Factorial results are affected by selection of subjects and by selection of tests. It is shown that the addition of one or more tests which are linear combinations of tests already in a battery causes the addition of one or more incidental factors. If the given test battery reveals a simple structure, the addition of tests which are linear combinations of the given tests leaves the structure unaffected unless the number of incidental factors is so large that the common factors become indeterminate.

A study is made of the extent to which correlations between items and between tests are affected by the difficulties of the items involved and by chance success through guessing. The Pearsonian product-moment coefficient does not necessarily give a correct indication of the relation between items or sets of items, since it tends to decrease as the items or tests become less similar in difficulty. It is suggested that the tetrachoric correlation coefficient can properly be used for estimating the correlation between the continua underlying items or sets of items even though they differ in difficulty, and a method for correcting a 2 × 2 table for the effect of chance is proposed.

There are a number of methods of factoring the correlation matrix which require the calculation of a table of residual correlations after each factor has been extracted. This is perhaps the most laborious part of factoring. The method to be described here avoids the computation of residuals after each factor has been computed. Since the method turns on the selection of a set of constellations or clusters of test vectors, it will be called a multiple group method of factoring. The method can be used for extracting one factor at a time if that is desired but it will be considered here for the more interesting case in which a number of constellations are selected from the correlation matrix at the start. The result of this method of factoring is a factor matrix F which satisfies the fundamental relation FF'=R.

The development of a criterion suitable for testing the significance of a correlation or regression coefficient is used as an illustration of the manner in which a research problem is bound to the selection of the particular data appropriate to collect and a fitting type of statistical analysis of the latter. The translation of the original inquiry into a problem of “testing linear hypotheses” is the means by which these two aspects of an investigation are held together. This presentation is offered as a plan which might be useful for some research workers in determining appropriate criteria for testing their particular hypotheses.

It is shown that a “second-order” factor pattern is equivalent to the transformation employed in rotating an orthogonal factor pattern to an oblique form. The correlation among the second-order factors may then be interpreted as due to the original first-order factors.

Under assumptions that will hold for the usual test situation, it is proved that test reliability and variance increase (a) as the average inter-item correlation increases, and (b) as the variance of the item difficulty distribution decreases. As the average item variance increases, the test variance will increase, but the test reliability will not be affected. (It is noted that as the average item variance increases, the average item difficulty approaches .50). In this development, no account is taken of the effect of chance success, or the possible effect on student attitude of different item difficulty distributions. In order to maximize the reliability and variance of a test, the items should have high intercorrelations, all items should be of the same difficulty level, and the level should be as near to 50% as possible.

This paper is concerned with a central concept in social measurement such as opinion polls, the measurement of attitudes, the prediction of political elections, the measurement of moral values, the measurement of consumer preferences, the measurement of utility, and the measurement of aesthetic values. The concept is that of the discriminal dispersion and its interesting effects in the prediction of choice.

Approximate methods of solving for discriminant functions have been tried on three sets of data. The principal illustration is the problem of finding a weighted sum of scores, on four psychological tests, so that men and women may be distinguished most clearly. The work starts from the complete solution, due to R. A. Fisher, where it is necessary to solve as many simultaneous equations, dependent on the standard deviations of the tests and their mutual correlations, as there are tests. It is proposed, by way of numerical simplification, that a set of equations be substituted where some one quantity replaces all the correlations. A solution is obtained where the weights to be assigned the tests are very simply expressed in terms of differences between the mean values of tests, the standard deviations of tests, and the said quantity. The difficulty remains of finding an estimate of the arbitrary constant that will give good discrimination. If an optimal solution is made a result is obtained which, in the three sets of data considered, is almost indistinguishable from that yielded by the complete solution. The calculation of this optimal common quantity is, however, itself so considerable that another estimate, previously suggested by R. W. B. Jackson, appears more profitable. This estimate is derived simply from the variability between the total scores for each subject and the variability of each test. Using this estimate, the discriminant functions can be rapidly calculated; the results compare very favorably, in the case of the data considered, with those from the complete solution.

I.B.M. tabulating equipment can be of considerable help in reducing the time and increasing the accuracy of multiple factor analysis, even if used for only a part of the calculations. Once the plugboard is wired and those cards punched which are used over and over again, problems involving any number of variables can be handled with dispatch. The correlation matrix is listed, the totals verified, and the signs changed on the tabulator. Then the factors and the residual coefficients are calculated by means of a calculator. Tucker's procedure has been modified by using a calculator instead of a multiplying punch, by reducing the number of cards used, by simplifying checks on calculations, by simplifying plugboard wiring, and by preparing work sheets on tabulator paper. Extraction of factors from 24 variables at the rate of one in four hour's time seems to justify the use of the tabulating equipment on small problems.

The steady state of a simple reaction system has been shown to have some of the properties of a psychophysical discrimination system, including the possibility of deducing a generalized Weber-Fechner Law, both in integral form and in difference form. The Weber ratio so deduced is not constant, and its dependence on stimulus intensity is exhibited. The dependence of the difference limen on the internal threshold is discussed; it is found that in general there is a finite value of this threshold for which response is impossible. This critical threshold is lower for higher values of the reference stimulus intensity. Similarly, it is shown that the difference limen and the Weber ratio, for a fixed value of the threshold, become infinite (i.e., discrimination is impossible) for a value of the stimulus intensity which in general is finite.

