A procedure for maximizing selective efficiency is developed for application to situations in which it is desired to select from a single group of applicants for several possible assignments. The problem of comparable units for the several criteria whose values must be compared to each other for differential assignment purposes is discussed. It is demonstrated that, assuming linear regressions, maximal selection is obtained if individuals in any given assignment are differentiated from those rejected according to critical rejection scores on the multiple weighted sum of the predictors and from another possible assignment by critical difference scores which are merely the differences between the two critical rejection scores. Since the relationships just indicated give no way of determining the magnitude of the critical scores required to select the required number of persons for each assignment, a successive approximation procedure for accomplishing this purpose has been devised and a computational example is worked out.

It is assumed that a scale of true scores on a function exists and that the probability of answering an item correctly is a curve of the type of the integral of the normal curve. The product moment correlation between the test score and true score is derived for a normal distribution of subjects and a test composed of equivalent items. Numerical examples demonstrate that the maximum correlation between test scores and true scores occurs for a one hundred item test when the point correlation between items is less than three tenths.

The method of rotation described here is applied to one hyperplane at a time. The method seems to be simple and quite effective and it can be applied by a relatively inexperienced computer. The method does not postulate a positive manifold, and hence it is applicable also to bipolar factors.

The test-retest reliability of qualitative items, such as occur in achievement tests, attitude questionnaires, public opinion surveys, and elsewhere, requires a different technique of analysis from that of quantitative variables. Definitions appropriate to the qualitative case are made both for the reliability coefficient of an individual on an item and for the reliability coefficient of a population on the item. From but a single trial of a large population on the item, it is possible to compute a lower bound to the group reliability coefficient. Two kinds of lower bounds are presented. From two experimentally independent trials of the population on the item, it is possible to compute an upper bound to the group reliability coefficient. Two upper bounds are presented. The computations for the lower and upper bounds are all very simple. Numerical examples are given.

The relation between item difficulty distributions and the “validity” and reliability of tests is computed through use of normal correlation surfaces for varying numbers of items and varying degrees of item intercorrelations. Optimal or near optimal item difficulty distributions are thus identified for various possible item difficulty distributions. The results indicate that, if a test is of conventional length, is homogeneous as to content, and has a symmetrical distribution of item difficulties, correlation with a normally distributed perfect measure of the attribute common to the items does not vary appreciably with variation in the item difficulty distribution. Greater variation was evident in correlation with a second duplicate test (reliability). The general implications of these findings and their particular significance for evaluating techniques aimed at increasing reliability are considered.

Ordinary product-moment correlation and regression methods are frequently not immediately applicable to qualitative data, whereas multiserial r, point-multiserial r, and multiserial eta can be easily applied. The multiserial r is rejected for prediction since it tells us only what the correlation might be if certain assumptions were true and if we could measure what is not now measured. The point-multiserial r and multiserial eta are identical when the number of categories is two but differ when it is three or greater. The multiserial eta is identical with the product-moment r when categories are assigned scale values equal to their means on the continuous variable. With three or more categories, the point-multiserial r, which assumes linearity with equal step intervals, is always lower than the multiserial eta, which forces linearity by adoption of unequal step intervals based upon difference in criterion attainment. While the multiserial eta expends one degree of freedom with the weighting of each category, this is known and correctable, whereas the vague partial loss of degrees of freedom due to the ordering of categories in the point-multiserial is not correctable.

A factorial analysis was made of a small battery of twelve anthropometric measurements. The correlations can be accounted for by four factors in a simple structure. This small battery has been used by the author for teaching purposes. Several of these factors seem to be meaningful, but their acceptance must depend on more comprehensive studies of body measurements with a larger number of measurements.

Formulas are presented for triserial correlation, quadriserial correlation, etc., and for serial correlation in general. These formulas are based on well-known procedures outlined by Kelley, Peters and Van Voorhis, and others, and involve Pearson's correction for “broad categories.” The formula for biserial correlation also may be developed following these procedures. The assumptions underlying serial correlation are that the segmented variable is basically continuous and normally distributed, and that all the segments which together would form a whole normal distribution are present.

