The theory of boolean algebras with an additional binary operation

This paper deals with Boolean algebras supplied with an additional binary operation, calledB-algebras for short.The aim of the paper is to generalize some theorems concerning topological Boolean algebras to more comprehensive classes ofB-algebras, to formulate fundamental properties ofB-algebras, and to find more important relationships of these algebras to other known algebras.The paper consists of two parts. At the beginning of the first one, several subclasses ofB-algebras are distinguished, and then, their basic properties, connections between them as well as certain relationships with other algebras, are investigated. In particular, it is shown that the class of Boolean algebras together with an arbitrary unary operation is polynomially equivalent to the class ofB ₁-algebras.The second part of the paper is concerned with the theory of filters and congruences inB-algebras.     

General estimators for the reliability of qualitative data

We study a proportional reduction in loss (PRL) measure for the reliability of categorical data and consider the general case in which each ofN judges assigns a subject to one ofK categories. This measure has been shown to be equivalent to a measure proposed by Perreault and Leigh for a special case when there are two equally competent judges, and the correct category has a uniform prior distribution. We consider a general framework where the correct category is assumed to have an arbitrary prior distribution, and where classification probabilities vary by correct category, judge, and category of classification. In this setting, we consider PRL reliability measures based on two estimators of the correct category—the empirical Bayes estimator and an estimator based on the judges' consensus choice. We also discuss four important special cases of the general model and study several types of lower bounds for PRL reliability.Bruce Cooil is Associate Professor of Statistics, and Roland T. Rust is Professor and area head for Marketing, Owen Graduate School of Management, Vanderbilt University. The authors thank three anonymous reviewers and an Associate Editor for their helpful comments and suggestions. This work was supported in part by the Dean's Fund for Faculty Research of the Owen Graduate School of Management, Vanderbilt University.     

Worlds are colliding! Explaining the fictional in terms of the real

I discuss Gregory Currie’s taxonomy of explanations of the fictional. On the one hand, there is an important kind of relation between internal and external explanations of some fictional truths that Currie leaves out, where both are salient and yet in a relation of harmony with each other. On the other hand, I do not see that he has established that there is a genuine relation of tension between some pairs of internal and external explanations, and thus I question the usefulness of the category of collapse. I also consider a further kind of explanation: the exterior explanation.

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.     

Some properties of the communality in multiple factor theory

Several theorems concerning properties of the communaltiy of a test in the Thurstone multiple factor theory are established. The following theorems are applicable to a battery ofn tests which are describable in terms ofr common factors, with orthogonal reference vectors.1. The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors.2. The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors and then—1 remaining tests. Corollary: The square of the multiple correlation of a testj with then—1 remaining tests is equal to or less than the communality of testj. It cannot exceed the communality.3. The square of the multiple correlation of a testj with then—1 remaining tests equals the communality of testj if the group of tests containsr statistically independent ests teach with a communality of unity.4. With correlation coefficients corrected for attenuation, when the number of tests increases indefinitely while the rank of the correlational matrix remains unchanged, the communality of a testj equals the square of the multiple correlation of testj with then—1 remaining tests.5. With raw correlation coefficients, it is shown in a special case that the square of the multiple correlation of a testj with then—1 remaining tests approaches the communality of testj as a limit when the number of tests increases indefinitely while the rank of correlational matrix remains the same. This has not yet been proved for the general case.The author wishes to express his appreciation of the encouragement and assistance given him by Dr. L. L. Thurstone.     

Concerning the determination of trait variability

An analytic technique for the study of trait variability is presented. An expression for the average variance from test to test and an expression for the variance of these variances are derived in terms of the number of tests and the intercorrelations between them, and limiting cases are examined. The question of the true relationship between the nature of the distribution of test scores in a sample ofN persons and the nature of the distribution ofn traits in a single individual is discussed, and other problems are introduced.     

Estimation ofJ scales for unidimensional unfolding

In Coombs' unidimensional unfolding theory each individual ranks all stimuli, often in terms of preference judgments. These ranks, calledI scales, are used to infer the latent continuum called aJ scale which is presumed to have generated theI scale rankings. A major problem concerns the inference of theJ scale given a set ofI scales because anI scale is not unique to any oneJ scale. This paper presents a procedure for estimatingJ scale probabilities given a set ofI scales. Models for three and four or more stimuli are presented. Results of computer tests are reported.     

