Noniterative estimation and the choice of the number of factors in exploratory factor analysis

Based on the usual factor analysis model, this paper investigates the relationship between improper solutions and the number of factors, and discusses the properties of the noniterative estimation method of Ihara and Kano in exploratory factor analysis. The consistency of the Ihara and Kano estimator is shown to hold even for an overestimated number of factors, which provides a theoretical basis for the rare occurrence of improper solutions and for a new method of choosing the number of factors. The comparative study of their estimator and that based on maximum likelihood is carried out by a Monte Carlo experiment.The author would like to express his thanks to Masashi Okamoto and Masamori Ihara for helpful comments and to the editor and referees for critically reading the earlier versions and making many valuable suggestions. He also thanks Shigeo Aki for his comments on physical random numbers.     

MODIFYING LETTER AND NUMBER REVERSALS IN ELEMENTARY SCHOOL CHILDREN1

Children from regular and special-education classes were exposed to tutoring procedures designed to modify letter and number reversals. In Experiment I, two students showed reversals during letter-naming (e.g., saying p for q) and two-digit number (e.g., writing 31 for 13) exercises. In Experiment II, one subject made two-digit reversals, while the other child reversed single-digit numbers (e.g., 9 written with circle on right side of stem). The subject in Experiment III emitted letter reversals under naming, dictation, and copying exercises. The child in Experiment IV showed written letter reversals on both sides of the “body midline” under dictation procedures. Basic treatment procedures during all experiments involved modelling of correct and incorrect symbol formation, and differential experimenter feedback following student responses. During Experiments I and II, subjects were praised for each correct response, and informed when a reversal occurred. Experiment III included a Modelling Only Phase before these feedback procedures. Besides praise, treatment conditions in Experiment IV consisted of additional feedback following correct responses (e.g., charting correct responses), and temporal delays imposed between dictated letters and student responses. Except for occasional letter-naming reversals of one subject (Experiment I), and letter dictation errors of another (Experiment IV), reversals were eliminated when all experiments terminated. The Modelling Only Phase in Experiment III also reduced reversals to a low level. These findings suggest that a variety of reversal problems may be effectively assessed and remediated via simple modelling and reinforcement procedures. In addition, postcheck observations indicated that the effects were enduring. The present procedures should be easily implemented by school personnel.     

FACAIC: Model selection algorithm for the orthogonal factor model using AIC and CAIC

This paper describes the authors' FORTRAN algorithm FACAIC for choosing the number of factors for an orthogonal factor model using Akaike's Information Criterion. FACAIC utilizes the IMSL subroutine OFCOMM.The authors dedicate this algorithm to Professor Hirotugu Akaike in appreciation of his pioneering work on AIC which was originally intended for the factor analysis and other statistical model identification problems.     

4~8年级学生分数数量表征的准确性及形式

在分数学习中,分数数量表征是非常重要的方面。本研究利用数字线估计任务对四到八年级学生分数数量表征的情况进行探索。研究结果表明：随着年级升高,被试分数数量表征的准确性也随之提高;被试对单位分数和非单位分数表征的准确性存在显著差异,表现为对单位分数表征的准确性显著高于对非单位分数的表征,这种差异在低年级显著,随着年级的增长,差异逐渐消失;四到八年级学生在0~3数字线上,对分数数量的表征表现为线性形式而非对数形式,且这种线性表征形式是在六年级开始出现并随着年龄增长逐渐发展起来的。中西方儿童在分数数量表征的准确性和形式上有相似的发展路径,但是在表征准确性上中国儿童更高、线性形式出现年级上中国儿童可能更早。     

小学生近似数量系统敏锐度的发展趋势及其与数学能力的关系：抑制控制的中介作用

采用测量法和问卷法考察了172名小学生近似数量系统敏锐度的发展,以及抑制控制在近似数量系统敏锐度与数学能力关系中的中介效应。结果表明：(1)随着年龄增长,小学生的近似数量系统敏锐度逐渐提高;(2)近似数量系统敏锐度(负相关序列中的韦伯系数)和抑制控制均能显著正向预测小学生的数学能力;(3)抑制控制在小学生近似数量系统敏锐度(负相关序列中的韦伯系数)与数学能力的关系中起部分中介作用。     

