A factor analysis of intra-task performance on two psychomotor tests

There are indications that even during the short time of administration of a single psychomotor test, the ability or abilities sampled may shift materially in importance. It then becomes important to know the stages in which these fluctuations occur, the stage at which the test is most complex, and the stage at which the test most nearly measures one ability at a time. This paper describes an application of factorial methods to this problem. Factor analysis of inter-trial correlations on two models of the Rudder Control Test revealed three factors, Steadiness-Control, Precision of Movement, and Strength. The same factor pattern was confirmed in a separate factor analysis on another sample in which the order of administration of the tests was reversed. Implications are pointed out for future psychomotor test development.Perceptual and Motor Skills Research Laboratory, Lackland Air Force Base, San Antonio, Texas. The opinions or conclusions contained in this report are those of the author. They are not to be construed as reflecting the views or indorsement of the Department of the Air Force.     

General solution of the analysis of variance and covariance in the case of unequal or disproportionate numbers of observations in the subclasses

In this paper a preview of the problem is given. Then the mathematical solutions of estimating the sums of squares and products of different sources of variation under different assumptions are presented. Two kinds of populations from which our samples are supposed to be drawn are specified. One is defined as possessing approximately the same stratification as our sample; while the other is defined as having equal frequencies in the subclasses. For the first kind of population, we should use the restrictions of the weighted means. For the second kind, we should use the restrictions of the unweighted means. The assumptions of zero interactions and significant interactions are also considered. After working out the exact method, two approximate methods with appropriate statistical assumptions to be fulfilled are given.For a more complete account, see:Fei Tsao, General solution of the analysis of variance and covariance in the case of unequal or disproportionate numbers of observations in the subclasses. Ph.D. Thesis, University of Minnesota, 1945. Pp. 120.     

An I.B.M. technique for the computation of ΣX 2 and ΣXY

Given I.B.M. cards punched with scores (or any numbers)—but not their squares—a method is presented of tabulating them (on the No. 405 alphameric I.B.M. tabulator) so as to obtain the sum of squares. The technique is also adaptable to summation of cross-products. The principle is an extension of the Mendenhall-Warren-Hollerith technique of vertical progressive digiting, without the necessity of manual addition or summary-punching, and is designed for machines not equipped with the card cycle total transfer device or progressive total device. Use is made of counter rolling. Efficient use of machine capacity is made only when intercorrelations between no more than two variables are requiredin addition to sums of squares. A resumé of some techniques now commonly employed is included.The author is indebted to Dr. Paul Dwyer, Associate Professor of Mathematics, University of Michigan, for valuable criticism of the original draft; and to Mr. Alan Meacham, in charge of the University's Tabulating Station, for testing the method.     

Synthesis of variance

The distribution of a linear combination of two statistics distributed as is Chi-square is studied. The degree of approximation involved in assuming a Chi-square distribution is illustrated for several representative cases. It is concluded that the approximation is sufficiently accurate to use in many practical applications. Illustrations are given of its use in extending the Chi-square, the Student t and the Fisher z tests to a wider range of problems.     

Points of neutrality in social attitudes of delinquents and non-delinquents

Delinquent boys are compared with non-delinquents with respect to their attitudes towards a series of good and bad social acts, by the use of scales having rational origins of measurement. A new technique, essentially an extension of Thurstone's Method of Successive Intervals, is found to give results similar to Horst's Method of Balanced Values. Significant differences in mean attitude between the two groups are not found.The author wishes to thank Dr. M. W. Richardson for his invaluable encouragement and counsel in this study. This paper is a part of a dissertation accepted by the faculty of the Department of Psychology, The University of Chicago, for the degree of Doctor of Philosophy.     

A technique for facilitating the rotation of factor axes,based on an equivalence between persons and tests

A technique is outlined which may facilitate the rotation of factor axes to a meaningful position. It is based on certain relationships between the results of test and person factor analysis, and consists essentially of supplementing the test factor space with tests which are thetest-equivalents of persons or groups of persons. These persons may be, for instance, well-known types in the domain being investigated, or even freaks. The ways in which these persons may be selected and used to determine the final rotated position of the factor axes is discussed.     

Alternative views on correlations between persons

The article discusses the general principles involved in factorizing correlations between persons, and compares the special techniques put forward by Burt and Stephenson respectively. The chief points of agreement and disagreement are summarized. In statistical procedure the two writers prove to be largely in accord; but they differ over the psychological and the methodological implications, the chief source of divergence being the so-called reciprocity principle.     

A re-analysis of a test of the theory of two factors

The study of William Brown and William Stephenson, A Test of the Theory of Two Factors, is re-analyzed by means of the Thurstone multiple factor methods. No tests or correlations are left out of the original table of correlations as is done in the original analysis in an attempt to validate the two-factor theory. Space, verbal, and perceptual speed factors similar to those found by Thurstone, Wright, and Garrett are identified. A common factor of Maturation is postulated to account for the remaining communality of the tests. A fifth factor is considered to have no significance due to the small amount of variance which it contributes to the total.Master's Thesis, University of Chicago, Department of Psychology, 1939.     

A note on correcting for chance success in objective tests

The conventional scoring formula to correct for guessing is derived and is compared with a regression method for scoring which has been recently proposed by Hamilton. It is shown that the usual formula,S=R–W/(n–1), yields a close approximation (correct within one point) to the maximum-likelihood estimate of an individual's true score on the test, if we assume that the individual knows or does not know the answer to each item, that guessing at unknown items is random, and that success at guessing is governed by the binomial law. It is also shown that the usual scoring formula yields an unbiased estimate of the individual's true score, when the true score is defined as the mean score over an indefinitely large number of independent attempts at the test or at equivalent (parallel) tests.     