Three sources of variation in experimental results for a test are distinguished: trials, persons, and items. Unreliability is defined only in terms of variation over trials. This definition leads to a more complete analysis than does the conventional one; Spearman's contention is verified that the conventional approach—which was formulated by Yule—introduces unnecessary hypotheses. It is emphasized that at least two trials are necessary to estimate the reliability coefficient. This paper is devoted largely to developing lower bounds to the reliability coefficient that can be computed from but a single trial; these avoid the experimental difficulties of making two independent trials. Six different lower bounds are established, appropriate for different situations. Some of the bounds are easier to compute than are conventional formulas, and all the bounds assume less than do conventional formulas. The terminology used is that of psychological and sociological testing, but the discussion actually provides a general analysis of the reliability of the sum of n variables.

A formula for internal consistency reliability is developed within the framework of the analysis of variance. The test items are assumed to be homogeneous, but may have any weights. Data needed for computation are the student test scores, and the total number of items answered so as to have the same weight. It is shown that this formula reduces to the Kuder-Richardson (21) for item weights of one and zero. Some empirical validation is offered.

For the purpose of determining whether the trait concept of “prejudice” is usable in the communication of meaning, representative samples of the responses of 101 ninth-grade children were submitted to a diverse group of 20 judges who were requested to rank 11 series of the responses in accordance with the amount of prejudice they were judged to exhibit. The usability, or use-value, of the concept is conceived as the extent to which the judges can agree in their ratings and is expressed in terms of the average intercorrelation of such ratings. It is shown that a null hypothesis of no use-value (\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tilde \gamma= 0$$\end{document}) is untenable. The data further suggest that the concept of “prejudice” tends to have its highest use-value in situations where the factor of prejudice is commonly considered to be a matter of serious social concern.

This paper describes a simple orthogonal multiple factor approximation procedure that involves no inversion of the sign of negative residuals, the estimation of only as many communalities as there are factors, and none or only a few minor rotations of the axes in an attempt to obtain a “meaningful” solution. It also suggests a technique for the estimation of those communalities that must be estimated. The factor loadings obtained by means of this procedure, which we shall designate as the pre-selection procedure, are affected by the order in which the factors are obtained, showing a reduction in variance accounted for by each successive factor, as is characteristic of the centroid, bi-factor, and principal factor solutions. The entire procedure takes considerably less time than that involved in the orthodox centroid method alone.

Fifty-two subjects differing in sex, age, education and domicile (rural or urban) were given the problem of judging the height of an upright board in a natural setting. A preliminary analysis was made on the basis of the simple initial ratio method, both for the original data in feet and for original data converted to log units. Because the effects of interaction of the several variables made the results of this method inconclusive, the analysis of variance technique, as described by Yates (11) for data where the classes are not equally represented, was applied. This technique showed that, while together the four factors markedly affected judgment, sex had no significant individual effect, age had the biggest individual effect but possibly a spurious one, education and domicile had suspiciously large individual effects, and the effect of the four factors may be regarded as simply additive. The relation of the findings to those of previous investigators is discussed. The authors regard as an important result of the analysis the guidance it offers in the design of further experiments, since it demonstrates the value of equal representation for all classes into which data are to be segregated.

A method is given for determining the reliability of each of the components resulting from a factor analysis by the principal axis method.

For testing the significance of differences between frequencies from different samples, an ellipse can easily be constructed on the basis of a formula developed on the assumption that both observed samples are random samples from the same parent population and that the best estimate of the true proportion is the weighted mean proportion of the two samples. The ellipse provides a very rapid method for testing pairs of frequencies.

This is the report of the application of the principles of factorial design to an investigation of individual educational development. The specific type of factorial design formulated was a 2 Χ 3 Χ 3 Χ 3 arrangement, that is, the effect of sex, grade location, scholastic standing, and individual order, singly and in all possible combinations was studied in relation to educational development as measured by the Iowa Tests of Educational Development. An application of the covariance method was introduced which resulted in increased precision of this type of experimental design by significantly reducing experimental error. The two concomitant measures used to increase the sensitiveness of the experiment were initial status of individual development and mental age. Without these statistical controls all main effects and two first-order interactions would have been accepted as significant. With their use only sex (doubtful), scholastic standing, and individual order demonstrated significant effects. The chief beauty of the analysis of variance and covariance as an integral part of a self-contained experiment is demonstrated in the complete single analysis of the data. The statistical utilization of the experimental results has also been developed for purposes of estimation and prediction. The mathematical statistician is being continuously required to develop and analyze experimental designs of increasing complexity since the introduction of the analysis of variance and covariance. The mathematical formulation and solution of the problem of this investigation is carried out. The methods illustrated and explained in this study, and modifications and extensions of them are capable of very wide application. The general principles can be used to various degrees and in a number of ways.

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 required in addition to sums of squares. A resumé of some techniques now commonly employed is included.