On robusiness of the normal-theory based asymptotic distributions of three reliability coefficient estimates

This paper studies the asymptotic distributions of three reliability coefficient estimates: Sample coefficient alpha, the reliability estimate of a composite score following a factor analysis, and the estimate of the maximal reliability of a linear combination of item scores following a factor analysis. Results indicate that the asymptotic distribution for each of the coefficient estimates, obtained based on a normal sampling distribution, is still valid within a large class of nonnormal distributions. Therefore, a formula for calculating the standard error of the sample coefficient alpha, recently obtained by van Zyl, Neudecker and Nel, applies to other reliability coefficients and can still be used even with skewed and kurtotic data such as are typical in the social and behavioral sciences.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.     

Coefficients alpha and reliabilities of unrotated and rotated components

It has been shown by Kaiser that the sum of coefficients alpha of a set of principal components does not change when the components are transformed by an orthogonal rotation. In this paper, Kaiser's result is generalized. First, the invariance property is shown to hold for any set of orthogonal components. Next, a similar invariance property is derived for the reliability of any set of components. Both generalizations are established by considering simultaneously optimal weights for components with maximum alpha and with maximum reliability, respectively. A short-cut formula is offered to evaluate the coefficients alpha for orthogonally rotated principal components from rotation weights and eigenvalues of the correlation matrix. Finally, the greatest lower bound to reliability and a weighted version are discussed.Comments by Henk A.L. Kiers and by anonymous referees are gratefully acknowledged.     

Trait lasting alteration of the brain default mode network in experienced meditators and the experiential selfhood

Based on the finding in novices that four months of meditation training significantly increases frontal default mode network (DMN) module/subnet synchrony while decreasing left and right posterior DMN modules synchrony, the current study tested the prediction whether experienced meditators (those who are practising meditation intensively for several years) had a change in the DMN “trinity” of modules as a baseline trait characteristic and whether this change is in a similar direction as in the novice trainees who practised meditation for only four months. Comparison of functional connectivity within DMN subnets (measured by electroencephalogram operational synchrony in the three separate DMN modules) between five experienced meditators and 10 naïve participants (who were about to start the meditation training) fully support the prediction. Interpretation that links such DMN subnets changes to the three-dimensional components of the experiential selfhood was proposed.     

Internal Consistency and Power When Comparing Total Scores from Two Groups

Researchers now know that when theoretical reliability increases, power can increase, decrease, or stay the same. However, no analytic research has examined the relationship of power to the most commonly used type of reliability—internal consistency—and the most commonly used measures of internal consistency, coefficient alpha and ICC(A,k). We examine the relationship between the power of independent samples t tests and internal consistency. We explicate the mathematical model upon which researchers usually calculate internal consistency, one in which total scores are calculated as the sum of observed scores on K measures. Using this model, we derive a new formula for effect size to show that power and internal consistency are influenced by many of the same parameters but not always in the same direction. Changing an experiment in one way (e.g., lengthening the measure) is likely to influence multiple parameters simultaneously; thus, there are no simple relationships between such changes and internal consistency or power. If researchers revise measures to increase internal consistency, this might not increase power. To increase power, researchers should increase sample size, select measures that assess areas where group differences are largest, and use more powerful statistical procedures (e.g., ANCOVA).     

Extension of Feldt's approach to testing homogeneity of coefficients of reliability

Limitations and extensions of Feldt's approach to testing the equality of Cronbach's alpha coefficients in independent and matched samples are discussed. In particular, this approach is used to test equality of intraclass correlation coefficients.This work was supported by the National Institute of Mental Health Specialized Research Center Grant # MH-30854.     

Simultaneous optimization of factor assessibility and representativeness

P. M. Bentler has shown that Rao's canonical factor analysis is in effect a psychometric analysis, leading to factors that are maximally assessible from the data. He contrasts this with Kaiser and Caffrey's alpha factor analysis that leads to factors that maximally represent the true factors in the content domain. Noting the problems associated with factors that may be highly assessible, but not very representative, or vice versa, Bentler suggests the need for a technique that would, insofar as possible, be optimal with respect to both criteria. Such a technique is presented here, and is shown to resolve into a traditional scaling method, which in turn acquires a richer psychometric interpretation.The author would like to gratefully acknowledge support for this research from the Workshop in Political Theory and Policy Analysis (Indiana University) and Esco Research Associates. The paper benefitted substantially from the criticisms and suggestions of an anonymous referee.     

A note on the relation between coefficient alpha and guttman's “split-half” lower bounds

Use of the same term split-half for division of ann-item test into two subtests containing equal [Cronbach], and possibly unequal [Guttman], numbers of items sometimes leads to a misunderstanding about the relation between Guttman's maximum split-half bound and Cronbach's coefficient alpha.Coefficient alpha is the average of split-half bounds in the Cronbach sense and so is not larger than the maximum split-half bound in either sense whenn is even. Whenn is odd, however, splithalf bounds exist only in the Guttman sense and the largest of these may be smaller than coefficient alpha.     

A modification of Feldt's test of the equality of two dependent alpha coefficients

The available statistical tests of the equality of nonindependent alpha reliability coefficients require that the product of the number of test parts times the number of subjects be quite large—1000 or more. A modification of one of these tests is derived which avoids this limitation. Monte Carlo studies indicate that the modified test effectively controls the Type I error rate with as few as 2 or 3 test parts and 50 subjects. This means the modified test can be safely employed in comparisons between interrater reliabilities.     

6～12岁儿童脑电α波频率分布特点与信息加工速度的关系

本研究以 2 2 8名 6～ 12岁儿童为研究对象 ,采用脑电波超慢涨落分析技术 (简称ET) ,重点探讨了与认知功能有关的α波 ( 8～ 13Hz)的分布特点。研究结果表明 :①α波在全脑 12个脑区的比率 ,因频率成分不同而表现出不同的分布特点 ;②α波在左右对称脑区的分布比率也因频率不同而表现出不同的左右脑优势。除左前颞叶的平均频率显著大于右前颞叶外 ,其他左右脑对称区域的平均频率的差异均不显著。α波平均频率的从高到低依次为枕叶、中央区、顶叶、左前颞叶、后颞叶、额叶、右前颞叶 ;③信息加工速度与α波的关系并不明显 ,快组和慢组被试的 6种频率 ( 8～13Hz)α波的比率均无显著性差异     

Commentary on Coefficient Alpha: A Cautionary Tale

The general use of coefficient alpha to assess reliability should be discouraged on a number of grounds. The assumptions underlying coefficient alpha are unlikely to hold in practice, and violation of these assumptions can result in nontrivial negative or positive bias. Structural equation modeling was discussed as an informative process both to assess the assumptions underlying coefficient alpha and to estimate reliability