A tutorial on Bayes factor estimation with the product space method

The Bayes factor is an intuitive and principled model selection tool from Bayesian statistics. The Bayes factor quantifies the relative likelihood of the observed data under two competing models, and as such, it measures the evidence that the data provides for one model versus the other. Unfortunately, computation of the Bayes factor often requires sampling-based procedures that are not trivial to implement. In this tutorial, we explain and illustrate the use of one such procedure, known as the product space method (Carlin & Chib, 1995). This is a transdimensional Markov chain Monte Carlo method requiring the construction of a “supermodel” encompassing the models under consideration. A model index measures the proportion of times that either model is visited to account for the observed data. This proportion can then be transformed to yield a Bayes factor. We discuss the theory behind the product space method and illustrate, by means of applied examples from psychological research, how the method can be implemented in practice.     

Constructing informative model priors using hierarchical methods

Despite their negative reputation, informative priors are very useful in inference. Priors that express psychologically meaningful intuitions damp out random fluctuations in the data due to sampling variability, without sacrificing flexibility. This article focuses on how an intuitively satisfying informative prior distribution can be constructed. In particular, it demonstrates how the hierarchical introduction of a parameterized generative account of the set of models under consideration naturally imposes a non-uniform prior distribution over the models, encoding existing intuitions about the models. The hierarchical approach for constructing informative model priors is made concrete using a worked example, the Varying Abstraction Model (VAM), a family of categorization models including and expanding the exemplar and prototype models. It is shown how psychological intuitions about the relative plausibilities of the models in the VAM can be formally captured in an informative prior distribution over these models, by specifying a theoretically informed process for generating the models in the VAM. The smoothing effect of the informative prior in estimation is demonstrated by considering ten previously published data sets from the category learning literature.     

儿童科学概念的认知发展

综述了作者在儿童科学概念认知领域的研究,其中包括儿童的朴素生物学、朴素物理学认知及其与心理理论发展关系的研究。作者研究了儿童在生命科学领域的自发概念,包括对生长、衰老、疾病和死亡等生命现象的认知,探查儿童对日常生活中物理学概念（如力等）的朴素认知,力图发现儿童早期认知中的潜力,同时探明他们相应的错误概念,并采用教育干预实验,促进儿童科学概念的转化。还探查影响儿童科学概念发展的因素,包括外部教育条件以及个体认知能力（如心理理论发展水平,推理能力等）。研究发现,儿童在接受正规的科学教育前就对人类重要的科学领域形成了自己的“朴素理论”,他们用这种理论来解释现实世界的现象。作者主张儿童的科学教育（包括健康教育）应该以儿童的“朴素理论”为基础,从而真正做到因材施教     

Easy bayes estimation for rasch-type models

A Bayes estimation procedure is introduced that allows the nature and strength of prior beliefs to be easily specified and modal posterior estimates to be obtained as easily as maximum likelihood estimates. The procedure is based on constructing posterior distributions that are formally identical to likelihoods, but are based on sampled data as well as artificial data reflecting prior information. Improvements in performance of modal Bayes procedures relative to maximum likelihood estimation are illustrated for Rasch-type models. Improvements range from modest to dramatic, depending on the model and the number of items being considered.This research was supported by ORN Contact #00014-86-K0087. We wish to thank Sheng-Hui Chu and Dzung-Ji Lii for providing intelligent and energetic programming support for this article. We also thank one of the reviewers for pointing out several interesting and useful perspectives.     

The riddle of Siegfried: exploring methods and psychological perspectives in analytical psychology

Jung's dream of the killing of Siegfried poses a riddle: why did the unconscious choose precisely Siegfried as the hero to be murdered? Jung himself declares that he does not know. This paper attempts to decipher this riddle using three distinct methodological approaches accepted by Jung, two of them in fact grounded in his theories of dream interpretation. Besides presenting some possible answers to the riddle of Siegfried, this interpretative reflection brings to light the discrepancy of the psychological perspectives created by the heterogeneity of methods within analytical psychology.     

