A simplified method for approximating multiple regression coefficients

A method of exhaustion has been described for calculating regression coefficients. This method dispenses with the solution of simultaneous equations but utilizes a process of successive extraction in obtaining's, where each successive is maximized. This procedure permits the worker to discard as he goes along those weights which are deemed unsatisfactory for purposes of prediction. The coefficients and theR in a problem involving a criterion and six independent variables were calculated in sixty minutes. TheR's obtained by this method are smaller than those yielded by the Doolittle technique, but in problems which have been considered this discrepancy has not exceeded .05.     

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.     

The measurement of conformity

Allport'sJ-curve hypothesis of conforming behavior and its attendant treatment of appropriate data are criticized on the following points: (1) narrowness of application; (2) flexibility of interpretation of results; (3) arbitrary selection of a criterion of conformity; (4) lack of a means by which the extent of conformity in one situation can be compared with that in another; (5) inequality of telic units. As an alternative treatment of such data, the method of higher moments is suggested and rationalized. Data from Allport and Solomon are reworked by this method and results compared.     

Testing linear hypotheses illustrated by a simple example in correlation

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.     

A simple orthogonal multiple factor approximation procedure

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.     

A table for the rapid determination of the tetrachoric correlation coefficient

A table is developed and presented to facilitate the computation of the PearsonQ ₃ (cosine method) estimate of the tetrachoric correlation coefficient. Data are presented concerning the accuracy ofQ ₃ as an estimate of the tetrachoric correlation coefficient, and it is compared with the results obtainable from the Chesire, Saffir, and Thurstone tables for the same four-fold frequency tables.The authors are indebted to Mr. John Scott, Chief of the Test Development Section of the U.S. Civil Service Commission, for his encouragement and to Miss Elaine Ambrifi and Mrs. Elaine Nixon for the large amount of computational work involved in this paper.     

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.     

Regression tests for several samples

In many situations it is desirable or necessary to administer a set of tests to several different groups, and to ask if the results obtained in the different groups may be regarded as being essentially the same in some sense. In the case of two variables (one dependent and one independent) one may, for instance, ask if the errors of estimate and the regression lines may be regarded as being the same for the populations from which the different groups are drawn. For this case, the present article considers tests for three hypotheses regarding the populations from which the different groups are drawn: (a)H _A, the hypothesis that all standard errors of estimate are equal; (b)H _B, the hypothesis that all regression lines are parallel, (assumingH _A); and (c)H _C, the hypothesis that the regression lines are identical, (assumingH _B). Test criteria for these three hypotheses and their sampling theory for large samples are presented. The results are extended to the case of several independent variables. An illustrative problem is presented for two groups, two independent and one dependent variable.     

Estimation of parameters in a truncated trivariate normal distribution

This paper gives maximum-likelihood estimators for certain parameters in a truncated trivariate normal distribution when the values of the other parameters are known. The estimators are functions of a random sample. Approximate variances and covariances of the estimators, when the sample size is large, are also given. The type of truncation considered is merely restriction of the range of one of the variates, whose true mean and variance are assumed to be known. Two cases of such restriction are treated: (a) ( x < + ); (b) (- <x '), where and ' are arbitrary cutoff points which are assumed to be known. A precise statement of the estimation problem is given in Section 1. Section 2 contains preliminary calculations. The estimators appear in Section 3. The asymptotic variances and covariances of the estimators are given in Section 4. The estimators and their asymptotic variances and covariances can be easily specialized to be suitable for the case of a certain truncated bivariate normal distribution (Sections 3 and 4).     

Percentage points of Wilks'L mvc andL vc criteria

Likelihood ratio tests have been proposed by Wilks for testing the hypothesis of equal means, variances, and covariances (H _mvc) and the hypothesis of equal variances and covariances (H _vc) in ap-variate normal distribution. Using exact distributions of the appropriate likelihood ratio statistics, tables of the .05 and .01 points of these distributions are constructed forp = 4, 5, 6, 7 and sample sizen = 25 (5) 60 (10) 100. A correction factor is recommended for largern. Two numerical examples illustrate use of the tables. A nonparametric test is proposed forH _mvc when the multivariate parent population is known to be non-normal.This research was supported partly by the Office of Naval Research under Contract No. Nonr-855(06) and partly by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Contract No. 18(600)-83. Reproduction in whole or in part for any purpose of the United States Government is permitted.     