Contributions to the mathematical theory of human relations III

In continuation of a previous study, somewhat more complex relations are considered for the case of interaction of two classes, of which one organizes the activities of the other. By way of illustration it is shown how different assumptions concerning the attitudes of the individuals to such an interaction of classes lead to different mathematical expressions.     

Contribution to the mathematical theory of human relations: VI. Periodic fluctuations in the behavior of social groups

A case of interaction between two groups of active and one group of passive individuals, in which the efforts of the influencing active groups decrease with increasing total success in the past, is studied. In that case the numbers of passive individuals, exhibiting respectively the two opposite behaviors, fluctuate periodically, with a positive damping.     

A technique for criterion-keying and selecting test items

For multiple-choice tests where noa priori key exists, the initial selection of a key for maximum validity may be made on the basis of the number of persons choosing each alternative and their mean criterion score. The keying formula is derived. Once the initial keying has been done, further precision in keying and item selection may use, in addition, the mean total test score for persons choosing each alternative. Item-selection formulas suggested by Horst and by Gulliksen for maximizing test validity are both in the form of a ratio, an item-validity index divided by an item-reliability index. The formula derived here is shown to be equivalent to the numerators of these formulas. The expression in the denominators uses the total test score. Although a radical appears in the denominator of Horst's formula and not in the denominator of Gulliksen's formula, both of them select the same items in practice.The author gratefully acknowledges the suggestions and criticisms of Dr. Harold Gulliksen, Research Adviser at the Educational Testing Service.     

A factor analysis of mechanical ability tests

The intercorrelations of thirty-seven variables, including the Minnesota battery of mechanical ability tests, the seven MacQuarrie tests of mechanical ability, O'Connor's Wiggly blocks, and the Stenquist picture-matching test, were analyzed by Thurstone's centroid method. Five factors, Perceptual, Verbal, Youth, Manual Agility, and Spatial, were taken out. Factors prominent in so-called mechanical ability tests are the Spatial and Perceptual ones with Mac-Quarrie's dotting test significantly high in the Manual Agility factor. Each of the factors can be measured with group pencil-and-paper tests.Acknowledgment is gratefully made to the State Engineering Experiment Station at the Georgia School of Technology for sponsoring and financially supporting the studies; to the Graduate Research Committee of the University of Illinois for providing funds for the purchase of tests and the tabulation of data; and to Dr. E. L. Welker of the University of Illinois Mathematics Department for assistance with statistical problems.     

On the mutual influence of individuals in a social group

A previous mathematical study of a situation, in which the behavior of a larger group of individuals is controlled by a smaller group, is generalized for the case when the activity of the individuals in the group is continuously graded. The existence of two possible social configurations and of sudden transitions from one configuration to another are found in this case also.     

An analytical solution for approximating simple structure in factor analysis

It is proposed that a satisfactory criterion for an approximation to simple structure is the minimization of the sums of cross-products (across factors) ofsquares of factor loadings. This criterion is completely analytical and yields a unique solution; it requires no plotting, nor any decisions as to the clustering of variables into subgroups. The equations involved appear to be capable only of iterative solution; for more than three or four factors the computations become extremely laborious but may be feasible for high-speed electronic equipment. Either orthogonal or oblique solutions may be achieved. For illustrations, the Johnson-Reynolds study of flow and selection factors and the Thurstone box problem are reanalyzed. The presence of factorially complex tests produces a type of hyperplanar fit which the investigator may desire to adjust by graphical rotations; the smaller the number of such tests, the closer the criterion comes to approximating simple structure.     

Note on the relationship between internal consistency and test-retest estimates of the reliability of a test

A comparison of the test-retest and internal consistency estimates of reliability coefficients is given, and it is shown that the two methods give different results. Application of the analysis of variance and covariance method reveals that there is not just one but a number of reliability coefficients involved, and that an estimate of each of these may be obtained. The analysis shows that in using the test-retest method the error or remainder effects are not independent on the two trials, possibly because the individuals remember the items and their responses to them on the previous trial.     

The contribution of an orthogonal multiple factor solution to multiple correlation

A method is indicated by which multiple factor analysis may be used in determining a number,r, and then in selectingr predicting variables out ofn variables so that each of the remainingn-r variables may be predicted almost as well from ther variables as it could be predicted from all then—1 variables.     

TheL-method

It is shown that theL-method is basic to test-building and to all combining of scores where the several sub-parts of a composite are weighted according to their respective standard deviations, i.e., with equal gross score weights or by adding the several sub-scores. With chiefly a listing adding machine, a few celluloid strips, and a master matrix table ofL's, one may, in a fraction of the time and with equally good or even better practical results, easily duplicate, with test-building material most of the feats obtainable by multiple regression equations.     

“Parallel proportional profiles” and other principles for determining the choice of factors by rotation

The choosing of a set of factors likely to correspond to the real psychological unitary traits in a situation usually reduces to finding a satisfactory rotation in a Thurstone centroid analysis. Seven principles, three of which are new, are described whereby rotation may be determined and/or judged. It is argued that the most fundamental is the principle of parallel proportional profiles or simultaneous simple structure. A mathematical proof of the uniqueness of determination by this means is attempted and equations are suggested for discovering the unique position.