Showing papers in "Psychometrika in 1953"

A new status index derived from sociometric analysis.

Abstract: For the purpose of evaluating status in a manner free from the deficiencies of popularity contest procedures, this paper presents a new method of computation which takes into accountwho chooses as well ashow many choose. It is necessary to introduce, in this connection, the concept of attenuation in influence transmitted through intermediaries.

Who belongs in the family

Abstract: I was sitting before my TV set, a while back, watching Captain Video and pondering the organizational problems of psychologists, psychometricians, psychodiagnosticians, psycho-somatists, psychosomnabulists, and psychoceramics (crack-pots to you). Wondering what I might do, in my small way, to help out, I decided to enlist Captain Video's help to bring me from the Black Planet that superogalactian hypermetrician, Dr. Idnozs HcahscrorTenib, cosmos-famous discoverer of Serutan. Why delay? The Galaxy was on its way. and in half a light year Dr. Tenib was at my side prepared to devote his gargantuan talents to the task. Seeing no point in confusing the good doctor by trying to describe to him the present administrative hodgepodge, I said, "Doctor, let's start from scratch. I want you to find out for me how these good people who are present at the annual meeting of the APA structure themselves? What families are represented? How many, or better, how few? And who belongs to each?" "We proceed," said the Doctor. "Bring sample of population; I measure." So we set out to design a sample. The problem presented some interesting theoretical aspects, but the final solution was relatively simple. We stationed representatives at each of the three state beverage stores and followed every third badge-wearing individual who came out of a store. We selected only outgoing patrons for obvious reasons. After assisting each respondent to unburden himself, we brought him to Dr. Idnozs (as we came to call him among ourselves) for study. "Now," murmured the Doctor, "we give tests. First is 'Draw-a-Psychiatrist Test.' " "We score this," he confided, "by if it gives horns." Presently we started on the physiological test battery. "We draw off saliva drop by drop," explained our idiot savant, "and see does he drool when we bring in Skinner Box." Later came the Peculiar Preference Blank. "Forced-choice, you know," whispered the Doctor. "Would you rather make mud pies or kiss gorgeous blonde?"

Image theory for the structure of quantitative variates

Abstract: A universe of infinitely many quantitative variables is considered, from which a sample ofn variables is arbitrarily selected. Only linear least-squares regressions are considered, based on an infinitely large population of individuals or respondents. In the sample of variables, the predicted value of a variablex from the remainingn − 1 variables is called the partial image ofx, and the error of prediction is called the partial anti-image ofx. The predicted value ofx from the entire universe, or the limit of its partial images asn → ∞, is called the total image ofx, and the corresponding error is called the total anti-image. Images and anti-images can be used to explain “why” any two variablesx j andx k are correlated with each other, or to reveal the structure of the intercorrelations of the sample and of the universe. It is demonstrated that image theory is related to common-factor theory but has greater generality than common-factor theory, being able to deal with structures other than those describable in a Spearman-Thurstone factor space. A universal computing procedure is suggested, based upon the inverse of the correlation matrix.

An analytical solution for approximating simple structure in factor analysis

Abstract: 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.

An application of confidence intervals and of maximum likelihood to the estimation of an examinee's ability

Abstract: A mathematical definition of the theoretical relation between the examinee's actual responses to the test items and his “true ability” is selected. A maximum-likelihood solution is obtained for estimating the examinee's “true ability” from his responses to the items. The standard error of the maximum-likelihood estimate is obtained, its relation to the discriminating power of the test is pointed out, and some generalizations are drawn as to the optimum level of item difficulty. The Neyman-Pearson power function is applied to determine which of two psychological tests is the most powerful for the selection of “successful” examinees.

Test reliability and effective test length

Abstract: Measures of effective test length are developed for speeded and power tests, which are independent of the number of items in the test or of the time required for administration. These measures are used in determining reliability for (1) speeded and power tests, where a separately timed short parallel form is administered in addition to the full-length test; (2) power tests, where a subset of items is imbedded within the total test, parallel to the total test; and (3) power tests, where the subset of items is correlated with the complementary parallel subset in the test.

Maximizing the discriminating power of a multiple-score test.

Abstract: Maximizing the discriminating power of a multiple-score test involves maximizing the homogeneity of each subtest and minimizing the correlations between subtests. A method is presented for constructing such tests from items whose intercorrelations are not too high. Under certain restrictions the saturation, defined as the proportion of inter-item covariance to total variance, is maximized for each subtest. The nucleus of each subtest is three items with high covariancesinter se. All items which will lower the saturation are discarded; the one item is added which will maximize the saturation of the resultant test. This process is repeated until all the items are included or discarded for that subtest. If the correlation between any such subtests approaches the geometric mean of their saturations, their items form a new pool for one or more subtests. Formulas are presented for deciding which items to eliminate in order to reduce further the correlations between subtests.

