The asymptotic posterior normality of the latent trait in an IRT model

It has long been part of the item response theory (IRT) folklore that under the usual empirical Bayes unidimensional IRT modeling approach, the posterior distribution of examinee ability given test response is approximately normal for a long test. Under very general and nonrestrictive nonparametric assumptions, we make this claim rigorous for a broad class of latent models.This research was partially supported by Office of Naval Research Cognitive and Neural Sciences Grant N0014-J-90-1940, 442-1548, National Science Foundation Mathematics Grant NSF-DMS-91-01436, and the National Center for Supercomputing Applications. We wish to thank Kumar Joag-dev and Zhiliang Ying for enlightening suggestions concerning the proof of the basic result.The authors wish to thank Kumar Joag-Dev, Brian Junker, Bert Green, Paul Holland, Robert Mislevy, and especially Zhiliang Ying for their useful comments and discussions.     

A compensatory approach to optimal selection with mastery scores

A Bayesian approach for simultaneous optimization of test-based decisions is presented using the example of a selection decision for a treatment followed by a mastery decision. A distinction is made between weak and strong rules where, as opposed to strong rules, weak rules use prior test scores as collateral data. Conditions for monotonicity of optimal weak and strong rules are presented. It is shown that under mild conditions on the test score distributions and utility functions, weak rules are always compensatory by nature. The authors are indebted to Wilbert Kallenberg for his valuable comments and to Jan Gulmans for providing the data for the empirical example. The names of the authors are alphabetical; they are equally responsible for the contents of this paper.     

Multiple-answer multiple-choice test items: Responding and scoring through bayes and minimax strategies

A multiple-answer multiple-choice test item has a certain number of alternatives,any number of which might be keyed. The examinee is also allowed to mark any number of alternatives. This increased flexibility over the one keyed alternative case is useful in practice but raises questions about appropriate scoring rules. In this article a certain class of item scoring rules called thebinary class is considered. The concepts ofstandard scoring rules and equivalence among these scoring rules are introduced in the misinformation model for which the traditional knowledge model is a special case. The examinee's strategy with respect to a scoring rule is examined. The critical role of a quantity called the scoring ratio is emphasized. In the case of examinee uncertainty about the number of correct alternatives on an item, a Bayes and a minimax strategy for the examinee are developed. Also an appropriate response for the examiner to the minimax strategy is outlined.Research partially supported under Grants N00014-67-A-0314-0022 from the Office of Naval Research and GS-32514 and MPS 75-07539 from the National Science Foundation.     

Estimation of stimulus-response curves by bayesian least squares

Bayesian least squares techniques are adapted to estimation of stimulus-response curves, rather broadly conceived. Illustrative examples deal with estimation of person characteristic curves and item characteristic curves in the context of mental testing, and estimation of a stimulus-response curve using data from a psychophysical experiment.The present paper is a version of an invited address of the same title, presented to the meetings of The Psychometric Society in Monterey, California on June 8, 1979, expanded by adding Example 1.The writer gratefully acknowledges the benefit of conversations with Don Pierce, and with John Shih, whose related work on density estimation pointed up an error in the original version of Section 2. He acknowledges, gratefully, support by the National Science Foundation through Grants MCS 76-02166A01 and MCS 78-01932. And he wishes to express his gratitude for the cordial hospitality and gracious cooperation offered by Professor Donald M. MacKay and the staff of the Department of Communication, University of Keele, during spring and summer, 1976; in particular, by Professor Dennis P. Andrews and Keichi Uchikawa, who kindly furnished data, a small portion of which is used in the illustrative example, Example 2.     

Parameter estimation in latent trait models

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parameters=( ₁, , _n ) and item parameters=( ₁, , _k ), where _j may be vector valued. With considered a random sample from a prior distribution with parameter, the estimation of (, ) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.This work was supported by Contract No. N00014-81-K-0265, Modification No. P00002, from Personnel and Training Research Programs, Psychological Sciences Division, Office of Naval Research. The authors wish to thank an anonymous reviewer for several valuable suggestions.     

