Showing papers in "Psychometrika in 1936"
TL;DR: In this paper, the problem of approximating one matrix by another of lower rank is formulated as a least-squares problem, and the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another.
Abstract: The mathematical problem of approximating one matrix by another of lower rank is closely related to the fundamental postulate of factor-theory. When formulated as a least-squares problem, the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another. The solution of the problem is simplified by first expressing the matrices in a canonic form. It is found that the problem always has a solution which is usually unique. Several conclusions can be drawn from the form of this solution. A hypothetical interpretation of the canonic components of a score matrix is discussed.
3,576 citations
TL;DR: In this article, a modification of the iterative scheme of calculating principal components there presented, in a fashion that materially accelerates convergence, is described, which can be used to obtain the magnitudes and orientations of the principal axes of a quadric or hyper-quadric in a manner which will ordinarily be far less laborious than those given in books on geometry.
Abstract: The resolution of a set of n tests or other variates into components 7~, each of which accounts for the greatest possible portion 71, ~,~,-., of the total variance of the tests unaccounted for by the previous components, has been dealt with by the author in a previous paper (2). Such \"factors,\" on account of their analogy with the principal axes of a quadric, have been called principal components. The present paper describes a modification of the iterative scheme of calculating principal components there presented, in a fashion that materially accelerates convergence. The application of the iterative process is not confined to statistics, but may be used to obtain the magnitudes and orientations of the principal axes of a quadric or hyper-quadric in a manner which will ordinarily be far less laborious than those given in books on geometry. This is true whether the quadrics are ellipsoids or hyperboloids; the proof of convergence given in an earlier paper is applicable to all kinds of central quadrics. For hyper-boloids some of the roots k~ of the characteristic equation would be negative, while for ellipsoids all are positive. If in a statistical problem some of the roots should come out negative, this would indicate either an error in calculation, or that, if correlations corrected for attenuation had been used, the same type of inconsistency had crept in that sometimes causes such correlations to exceed unity. Another method of calculating principal components has been discovered by Professor Truman L. Kelley, which involves less labor than the original iterative method, at least in the examples to which he has applied it (5). How it would compare with the present accelerated method is not clear, except that some experience at Columbia University has suggested that the method here set forth is the more efficient. It is possible that Kelley's method is more suitable when all the characteristic roots are desired, but not the corresponding correlations of the variates with the components. The present method seems to the computers who have tried both to be superior when the components themselves, as well as their contributions to the total variance , are to be specified. The advantage of the present method is enhanced when, as will often be the case in dealing with numerous vari-ates, not all the characteristic roots but only a few of the largest are required.
206 citations
TL;DR: In this article, four methods of factorizing the fundamental matrices used in factor analysis are described and illustrated, and the implications of each method for different schools of psychology are pointed out.
Abstract: Four methods of factorizing the fundamental matrices used in factor analysis are described and illustrated. The first is represented by the techniques already developed. The second is the obverse factor technique. The third and fourth methods are variants of the first and second. The implications of each method for different schools of psychology are pointed out. The methods are complementary, not competitive.
80 citations
TL;DR: In this paper, a 50-item test element is scored on an all-or-none basis with different standards for passing, and the percentage of passes for successive points on the criterion variable is computed.
Abstract: Using scores of 1200 students on a long test as a criterion, each of five subtests of different difficulty has maximum correlation with the criterion when the criterion is dichotomized at a value appropriate to the difficulty of the subtest. A 50-item test element is scored on an all-or-none basis with different standards for passing, and the percentage of passes for successive points on the criterion variable is computed. The Constant Method is applied to this relationship. The limen thus computed is a measure of difficulty, the dispersion is a measure of average (or total) validity, and the slope of the curve is a measure of differential validity. The difficulty of a test element is thus directly related to the maximum differential validity.
69 citations
61 citations
TL;DR: In this paper, a table of corrected proportions for different numbers of alternatives varying from two to eight is given for a test item with a limited number of alternative responses to make allowance for chance success.
Abstract: The evaluation of the level of difficulty of a test item is ordinarily derived from the proportion of a specified population passing or failing the item. With items that have a limited number of alternative responses there must be a correction in this proportion to make allowance for chance success. A table of corrected proportions is given for different numbers of alternatives varying from two to eight.
45 citations
TL;DR: Several theorems concerning properties of the communaltiy of a test in the Thurstone multiple factor theory are established as mentioned in this paper, which are applicable to a battery of tests which are describable in terms of common factors, with orthogonal reference vectors.