At the forefront of thought: The effect of media exposure on airplane typicality

The terrorist attacks of September 11, 2001 provided a unique opportunity to investigate the causal status of frequency on typicality for one exemplar of a common conceptual category-namely, the typicality ofairplane as a member of the category ofvehicles. The extensive media coverage following the attacks included numerous references to the hijacked airplanes and to the consequences of suspending air travel to and from the United States for several days. The present study, involving 152 undergraduates, assessed airplane typicality at three time points ranging from 5 h to 1 month after the attacks and then again at 4.5 months after the attacks. Airplane was judged to be a more typical vehicle for 1 month following the attacks, relative to a baseline calculated from data collected yearly for 5 years preceding the attacks. By 4.5 months, however, typicality was back to baseline.     

CONSTRUCTING AND DERIVING RECIPROCAL TRIGONOMETRIC RELATIONS: A FUNCTIONAL ANALYTIC APPROACH

Participants were pretrained and tested on mutually entailed trigonometric relations and combinatorially entailed relations as they pertained to positive and negative forms of sine, cosine, secant, and cosecant. Experiment 1 focused on training and testing transformations of these mathematical functions in terms of amplitude and frequency followed by tests of novel relations. Experiment 2 addressed training in accordance with frames of coordination (same as) and frames of opposition (reciprocal of) followed by more tests of novel relations. All assessments of derived and novel formula‐to‐graph relations, including reciprocal functions with diversified amplitude and frequency transformations, indicated that all 4 participants demonstrated substantial improvement in their ability to identify increasingly complex trigonometric formula‐to‐graph relations pertaining to same as and reciprocal of to establish mathematically complex repertoires.     

The mind‐body relationship in Pāli Buddhism: A philosophical investigation

The Suttas indicate physical conditions for success in meditation, and also acceptance of a not‐Self life‐principle (primarily viññana) which is (usually) dependent on the mortal physical body. In the Abhidhamma and commentaries, the physical acts on the mental through the senses and through the ‘basis’ for mind‐organ and mind‐consciousness, which came to be seen as the ‘heart‐basis’. Mind acts on the body through two ‘intimations’: fleeting modulations in the primary physical elements. Various forms of rūpa are also said to originate dependent on citta and other types of rūpa. Meditation makes possible the development of a ‘mind‐made body’ and control over physical elements through psychic powers. The formless rebirths and the state of cessation are anomalous states of mind‐without‐body, or body‐without‐mind, with the latter presenting the problem of how mental phenomena can arise after being completely absent. Does this twin‐category process pluralism avoid the problems of substance‐dualism?     

Multiple rectilinear prediction and the resolution into components

It is assumed that a battery ofn tests has been resolved into components in a common factor space ofr dimensions and a unique factor space of at mostn dimensions, wherer is much less thann. Simplified formulas for ordinary multiple and partial correlation of tests are derived directly in terms of the components. The best (in the sense of least squares) linear regression equations for predicting factor scores from test scores are derived also in terms of the components. Spearman's single factor prediction formulas emerge as special cases. The last part of the paper shows how the communality is an upper bound for multiple correlation. A necessary and sufficient condition is established for the square of the multiple correlation coefficient of testj on the remainingn—1 tests to approach the communality of testj as a limit asn increases indefinitely whiler remains constant. Limits are established for partial correlation and regression coefficients and for the prediction of factor scores.I am indebted to Professor Dunham Jackson for helpful criticism of most of this paper.     

Using the conditional grade-of-membership model to assess judgment accuracy

Consider the case whereJ instruments are used to classify each ofI objects relative toK nominal categories. The conditional grade-of-membership (GoM) model provides a method of estimating the classification probabilities of each instrument (or judge) when the objects being classified consist of both pure types that lie exclusively in one ofK nominal categories, and mixtures that lie in more than one category. Classification probabilities are identifiable whenever the sample of GoM vectors includes pure types from each category. When additional, relatively mild, assumptions are made about judgment accuracy, the identifiable correct classification probabilities are the greatest lower bounds among all solutions that might correspond to the observed multinomial process, even when the unobserved GoM vectors do not include pure types from each category. Estimation using the conditional GoM model is illustrated on a simulated data set. Further simulations show that the estimates of the classification probabilities are relatively accurate, even when the sample contains only a small percentage of approximately pure objects.The authors thank Max A. Woodbury, Kenneth G. Manton and H. Dennis Tolley for their help and four anonymous Psychometrika reviewers (including an associate editor) for their beneficial expository and technical suggestions. This work was supported by the Dean's Fund for Summer Research, Owen Graduate School of Management, Vanderbilt University.     