An analytic method for rotation to orthogonal simple structure

Abstract: The objective of simple structure rotation is to facilitate the interpretation and use of factor analytic results. However, a given factor analysis may fall short of this goal for many reasons other than ineffective rotation, It is therefore desirable to have a means for obtaining and/or recognizing a reliably good approximate rotation to simple structure in a given situation, when this rotation is uniquely determined. One basis is provided by identifying simple structure with the maximized kurtosis of the distribution of factor loadings, when each variable is tallied in both its normal and reflected form to eliminate the artifact of scoring. For orthogonal rotation this only requires maximization of the sum of fourth powers of all factor loadings, Q, where . Differentiation shows that Figure 1. Download figure to PowerPoint must necessarily hold for each pair of factors, j and k. It is further shown that these functions vary sinusoidally with rotation in any plane, with period of ¼. These facts are used to develop two convenient computational routines – for hand computing and for high-speed electronic computing – and to demonstrate the convergence of these routines to a stable position. The method is illustrated with two real and two hypothetical examples, and it is shown that the real data tend to yield a simple structure better than that obtainable when the same vectors are placed into random configurations. Certain difficulties of the method are discussed.

A method for factoring large numbers of items

Abstract: The computation of intercorrelation matrices involving large numbers of variables and the subsequent factoring of these matrices present a formidable task. A method for estimating factor loadings without computing the intercorrelation matrix is developed. The estimation procedure is derived from a theoretical model which is shown to be a special case of the multiple-group centroid method of factoring. Empirical checks have indicated that the model, even though it makes some stringent assumptions, can be applied to a variety of variables found in psychological factoring problems. It has been found to be particularly useful in factoring test items.

Reliability formulas that do not assume experimental independence

Abstract: Tautologies are established for the reliability coefficientρ2t 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 toρ2t, four of which are developed.

A proposed index of the conformity of one sociometric measurement to another

Abstract: An index is proposed to measure the extent of agreement of the data of a sociometric test with another test made at an earlier time or on another test criterion. The index is used to define an index of concordancebetween the two tests. It is shown how the index may be used for either individuals or groups. Tests of the hypothesis that agreement is random are given for all cases and applied to an example.

Note on “A Table for the Rapid Determination of the Tetrachoric Correlation Coefficient”

Abstract: This note revises the figures in the original article concerning the accuracy of the tetrachoric estimates involved. These estimates are better than previously noted and are very satisfactory. Some minor known errors in the original article are also noted.

A factor-analytic study of reasoning abilities

Abstract: A battery of 32 tests was administered to a sample including 144 Air Force Officer Candidates and 139 Air Cadets. The factor analysis, using Thurstone's complete centroid method and Zimmerman's graphic method of orthogonal rotations, revealed 12 interpretable factors. The non-reasoning factors were interpreted asverbal comprehension, numerical facility, perceptual speed, visualization, andspatial orientation. The factors derived from reasoning tests were identified asgeneral reasoning, logical reasoning, education of perceptual relations, education of conceptual relations, education of conceptual patterns, education of correlates, andsymbol substitution. The logical-reasoning factor corresponds to what has been called deduction, but eduction of correlates is perhaps closer to an ability actually to make deductions. The area called induction appears to resolve into three eduction-of-relations factors. Reasoning factors do not appear always to transcend the type of test material used.

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

Abstract: 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.

A note on the Neyman-Johnson technique

Abstract: A statistical problem which frequently arises in educational and psychological experimentation is that of testing the significance of the difference of the mean scores of two groups on some criterion variable, where the differential effects of one or more variables which are correlated with the criterion must be statistically eliminated. The usual analytical technique for this type of problem is the analysis of covariance (9). The Neyman-Johnson technique (7) provides another, and substantially different, approach. A computational procedure is suggested here which utilizes the advantages of both techniques without an undue increase in computational labor. In addition, the Neyman-Johnson technique is generalized to the case ofn predictor variables. Its application has heretofore been limited to a maximum of three predictor variables.

A revised orthogonal rotational solution for Thurstone's original primary mental abilities test battery

Abstract: By extension of the rotational process, meaningful orthogonally related positions were found for all of the thirteen centroid factors which Thurstone extracted from his original PMA intercorrelations Most of the original primary ability factors were more sharply delineated and corresponded more closely to the Army Air Force factors that bear similar names (demonstrating greater invariance from analysis to analysis) While such different results obtained by two investigators applying the same methods on the same data may initiate some concern, the results strengthen rather than weaken the idea that more psychological meaningfulness and greater invariance will result if centroid axes are rotated, using the concepts of a simple structure and positive manifold

A generalization of Thurstone's learning function

Abstract: Thurstone's equation giving the probability of a correct response (p) as a function of practice time (t) when punishment and reward have equal effects has been generalized to the case where the effect of punishment is not necessarily equal to the effect of reward. Since the general equation is somewhat unwieldy, three special cases are considered, where reward has no effect, where punishment has no effect, and where these effects are equal. Equations are given together with tables for making a rectified plot for each of the three special cases.

