A distinction is drawn between the method of principal components developed by Hotelling and the common factor analysis discussed in psychological literature both from the point of view of stochastic models involved and problems of statistical inference. The appropriate statistical techniques are briefly reviewed in the first case and detailed in the second. A new method of analysis called the canonical factor analysis, explaining the correlations between rather than the variances of the measurements, is developed. This analysis furnishes one out of a number of possible solutions to the maximum likelihood equations of Lawley. It admits an iterative procedure for estimating the factor loadings and also for constructing the likelihood criterion useful in testing a specified hypothesis on the number of factors and in determining a lower confidence limit to the number of factors.

Sampling fluctuations resulting from the sampling of test items rather than of examinees are discussed. It is shown that the Kuder-Richardson reliability coefficients actually are measures of this type of sampling fluctuation. Formulas for certain standard errors are derived; in particular, a simple formula is given for the standard error of measurement of an individual examinee's score. A common misapplication of the Wilks-Votaw criterion for parallel tests is pointed out. It is shown that the Kuder-Richardson formula-21 reliability coefficient should be used instead of the formula-20 coefficient in certain common practical situations.

Fellow members of the Psychometric Society, and our colleagues, members of Division 5 of the American Psychological Association, I understand my mission tonight is to provide you with both humor and a mild message. Precedent has established an eminently high standard. Maybe I can explain the humbleness with which I approach this task by relating the account of an episode. This occurrence, like much illustrative data in Psychometrika, is fictitious, of course.

During my trip from the splendorous East Coast to the beauteous West Coast, I took vacation time to tour part of the magnificent country in between, as many of you, undoubtedly, did also. 0he afternoon I came upon a small tent village in a fairly remote mountain section. Not that such camps were so uncommon as to be remarkable in themselves; something about this particular camp caught my eye. In the center of this camp was a rustic conference table flanked on one side by a large blackboard mounted on cedar posts.

Two systems of factor analysis—factoring correlations with units in the diagonal cells and factoring correlations with communalities in the diagonal cells—are considered in relation to the commonly used statistical procedure of separating a set of data (scores) into two or more parts. It is shown that both systems of factor analysis imply the separation of the observed data into two orthogonal parts. The matrices used to achieve the separation differ for the two systems of factor analysis.

New formulas are developed to give lower bounds to the reliability of a test, whether or not all respondents attempt all items. The formulas apply in particular, then, to completed tests, pure speed tests, pure power tests, and any mixture of speed and power. For the case of completed tests, the formulas give the same answer as certain standard ones; for noncompleted tests the formulas give a correct answer where previous standard formulas are inappropriate. The formulas hold both in the sense of retest reliability and of parallel tests.

By a simplex is meant a set of statistical variables whose interrleations reveal a simple order pattern. For the case of quantitative variables, an order model was analyzed previously which allowed only for positive correlations among the variables and a limited type of gradient among the correlation coefficients. The present paper analyzes a more general model and shows how it is more appropriate to empirical data. Among the novel features emerging from the analysis are: (a) the “factoring” implied of the correlation matrix; (b) the use of a non-Euclidean distance function; and (c) the possible underlying psychological theories.

It is postulated that occurrence of a conditioned response depends on recurrence of one of a finite number of specific vigilance reactions. Number of trial on which a conditioned response occurs is shown to be a sufficient statistic for estimating the number of such vigilance reactions. The hypothesis is tested by noting whether numbers of trials on which conditioned responses occur fall within confidence intervals determined on the basis of a selected sufficient statistic. Applications of the model to psychological research are suggested.

This article presents a survey of the assumptions which may be made in variance designs, a description of the mathematical models which reflect these assumptions, and a discussion of the ways in which various experimental conditions affect the choice of an error mean square. Particular emphasis is laid upon the principles, purposes, and dangers of pooling error mean squares in order to raise the power of a test. Specific recommendations are made for the rules of procedure for pooling (under various conditions) which produce tests with optimum power and error characteristics.

In an earlier paper, a method of analysis, due to Neyman and now known generally as variance component analysis, was used to examine F-test bias for experimental designs in education of the randomized block type. The same method is now applied to study F-test bias for designs of the Latin square type. The results, in general, disprove the view that, for a valid application of Latin square techniques, it is necessary that all interactions are zero.

Certain problems of equating are discussed. The maximum likelihood solution is presented for the following special equating problem: Two tests, U and V, are to be equated, making use of a third “anchor” test, W. The examinees are divided into two random halves. Tests U and W are administered to one half; tests V and W are administered to the other half. It is assumed that any practice effect or other effect, exerted by U and V on W, is the same for U and for V.

