A generalized expression for the reliability of measures

In certain situations it is important to obtain as many measures as possible, all presumably measuring the same function, for each of a group of persons. In general the number and source of the measures may vary from one member of the group to another. We take the mean of the measures for each person as the best estimate of the function for that person. The conventional formulas can not be used to determine the reliability of a set of means so obtained. A formula is developed which provides a unique estimate of the reliability of such a set of means. The formula is more general than some of the well-known reliability formulas, so that these formulas are shown to be special cases of the more general formula.     

可满足性问题的归约技术

通过一个恰当的归约变换,可以将一个CNF公式变换为另一个具有某种特殊结构或性质的公式,使其两者具有相同的可满足性。一个典型的归约是将一般的CNF公式变换为3．GVF公式。通过构造一些恰当的工具,可以将公式类变换为所要求的正则类。极小不可满足公式具有一个临界特征,公式本身不可满足,从原始公式中删去任意一个子句后得到的公式可满足。我们提供了一种归约技术,通过构造恰当的极小不可满足公式作为工具,将公式类变换为具有正则结构的公式类。研究正则结构的公式的复杂性及性质很有意义。如,将一个从3．CNF公式变换为（3,4）一CNF公式,这里（3,4）一CNF公式是指公式中每个子句的长度恰为3,每个变元出现的次数恰为4。因此,（3,4）．SAT是一个NP．完全问题,并且正N-部图的诸多良好性质对于研究正则结构的SAT问题具有许多潜在的作用。     

A method for combining occurrence and nonoccurrence interobserver agreement scores

Various statistics have been proposed as standard methods for calculating and reporting interobserver agreement scores. The advantages and disadvantages of each have been discussed in this journal recently but without resolution. A formula is presented that combines separate measures of occurrence and nonoccurrence percentages of agreement, with weight assigned to each measure, varying according to the observed rate of behavior. This formula, which is a modification of a formula proposed by Clement (1976), appears to reduce distortions due to "chance" agreement encountered with very high or low observed rates of behavior while maintaining the mathematical and conceptual simplicity of the conventional method for calculating occurrence and nonoccurrence agreement.     

Coinductive models and normal forms for modal logics (or how we learned to stop worrying and love coinduction)

We present a coinductive definition of models for modal logics and show that it provides a homogeneous framework in which it is possible to include different modal languages ranging from classical modalities to operators from hybrid and memory logics. Moreover, results that had to be proved separately for each different language—but whose proofs were known to be mere routine—now can be proved in a general way. We show, for example, that we can have a unique definition of bisimulation for all these languages, and prove a single invariance-under-bisimulation theorem.We then use the new framework to investigate normal forms for modal logics. The normal form we introduce may have a smaller modal depth than the original formula, and it is inspired by global modalities like the universal modality and the satisfiability operator from hybrid logics. These modalities can be extracted from under the scope of other operators. We provide a general definition of extractable modalities and show how to compute extracted normal forms. As it is the case with other classical normal forms—e.g., the conjunctive normal form of propositional logic—the extracted normal form of a formula can be exponentially bigger than the original formula, if we require the two formulas to be equivalent. If we only require equi-satisfiability, then every modal formula has an extracted normal form which is only polynomially bigger than the original formula, and it can be computed in polynomial time.     

On Sahlqvist Formulas in Relevant Logic

This paper defines a Sahlqvist fragment for relevant logic and establishes that each class of frames in the Routley-Meyer semantics which is definable by a Sahlqvist formula is also elementary, that is, it coincides with the class of structures satisfying a given first order property calculable by a Sahlqvist-van Benthem algorithm. Furthermore, we show that some classes of Routley-Meyer frames definable by a relevant formula are not elementary.     

A Meta-Analysis of the Reliability of Free and For-Pay Big Five Scales

The present study meta-analytically compared coefficient alpha reliabilities reported for free and for-pay Big Five scales. We collected 288 studies from five previous meta-analyses of Big Five traits and harvested 1,317 alphas from these studies. We found that free and for-pay scales measuring Big Five traits possessed comparable reliabilities. However, after we controlled for the numbers of items in the scales with the Spearman-Brown formula, we found that free scales possessed significantly higher alpha coefficients than for-pay scales for each of the Big Five traits. Thus, the study offers initial evidence that Big Five scales that are free more efficiently measure these traits for research purposes than do for-pay scales.     

A method for approximating the average intercorrelation coefficient by correlating the parts with the sum of the parts

It is noted that the average inter-item correlation, which represents the internal consistency of a test, yields a unique estimate of test reliability. A close approximation to this average is given by a formula which requires the correlation of each item with the total score and the standard deviation of each item. The formula is especially useful in those instances where the number of items is small and where the variation in item sigmas should not be neglected.     