The impact of maternal employment on children's perceptions of parents and personal development

This study assessed the impact of maternal employment on young adolescents' self-concept, school achievement, vocational development, and the perceptions of their parents. It also assessed the impact of maternal employment in nonintact homes. Multivariate F tests indicated no significant differences in the subjects' perceptions of either parent or in their educational and vocational development that could be attributed to maternal employment. Similar results were reported in nonintact families.     

A more powerful method for testing for agreement between a judge and a known standard

We assume that a judge's task is to categorize each ofN subjects into one ofr known classes. The design of primary interest is employed if the judge is presented withs groups, each containingr subjects, such that each group of sizer consists of exactly one subject of each of ther types. The probability distribution for the total number of correct choices is developed and used to test the null hypothesis that the judge is guessing in favor of the alternative that he or she is operating at a better than chance level. The power of the procedure is shown to be superior to two other procedures which appear in the literature.The authors are grateful for the suggestions of the referees and for computer funding provided by the Northeast Regional Data Center at the University of Florida.     

On the theory of test discrimination

This paper discusses the properties of distributions of test scores and advances the view that the properties of the distribution should depend on the function which the test is intended to perform. A theory of test discrimination is developed which defines discriminatory capacity in terms of the number of relations of difference established by the operation of administering a test ofk items to a sample ofn individuals. A simple proof is presented which indicates that maximum discrimination between individuals is achieved when tests are constructed to yield distributions of the rectangular form. A coefficient of test discrimination is developed. The problem of obtaining in practice distributions approximating to the rectangular form is briefly discussed.This paper was prepared under the auspices of the Defence Research Board, Ottawa, Canada.     

Sample size determination for at test given at value from a previous study: A FORTRAN 77 program

When uncertain about the magnitude of an effect, researchers commonly substitute in the standard sample-size-determination formula an estimate of effect size derived from a previous experiment. A problem with this approach is that the traditional sample-size-determination formula was not designed to deal with the uncertainty inherent in an effect-size estimate. Consequently, estimate-substitution in the traditional sample-size-determination formula can lead to a substantial loss of power. A method of sample-size determination designed to handle uncertainty in effect-size estimates is described. The procedure uses thet value and sample size from a previous study, which might be a pilot study or a related study in the same area, to establish a distribution of probable effect sizes. The sample size to be employed in the new study is that which supplies an expected power of the desired amount over the distribution of probable effect sizes. A FORTRAN 77 program is presented that permits swift calculation of sample size for a variety oft tests, including independentt tests, relatedt tests,t tests of correlation coefficients, andt tests of multiple regressionb coefficients.     

Idiosyncratic deals: Testing propositions on timing, content, and the employment relationship

This study tests propositions regarding idiosyncratic deals (i-deals) in a sample of N = 265 hospital employees using structural equation modeling. Timing and content of idiosyncratic employment arrangements are postulated to have differential consequences for the nature of the employment relationship. Results confirm that i-deals made after hire have greater impact on the employment relationship than those made ex ante. Developmental i-deals are positively related to perceiving employment as a social exchange rather than an economic exchange, whereas work hour i-deals show the opposite pattern. Implications for research and practice are discussed.     

Employment test usage as a predictor of gross domestic product

The relationship between the usage of employment tests—particularly ability tests-and job performance criteria has been extensively studied. The usage of testing was reported to have declined after the landmark personnel selection caseGriggs (1971). Models of selection utility were reviewed, and an ecological systems theory of test incidence and the American economy was developed. For both ability tests and non-personality employment tests, a positive relationship was hypothesized between the historical incidence of firms using tests and American Gross Domestic Product. Using historical survey and economic data, results were consistent with the hypothesis involving ability testing incidence.The views expressed are those of the author and not necessarily those of the U.S. Air Force, the Department of Defense, or the U.S. government.     

The concept of Cheng and its western translations

Conclusion The main reasons for the difficulty in understanding and translatingcheng may be summarized as follows. First, its prehistory is not always clear. This makes it troublesome to identify its original meaning. Second, the multiple sources from the three schools, Confucianism, Daoism, and Buddhism, often causecheng to be entangled with various concepts specifically affiliated to certain schools. The particular meanings of these concepts and their connections withcheng possibly mislead our effort to explore the core content ofcheng as such. Finally,cheng has been described as a force causing either “transformation” or “change.” In the former case, its influence is often exaggerated, while in the latter, it appears, more or less, mystical. However, despite all the complexity, the core content ofcheng is still identifiable. It consists of what two English terms-“sincerity” and “reality”-convey. In general, the termcheng can be properly applied to a person or thing so long as either one of them or the unity of them is present.