时间赋义对数字空间表征的影响：来自数字线估计任务的证据

数字线估计任务的大量研究以纯数字为研究对象而忽视了赋义数字。本研究以Siegler等的数字线估计任务为原型,探讨在对数字时间赋义条件下小学二、四和六年级儿童的数字表征形式是否发生变化。结果表明,小学二年级是0-1000范围内数字表征从对数形式转换为线性形式的转折点,对数字赋予时间含义后,三个年级均出现了赋义效应。在线性模型和对数模型中时间赋义的作用相反,时间赋义表现出抑线升对（抑制线性模型提升对数模型解释力）的效果。     

Orienting numbers in mental space: Horizontal organization trumps vertical

While research on the spatial representation of number has provided substantial evidence for a horizontally oriented mental number line, recent studies suggest vertical organization as well. Directly comparing the relative strength of horizontal and vertical organization, however, we found no evidence of spontaneous vertical orientation (upward or downward), and horizontal trumped vertical when pitted against each other (Experiment 1). Only when numbers were conceptualized as magnitudes (as opposed to nonmagnitude ordinal sequences) did reliable vertical organization emerge, with upward orientation preferred (Experiment 2). Altogether, these findings suggest that horizontal representations predominate, and that vertical representations, when elicited, may be relatively inflexible. Implications for spatial organization beyond number, and its ontogenetic basis, are discussed.     

Evaluation of drug toxicity in clinical trials

An increasing number of drugs removed from the market because of unacceptable toxicity raises concerns regarding preapproval testing of drug safety. In the present paper it is postulated that the non-inferiority type of trial should be abandoned in favor of the superiority trial with active controls and less stringent (p<0.1, both for efficacy and toxicity) statistics. This approach will increase sensitivity of detection of drug-induced adverse effects at the expense of increasing false positive results regarding the difference in efficacy between the tested and reference drug. Such a move will increase the protection of future patients. In addition, the proposed design is far more acceptable from the clinical (e.g. no need to specify the statistically expected “unimportant” number of deaths) and ethical points of view, as well as being favored by the strong incentive of involved parties. In the second part of this paper arguments are presented in favor of the hypothesis that placebo (still used in some superiority trials) does not induce adverse effects. The assertion that placebo may induce adverse effects is probably biased by the nature of the clinical experiment. Such a conclusion is supported by studies indicating that placebo-induced adverse effects are disease — and treatment — specific. The modification of clinical trials according to the proposed changes may increase the trials’ sensitivity at detecting adverse effects of drugs. A lecture on the subject of this paper was presented at the 6th International Bioethics Conference on the subject of ‘The Responsible Conduct of Basic and Clinical Research’, held in Warsaw, Poland, 3–4 June 2005.     

Never getting to zero: Elementary school students' understanding of the infinite divisibility of number and matter

Clinical interviews administered to third- to sixth-graders explored children's conceptualizations of rational number and of certain extensive physical quantities. We found within child consistency in reasoning about diverse aspects of rational number. Children's spontaneous acknowledgement of the existence of numbers between 0 and 1 was strongly related to their induction that numbers are infinitely divisible in the sense that they can be repeatedly divided without ever getting to zero. Their conceptualizing number as infinitely divisible was strongly related to their having a model of fraction notation based on division and to their successful judgment of the relative magnitudes of fractions and decimals. In addition, their understanding number as infinitely divisible was strongly related to their understanding physical quantities as infinitely divisible. These results support a conceptual change account of knowledge acquisition, involving two-way mappings between the domains of number and physical quantity.     

The construction of large number representations in adults

What is the nature of our mental representation of quantity? We find that human adults show no performance cost of comparing numerosities across vs. within visual and auditory stimulus sets, or across vs. within simultaneous and sequential sets. In addition, reaction time and performance in such tasks are determined by the ratio of the numerosities to be compared; absolute set size has no effect. These findings suggest that modality-specific stimulus properties undergo a non-iterative transformation into representations of quantity that are independent of the modality or format of the stimulus.