As an alternative to the analysis of variance approach to reliability a multiple-factor analysis approach is illustrated. The one-factor and the multiple-factor models for reliability are compared. Tests on the latent roots associated with the principal components of intercorrelation matrices are used to determine the number of components to be retained. Conditions under which one or more of the principal components should be utilized are discussed.

The first contribution of this paper is to demonstrate that the paired-comparison model may be derived from a variety of different initial assumptions about the nature of paired comparisons, some of these assumptions appearing to the author to be more appropriate to a description of the preference-decision process than others. The second contribution of this paper is to note that the generalization of the model to triple comparisons chosen earlier is not the one compatible with the Lehmann model even though it possesses other desirable properties. Limited numerical calculations suggest that both models for triple comparisons give comparable results in applications. Their asymptotic properties should be similar.

This paper seeks to meet the need for a general treatment of the problem of error in classification. Within an m-attribute classificatory system, an object's typical subclass is that subclass to which it is most often allocated under repeated experimentally independent applications of the classificatory criteria. In these terms, an error of classification is an atypical subclass allocation. This leads to definition of probabilities O of occasional subclass membership, probabilities T of typical subclass membership, and probabilities E of error or, more generally, occasional subclass membership conditional upon typical subclass membership. In the relationship f: (O, T, E) the relative incidence of independent O, T, and E values is such that generally one can specify O values given T and E, but one cannot generally specify T and E values given O. Under the restrictions of homogeneity of E values for all members of a given typical subclass, mutual stochastic independence of errors of classification, and suitable conditions of replication, one can find particular systems O = f (T, E) which are solvable for T and E given O. A minimum of three replications of occasional classification is necessary for a solution of systems for marginal attributes, and a minimum of two replications is needed with any cross-classification. Although for such systems one can always specify T and E values given O values, the solution is unique for dichotomous systems only.

A set of inequalities on communalities developed by Darroch [1] is used to provide some rules for determining admissible sets of communalities in factor analysis.

The correct matching method is treated in a new way which increases its power and enables the results to be expressed by a correlation coefficient. Tables for 3 ≤ n ≤ 7 are included.

When an arbitrary positive scalar matrix is added to a correlation matrix the latent roots of the sum are equal to the corresponding roots of the correlation matrix plus an amount equal to the scalar number of the scalar matrix. The latent vectors of the sum are identical with those of the correlation matrix. An approximation to these relationships is suggested for the case in which the sum is of a correlation matrix and of a positive semidefinite diagonal matrix. The approximation is used to allow the solution of a characteristic problem for a correlation matrix with unities in the main diagonal to provide a family of solutions for the same correlation matrix.

By taking advantage of an experimental design which increases the number of observations per subject, the sensitivity of the proposed test is increased. In the first of a series of subexperiments, the score of a randomly selected experimentalS is ranked with the scores of the remaining (control) Ss, and the experimental S is discarded. This procedure is repeated until one S remains. The test criterion, Rn, is the sum of ranks of the experimental Ss; its exact sampling distribution, a normal approximation, and the sensitivity of the test relative to the Wilcoxon T, are described. Because of the large number of scores obtained, the test is particularly useful when preparing Ss is more costly than testing them, e.g., in tests following involved conditioning procedures. Since each S is discarded after experimental treatment, the test fills a real need in studies involving radiation, ablation, or other irreversible treatments.

A model for analyzing the learning process with a special emphasis on serial-position effect is proposed. This model consists of two analyses, one being an analysis of the learning process of each item in a list by a stochastic method, and the other being an analysis of serial-position effect in terms of pro- and retroactive inhibitions, and of forgetting. The model is experimentally verified, and moreover, it is found that the model permits prediction of the results of many experiments with lists of various lengths and varying difficulty.

A reliability study is assumed to be carried out with each of a number of observers making a dichotomous judgment concerning each of a sample of subjects. A nonparametric model is proposed for the errors underlying such judgments, and conditions are given under which Cochran's Q statistic is valid for testing the hypothesis of no systematic differences among the judgments of the different observers. Inferences concerning the probabilities of error are shown to be possible in terms of the intraclass correlation coefficient. A numerical example is given.

A measure of the discrepancy between observed transition frequencies and those predicted by a learning model, an “average error” of a learning model, is presented. The maximum-likelihood estimator of the average error is derived and its use in a modified test of goodness of fit is demonstrated.

Two common methods of measuring personality variables are examined with respect to the ways in which they control respondent biases. A form of the mental-test theory model is developed which makes explicit differences in score components that result from different methodological constraints imposed by the two methods. From the model, it is possible to specify the operations necessary to provide equivalent information from the two sets of data. Performance of these operations in an empirical test supports the model by producing generally higher correlations between the variables having score components which are more closely matched.

Explicit solutions are obtained for a sequence of limiting distributions of response probabilities for the two experimenter-controlled events learning model of Bush and Mosteller [2]. A generalization to the s experimenter-controlled events model is found.

General formulas for part and partial correlation of any order are derived in terms of multiple correlation coefficients, standard partial regression weights, and validities. The relationship between part correlation and the “independent contribution of a predictor” is discussed.

An approximation to the sampling distribution of Kuder-Richardson reliability formula 20 is derived, using its algebraic equivalent obtained through an items-by-subjects analysis of variance. The theoretical distribution is compared to empirical estimates of the sampling distribution to assess how crucial certain assumptions are. The use of the theoretical distribution for testing hypotheses and deriving confidence intervals is illustrated. A table of equations for approximating 80, 90, and 95 per cent confidence intervals is presented with N ranging from 40 to 500.

When the purpose of the experiment is to compare treatments, the Sequences × Positions Latin Square has been employed to control unwanted effects attributable to individuals, position, and sequence. This particular Latin Square has been subjected to criticism on the grounds there is confounding due to structure, random variables, and subject interactions. Special Latin Square, a subclass of the Sequences × Positions Latin Square, is basically a p ×p factorial design in blocks of size p. The two factors are treatments (T) and positions (P). Sequence is one component of the TP interaction, and square uniqueness is the sum of the remaining components. This completely replicated factorial design has no structural or random variable confounding; if subject interactions are present, square uniqueness may be used as the error term and the bias in the test of treatments will be conservative.

It is common to assume that the proportion of correct answers to an item has a normal-ogive or logistic relationship to total test score. However, this is shown to be a mistaken and an undesirable notion.

An objective method for the orthogonal rotation of factors which gives results closer to the graphic method is proposed. First, the fact that the varimax method does not always satisfy simple-structure criteria, e.g., the positive manifold and the level contributions of all factors, is pointed out. Next, the principles of our method which are based on “geometric vector” are discussed, and the computational procedures for this method are explained using Harman and Holzinger's eight physical variables. Finally, six numerical examples by our method are presented, and it is shown that they are very close to the factors obtained from empirical studies both in values and in signs.