × E and A × C interactions; and (3) extending the univariate approach to a multivariate approach. By means of simulations, we study the power of the univariate method to detect the different G × E interactions in varying situations. In addition, we study how well we could distinguish between A × E, A × C, and C × E. We apply a multivariate version of the extended model to an empirical data set on cognitive abilities.

Potential application of item-response theory to interpretation of medical codes in electronic patient records

BackgroundElectronic patient records are generally coded using extensive sets of codes but the significance of the utilisation of individual codes may be unclear. Item response theory (IRT) models are used to characterise the psychometric properties of items included in tests and questionnaires. This study asked whether the properties of medical codes in electronic patient records may be characterised through the application of item response theory models.MethodsData were provided by a cohort of 47,845 participants from 414 family practices in the UK General Practice Research Database (GPRD) with a first stroke between 1997 and 2006. Each eligible stroke code, out of a set of 202 OXMIS and Read codes, was coded as either recorded or not recorded for each participant. A two parameter IRT model was fitted using marginal maximum likelihood estimation. Estimated parameters from the model were considered to characterise each code with respect to the latent trait of stroke diagnosis. The location parameter is referred to as a calibration parameter, while the slope parameter is referred to as a discrimination parameter.ResultsThere were 79,874 stroke code occurrences available for analysis. Utilisation of codes varied between family practices with intraclass correlation coefficients of up to 0.25 for the most frequently used codes. IRT analyses were restricted to 110 Read codes. Calibration and discrimination parameters were estimated for 77 (70%) codes that were endorsed for 1,942 stroke patients. Parameters were not estimated for the remaining more frequently used codes. Discrimination parameter values ranged from 0.67 to 2.78, while calibration parameters values ranged from 4.47 to 11.58. The two parameter model gave a better fit to the data than either the one- or three-parameter models. However, high chi-square values for about a fifth of the stroke codes were suggestive of poor item fit.ConclusionThe application of item response theory models to coded electronic patient records might potentially contribute to identifying medical codes that offer poor discrimination or low calibration. This might indicate the need for improved coding sets or a requirement for improved clinical coding practice. However, in this study estimates were only obtained for a small proportion of participants and there was some evidence of poor model fit. There was also evidence of variation in the utilisation of codes between family practices raising the possibility that, in practice, properties of codes may vary for different coders.

A flexible latent trait model for response times in tests

Latent trait models for response times in tests have become popular recently. One challenge for response time modeling is the fact that the distribution of response times can differ considerably even in similar tests. In order to reduce the need for tailor-made models, a model is proposed that unifies two popular approaches to response time modeling: Proportional hazard models and the accelerated failure time model with log–normally distributed response times. This is accomplished by resorting to discrete time. The categorization of response time allows the formulation of a response time model within the framework of generalized linear models by using a flexible link function. Item parameters of the proposed model can be estimated with marginal maximum likelihood estimation. Applicability of the proposed approach is demonstrated with a simulation study and an empirical application. Additionally, means for the evaluation of model fit are suggested.