A new proof for monotone likelihood ratio for the sum of independent bernoulli random variables

By use of an inequality of Marcus and Lopes for elementary symmetric functions, a new proof is presented for the following result by Ghurye and Wallace: Given that the independent random variablesX _j are Bernoulli with success probabilityp _j () strictly between 0 and 1 and nondecreasing in, the sum X _j has monotone likelihood ratio.     

A latent trait model for dichotomous choice data

The PARELLA model is a probabilistic parallelogram model that can be used for the measurement of latent attitudes or latent preferences. The data analyzed are the dichotomous responses of persons to stimuli, with a one (zero) indicating agreement (disagreement) with the content of the stimulus. The model provides a unidimensional representation of persons and items. The response probabilities are a function of the distance between person and stimulus: the smaller the distance, the larger the probability that a person will agree with the content of the stimulus. An estimation procedure based on expectation maximization and marginal maximum likelihood is developed and the quality of the resulting parameter estimates evaluated.I gratefully acknowledge Ivo Molenaar and Wijbrandt van Schuur for their advice and encouragement during the course of the investigation, Derk-Jan Kiewiet who constructed the program for the ML estimator for the person parameter and Anne Boomsma, Wendy Post, Tom Snijders, and David Thissen for their comments on smaller aspects of the investigation.     

Penalization approaches in the conditional maximum likelihood and Rasch modelling context

Recent detection methods for Differential Item Functioning (DIF) include approaches like Rasch Trees, DIFlasso, GPCMlasso and Item Focussed Trees, all of which - in contrast to well established methods - can handle metric covariates inducing DIF. A new estimation method shall address their downsides by mainly aiming at combining three central virtues: the use of conditional likelihood for estimation, the incorporation of linear influence of metric covariates on item difficulty and the possibility to detect different DIF types: certain items showing DIF, certain covariates inducing DIF, or certain covariates inducing DIF in certain items. Each of the approaches mentioned lacks in two of these aspects. We introduce a method for DIF detection, which firstly utilizes the conditional likelihood for estimation combined with group Lasso-penalization for item or variable selection and L1-penalization for interaction selection, secondly incorporates linear effects instead of approximation through step functions, and thirdly provides the possibility to investigate any of the three DIF types. The method is described theoretically, challenges in implementation are discussed. A dataset is analysed for all DIF types and shows comparable results between methods. Simulation studies per DIF type reveal competitive performance of cmlDIFlasso, particularly when selecting interactions in case of large sample sizes and numbers of parameters. Coupled with low computation times, cmlDIFlasso seems a worthwhile option for applied DIF detection.     

项目反应理论中潜在心理特质“填补”的参数估计方法及其演变

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” (data augmentation)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。     

Contingent approaches to making likelihood judgments about polychotomous cases: the influence of task factors

Two experiments tested the influence of three task factors on respondents' tendency to use normative, heuristic, and random approaches to making likelihood judgments about polychotomous cases (i.e., cases in which there is more than one alternative to a focal hypothesis). Participants estimated their likelihood of winning hypothetical raffles in which they and other players held various numbers of tickets. Responding on non‐numeric scales (vs. numeric ones) and responding under time pressure (vs. self‐paced) increased participants' use of a comparison‐heuristic approach, resulting in non‐normative judgment patterns. A manipulation of evidence representation (whether ticket quantities were represented by numbers or more graphically by bars) did not have reliably detectable effects on processing approaches to likelihood judgment. The authors discuss the implications of these findings for the further development of likelihood judgment theories, and they discuss parallels between contingent processing in choice and contingent processing in likelihood judgment. Copyright © 2005 John Wiley & Sons, Ltd.     

Order‐constrained linear optimization

Despite the fact that data and theories in the social, behavioural, and health sciences are often represented on an ordinal scale, there has been relatively little emphasis on modelling ordinal properties. The most common analytic framework used in psychological science is the general linear model, whose variants include ANOVA, MANOVA, and ordinary linear regression. While these methods are designed to provide the best fit to the metric properties of the data, they are not designed to maximally model ordinal properties. In this paper, we develop an order‐constrained linear least‐squares (OCLO) optimization algorithm that maximizes the linear least‐squares fit to the data conditional on maximizing the ordinal fit based on Kendall's τ. The algorithm builds on the maximum rank correlation estimator (Han, 1987, Journal of Econometrics, 35, 303) and the general monotone model (Dougherty & Thomas, 2012, Psychological Review, 119, 321). Analyses of simulated data indicate that when modelling data that adhere to the assumptions of ordinary least squares, OCLO shows minimal bias, little increase in variance, and almost no loss in out‐of‐sample predictive accuracy. In contrast, under conditions in which data include a small number of extreme scores (fat‐tailed distributions), OCLO shows less bias and variance, and substantially better out‐of‐sample predictive accuracy, even when the outliers are removed. We show that the advantages of OCLO over ordinary least squares in predicting new observations hold across a variety of scenarios in which researchers must decide to retain or eliminate extreme scores when fitting data.     

Full Information Maximum Likelihood Estimation for Latent Variable Interactions With Incomplete Indicators

Researchers have developed missing data handling techniques for estimating interaction effects in multiple regression. Extending to latent variable interactions, we investigated full information maximum likelihood (FIML) estimation to handle incompletely observed indicators for product indicator (PI) and latent moderated structural equations (LMS) methods. Drawing on the analytic work on missing data handling techniques in multiple regression with interaction effects, we compared the performance of FIML for PI and LMS analytically. We performed a simulation study to compare FIML for PI and LMS. We recommend using FIML for LMS when the indicators are missing completely at random (MCAR) or missing at random (MAR) and when they are normally distributed. FIML for LMS produces unbiased parameter estimates with small variances, correct Type I error rates, and high statistical power of interaction effects. We illustrated the use of these methods by analyzing the interaction effect between advanced cancer patients’ depression and change of inner peace well-being on future hopelessness levels.     

基于等级反应模型的属性层级方法

给出基于等级反应模型的属性层级方法（Attribute Hierarchy Method, AHM）,并简记为GRM-AHM,提出了相应的确定GRM-AHM的期望项目反应模式全集的方法和一种新的归类法LL。用蒙特卡洛模拟实验比较GRM-AHM的几种归类法的归准率（属性模式归准率和单个属性的平均判准率）。结果发现,新归类法的归准率与AHM中的方法A差不多,但比方法B高很多;随着被试作答失误率的提高,它们的归准率都有所下降。在归类精度和简单性方面,GRM-AHM都比Bolt等(2004)提出的多级评分融合模型（Fusion Model）好     

Rejoinder on the normalized maximum likelihood and Bayesian decision theory: Reply to Grünwald and Navarro (2009)

We provide a short reply to [Grünwald, P., & Navarro, D. (2009). NML, Bayes and true distributions: A comment on Karabatsos and Walker. Journal of Mathematical Psychology, in press (doi:10.1016/j.jmp.2008.11.005)] comment on the article by [Karabatsos, G., & Walker, S. (2006). On the normalized maximum likelihood and Bayesian decision theory. Journal of Mathematical Psychology, 50, 517-520].