Four bootstrap confidence intervals for the binomial-error model

Confidence intervals for the mean function of the true proportion score ( _x ), where andx respectively denote the true proportion and observed test scores, can be approximated by the Efron, Bayesian, and parametric empirical Bayes (PEB) bootstrap procedures. The similarity of results yielded by all the bootstrap methods suggests the following: the unidentifiability problem of the prior distributiong() can be bypassed with respect to the construction of confidence intervals for the mean function, and a beta distribution forg() is a reasonable assumption for the test scores in compliance with a negative hypergeometric distribution. The PEB bootstrap, which reflects the construction of Morris intervals, is introduced for computing predictive confidence bands for x. It is noted that the effect of test reliability on the precision of interval estimates varies with the two types of confidence statements concerned.The Authors are indebted to the Editor and anonymous reviewers for constructive suggestions and comments. The authors wish to thank Min-Te Chao and Cheng-Der Fuh for some useful suggestions at earlier stages of writing this paper.     

On the misuse of manifest variables in the detection of measurement bias

Measurement invariance (lack of bias) of a manifest variableY with respect to a latent variableW is defined as invariance of the conditional distribution ofY givenW over selected subpopulations. Invariance is commonly assessed by studying subpopulation differences in the conditional distribution ofY given a manifest variableZ, chosen to substitute forW. A unified treatment of conditions that may allow the detection of measurement bias using statistical procedures involving only observed or manifest variables is presented. Theorems are provided that give conditions for measurement invariance, and for invariance of the conditional distribution ofY givenZ. Additional theorems and examples explore the Bayes sufficiency ofZ, stochastic ordering inW, local independence ofY andZ, exponential families, and the reliability ofZ. It is shown that when Bayes sufficiency ofZ fails, the two forms of invariance will often not be equivalent in practice. Bayes sufficiency holds under Rasch model assumptions, and in long tests under certain conditions. It is concluded that bias detection procedures that rely strictly on observed variables are not in general diagnostic of measurement bias, or the lack of bias.Preparation of this article was supported in part by PSC-CUNY grant #661282 to Roger E. Millsap.     

The case of Piaget's group INRC

What is it that distinguishes Piaget's transformations N, R, and C from the rest of the 16! transformations of the 16 binary propositional operations? Here Piaget's INRC is considered as a subgroup of the group $\text{M}$₂ of all automorphisms and dual automorphisms of the free Boolean algebra with two generators. This group is isomorphic to S₄ × C₂. Its elements are given explicitly. Many other psychologically relevant subgroups of $\text{M}$₂ play an important role. They are discussed and their connections shown. Particular attention is given to involutions, even if the view that they constitute the sole representation of reversibility is abandoned. Piaget's transformation R turns out not to be the inverse operation of relations. The group of automorphisms, dual automorphisms, anti-automorphisms of the algebra of binary relations on a finite set is found. A crystallographic presentation of these groups is given and related work by Bart (Journal of Mathematical Psychology, 1971, 5, 539–553), Leresche (Revue Européenne des Sciences Sociales, 1976, 14, 219–241), and Pólya (The Journal of Symbolic Logic, 1940, 5, 98–103) is discussed.     

A Gibbs sampler for the multidimensional four-parameter logistic item response model via a data augmentation scheme

The four-parameter logistic (4PL) item response model, which includes an upper asymptote for the correct response probability, has drawn increasing interest due to its suitability for many practical scenarios. This paper proposes a new Gibbs sampling algorithm for estimation of the multidimensional 4PL model based on an efficient data augmentation scheme (DAGS). With the introduction of three continuous latent variables, the full conditional distributions are tractable, allowing easy implementation of a Gibbs sampler. Simulation studies are conducted to evaluate the proposed method and several popular alternatives. An empirical data set was analysed using the 4PL model to show its improved performance over the three-parameter and two-parameter logistic models. The proposed estimation scheme is easily accessible to practitioners through the open-source IRTlogit package.