Through an extension of work by Guttman, common factor theory, image theory, and component theory are derived from distinct minimum subsets of assumptions chosen out of a set of five possible assumptions. It is thence shown that the problem of indeterminacy of factor scores in the common factor model is precisely reflected in the problem of the non-orthogonality of anti-images. Indeed, image scores are determinate for the same reason that the usual estimates of factor scores are determinate, and image scores cannot be used as though they were factor scores for the same reason that factor score estimates cannot be used as though they were factor scores.

McNemar’s problem concerns the hypothesis of equal probabilities for the unlike pairs of correlated binary variables.

We consider four different extensions to this problem, each for testing simultaneous equality of proportions of unlike pairs in c independent populations of correlated binary variables, but each under different assumptions and/or additional hypotheses. For each extension both the likelihood ratio test and the goodness-of-fit chi-square test are given. When c = 1, all cases reduce to McNemar’s problem. Forc ≥ 2, however, the tests are quite different, depending on exactly how the hypothesis and alternatives of McNemar are extended.

An example illustrates how widely the results may differ, depending on which extended framework is appropriate.

A simple method of monotone regression is described based on the principle of minimizing pairwise departures from monotonicity.

Following a general introduction to Tukey’s Jackknife technique and the construction of approximate confidence intervals, ten variants of the procedure are defined for giving interval estimates for coefficient alpha. A data base of known population characteristics was generated and used to compare the robustness of the ten Jackknife alternatives with Feldt’s well-known sampling theory based upon the assumptions of normality. The empirical results indicate that out of the ten variants, defined by five transformations and two methods of data subdivision, only two are justifiable competitors to Feldt’s approach.

Kaiser’s iterative algorithm for the varimax rotation fails when (a) there is a substantial cluster of test vectors near the middle of each bounding hyperplane, leading to non-bounding hyperplanes more heavily overdetermined than those at the boundaries of the configuration of test vectors, and/or (b) there are appreciably more than m (m factors) tests whose loadings on one of the factors of the initial F- matrix, usually the first, are near-zero, leading to overdetermination of the hyperplane orthogonal to this initial F- axis before rotation. These difficulties are overcome by weighting the test vectors, giving maximum weights to those likely to be near the primary axes, intermediate weights to those likely to be near hyperplanes but not near primary axes, and near-zero weights to those almost collinear with or almost orthogonal to the first initial F- axis. Applications to the Promax rotation are discussed, and it is shown that these procedures solve Thurstone’s hitherto intractable “invariant” box problem as well as other more common problems based on real data.

Using an approach nearly identical to one adopted by Guttman, it is established that within the framework of classical test theory the squared multiple correlation for predicting an element of a composite measure from the n − 1 remaining elements is a lower-bound to the reliability of the element. The relationship existing between the reliabilities of the elements of a composite measure and their squared-multiple correlations with remaining elements is used to derive Guttman’s sixth lower bound (λ6) to the reliability of a composite measure. It is shown that Harris factors of a correlation matrix R are associated with a set of (observable) uncorrelated latent variables having maximum coefficients λ6.

For the six available sets of empirical data, the discrimination (slope) parameter of the logistic item characteristic curve was found to have a significant positive correlation over items with the difficulty (location) parameter. This unpleasant situation can be eliminated by a suitably chosen transformation of the ability scale. An example of the transformation is given and some empirical results of using the transformed scale are shown.

The small-world problem revolves around the tracing of a line of acquaintances linking any two persons chosen at random. This paper discusses the statistical analysis of data arising out of two experimental studies of the small-world problem. The estimation of parameters in two previously proposed models is discussed, and the goodness-of-fit of these models is considered.

A method for dealing with the problem of missing observations in multivariate data is developed and evaluated. The method uses a transformation of the principal components of the data to estimate missing entries. The properties of this method and four alternative methods are investigated by means of a Monte Carlo study of 42 computer-generated data matrices. The methods are compared with respect to their ability to predict correlation matrices as well as missing entries.

The results indicate that whenever there exists modest intercorrelations among the variables (i.e., average off diagonal correlation above .2) the proposed method is at least as good as the best alternative (a regression method) while being considerably faster and simpler computationally. Models for determining the best alternative based upon easily calculated characteristics of the matrix are given. The generality of these models is demonstrated using the previously published results of Timm.

McDonald [1974] obtained Maximum Likelihood (ML) estimates of the free parameters, and an asymptotic likelihood-ratio test, for the hypothesis that one or more elements of a covariance matrix are zero, and/or that groups of two or more of its elements are equal. Estimation was by Newton's method, starting from a closed-form Least Squares (LS) solution that is typically close to the ML solution point. The hypothesis can also be tested using the general model for the analysis of covariance structures given by Jöreskog [1970], but the combination of a closed-form LS starting point and the classical Newton method given by McDonald yields estimates in about a quarter of the computer time needed by JOreskog's program, ACØVS.