A factor analysis of verbal abilities

A multiple-factor analysis was made of a battery of 42 tests of verbal abilities administered to 119 college adults. Where necessary, the distributions of test scores were normalized before the inter-test correlations were computed. Thurstone'sM (Memory or Rote Learning) factor has been confirmed, but hisV (Verbal Relations) factor seems to have been split into two or possibly three factors,C,J, andG; and hisW (Word Fluency) factor has been split into two factors,A andE. TheC factor seems to represent the richness of the individual's stock of linguistic responses, and theJ factor seems to involve the ability to handle semantic relationships. No satisfactory interpretation can as yet be made of theG factor. TheA factor seems to correspond to the speed of association for common words where there is a high degree of restriction as to appropriate responses. TheE factor is described as an associational facility with verbal material where the only restriction is that the responses must be syntactically coherent. The new factors are:F, facility and fluency in oral speech;H, facility in attaching appropriate names or symbols to stimuli; andD, speed of articulatory movements.This paper is a condensation of the writer's doctoral dissertation, A Factor Analysis of Verbal Abilities, on file at the library of the University of Minnesota.     

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.     

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.     

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.     

Reliability formulas that do not assume experimental independence

Tautologies are established for the reliability coefficient ² _t of the sum ofn part scores. It is not assumed that the part scores are experimentally independent of each other nor that the parts are equivalent to each other. The tautologies show the exact role played by experimental dependence and nonequivalence of parts, respectively, in the reliability of the sum. The formal algebra is appropriate to reliability in the sense of repeated trials of the same test, as well as in the sense of a universe of parallel tests, although the empirical meanings are different. Emphasis is on practical formulas that require information from only a single experiment (or test). These can take the form only of lower bounds to ² _t , four of which are developed.     

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.     

The act of surrender in the treatment of the alcoholic

The success of A. A. in the alleviation of alcoholism has provided psychiatrists with a rich source of clinical material for a study of the religious process. The paper which follows is a discussion of one aspect of that process, namely, the act of surrender. It was prepared for a psychiatric audience; it may have some pertinence for the pastoral group since it represents at least an effort at understanding the religious life. Fully aware of its deficiencies, I offer the paper in the hope that it may seem to contain a measure of truth and be of some value to pastors in their relationships with their parishioners.This article is reprinted from Quarterly Journal of Studies on Alcohol, June, 1949 where it appeared under the title The Act of Surrender in the Therapeutic Process, by permission of author and publisher.s     

The concept of test and item reliability in relation to factor pattern

It is shown that approaches other than the internal consistency method of estimating test reliability are either less satisfactory or lead to the same general results. The commonly attendant assumption of a single factor throughout the test items is challenged, however. The consideration of a test made up ofK sub-tests each composed of a different orthogonal factor disclosed that the assumption of a single factor produced an erroneous estimate of reliability with a ratio of (n–K)/(n–1) to the correct estimate. Special difficulties arising from this error in application of current techniques to short tests or to test batteries are discussed. Application of this same multi-factor concept to item-analysis discloses similar difficulties in that field. The item-test coefficient approaches 1/K as an upper limit rather than 1.00 and approaches 1/n as a lower limit rather than .00. This latter finding accounts for an over-estimation error in the Kuder-Richardson formula (8). A new method of isolating sub-tests based upon the item-test coefficient is proposed and tentatively outlined. Either this new method or a complete factor analysis is regarded as the only proper approach to the problem of test reliability, and the item-sub-est coefficient is similarly recommended as the proper approach for item analysis.     

A statistical test for means of samples from skew populations

This paper presents a test for determining significance of differences between means of samples which are drawn from positively skewed populations, more specifically, those having a Pearson Type III distribution function. The quantity 2npx _g /x _p (wherep equals the mean squared divided by the variance andn is the number of cases in the sample), which distributes itself as Chi Square for 2np degrees of freedom, may be referred to the tables of Chi Square for testing hypotheses about the value of the true mean. For two independent samples, the larger mean divided by the smaller mean, which distributes itself asF for 2n ₁ p ₁ and 2n ₂ p ₂ degrees of freedom, may be referred to theF distribution tables for testing significance of difference between means. The test assumes that the range of possible scores is from zero to infinity. When a lower theoretical score limit (c) exists which is not zero, the quantity (Mean —c) should be used instead of the mean in all calculations.     

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.