This paper explores certain problems which arise within the context of the theory of generalizability put forward by Cornbach, Rajaratnam, and Gleser. In particular, a formal explication of their theory for the single observation is given, and the various coefficients of generalizability which they define are related to the estimation of universe scores.

After I had started to put some of my thoughts into a first draft for this talk, I selected “The Error” as a title. This selection gave me a great deal of flexibility since, no matter how I went wrong, I was still well within the limits of the title. After additional deliberation, I had more than a slight feeling that the title was more appropriate for a potential best-seller in the field of popular fiction than it was for a potential presidential address to this society. I have, on occasion, thought that I could write–in fact, I started my academic life in a School of Journalism. At least at the academic level, the journalistic style of writing rather quickly suppressed my interest therein as a means of making a livelihood.

Errors, random and otherwise, are often the heart of some rather undercover plots and subplots in best-sellers. But just as in the case of novels, in experimental designs errors also play a dominant role–unfortunately the experimental units, which are the players, often do not understand their roles as set forth in the design script. The latter is usually in the form of a mathematical model. A question might be raised in some instances as to whether it is more difficult to teach the experimental units what the model calls for, or to teach the designer what types of designs are potentially appropriate for expressing the behavior of experimental units, under specified experimental conditions.

It is customary in multivariate analysis to search out relational structure by partitioning measures of total variation or overall relatedness into components variously attributable to different sources. Although the multivariate systems most familiar today–notably, linear correlational analysis, analysis of variance, and Information theory–base their decompositions upon very different substantive quantities, they all embody the very same abstract mathematical structure which, moreover, is capable of generating potentially fruitful patterns of data analysis in many other substantive applications as well.

Classical item analysis procedures were developed for dichotomously scored items and do not apply to items allowing multiple correct responses. Maximum likelihood procedures analogous to those employed in polychotomous bio-assay are presented which yield estimates of the sets of parameters for items having multiple nonordered responses. Expressions for the estimates of the asymptotic variances of the item parameters and on overall chi-square goodness of fit test are also provided.

Statistical properties of several methods for obtaining estimates of factor loadings and procedures for estimating the number of factors are compared by means of random sampling experiments. The effect of increasing the ratio of the number of observed variables to the number of factors, and of increasing sample size, is examined. A description is given of a procedure which makes use of the Bartlett decomposition of a Wishart matrix to generate random correlation matrices.

A least squares method for approximating a given symmetric matrix B by another matrix B which is orthogonally similar to a second given matrix A is derived and then generalized to nonsymmetric (but square) A and B. A possible application to ordering problems is discussed.

When knowledge of the response sequence permits one to specify the sequence of theoretical states that occurred, we propose to say that the theory has observable states. Our earlier usage of “identifiable” conflicts with established usage in econometrics. We also present a revised definition of implication between theories that avoids a technical flaw in the earlier definition, and we correct a mistake in an example in our earlier paper.

A Markov chain with transition probabilities pij(θ) that are functions of a parameter vector θ is defined. One of t input values is delivered to the chain on each trial. Under the hypothesis H0 the parameter vector is independent of the input; under the hypothesis H1 the vector is in general different for each different input. A likelihood ratio test for a single observation on a chain of great length is given for testing H0 against H1, given that the distribution of the inputs depends at most on the previous input and the present state of the chain. The test is therefore one of stationarity of the transition probabilities against a specific alternative form of nonstationarity. Application of the approach to a statistical test of lumpability of the states of a chain is indicated. Tests for other related hypotheses are suggested. Application of the test to “within-subject effects” for individual subjects is considered. Finally, some applications of the results in psychophysical contexts are suggested.

An alpha-O coefficient of internal consistency is defined for an observed score composite. Maximizing alpha-O leads to a system of psychometric (vs. statistical) factor analysis in which successive factors describe dimensions of successively less internal-consistency. Factoring stops when alpha-O is zero or less. In contrast to Kaiser-Caffrey's alpha-C analysis, when the factored matrix is rank 1, alpha-O does not reach unity; it can approach unity only as the number of variables reach infinity. The relative usefulness and domains of generalization of alpha-C and alpha-O are compared. Basically, alpha-C analysis is concerned with the representativeness of factors while alpha-O analysis is concerned with the assessibility of factors. Consequently, either system of factoring can and should be summarized by both the alpha-C and alpha-O coefficients. Not surprisingly, alpha-O analysis is computationally analogous to Rao's canonical factor analysis.