On the axiomatization of finiteK-frames

We find a short way to construct a formula which axiomatizes a given finite frame of the modal logicK, in the sense that for each finite frameA, we construct a formula ωA which holds in those and only those frames in which every formula true inA holds. To obtain this result we find, for each finite model \(\mathfrak{A}\) and each natural numbern, a formula ω \(\mathfrak{A}\) which holds in those and only those models in which every formula true in \(\mathfrak{A}\) , and involving the firstn propositional letters, holds.     

Remarks on an Avoidable Error in a Posteriori Testing

This article discusses a widespread error that occurs with a posteriori testing in factorial analyses of variance. Frequently, an incorrect sample size is used under the radical in the formula for computing a critical difference such as the Fisher least significance difference. This usually leads to an inappropriately large critical difference; thus, few if any significant differences are uncovered between the means. An interesting example of this error, from a journal article by Van Berschot (1979), is examined. Researchers are urged always to use under the radical the number of observations on which each mean being compared is based.     

Proportionality,linearity, and parameter constancy: Messick and Sentis reconsidered

Messick and Sentis found greater support for weak proportionality in subjects' outcome allocations (suggesting that subjects wish to maintain a constant ratio between outcomes across situations when inputs are held constant) than for constant differences between outcomes. They claimed that their data thus provided stronger support for Adams' ratio definition of equity than for Harris' linear formula. A dialogue between the two sets of authors establishes that (1) so far as their specification of fair outcomes is concerned, Adams' formula is a special case of the linear formula; (2) weak proportionality does not necessarily follow from Adams' formula and can be derived from the linear formula; (3) Adams' formula applied to subjective (rather than experimenter-defined) inputs nevertheless provides a more natural base from which to derive weak proportionality than does the linear formula in its general form; (4) excellent fits to several sets of data, including much better fit to Messick and Sentis' data than that provided by weak proportionality, are obtained if we assume that subjects use allocations which are a weighted average of those implied by Adams' formula, the Equality principle, and Komorita and Kravitz' Equal Excess Norm; and (5) a fit to Messick and Sentis' data that is almost as good as that provided by the weighted solutions model is obtained by assuming that subjects apply Adams' formula to subjective inputs and subjective outcomes that are different from the corresponding experimenter-defined values. It is agreed that the delineation of the relationships among observable properties of outcome distributions, alternative equity formulae, and different psychological processes is as important as it can be devious.     

Effect of meaningfulness on judgments of computability

The University of Michigan, Ann Arbor, Michigan 48104 After memorizing a set. of four equations, subjects were given problem frames specifying an unknown variable and two, three, or four given variables. The task on each item was to judge whether the presented variables corresponded to a problem that could be computed. (No values were given and no computation was carried out.) The main finding was a strong interaction: With formulas consisting of uninterpreted letters, judgments of uncomputable items were much slower than judgments of computable items; however, the difference between positive and negative items was much smaller when formulas consisted of meaningful concepts. Attempts to remove the interaction by presenting the nonsense formulas as an integrated network or by using a format allowing easy chunking of the variables in each formula did not succeed. At this point it appears most promising to investigate effects of meaningfulness on basic component processes, such as decisions about whether pairs of items come from the same formula.     

The probability of a given 1-choice structure

A 1-choice structure arises whenever each person in a group chooses exactly one other person according to some criterion. Our purpose is to study the situation in which these choices are made at random. As noted in Harary, Norman and Cartwright [2], such a structure can be represented by a directed graph in which the points represent people and the directed lines their choices. We first describe the shape of such a 1-choice structure, and define its symmetry number. With the help of these properties we are then able to develop and prove a formula which gives a probability of obtaining a given structure in the random choice situation. In order to supply data for these results, we include in the Appendix the diagrams of all 1-choice structures with at most 7 points and the probability of each.The preparation of this article was supported by Grant MH 10834 from the National Institute of Mental Health and Grant 1026-66 from the U. S. Air Force Office of Scientific Research.     

The Secret of My Success

In an information state where various agents have both factual knowledge and knowledge about each other, announcements can be made that change the state of information. Such informative announcements can have the curious property that they become false because they are announced. The most typical example of that is ‘fact p is true and you don’t know that’, after which you know that p, which entails the negation of the announcement formula. The announcement of such a formula in a given information state is called an unsuccessful update. A successful formula is a formula that always becomes common knowledge after being announced. Analysis of information systems and ‘philosophical puzzles’ reveals a growing number of dynamic phenomena that can be described or explained by unsuccessful updates. This increases our understanding of such philosophical problems. We also investigate the syntactic characterization of the successful formulas. An erratum to this article is available at .     

