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.

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 estimate G of ρ. An empirical investigation was undertaken with the object of obtaining some information about the distribution of G for large sample size. Methods for determining the “pass-fail” cut-off are considered.

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 Hcahscror-Tenib, 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?”

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.

A universe of infinitely many quantitative variables is considered, from which a sample of n 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 variable x from the remaining n − 1 variables is called the partial image of x, and the error of prediction is called the partial anti-image of x. The predicted value of x from the entire universe, or the limit of its partial images as n → ∞, is called the total image of x, and the corresponding error is called the total anti-image. Images and anti-images can be used to explain “why” any two variables xj and xk 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.

Case IV of Thurstone's Law of Comparative Judgment is displayed as a system of homogeneous linear equations for which a least-squares solution is presented, using various conditional equations which fix the origin and the unit of measurement. The computational load is, however, quite heavy.

The Gleser-DuBois conditions for selecting from a number of test items those which will maximize the correlation between total test score and criterion will degenerate into expressions requiring only item counts on total distributions and the upper halves of distributions. A grouping convention for scores near medians is recommended. The inefficiency of the method is easily compensated for, because, regardless of the size of the sample, only standard test-scoring equipment and brief computations are required. A procedure is outlined, and some applications are discussed.

Guttman's scalogram board technique for reordering the columns and rows of a matrix is described and its disadvantages are pointed out. A simple and inexpensive procedure for doing the same job without these disadvantages is outlined.

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 of n predictor variables. Its application has heretofore been limited to a maximum of three predictor variables.

A table is developed and presented to facilitate the computation of the Pearson Q3 (“cosine method”) estimate of the tetrachoric correlation coefficient. Data are presented concerning the accuracy of Q3 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.

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.

It is proposed that a satisfactory criterion for an approximation to simple structure is the minimization of the sums of cross-products (across factors) of squares 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.

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 account who chooses as well as how many choose. It is necessary to introduce, in this connection, the concept of attenuation in influence transmitted through intermediaries.

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 covariances inter 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.

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.

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.

The following development is a coordination of certain aspects of mental test methods and psychophysics. The biserial and triserial prediction equations of the Katzell-Cureton theory are utilized to reformulate the determination of absolute and differential limens from data produced by the constant method.

Tautologies are established for the reliability coefficient ρ2t of the sum of n 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 ρ2t, four of which are developed.