Abstract: Several theorems concerning properties of the communaltiy of a test in the Thurstone multiple factor theory are established The following theorems are applicable to a battery ofn tests which are describable in terms ofr common factors, with orthogonal reference vectors 1 The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors 2 The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors and then—1 remaining tests
Corollary: The square of the multiple correlation of a testj with then—1 remaining tests is equal to or less than the communality of testj It cannot exceed the communality 3 The square of the multiple correlation of a testj with then—1 remaining tests equals the communality of testj if the group of tests containsr statistically independent ests teach with a communality of unity 4 With correlation coefficients corrected for attenuation, when the number of tests increases indefinitely while the rank of the correlational matrix remains unchanged, the communality of a testj equals the square of the multiple correlation of testj with then—1 remaining tests 5 With raw correlation coefficients, it is shown in a special case that the square of the multiple correlation of a testj with then—1 remaining tests approaches the communality of testj as a limit when the number of tests increases indefinitely while the rank of correlational matrix remains the same This has not yet been proved for the general case
42 citations
40 citations
TL;DR: In this paper, the problem of weighting separate criterion variates is solved by minimizing the differences among the standard scores of the individual upon the various measures, and the method is compared with Horst's procedure of maximizing the inter-individual differences.
Abstract: The problem of weighting separate criterion variates is solved by minimizing the differences among the standard scores of the individual upon the various measures. The method is compared with Horst's procedure of maximizing the inter-individual differences. An application is made to personnel data.
40 citations
TL;DR: In this article, an experimental study of the isolation, by factor methods, of primary abilities from a battery of tests given to 240 students is presented, and Tentative interpretations of the twelve orthogonal primary factors are given.
Abstract: This is an experimental study of the isolation, by factor methods, of primary abilities from a battery of tests given to 240 students. The range and nature of the fifty-six tests is briefly described. Tentative interpretations of the twelve orthogonal primary factors are given.
30 citations
TL;DR: In this paper, a derivation of equations fundamental to the technique of path coefficients is given, and the relations of the technique to those of partial correlation, semi-partial correlation, part correlation, multiple correlation, and factor analysis are discussed.
Abstract: A derivation of equations fundamental to the technique of path coefficients is given. Suggestions are made with respect to the calculations required in the use of the technique. The relations of the technique to those of partial correlation, semi-partial correlation, part correlation, multiple correlation, and factor analysis are discussed. Some consideration is given to the merits and limitations of the technique of path coefficients.
TL;DR: In this paper, a new item selection technique is presented which takes into account the intercorrelations of the items as well as their correlations with the criterion, and an approximate solution of the mathematical equations is suggested, based on the mathematical solution, but much simpler in procedure.
Abstract: A new item selection technique is presented which takes into account the intercorrelations of the items as well as their correlations with the criterion. The technique is regarded as superior to comparable techniques in that it is considered to achieve greater economy of time, greater objectivity of procedure, higher validity, and higher reliability. The mathematical theory underlying the method is developed. An approximate solution of the mathematical equations is suggested. An approximation procedure for the complete item selection technique is presented, based on the mathematical solution, but much simpler in procedure. The clerical operations involved in the approximation procedure are outlined and illustrated on a sample worksheet.
TL;DR: The implications contained in Richardson's article on item analysis in the March 1936 issue ofPsychometrika are examined in the light of multiple factor theory as discussed by the authors, and it is shown that item analysis is a necessary, but not a sufficient condition for the construction of a test which shall measure a single trait.
Abstract: The implications contained in Richardson's article on item analysis in March 1936 issue ofPsychometrika are examined in the light of multiple factor theory. It is shown that item analysis is a necessary, but not a sufficient condition for the construction of a test which shall measure a single trait. The intercorrelations of certain items selected by a method of item analysis are examined, found to contain many zero and some negative correlations. Multiple factor analysis showed that eight traits were measured by the items which had been asserted to measure only one.
TL;DR: In this article, a time-saving systematic procedure for the computation of bi-serial correlation co-efficients is outlined for application to the evaluation of items of a test and a table of z for arguments ofp =.000 top=.999 is given.
Abstract: By the use of an algebraic variant of the ordinary formula for bi-serial correlation, tables, and graphic devices, a time-saving systematic procedure for the computation of bi-serial correlation co-efficients is outlined for application to the evaluation of items of a test. A table of z for arguments ofp=.000 top=.999 is given.
TL;DR: The content unreliability of an essay test is the error due to the items used or the content of the test as discussed by the authors, while the reader unreliability is due to variation in judgment of the persons who read and score the essay test.
Abstract: The content unreliability of an essay test is the error due to the items used or the content of the test. The reader unreliability is due to variation in judgment of the persons who read and score the essay test. The content reliability of an essay test is accordingly defined as being independent of the reader reliability. Formulae are derived for the reader reliability and for the content reliability. The content reliability is found to be equal to the geometric mean of the test reliabilities computed from the scores assigned by the two readers, divided by the reader reliability.
TL;DR: In this article, a correlation determinant Δ′ is derived containing parallel forms instead of unity as an element on the principal diagonal, where the average correlations between a varying number of parallel forms contained within each test are derived.