Sentential constants in systems near R

An Ackermann constant is a formula of sentential logic built up from the sentential constant t by closing under connectives. It is known that there are only finitely many non-equivalent Ackermann constants in the relevant logic R. In this paper it is shown that the most natural systems close to R but weaker than it-in particular the non-distributive system LR and the modalised system NR-allow infinitely many Ackermann constants to be distinguished. The argument in each case proceeds by construction of an algebraic model, infinite in the case of LR and of arbitrary finite size in the case of NR. The search for these models was aided by the computer program MaGIC (Matrix Generator for Implication Connectives) developed by the author at the Australian National University.     

Simplified formulas for item selection and construction

The formula for the Pearson correlation coefficient of a dichotomous variable with a multiple-categoried variable is simplified for computational purposes by effecting in the multiple-categoried variable two types of arbitrary distributions: (1) rectangular and (2) proportional to binomial expansion coefficients. The formulas which result are convenient for the selection of test items and are applicable to the objective estimation of the comparative merits of the alternatives in multiple-choice test items. It is shown that the authoritative answer should have a high positive criterion coefficient, while the omissions and several wrong-answer alternatives should each have low (algebraic) negative criterion coefficients.     

两水平研究中单维测验信度的估计

在心理、教育和管理等研究领域中,经常会碰到两水平（两层）的数据结构,如学生嵌套在班级中,员工嵌套在企业中。在两水平研究中,被试通常不是独立的,如果直接用单水平信度公式进行估计,会高估测验信度。文献上已有研究讨论如何更准确地估计两水平研究中单维测验的信度。本研究指出了现有的估计公式的不足之处,用两水平验证性因子分析推导出一个新的信度公式,举例演示如何计算,并给出简单的计算程序。     

Sociometric typologies during adolescence: contrasting different scoring techniques and formulas

Our aim in this work is to compare the different sociometric classification formulae and to analyze possible connections between each type of formula and each behaviour assessed by peers during adolescence. We classified a total of 1,009 Spanish fourth-grade secondary education students (15-year-olds) using four different systems: two were based on a nomination technique by Coie and Dodge, Newcomb and Bukowski, the third system was a modified version of the latter based on the revision of the neglected status, and a fourth system was based on the qualification scale by Maassen and Laandsheer. Results show a significant relation between all four classifications, but there are also differences among them. The classifications proceeding from the nomination technique share more similarities and also offer a better behavioural characterization of sociometric categories.     

A technique for criterion-keying and selecting test items

For multiple-choice tests where noa priori key exists, the initial selection of a key for maximum validity may be made on the basis of the number of persons choosing each alternative and their mean criterion score. The keying formula is derived. Once the initial keying has been done, further precision in keying and item selection may use, in addition, the mean total test score for persons choosing each alternative. Item-selection formulas suggested by Horst and by Gulliksen for maximizing test validity are both in the form of a ratio, an item-validity index divided by an item-reliability index. The formula derived here is shown to be equivalent to the numerators of these formulas. The expression in the denominators uses the total test score. Although a radical appears in the denominator of Horst's formula and not in the denominator of Gulliksen's formula, both of them select the same items in practice.The author gratefully acknowledges the suggestions and criticisms of Dr. Harold Gulliksen, Research Adviser at the Educational Testing Service.     

The benefit of interleaved mathematics practice is not limited to superficially similar kinds of problems

Most mathematics assignments consist of a group of problems requiring the same strategy. For example, a lesson on the quadratic formula is typically followed by a block of problems requiring students to use that formula, which means that students know the appropriate strategy before they read each problem. In an alternative approach, different kinds of problems appear in an interleaved order, which requires students to choose the strategy on the basis of the problem itself. In the classroom-based experiment reported here, grade 7 students (n = 140) received blocked or interleaved practice over a nine-week period, followed two weeks later by an unannounced test. The mean test scores were greater for material learned by interleaved practice rather than by blocked practice (72 % vs. 38 %, d = 1.05). This interleaving effect was observed even though the different kinds of problems were superficially dissimilar from each other, whereas previous interleaved mathematics studies had required students to learn nearly identical kinds of problems. We conclude that interleaving improves mathematics learning not only by improving discrimination between different kinds of problems, but also by strengthening the association between each kind of problem and its corresponding strategy.