I did not expect to be dreaming: Explaining realization in lucid dreams with a Bayesian framework

The commonsense view is that a lucid dream starts when the dreamer realizes that they are currently dreaming. The notion of realization, however, has been accepted at face value, with little consideration of whether the dreamer realizes that they are dreaming in the sense of actual reasoning, or if it is a mere epiphenomenon of lucid dream initiation. This article offers a solution to this problem by, first, arguing that the transition to lucidity can occur as a result of successful reasoning, and second, building a model of this reasoning in terms of probabilistic reasoning. The established Bayesian model explains realization in lucid dreams taking under consideration two factors: the beliefs that the dreamer holds on what is generally probable and improbable, and the dreamer’s admissibility of being in a dream. Defended against important objections, the model offers an explanation of lucid dream initiation, relevant for future research on dreaming.     

解读不显著结果：基于500个实证研究的量化分析

不显著结果(如, p > 0.05)在心理学研究中十分常见, 且容易被误解为接受零假设的证据, 并可能导致分组匹配研究的错误推断或者忽视被小样本的不显著结果掩盖的真实效应。但国内目前尚无实证研究对不显著结果的普遍性及其解读进行调查。本研究调查500篇中文心理学实证研究, 统计其摘要中出现与不显著结果相关的阴性陈述的频率, 判断并统计基于阴性陈述的推断准确性, 并使用贝叶斯因子对不显著结果中包含t值的研究进行重新评估。结果表明, 36%的摘要提及不显著结果, 共包含236个阴性陈述。其中, 41%的阴性陈述对不显著结果的解读出现偏差(如, 解读为支持了零假设)。对包含t值的研究进行贝叶斯因子分析, 结果显示仅有5.1%的不显著结果可以提供强证据支持零假设(BF₀₁ > 10)。与先前对国际心理学期刊的调查结果相比(32%的摘要包含阴性陈述; 72%的阴性陈述对不显著结果的解读错误), 中文心理学期刊中报告不显著结果的比例更高, 且对不显著结果解读错误的比例更低。但国内研究者仍需进一步加强对不显著结果的认识, 推广适于评估不显著结果的统计方法。     

Model comparison of nonlinear structural equation models with fixed covariates

Recently, it has been recognized that the commonly used linear structural equation model is inadequate to deal with some complicated substantive theory. A new nonlinear structural equation model with fixed covariates is proposed in this article. A procedure, which utilizes the powerful path sampling for computing the Bayes factor, is developed for model comparison. In the implementation, the required random observations are simulated via a hybrid algorithm that combines the Gibbs sampler and the Metropolis-Hastings algorithm. It is shown that the proposed procedure is efficient and flexible; and it produces Bayesian estimates of the parameters, latent variables, and their highest posterior density intervals as by-products. Empirical performances of the proposed procedure such as sensitivity to prior inputs are illustrated by a simulation study and a real example.This research is fully supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 4346/01H). The authors are thankful to the Editor, the Associate Editor, and anonymous reviewers for valuable comments which improve the paper significantly, and grateful to ICPSR and the relevant funding agency for allowing use of the data in the example. The assistance of Michael K.H. Leung and Esther L.S. Tam is gratefully acknowledged.     

Computation and Causation

The computer's effect on our understanding of causation has been enormous. By the mid-1980s, philosophical and social-scientific work on the topic had left us with (1) no reasonable reductive account of causation and (2) a class of statistical causal models tied to linear regression. At this time, computer scientists were attacking the problem of equipping robots with models of the external that included probabilistic portrayals of uncertainty. To solve the problem of efficiently storing such knowledge, they introduced Bayes Networks and directed graphs. By attaching a causal interpretation to Bayes Networks, the philosophy of causation changed dramatically. We are now able to be extremely general about how causal structure connects to data, and systematic about when causal structures are empirically indistinguishable. In this essay I try to motivate and describe this synthesis.     

Moving in Parallel Toward a Modern Modeling Epistemology: Bayes Factors and Frequentist Modeling Methods

The Bayesian-frequentist debate typically portrays these statistical perspectives as opposing views. However, both Bayesian and frequentist statisticians have expanded their epistemological basis away from a singular focus on the null hypothesis, to a broader perspective involving the development and comparison of competing statistical/mathematical models. For frequentists, statistical developments such as structural equation modeling and multilevel modeling have facilitated this transition. For Bayesians, the Bayes factor has facilitated this transition. The Bayes factor is treated in articles within this issue of Multivariate Behavioral Research. The current presentation provides brief commentary on those articles and more extended discussion of the transition toward a modern modeling epistemology. In certain respects, Bayesians and frequentists share common goals.