Properties of the Hubert-Arabie adjusted Rand index

This article provides an investigation of cluster validation indices that relates 4 of the indices to the L. Hubert and P. Arabie (1985) adjusted Rand index--the cluster validation measure of choice (G. W. Milligan & M. C. Cooper, 1986). It is shown how these other indices can be "roughly" transformed into the same scale as the adjusted Rand index. Furthermore, in-depth explanations are given of why classification rates should not be used in cluster validation research. The article concludes by summarizing several properties of the adjusted Rand index across many conditions and provides a method for testing the significance of observed adjusted Rand indices.     

A ROTATIONAL PROCEDURE FOR SEPARATION OF TRAIT,METHOD AND INTERACTION FACTORS IN MULTTTRAIT-MULTIMETHOD MATRICES

Cluster recovery indices are more important than ever, because of the necessity for comparing the large number of clustering procedures available today. Of the cluster recovery indices prominent in contemporary literature, the Hubert and Arabie (1985) adjustment to the Rand index (1971) has been demonstrated to have the most desirable properties (Milligan &; Cooper, 1986). However, use of the Hubert and Arabie adjustment to the Rand index is limited to cluster solutions involving non-overlapping, or disjoint, clusters. The present paper introduces a generalization of the Hubert and Arabie adjusted Rand index. This generalization, called the Omega index, can be applied to situations where both, one, or neither of the solutions being compared is non-disjoint. In the special case where both solutions are disjoint, the Omega index is equivalent to the Hubert and Arabie adjusted Rand index.     

Inference using categories

How do people use category membership and similarity for making inductive inferences? The authors addressed this question by examining the impact of category labels and category features on inference and classification tasks that were designed to be comparable. In the inference task, participants predicted the value of a missing feature of an item given its category label and other feature values. In the classification task, participants predicted the category label of an item given its feature values. The results from 4 experiments suggest that category membership influences inference even when similarity information contradicts the category label. This tendency was stronger when the category label conveyed class inclusion information than when the label reflected a feature of the category. These findings suggest that category membership affects inference beyond similarity and that category labels and category features are 2 different things.     

A Bayesian Approach for Estimating Mediation Effects With Missing Data

Methodologists have developed mediation analysis techniques for a broad range of substantive applications, yet methods for estimating mediating mechanisms with missing data have been understudied. This study outlined a general Bayesian missing data handling approach that can accommodate mediation analyses with any number of manifest variables. Computer simulation studies showed that the Bayesian approach produced frequentist coverage rates and power estimates that were comparable to those of maximum likelihood with the bias-corrected bootstrap. We share an SAS macro that implements Bayesian estimation and use 2 data analysis examples to demonstrate its use.     

Young children's discovery of spatial classification

Around the age of 18 months, children begin to classify objects spatially by kind, placing objects of the same kind close together in space and placing unlike objects apart. This behavior may be symbolic in the sense that children use spatial proximity to represent similarity. We examined the possibility that spatial classification is discovered during play—that the external products of play lead children to use space to represent similarity. Experiment 1 was a longitudinal study of four children's classification behaviors, observed from the age of 16 to 21 months. Results suggest that play with one kind of object to the exclusion of another kind leads to the discovery of spatial classification. Experiment 2 examined how children's tendencies to interact with one category might promote spatial classification of multiple categories. Twenty-four 18-month-old children who did not yet spatially classify objects by kind participated. Children who were given the experience of playing with two kinds of objects in a context that promoted interaction with only one kind were more likely to demonstrate spontaneous spatial classification of multiple kinds in a subsequent test period. Children who played equally with both kinds did not show heightened spontaneous classification. The results further suggest that comparison of different kinds during play is critical to the spontaneous occurrence of spatial classification.     

两种学习模式下类别学习的结果：原型和样例

利用“学习－迁移”的任务范式和单一特征类别判断技术,探讨了分类和推理两种类别学习模式的结果,比较了两种学习模式的效果和策略。研究表明：两种学习模式产生了不同的结果,分类学习的结果是样例,推理学习的结果是原型;在学习效果方面,分类学习比推理学习在达标比例上更高,但在进度上差异不显著;在策略运用方面,分类学习比推理学习更快地使用单维度策略,而在高水平策略的运用上,两者差异不显著     

Action information from classification learning

University of Illinois at Urbana-Champaign, Urbana, Illinois Much of our learning comes from interacting with objects. Two experiments investigated whether or not arbitrary actions used during category learning with objects might be incorporated into object representations and influence later recognition judgments. In a virtual-reality chamber, participants used distinct arm movements to make different classification responses. During a recognition test phase, these same objects required arm movements that were consistent or inconsistent with the classification movement. In both experiments, consistent movements were facilitated relative to inconsistent movements, suggesting that arbitrary action information is incorporated into the representations.     