The effects of N and communality on the variability of zero and nonzero factor loadings were assessed using a Monte Carlo approach. It was found that increasing N or communality resulted in decreased sampling error of individual factor loadings, but for zero loadings N was found to have the greatest influence. It was also found that distributions of factor loadings become relatively elongated as communality increases.

Usually, an iterative procedure based on two-dimensional rotations is employed to find the varimax solution in factor analysis. A matrix is given where this procedure does not yield the maximum value of the varimax criterion. However, random orthogonal transformations of some matrices and subsequent varimax-rotation using the iterative procedure seem to indicate that usually no local maxima exist.

A model is considered for learning where there is an initial stage with only errors, an intermediate state with errors and correct responses, and an absorbing state with only correct responses. A model with observable states is constructed, and sufficient identifiable parameters are given for the original model. Distribution statistics and other properties of data are derived as functions of the identifiable parameters. Informal estimates and sufficient statistics are given for the identifiable parameters. A number of testable hypotheses about the theoretical parameters are described. Some identifying restrictions for the model are given, along with the estimates for the unrestricted free parameters which follow. An application is made to data from a pairedassociate learning experiment.

Previous results of the application of Lawley's selection theorem to the common factor analysis model are extended to a revision of Tucker's three-mode principal components model. If the regression of the three-mode manifest variates on variates used to select subpopulations is both linear and homoscedastic, the two factor pattern matrices, the core matrix, and the residual variance-covariance matrix in the three-mode model can all be assumed to be invariant across subpopulations. The implication of this finding for simple structure is discussed.

Let θ > 0 be a measure of the average step size of a stochastic process {pn(θ) }n=1(∞). Conditions are given under which pn(θ) is approximately normally distributed when n is large and θ is small. This result is applied to a number of learning models where θ is a learning rate parameter and pn(θ) is the probability that the subject makes a certain response on the nth experimental trial. Both linear and stimulus sampling models are considered.

For the case of the one-element Markov learning model for which the guessing parameter p is assumed known the efficiency of Bower's method of moments estimator c+ for the learning parameter, c, is found. It seems that although c+ is not efficient, for some practical value of c it is not very inefficient. If p and c are small, ĉ, the maximum likelihood estimate, and c+ have approximately the same first term when expanded in powers of p.

The paper presents a general framework for the dependent factor method in which judgmental as well as analytic criteria may be employed. The procedure involves a semi-orthogonal transformation of an oblique solution comprising a number of reference factors and a number of experimental factors (composites). It determines analytically the residual factors in the experimental field keeping the reference field constant. The method is shown to be a generalization of the multiple factor procedure [Thurstone, 1947] in so far as it depends on the use of generalized inverse in deriving partial structure from total pattern. It is also shown to provide an example of the previously empty category (Case III) of the Harris-Kaiser generalization [1964]. A convenient computational procedure is provided. It is based on an extension of Aitken's [1937] method of pivotal condensation of a triple-product matrix to the evaluation of a matrix of the form H − VA−1U' (for a nonsingular A).

A general procedure is described for obtaining weighted linear combinations of variables. This includes as special cases, multiple regression weights, canonical variate analysis, principal components, maximizing composite reliability, canonical factor analysis, and certain other well-known methods. The general procedure is shown to yield certain desirable invariance properties, with respect to transformations of the variables.

It is shown that several variants of the Tucker-Messick points of view procedure are interpretable in terms of their “idealized individual” concepts. It is contended that the main function of the method is that it allows the computation of several vectors of mean judgments instead of only one and that each of these may be used to stand for the actual judgments of a subgroup of subjects. Vectors that do not correspond to the judgments of any subjects are to be avoided. It is alleged that Ross's interpretation of “points of view” is quite different from that intended by Tucker and Messick.

This paper describes a method of matrix decomposition which retains the ability of factor analytic techniques to summarize data in terms of a relatively low number of coordinates; but at the same time, does not sacrifice the useful analysis of variance heuristic of partitioning data matrices into independent sources of variation which are relatively simple to interpret. The basic model is essentially a two-way analysis of variance model which requires that the matrix of interaction parameters be decomposed by using factor analytic techniques. Problems of judging statistical significance are discussed; and an illustrative example is presented.