Application of a large sampling criterion to some sampling problems in factor analysis

Abstract: A technique is presented to test the completeness of factor solutions and also to test the significance of common-component loadings. The chisquare test involved is based upon the asymptotic normal properties of the residuals.

Estimation of the correlation coefficient in the case of a bivariate normal population when one of the variables is dichotomized

Abstract: It is shown that the problem of estimation of the correlation coefficient of a bivariate normal population when one of the variables is dichotomized may be attacked with “probit analysis” methods. This represents an extension of the work of Gillman and Goode (3), as it was possible to find by this approach an approximation to the large-sample variance of the resulting estimateG ofρ. An empirical investigation was undertaken with the object of obtaining some information about the distribution ofG for large sample size. Methods for determining the “pass-fail” cut-off are considered.

Differences in factor content of rights and wrongs scores

Abstract: The right-response scores and wrong-response scores of speeded aptitude tests were factor analyzed to determine whether they differ in factorial content The information thus obtained was used to derive scoring formulas that yield purer measures of a factor than do scoring formulas derived in other ways

Validation of new item types for the scholastic aptitude test of the college entrance examination board

Abstract: The College Board is looking forward to the possibility of adding to the Scholastic Aptitude Test several scores in addition to the present Verbal and Mathematics scores. Towards this end, eleven newly developed or newly adapted tests were tried out in combinations of four at ten colleges. The intercorrelations and validities for these tests have been computed for the groups of students taking the more popular freshman courses, a total of 79 groups. A composite college table and a list of cases where the experimental test raised the prediction above that already provided by SAT have been set up to aid in decisions with regard to the tests. Some very satisfactory gains in the multiple correlation were obtained. Corrections for restriction of range and for attenuation are regarded as unnecessary but were carried out in a couple of cases for illustrative purposes.

Spectral analysis and the study of individual differences in the performance of routine, repetitive tasks

Abstract: For the purpose of studying individual differences in the performance of routine, repetitive tasks, mathematical methods of analysis of data which is ordered along the time dimension are considered. Spectral analysis is chosen as the best method, since it gives a complete summarization of the stationary properties of time-ordered data, and the measures derived from it possess convenient and orderly statistical properties. Spectral analysis decomposes the total oscillation of a time-series into oscillations of varying rates, specifying the relative contribution of each of the components of oscillation. Spectral analysis as an analytical tool is applied to available psychological data; in one case, to a study of “mental blocking,” and in another, to a study of serial patterns of response in an auditory discrimination. In both cases, spectral analysis leads to a modification and clarification that had been reached by the previous authors. Spectral analysis is then applied to the experimental data of the present paper. The experimental task was that of jabbing a stylus repeatedly at a target line or lines. The deviations of the jabs from a reference line were measured in serial order, and the series were then subjected to a spectral analysis. Thirty-three subjects were tested on five variations of the main task. Fifteen of the subjects were retested a month later. Task variations proved unimportant, but reliable individual differences were found in three measures, two of which did not depend on spectral analysis and one of which did. These three measures were related to general personality characteristics by means of an inductive process. An individual who had had considerable contact with the experimental subjects in interview situations over a period of two years was asked to suggest personality factors which might have produced the differences found on each of the three measures. Without knowledge of the nature of the experimental measures, he suggested personality factors corresponding to them. These personality factors and their relation to routine, repetitive tasks are discussed.

The stability of the factorial pattern of aircrew classification tests in four analyses

Abstract: This study consists of four factor analyses of the Army Air Forces Aircrew Classification Batteries. The first was an analysis of the 1945 wartime battery, while the other three were analyses of the 1947 postwar battery, consisting of essentially the same variables, but using different samples. Eleven factors were found which had been identified and reported in previous analyses. An additional factor, possibly an artifact, was identified as an age-education doublet. The only factor which differed significantly in the analyses was pilot or flying interest. These factor analyses show that the factorial content of the tests remains quite similar in both wartime and postwar populations.

Moderator variables in prediction, with special reference to freshman engineering grades and the strong vocational interest blank

Abstract: The Accountant Scale of the Strong Interest Blank for Men bears little direct relation to a criterion of freshman engineering grades However, it may act as a suppressor variable in conjunction with the Banker or Psychologist Scale of the Strong It may also aot as a moderator variable in conjunction with the Math-Science Teacher or Real Estate Salesman Scale These effects are illustrated by results obtained in two engineering schools The basic algebra and geometry of moderator variables is given