Abstract: Givens fallible testst
1,t
2, ⋯t
s
, the problem is to express their intercorrelations in terms of the average correlations between a varying number of parallel forms contained within each test. A new correlation determinant Δ′ is derived containingd
ii
instead of unity as an element on the principal diagonal, where
$$d_{ii} = [1 + (m_i - 1)\bar r_{ii} ]/m_i ,$$
in whichm
i
is the number of parallel forms comprising thei'th test and
$$\bar r_{ii} $$
is the average intercorrelation of them
i
(m
i
− 1) / 2 parallel forms. Asm
i
→ ∞,d
ii
approaches the correlation “corrected for attenuation.” These results make explicit the assumptions, as to intrisic accuracy of all measures, which are implicit in the usual multiple and partial correlation analysis. These results also make possible a simple procedure for estimating the effect on various partial correlation measures of improving the accuracy of part or all of the measures by including additional parallel forms.
TL;DR: In continuation of a previous paper, a mechanism of delayed reflexes is considered more in detail and the formulae obtained describe the phenomenon of “concentration” of a conditioned reflex around a definite time interval after stimulation.
Abstract: In continuation of a previous paper, a mechanism of delayed reflexes is considered more in detail. Equations governing such a mechanism are established and approximately solved. The formulae thus obtained describe the phenomenon of “concentration” of a conditioned reflex around a definite time interval after stimulation. Applied along the lines discussed in the previous paper to some simple combinations of stimuli and responses the formulae lead to a description of the elimination of errors by trial. They give a relation between the number of repetitions, necessary to eliminate a wrong act, and other constants, describing the situation.
TL;DR: The degree of learning at the point when training ceased, plotted against errors made in transposition, brings out a clear relationship between degree of original learning and accuracy of transposition.
Abstract: An experiment was performed to determine the relationship between the accuracy of the original learning and the accuracy of transposition. The usual method of comparison of the average number of errors in the transposition test made by a group of rats trained to a criterion of 10 consecutive errorless trials with the average number of errors made by a group of rats trained to a criterion of 30 consecutive errorless trials reveals no clear difference between the groups.
TL;DR: In this paper it is shown how some of the parameters which enter in the equations governing the structure of the social group can be determined by means of those equations from actually observable data.
Abstract: In continuation of a previous paper, some consequences of the fundamental equations established there are studied. For some simple hypothetical cases it is shown how some of the parameters which enter in the equations governing the structure of the social group can be determined by means of those equations from actually observable data. Furthermore some general properties of the variation with respect to time of the fundamental distribution function, which enters in the equations, are derived.
TL;DR: In this article, the authors derived formulas for simplified computation of partial and multiple correlation coefficients, and generalized ton variables, and compared the time required for computation with other methods, and showed that the simplified method is faster than other methods.
Abstract: Formulas are derived for simplified computation of partial and multiple correlation coefficients, and generalized ton variables. Time required for computation is compared with other methods.
TL;DR: In this article, a variation of the centroid method is described and illustrated, which reduces to insignificance the factor loadings of tests showing insignificant correlation (original or residual) with clusters of tests having relatively high intercorrelations.
Abstract: A variation of the centroid method is described and illustrated. By the application of new rules for reflecting signs, it may be possible to reduce to insignificance the factor loadings of tests showing insignificant correlation (original or residual) with clusters of tests having relatively high intercorrelations. As a result, a factor common to any one of these clusters may be revealed by the centroid method itself with little or no need for rotation of axes or further calculations.
TL;DR: In this paper, a mathematical supplement to the preceding paper by Professor Godfrey H. Thomson is presented, which gives rigorous proofs of theorems enunciated by him and by Dr. J. Ridley Thompson, and extends them.
Abstract: This paper is a mathematical supplement to the preceding paper by Professor Godfrey H. Thomson. It gives rigorous proofs of theorems enunciated by him and by Dr. J. Ridley Thompson, and extends them. Its basic theorem is that if a matrix of correlations is to be factorized without the aid of higher factors thans-factors (withn-s zero loadings), then the largest latent root of the matrix must not exceed the sum of thes largest communalities on the diagonal.
TL;DR: In this article, a simple rule for determining when a matrix of correlations, with communalities reducing it to minimum rank, cannot be analyzed into factors such that every column of loadings has at least as many zeros as the number of common factors, as required by Thurstone.
Abstract: The author arrives at a simple rule for ascertaining when a matrix of correlations, with communalities reducing it to minimum rank, cannot be analyzed into factors such that every column of loadings has at least as many zeros as the number of common factors, as required by Thurstone. A more exact but arithmetically tedious rule is also deduced from Ridley Thompson's boundary conditions, and a correction is made to the latter.