It often happens that a theory specifies some variables or states which cannot be identified completely in an experiment. When this happens, there are important questions as to whether the experiment is relevant to certain assumptions of the theory. Some of these questions are taken up in the present article, where a method is developed for describing the implications of a theory for an experiment. The method consists of constructing a second theory with all of its states identifiable in the outcome-space of the experiment. The method can be applied (i.e., an equivalent identifiable theory exists) whenever a theory specifies a probability function on the sample-space of possible outcomes of the experiment. An interesting relationship between lumpability of states and recurrent events plays an important role in the development of the identifiable theory. An identifiable theory of an experiment can be used to investigate relationships among different theories of the experiment. As an example, an identifiable theory of all-or-none learning is developed, and it is shown that a large class of all-or-none theories are equivalent for experiments in which a task is learned to a strict criterion.

When several test forms are used interchangeably, they will not in practice, despite all efforts, be rigorously parallel. A definition of the standard error of measurement appropriate for this type of situation can be provided; however, it will be different from the definition in classical test theory. Appropriate formulas for the standard error of measurement and for other related quantities under both definitions are derived and compared; also sample statistics for estimating these quantities.

A general framework for obtaining all possible factor analytic solutions, orthogonal and oblique, for a given common factor space is developed in detail. Interestingly, and seemingly paradoxically, any one of these solutions may be obtained by orthogonal transformations of selected matrices; thus an oblique solution may be determined by orthogonal transformations. Within the possible oblique solutions, two distinct categories of solutions emerge, a special case of the simpler of which apparently provides a definitive solution to the problem of independent, but correlated, clusters. Possible further specializations of the general approach to specific problems are discussed.

A ratio scale of subjective magnitude is developed from paired-comparison data. The model attempts to combine arguments of Restle, Ekman, and Luce relating data obtained from paired comparisons and corresponding data from direct psychophysical scaling methods at the ratio level. The basic set-theoretical model involves the use of imperfectly nested sets. A numerical example illustrates the application of the theory.

The N-dimensional geometry of a Spearman-Thurstone factor solution reveals two sources for the indeterminancy of factor scores: indeterminancy of total factor space and a rotational indeterminancy within a given total factor space. The analytical papers of Ledermann [4] and Guttman [2] on indeterminancy of factor scores are related to these findings and a simple vector model is developed to reveal the properties of rotational indeterminancy. The significance of factor-score indeterminancy is discussed in light of these findings.

A method of estimating the product moment correlation from the polychoric series is developed. This method is shown to be a generalization of the method which uses the tetrachoric series to obtain the tetrachoric correlation. Although this new method involves more computational labor, it is shown to be superior to older methods for data grouped into a small number of classes.

A transformation of a set of partial regression equations to a potentially simple form is shown to be permissible for personnel classification purposes when all individuals are assigned to jobs. Simplification is achieved by maximizing the number of near zero partial regression weights.

An interesting problem in linear programming is the group assembly problem which is mathematically equivalent to the general transportation problem of economics. Computer programs designed for the determination of exact and approximate optimal group assemblies have been available for some time. This paper presents formulas for the mean and squared standard deviation of the distribution of all possible group assembly sums. Computational techniques are presented and the results are related to those of the analysis of variance of a k-factor problem with n levels of each factor.

Two small-sample tests (proposed by Tate and Clelland and by Chapanis respectively) of hypotheses about the parameters of the multinomial distribution, where

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(x|p) = n!\prod\limits_{i = 1}^k {\frac{{p_i^{x_i } }}{{x_i !}}} $$\end{document}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R = \frac{{\smallint f(x|p)g(p)dp}}{{f(x|p_0 )}}$$\end{document}