Analysis of an empirical theory into a formal system with specified primitive notions and axioms has the advantage of making it clear what deductions from the theory are permissible, and clarifying the internal structure of the theory. An example of such analysis is presented in this paper.

A rational function is developed relating the length of a rest period and length of subsequent work period in an ergographic situation. Simple energistic postulates are used for a critical organ or neuromuscular structure whose failure to perform adequately results in a stoppage of the work period. Experimental results for two subjects using a finger ergograph indicate that the function yields the general trend of the data but that there seem to be some systematic deviations of the data from the present rational function. One parameter determined from the data represents rate of recovery from moderate fatigue. It is hoped that this development will aid in studies of motor functions as related to such other variables as age, motivation, and effects of drugs.

A study by Shaw (7) some twenty years ago is frequently cited by social scientists to support the generalization that groups are superior to individuals in problem-solving. Shaw suggests that personal interaction within the group is responsible for the superior performance of groups. This article re-examines her data in the light of two models which propose that the difference in quality of solution between group and individual performance is solely a matter of ability. It is shown that Shaw's data may be considered to have been an outcome of behavior postulated by the models. Since Shaw's observations relate to a special population and to special kinds of problems, the proposed models may not be appropriate under differing experimental conditions. In fact, Lorge et al. (4) have indicated that experimental demonstration of the superiority of groups over individuals in problem-solving depends not only on the kind of group but also on the kind of problem to be solved. In addition, the diversity of transfer of training for groups and for individuals is considered.

Analogs of Pearson's coefficient of racial likeness and of Mahalanobis' distance measure have been proposed as descriptive statistics for comparing two individuals. This paper shows that two different definitions of “uncorrelated” variables—one associated with an inverse transformation and the other with a principal-axis transformation—give rise to these two descriptive statistics. The effects of putting the data into certain forms, such as equalizing the variances of the variables or equalizing the means of the persons, prior to using either of the two transformations, are discussed.

Requirements for an objective definition of simple structure are investigated and a number of proposed objective criteria are evaluated. A distinction is drawn between exploratory factorial studies and confirmatory factorial studies, with the conclusion drawn that objective definition of simple structure depends on study design as well as on objective criteria. A proposed definition of simple structure is described in terms of linear constellations. This definition lacks only a statistical test to compare with possible chance results. A computational procedure is also described for searching for linear constellations. This procedure is very laborious and might best be accomplished on high-speed automatic computers. There is no guarantee that the procedure will find all linear constellations, but it probably would yield satisfactory results for well-designed studies.

A coefficient of interrelationship between overlapping groups that avoids indeterminacies inherent in the construction of fourfold tables for such purposes and, at the same time, is relatively insensitive to the absolute magnitude of marginal totals of fourfold tables, is developed. Under assumptions that are consistent with the objectives of organizational analysis, this coefficient is shown to be equivalent to a product-moment correlation coefficient. The advantages and limitations of this coefficient are pointed out. A numerical example giving computational procedures is presented.

A new algebraic formula is derived for estimation of the discriminal dispersion in the method of successive intervals. The legitimate use of the formula requires that as many normal deviates as possible be present in the matrix. For this reason, it is recommended that deviates corresponding to the interval (0.01, 0.99) of the cumulative proportions be used, instead of those corresponding to (0.05, 0.95), the interval used by Edwards and Thurstone. Computations on data published by Edwards and Thurstone showed that when adjustment was made for variability in dispersions calculated by the formula of this paper, a reduction of fifty per cent in mean absolute discrepancy was produced. Since the formula is easy to use and avoids the disadvantages of its predecessors, it should have fairly wide applicability in psychological research.

In principle it is possible to design automata to display any explicitly described behavior. The McCulloch-Pitts “neuron” is a convenient elementary component for the control mechanisms of automata. Previously described techniques permit the design of an automaton which would arbitrarily well simulate human behavior. The difficulty of producing such a design lies primarily in formulating an explicit description of the required behavior. The control mechanism of such an automaton would be of very great logical complexity. Its mode of operation probably would not resemble that of a human brain. The brain is more plausibly represented by stochastic models as proposed by Hebb. Such models can more easily be designed or understood by reason of lesser logical complexity. A method of computational investigation of the functioning of such stochastic models is described. Several extremely simple models have been investigated. One is shown to have properties suggestive of learning ability.