On the distribution of the maximum likelihood estimator of Cronbach's alpha

The asymptotic normal distribution of the maximum likelihood estimator of Cronbach's alpha (under normality) is derived for the case when no assumptions are made about the covariances among items. The asymptotic distribution is also considered for the special case of compound symmetry and compared to the exact distribution.The authors would like to thank Willem J. Heiser, an associate editor and the reviewers for valuable and helpful comments to improve the quality of this work.     

Processing speed, intelligence, creativity, and school performance: Testing of causal hypotheses using structural equation models

According to mental speed theory of intelligence, the speed of information processing constitutes an important basis for cognitive abilities. However, the question, how mental speed relates to real world criteria, like school, academic, or job performance, is still unanswered. The aim of the study is to test an indirect speed-factor model in comparison to rivaling models explaining the relationships between different mental abilities and performance. In this speed-factor model, basic cognitive processing is assumed to influence higher mental abilities (IQ and creativity). Intelligence and creativity themselves should be valid predictors of school performance. We computed bivariate correlations and structural equation models to test this hypothesis, using indicators of processing speed [Zahlen-Verbindungs-Test (ZVT) and Coding Test], psychometric intelligence [Kognitiver Fähigkeits-Test (KFT) and Raven's Advanced Progressive Matrices (APM)], creativity [Verbaler Kreativitäts-Test (VKT) and Verwendungs-Test (VWT)] and school performance (grades). In a sample of 271 students from German gymnasiums (Class Levels 9 to 11) the speed-factor model can reproduce at best the empirical relationships between processing speed, intelligence, creativity, and school performance: It assumes that processing speed influences higher mental abilities (intelligence and creativity), which, in the sequel, influence school performance. Therefore, processing speed seems to have no direct effect on school performance; the effect is indirect as it operates via mediation through higher cognitive abilities.     

Testing differences between nested covariance structure models: Power analysis and null hypotheses

For comparing nested covariance structure models, the standard procedure is the likelihood ratio test of the difference in fit, where the null hypothesis is that the models fit identically in the population. A procedure for determining statistical power of this test is presented where effect size is based on a specified difference in overall fit of the models. A modification of the standard null hypothesis of zero difference in fit is proposed allowing for testing an interval hypothesis that the difference in fit between models is small, rather than zero. These developments are combined yielding a procedure for estimating power of a test of a null hypothesis of small difference in fit versus an alternative hypothesis of larger difference.     

Testing pattern hypotheses for covariance matrices

Maximum likelihood estimates of the free parameters, and an asymptotic likelihood-ratio test, are given for the hypothesis that one or more elements of a covariance matrix are zero, and/or that two or more of its elements are equal. The theory applies immediately to a transformation of the covariance matrix by a known nonsingular matrix. Estimation is by Newton's method, starting conveniently from a closed-form least-squares solution.Numerical illustrations include a test for equality of diagonal blocks of a covariance matrix, and estimation of quasi-simplex structures.     

Exact distributions of intraclass correlation and Cronbach's alpha with Gaussian data and general covariance

Intraclass correlation and Cronbach's alpha are widely used to describe reliability of tests and measurements. Even with Gaussian data, exact distributions are known only for compound symmetric covariance (equal variances and equal correlations). Recently, large sample Gaussian approximations were derived for the distribution functions. New exact results allow calculating the exact distribution function and other properties of intraclass correlation and Cronbach's alpha, for Gaussian data with any covariance pattern, not just compound symmetry. Probabilities are computed in terms of the distribution function of a weighted sum of independent chi-square random variables. NewF approximations for the distribution functions of intraclass correlation and Cronbach's alpha are much simpler and faster to compute than the exact forms. Assuming the covariance matrix is known, the approximations typically provide sufficient accuracy, even with as few as ten observations. Either the exact or approximate distributions may be used to create confidence intervals around an estimate of reliability. Monte Carlo simulations led to a number of conclusions. Correctly assuming that the covariance matrix is compound symmetric leads to accurate confidence intervals, as was expected from previously known results. However, assuming and estimating a general covariance matrix produces somewhat optimistically narrow confidence intervals with 10 observations. Increasing sample size to 100 gives essentially unbiased coverage. Incorrectly assuming compound symmetry leads to pessimistically large confidence intervals, with pessimism increasing with sample size. In contrast, incorrectly assuming general covariance introduces only a modest optimistic bias in small samples. Hence the new methods seem preferable for creating confidence intervals, except when compound symmetry definitely holds. An earlier version of this paper was submitted in partial fulfillment of the requirements for the M.S. in Biostatistics, and also summarized in a presentation at the meetings of the Eastern North American Region of the International Biometric Society in March, 2001. Kistner's work was supported in part by NIEHS training grant ES07018-24 and NCI program project grant P01 CA47 982-04. She gratefully acknowledges the inspiration of A. Calandra's “Scoring formulas and probability considerations” (Psychometrika, 6, 1–9). Muller's work supported in part by NCI program project grant P01 CA47 982-04.     

Testing global memory models using ROC curves.

