Finding the greatest lower bound for the reliability of the total score on a test comprising n non-homogenous items with dispersion matrix Σx is equivalent to maximizing the trace of a diagonal matrix ΣE with elements θI, subject to ΣE and ΣT=Σx − ΣE being non-negative definite. The cases n=2 and n=3 are solved explicity. A computer search in the space of the θi is developed for the general case. When Guttman's λ4 (maximum split-half coefficient alpha) is not the g.l.b., the maximizing set of θi makes the rank of ΣT less than n − 1. Numerical examples of various bounds are given.

Tucker has outlined an application of principal components analysis to a set of learning curves, for the purpose of identifying meaningful dimensions of individual differences in learning tasks. Since the principal components are defined in terms of a statistical criterion (maximum variance accounted for) rather than a substantive one, it is typically desirable to rotate the components to a more interpretable orientation. “Simple structure” is not a particularly appealing consideration for such a rotation; it is more reasonable to believe that any meaningful factor should form a (locally) smooth curve when the component loadings are plotted against trial number. Accordingly, this paper develops a procedure for transforming an arbitrary set of component reference curves to a new set which are mutually orthogonal and, subject to orthogonality, are as smooth as possible in a well defined (least squares) sense. Potential applications to learning data, electrophysiological responses, and growth data are indicated.

A variety of distributional assumptions for dissimilarity judgments are considered, with the lognormal distribution being favored for most situations. An implicit equation is discussed for the maximum likelihood estimation of the configuration with or without individual weighting of dimensions. A technique for solving this equation is described and a number of examples offered to indicate its performance in practice. The estimation of a power transformation of dissimilarity is also considered. A number of likelihood ratio hypothesis tests are discussed and a small Monte Carlo experiment described to illustrate the behavior of the test of dimensionality in small samples.

Three different software programs which contain hierarchical agglomerative cluster analysis procedures were shown to generate different solutions on the same data set using apparently the same options. The basis for the differences in the solutions was the formulae used to calculate Euclidean distance. Users of statistical software were warned of terminological confusion concerning cluster analysis.

A two-step weighted least squares estimator for multiple factor analysis of dichotomized variables is discussed. The estimator is based on the first and second order joint probabilities. Asymptotic standard errors and a model test are obtained by applying the Jackknife procedure.

Necessary and sufficient conditions for rotating matrices to maximal agreement in the least-squares sense are discussed. A theorem by Fischer and Roppert, which solves the case of two matrices, is given a more straightforward proof. A sufficient condition for a best least-squares fit for more than two matrices is formulated and shown to be not necessary. In addition, necessary conditions suggested by Kristof and Wingersky are shown to be not sufficient. A rotation procedure that is an alternative to the one by Kristof and Wingersky is presented. Upper bounds are derived for determining the extent to which the procedure falls short of attaining the best least-squares fit. The problem of scaling matrices to maximal agreement is discussed. Modifications of Gower’s method of generalized Procrustes analysis are suggested.

Browne provided a method for finding a solution to the normal equations derived by Mosier for rotating a factor matrix to a best least squares fit with a specified structure. Cramer showed that Browne's solution is not always valid, and proposed a modified algorithm. Both Browne and Cramer assumed the factor matrix to be of full rank. In this paper a general solution is derived, which takes care of rank deficient factor matrices as well. A new algorithm is offered.

Kruskal has proposed two modifications of monotone regression that can be applied if there are ties in nonmetric scaling data. In this note we prove Kruskal's conjecture that his algorithms give the optimal least squares solution of these modified monotone regression problems. We also propose another (third) approach for dealing with ties.

Two simple classes of mastery scores which are suitable for hand calculations are presented for beta-binomial test score distributions combined with linear and cubic referral success. The models provide a simple way to explore the consequences of selecting an arbitrary mastery score. Such assessment would be useful whenever the test user is not willing to post a priori a loss ratio, but wishes to look at the various consequences before aiming at a particular score.

Kaiser presented a method for finding a set of derived orthogonal variables which correlate maximally with a set of original variables. A simpler, more complete derivation of Kaiser's result is given and compared to related types of transformations. The transformation derived here suggests a direct method for finding the orthogonal factor solution which is maximally similar to a given oblique solution.

A scale-invariant index of factorial simplicity is proposed as a summary statistic for principal components and factor analysis. The index ranges from zero to one, and attains its maximum when all variables are simple rather than factorially complex. A factor scale-free oblique transformation method is developed to maximize the index. In addition, a new orthogonal rotation procedure is developed. These factor transformation methods are implemented using rapidly convergent computer programs. Observed results indicate that the procedures produce meaningfully simple factor pattern solutions.

A procedure that utilizes the sample multiple correlation to form a lower bound for the level of predictive precision of a fitted regression equation is suggested. The procedure is shown to yield probability statements which are true at least 100(1−α)% of the time.

Using the assumption of randomly parallel tests and concepts from generalizability theory, three signal/noise ratios for domain-referenced tests are developed, discussed, and compared. The three ratios have the same noise but different signals depending upon the kind of decision to be made as a result of measurement. It is also shown that these ratios incorporate a definition of noise or error which is different from the classical definition of noise typically used to characterize norm-referenced tests.

Parameter estimation for Keats generalization of the Rasch model that takes account of guessing behavior is investigated. It is shown that no minimal sufficient statistics for the ability parameters independent of the difficulty parameters exist. Thus Andersen's conditional inference technique for consistent estimation is not applicable to Keats' model. The notion of separability of the parameters is generalized.

This paper discusses the advantages and problems related to factor analysis by minimizing residuals (minres). It is shown that this method fails if the starting point of iterations is not well chosen. A suitable starting point is suggested.

The role of conditionality in the INDSCAL and ALSCAL procedures is explained. The effects of conditionality on subject weights produced by these procedures is illustrated via a single set of simulated data. Results emphasize the need for caution in interpreting subject weights provided by these techniques.

This paper presents a simple random procedure for selecting subsets of stimulus pairs for presentation to subjects. The resulting set of ratings from the group of subjects allows the construction of a group (average) space through the use of the computer program TORSCA-9 or equivalent programs. The procedure is applied to both real and simulated data. It is found that subjects need only make between 20% and 50% of the usual judgements to reconstruct a reasonably accurate group space.

A general formulation is presented for obtaining conditionally unbiased, univocal common-factor score estimates that have maximum validity for the true orthogonal factor scores. We note that although this expression is formally different from both Bartlett's formulation and Heermann's approximate expression, all three, while developed from very different rationales, yield identical results given that the common-factor model holds for the data. Although the true factor score validities can be raised by a different non-orthogonal transformation of orthogonalized regression estimates—as described by Mulaik—the resulting estimates lose their univocality.

Kaiser's measure of sampling adequacy is applied to a special Spearman matrix and a special q-cluster generalization. The result supports the contention that the measure should be no less than .5 for data to be appropriate for factor analysis.

Three properties of the binormamin criterion for analytic transformation in factor analysis are discussed. Particular reference is made to Carroll’s oblimin class of criteria.