A “strong” mathematical model for the relation between observed scores and true scores is developed. This model can be used

1. 1. To estimate the frequency distribution of observed scores that will result when a given test is lengthened.

2. 2. To equate true scores on two tests by the equipercentile method.

3. 3. To estimate the frequencies in the scatterplot between two parallel (nonparallel) tests of the same psychological trait, using only the information in a (the) marginal distribution(s).

4. 4. To estimate the frequency distribution of a test for a group that has taken only a short form of the test (this is useful for obtaining norms).

5. 5. To estimate the effects of selecting individuals on a fallible measure.

6. 6. To effect matching of groups with respect to true score when only a fallible measure is available.

7. 7. To investigate whether two tests really measure the same psychological function when they have a nonlinear relationship.

8. 8. To describe and evaluate the properties of a specific test considered as a measuring instrument.

The model has been tested empirically, using it to estimate bivariate distributions from univariate distributions, with good results, as checked by chi-square tests.

(1) An analysis of the method of paired comparisons shows that Case V and Case VI, the latter characterized by log-normal distributions and Weber’s law for subjective continua, are fundamentally indistinguishable. Case VI produces a log-arithmetic interval scale of subjective magnitude. (2) It is demonstrated that the difference between discrimination scales according to Case V and from category rating is due to the difference between intra- and interindividual variability yielding different Weber functions.

This paper considers the problem of computing estimates of factor loadings, specific variances, and communalities for a factor analytic model. The equations for maximum-likelihood estimators are discussed. Iterative formulas are developed to solve the maximum-likelihood equations and a simple and efficient method of implementation on a digital computer is described. Use of the iterative formulas and computing techniques for other estimators of factor loadings and communalities is also considered to provide a very general approach for this aspect of factor analysis.

A method of generating any number of score and correlation matrices with arbitrary population parameters is described. Either Z scores or stanines are sampled from a normal population to represent factor scores by an IBM 1620 program. These are converted to variates from a population with an a priori factor structure. The effectiveness of the method is illustrated from research data. Some further modifications and uses of the method 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.

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.

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 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.

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.

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.

Certain questions about the shape of item-test regressions are answered on the basis of data obtained from 103,275 examinees. Ten of the regressions obtained are shown in illustrative plots.