Global memory models are evaluated by using data from recognition memory experiments. For recognition, each of the models gives a value of familiarity as the output from matching a test item against memory. The experiments provide ROC (receiver operating characteristic) curves that give information about the standard deviations of familiarity values for old and new test items in the models. The experimental results are consistent with normal distributions of familiarity (a prediction of the models). However, the results also show that the new-item familiarity standard deviation is about 0.8 that of the old-item familiarity standard deviation and independent of the strength of the old items (under the assumption of normality). The models are inconsistent with these results because they predict either nearly equal old and new standard deviations or increasing values of old standard deviation with strength. Thus, the data provide the basis for revision of current models or development of new models.     

Testing the group polarization hypothesis by using logit models

This paper focuses on methodological aspects of group polarization research and has two well‐defined parts. The first part presents a methodological overview of group polarization research together with an examination of the inadequacy, under certain circumstances, of the traditional parametric approach usually used to test this phenomenon based on pre‐test/post‐test means comparison across groups. It is shown that this approach will produce masks effects when groups are heterogeneous with regard to the observed change from pre‐test to post‐test. The second part suggests an alternative methodological approach based on logit models for the analysis of contingency tables from a categorization of the variable ‘kind of shift’. This approach is illustrated and compared with the parametric approach with a simulated data set. Copyright © 2002 John Wiley & Sons, Ltd.     

Person-organization fit: Testing socialization and attraction-selection-attrition hypotheses

Using the socialization and attraction-selection-attrition (ASA) frameworks, this study examined the relation between employees’ work values and their organization’s values (person-organization fit). With a two year time interval, 140 respondents participated in a longitudinal study. After entry, socialization served to enhance homogeneity. The work values underwent small changes and the perception of fit with the organization grew. Despite high retention rates results confirmed the attrition effect. Results indicated that the lower the perceived match between own and organizational values at entry, the more likely it was that someone left the organization over time. We concluded that socialization as well as attrition mechanisms were present at the same time.     

Testing hypotheses on specific environmental causal effects on behavior

There have been strong critiques of the notion that environmental influences can have an important effect on psychological functioning. The substance of these criticisms is considered in order to infer the methodological challenges that have to be met. Concepts of cause and of the testing of causal effects are discussed with a particular focus on the need to consider sample selection and the value (and limitations) of longitudinal data. The designs that may be used to test hypotheses on specific environmental risk mechanisms for psychopathology are discussed in relation to a range of adoption strategies, twin designs, various types of "natural experiments," migration designs, the study of secular change, and intervention designs. In each case, consideration is given to the need for samples that "pull-apart" variables that ordinarily go together, specific hypotheses on possible causal processes, and the specification and testing of key assumptions. It is concluded that environmental risk hypotheses can be (and have been) put to the test but that it is usually necessary to use a combination of research strategies.     

Testing linear hypotheses illustrated by a simple example in correlation

The development of a criterion suitable for testing the significance of a correlation or regression coefficient is used as an illustration of the manner in which a research problem is bound to the selection of the particular data appropriate to collect and a fitting type of statistical analysis of the latter. The translation of the original inquiry into a problem of testing linear hypotheses is the means by which these two aspects of an investigation are held together. This presentation is offered as a plan which might be useful for some research workers in determining appropriate criteria for testing their particular hypotheses.     

Testing clinical hypotheses by means of "falsificationist epistemology"

A family therapist using the "falsificationist epistemology" can, during the first session, test the hypotheses formulated on the telephone intake chart and also gather further information to reexamine and redefine these hypotheses. We shall examine a case treated by Dr. G. Prata, analyzing the most important phases of the first session, and stressing what methodological and tactical choices were designed to test the leading hypothesis of the "game" played by this family.     

Testing hypotheses about the self: Assessments of job suitability1

This investigation examined the strategies employed by individuals when testing hypotheses about the self. Participants known to possess masculine, feminine, or androgynous sex role identities tested hypotheses about their suitability for one of two jobs: either a job characterized in terms of the attributes and aptitudes typically associated with a “masculine” personality, or one characterized in terms of those typically associated with a “feminine” personality. When testing these hypotheses about themselves, participants preferentially reported those aspects of themselves that would suggest their suitedness rather than their unsuitedness for the job under consideration regardless of their sex role identity. Moreover, subsequent judgments of job suitability were positively related to the amount of suitedness evidence reported, but unrelated to the amount of unsuitedness evidence reported. Some consequences of these strategies for testing hypotheses about the self are discussed.     

Testing hypotheses about other people: The use of historical knowledge

Three experiments were conducted to examine the use of historical knowledge to test contemporary hypotheses about the personal attributes of other people. In the first and second experiments, participants read an extensive account of events in one week of the life of a woman named Jane. Two days later, they used this previously learned information to test hypotheses about Jane's suitability for one of two jobs: either the rather extraverted job of real estate salesperson, or the rather introverted job of research librarian. In their hypothesis-testing activities, participants first reported all those previously learned facts that they regarded as relevant to assessing Jane's suitability for the job under consideration, and then reported their judgments of her job suitability. Participants reported greater amounts of hypothesis-confirming than hypothesis-disconfirming factual material. Moreover, having tested hypotheses about Jane's suitability for one job, participants judged her to be better suited for that job than for the other job. In the third experiment, participants framed hypotheses for assessing job suitability and defined the task of testing these hypotheses. Participants framed hypotheses in terms of those attributes whose presence would confirm the hypotheses and defined the hypothesis-testing task as one of preferentially collecting hypothesis-confirming evidence. The nature and consequences of confirmatory hypothesis-testing strategies are discussed.