Gaze and informativeness during category learning: Evidence for an inverse relation

In what follows, we explore the general relationship between eye gaze during a category learning task and the information conveyed by each member of the learned category. To understand the nature of this relationship empirically, we used eye tracking during a novel object classification paradigm. Results suggest that the average fixation time per object during learning is inversely proportional to the amount of information that object conveys about its category. This inverse relationship may seem counterintuitive; however, objects that have a high-information value are inherently more representative of their category. Therefore, their generality captures the essence of the category structure relative to less representative objects. As such, it takes relatively less time to process these objects than their less informative companions. We use a general information measure referred to as representational information theory (Vigo, 2011a, 2013a) to articulate and interpret the results from our experiment and compare its predictions to those of three models of prototypicality.     

Induction from a single instance: formation of a novel category

This study examines whether preschoolers can use information from a known category to induce a characteristic attribute of a novel, contrasting category based on a single instance. We showed 32 four-year-olds three instances of a Given Category and one instance of a Target Category. These objects could vary along two attribute dimensions, such as color and shape. All instances of the Given Category shared identical values of one attribute (e.g., all were blue), but could have different values of the other attribute (e.g., a circle, a square, and a triangle). The single instance of the Target Category was different from the Given on both attribute dimensions (e.g., a red diamond). Children gave yes/no judgements as to whether additional objects were instances of the Target Category. There were two possible sources of information about the relevance of an attribute to classification: explicit (labeling) and implicit (variation in the Given Category). There were four conditions such that each source of information was either available or not. Both types of information were effective in eliciting inductions of the relevant kind of attribute and the characteristic value of this attribute in the novel category (explicit: p = .0004; implicit: p = .031). This suggests that children use an inductive bias that the instances of two related but distinct categories tend to be alike in the same way.     

A note on the expected value of the Rand index

Two expectations of the adjusted Rand index (ARI) are compared. It is shown that the expectation derived by Morey and Agresti (1984, Educational and Psychological Measurement, 44, 33) under the multinomial distribution to approximate the exact expectation from the hypergeometric distribution (Hubert & Arabie, 1985, Journal of Classification, 2, 193) provides a poor approximation, and, in some cases, the difference between the two expectations can increase with the sample size. Proofs concerning the minimum and maximum difference between the two expectations are provided, and it is shown through simulation that the ARI can differ significantly depending on which expectation is used. Furthermore, when compared in a hypothesis testing framework, multinomial approximation overly favours the null hypothesis.     

Causal status as a determinant of feature centrality

One of the major problems in categorization research is the lack of systematic ways of constraining feature weights. We propose one method of operationalizing feature centrality, a causal status hypothesis which states that a cause feature is judged to be more central than its effect feature in categorization. In Experiment 1, participants learned a novel category with three characteristic features that were causally related into a single causal chain and judged the likelihood that new objects belong to the category. Likelihood ratings for items missing the most fundamental cause were lower than those for items missing the intermediate cause, which in turn were lower than those for items missing the terminal effect. The causal status effect was also obtained in goodness-of-exemplar judgments (Experiment 2) and in free-sorting tasks (Experiment 3), but it was weaker in similarity judgments than in categorization judgments (Experiment 4). Experiment 5 shows that the size of the causal status effect is moderated by plausibility of causal relations, and Experiment 6 shows that effect features can be useful in retrieving information about unknown causes. We discuss the scope of the causal status effect and its implications for categorization research.     

A modified procedure for parallel analysis of ordered categorical data

Parallel analysis has been well documented to be an effective and accurate method for determining the number of factors to retain in exploratory factor analysis. The O'Connor (2000) procedure for parallel analysis has many benefits and is widely applied, yet it has a few shortcomings in dealing with missing data and ordinal variables. To address these technical issues, we adapted and modified the O'Connor procedure to provide an alternative method that better approximates the ordinal data by factoring in the frequency distributions of the variables (e.g., the number of response categories and the frequency of each response category per variable). The theoretical and practical differences between the modified procedure and the O'Connor procedure are discussed. The SAS syntax for implementing this modified procedure is also provided.     

类别学习中两种学习模式的比较研究：分类学习与推理学习

采用学习迁移任务范式,使用基于单一特征的类别判断技术,比较了非线性分离结构下,分类学习和推理学习的学习效率、学习过程与策略和学习结果。结果表明：在学习效率上,分类学习比推理学习更好地习得了含有较多样例的类别知识,分类学习的速度上显著快于推理学习。在学习的过程与策略上,推理学习比分类学习更为关注类别内不同特征的相关,但在分类策略的运用上不如分类学习灵活。在学习的结果上,推理学习倾向于原型记忆,分类学习倾向于进行样例记忆,分类学习比推理学习更好地掌握了类别原型     

On normal theory based inference for multilevel models with distributional violations

Data in social and behavioral sciences are often hierarchically organized though seldom normal, yet normal theory based inference procedures are routinely used for analyzing multilevel models. Based on this observation, simple adjustments to normal theory based results are proposed to minimize the consequences of violating normality assumptions. For characterizing the distribution of parameter estimates, sandwich-type covariance matrices are derived. Standard errors based on these covariance matrices remain consistent under distributional violations. Implications of various covariance estimators are also discussed. For evaluating the quality of a multilevel model, a rescaled statistic is given for both the hierarchical linear model and the hierarchical structural equation model. The rescaled statistic, improving the likelihood ratio statistic by estimating one extra parameter, approaches the same mean as its reference distribution. A simulation study with a 2-level factor model implies that the rescaled statistic is preferable.This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of North Texas faculty research grant. We would like to thank the Associate Editor and two reviewers for suggestions that helped to improve the paper.     