Most existing tests for significance of difference between means require specific assumptions concerning the distribution function in the parent population. The need for a test which can be applied without making any such assumption is stressed. Such a statistical test is derived. The application of the test involves converting scores to rank orders. The exact probabilities may then be calculated for specified differences between samples by means of which the null hypothesis may be tested. The application of the test is simple and requires a minimum of calculation. The test loses in precision because of the conversion to rank orders but gains in generality since it may be safely used with any kind of distribution.

A rapid method of estimating a correlation coefficient is given. The method expresses the correlation coefficient as the ratio between two differences in sums (or means) of the dependent variable computed only for extremes of the bivariate distribution. A trial shows that this method gives results similar to the product-moment correlation coefficient. Extensions of the method to qualitative data are also suggested.

A description is given of diagrams (available separately) for computing tetrachoric correlation coefficients. The diagrams are entered with “per cent of combined groups above dividing point” and difference between groups in their per cents above the dividing point.

This paper presents a prepunched deck of cards to enable the extraction of square roots on standard punch card tabulating equipment. Such a deck is valuable in constructing mathematical tables which involve square roots or in obtaining standard deviations in connection with computing correlation coefficients. By using a deck of reciprocals in conjunction with the deck for square roots, correlations may be solved completely on IBM equipment.

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.

A new method of test selection, which attempts to combine the merits of the Toops L-Method with those of the Wherry-Doolittle Method, is presented. It results in integral (unit if desired) positive and/or negative (optional) weights. This flexibility makes the method applicable to all kinds of material and for both selecting items for tests and tests for batteries. An explicit solution of one test construction problem is presented. Necessary changes in method for the solution of five other types of test construction problem are presented. A few cautions are provided for potential users.

Twenty-one reagents were rated on a pleasantness scale by 182 subjects. The tetrachoric correlation coefficients were determined for the variables. The correlation matrix thus obtained was subjected to a Thurstone analysis which gave rise to a 6-dimensional structure. Two of these were not clear-cut, and the last one appeared to be a residual factor. One factor, the benzene-ketone, revealed the chemical property of unsaturation and the physiological quality of possible trigeminal stimulation. The other two factors were marked by the presence of oxygen and nitrogen, respectively, and were tentatively labelled as a plant factor and an animal factor accordingly. The intercorrelations of the primaries are high.

A theoretical discussion of the factor pattern of predictor tests and criterion shows that ordinary test selection methods break down under certain circumstances. It is shown that maximal results may not occur if suppressor variables are present among the predictors. Suggested solutions to the problem include: (1) prior item analysis of tests against the criterion, (2) selection of several trial batteries including some with suppressor variables on the basis of a factor analysis of tests and criterion, (3) modification of the usual test selection procedures to include separate solutions based upon each of several starting variables, or (4) the cumbersome and tedious solution of all possible combinations of predictors. The solutions are recommended in the order named above. Although all of the suggested solutions involve added labor and may not be necessary, the test or battery constructor should at least be aware of the problem.

Correlation data on nine reading tests originally analyzed by Frederick B. Davis by the principal axes method are reanalyzed by Spearman's uni-dimensional method. It is concluded that a single common factor (reading ability) accounts for the correlations among the tests with residuals remarkably small in view of the fact that the tests were designed to test nine supposedly different skills. Three of the tests showed additional specific variance not attributable to the common factor.

The aesthetic preferences of a group of persons are obtained from their orders of sets of pictures and patterns according to “liking.” The same pictures are ordered independently by a team of experts, according to certain artistic criteria such as naturalism, composition, color, rhythm, etc. The orders of preference and orders according to the criteria are compared by correlation and matrices of correlation formed from (1) correlations between the persons' orders of preference; (2) correlations between the orders of preference and orders according to artistic criteria; and (3) correlations between the criterion orders. These matrices are symbolised by Rp, R0, and Rc, respectively, and combined to form a single matrix

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left[ \begin{gathered}R_p R_o\hfill \\R'_o R_c\hfill \\ \end{gathered}\right]$$\end{document}

When one knows in advance the minimum percentage difference which will be of practical interest in an investigation, principles of economy suggest the desirability of an accurate estimate of the size of the sample to be used. Charts are presented which eliminate any computation where the two groups involved are to be equal. A table is included for the case where one group will be from one to three times the size of the other. The charts are also useful for determining whether or not obtained differences are statistically significant at the 5 per cent or the 1 per cent level.