A cautionary note on using internal cross validation to select the number of clusters

A highly popular method for examining the stability of a data clustering is to split the data into two parts, cluster the observations in Part A, assign the objects in Part B to their nearest centroid in Part A, and then independently cluster the Part B objects. One then examines how close the two partitions are (say, by the Rand measure). Another proposal is to split the data into k parts, and see how their centroids cluster. By means of synthetic data analyses, we demonstrate that these approaches fail to identify the appropriate number of clusters, particularly as sample size becomes large and the variables exhibit higher correlations.The authors express their thanks to the Sol C. Snider Entrepreneurial Center, Wharton School, for support of this project.     

The role of comparison in preschoolers’ novel object categorization

We examined the role of the comparison process and shared names on preschoolers’ categorization of novel objects. In our studies, 4-year-olds were presented with novel object sets consisting of either one or two standards and two test objects: a shape match and a texture match. When children were presented with one standard, they extended the category based on shape regardless of whether the objects were named. When children were presented with two standards that shared the same texture and the objects were named with the same noun, they extended the category based on texture. The opportunity to compare two standards, in the absence of shared names, led to an attenuation of the effect of shape. These findings demonstrate that comparison plays a critical role in the categorization of novel objects and that shared names enhance this process.     

Category effects on stimulus estimation: Shifting and skewed frequency distributions

The category adjustment model (CAM) proposes that estimates of inexactly remembered stimuli are adjusted toward the central value of the category of which the stimuli are members. Adjusting estimates toward the average value of all category instances, properly weighted for memory uncertainty, maximizes the average accuracy of estimates. Thus far, the CAM has been tested only with symmetrical category distributions in which the central stimulus value is also the mean. We report two experiments using asymmetric (skewed) distributions in which there is more than one possible central value: one where the frequency distribution shifts over the course of time, and the other where the frequency distribution is skewed. In both cases, we find that people adjust estimates toward the category’s running mean, which is consistent with the CAM but not with alternative explanations for the adjustment of stimuli toward a category’s central value.     

On the Interaction of Prior Knowledge and Stimulus Structure in Category Learning

Contemporary theories of categorization propose that concepts are coherent in virtue of being embedded in a network of theories about the world. Those theories function to pick out some of the many possible features of a set of objects as most salient for purposes of classification, a process that is complex and still poorly understood (Murphy & Medin, 1985). Part of what makes this account incomplete is a lack of information as to (1) what makes a feature salient on a given occasion and (2) how feature salience interacts with category structure to determine the course of learning. We report on the results of three studies of category learning using complex schematic drawings to show that (1) the contrast set defined by one's initial encounters with category exemplars can be a source of individual differences in feature salience assignments; (2) such effects are short-lived in the face of clear evidence about actual feature diagnosticity; and (3) more robust prior hypotheses interact with category structure to either enhance learning or impede it. The enhancement occurs when the hypothesis emphasizes category-relevant features, even if the hypothesis is in fact incorrect. A hypothesis that assigns high salience to irrelevant features impedes learning. Learning does occur as feedback concerning category structure leads to enhanced salience for relevant features. Salience of irrelevant features remains high, however, suggesting that such learning as occurs involves augmentation and not total revision of the (incorrect) prior hypothesis.     

类别学习和分类运用的神经机制

从类别学习和分类运用(包括非人类对象分类和社会分类)两个方面阐述了分类的神经机制。类别学习主要与新皮层、内侧颞叶、基底神经节、中脑多巴胺能系统有关, 不同类别的学习会激活这些神经系统间不同的连接。对非人类对象分类时, 不同类型、级别、熟悉度及相似度类别分类的神经机制不同, 分类对象的清晰度、类别不确定性会影响分类的神经机制, 在分类进程的不同时段会出现对应的ERP指标。社会分类时个体先注意到外群体再加工内群体, 且对内群体的加工更深, P200和N200是对内、外群体区分的特异性波, 内外群体分类时, 内群体激活梭状回和扣带回后部, 外群体激活杏仁核。文章最后比较了人类和灵长类动物分类神经机制的异同, 并指出社会分类和非人类对象分类神经机制的整合以及人类和灵长类动物分类神经机制的比较是今后研究需要关注的问题。