In large-scale psychological testing programs, it happens frequently that a group of items must be checked for accuracy with respect to certain characteristics. In many cases it is not feasible to check each item in the group separately, so that a decision concerning the entire group is usually reached by examining in detail only a small portion of the group. This paper proposes some sequential sampling plans for use in this type of work. A general discussion of the problem is followed by an illustrative example of the method applied to checking scores on groups of psychological test papers.

The topic that I have chosen, “A Superior Rotational Method in Factor Analysis or Psychometricians in Government Service,” may puzzle you at first. But there is an explanation. A study in progress at the University of North Carolina requires the interpretation of 16 factors obtained from a centroid analysis of 66 variables. The task that was approached with greatest apprehension was the location of a meaningful reference frame. With so many vectors protruding into a space of such a large number of dimensions, the problem appeared practically insoluble. Hence we sought a rotational method that would obviate the necessity for the numerous graphs and the matrix multiplication associated with the commonly accepted techniques.

Perhaps some of you have had the experience of solving a difficult mathematical problem in a state of sleep and later being able to transcribe the complete solution immediately. A relatively painless approach to mathematics, it is the one which enabled me to pass a course in Mathematical Probability years ago. I scarcely dared presume that I could evoke it in the present emergency. To my astonishment, however, it worked. One day the project staff had had a particularly trying session during which we tried in vain to stretch the available funds to cover the rotational work. Upon falling asleep in my office chair, I dreamed a solution so simple that an ordinary statistical clerk, upon mere inspection of the score matrix, could visualize the most interpretable reference frame in up to an infinite number of dimensions. This was beyond our most extravagant hopes, for it rendered obsolete both the correlation matrix and the centroid solution.

The personnel classification problem arises in its pure form when all job applicants must be used, being divided among a number of job categories. The use of tests for classification involves problems of two types: (1) problems concerning the design, choice, and weighting of tests into a battery, and (2) problems of establishing the optimum administrative procedure of using test results for assignment. A consideration of the first problem emphasizes the desirability of using simple, factorially pure tests which may be expected to have a wide range of validities for different job categories. In the use of test results for assignment, an initial problem is that of expressing predictions of success in different jobs in comparable score units. These units should take account of predictor validity and of job importance. Procedures are described for handling assignment either in terms of daily quotas or in terms of a stable predicted yield.

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) HA, the hypothesis that all standard errors of estimate are equal; (b) HB, the hypothesis that all regression lines are parallel, (assuming HA); and (c) HC, the hypothesis that the regression lines are identical, (assuming HB). 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.

Evidence is cited to show that specificity, or lack of equivalence, in the comparable forms of tests has a tendency to lower the value of reliability coefficients but has no tendency to lower the value of observed trait coefficients. This implies that the greater the lack of equivalence, the higher will be coefficients corrected for attenuation. Errors of measurement are supposed to reduce the magnitude of observed trait coefficients. Since specificity does not lower the correlation between two tests and since the split-half and equivalent-form reliability coefficients treat specificity as error, it follows that these two coefficients cannot legitimately be used in Spearman's correction-for-attenuation formula.

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) (δ ≤x < + ∞); (b) (- ∞ <x ≤ δ'), where δ and δ' 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).

A method is described for the calculation of the sum of squares for a second-order interaction. It is then shown that the method is general and can be used for the calculation of the sum of squares for any higher-order interaction.

Formulas are derived by which, given the factor loadings and the internal reliability of a test of unit length, the following estimates can be made: (1) the common-factor loadings for a similar (homogeneous) test of length n; (2) the number of times (n) that a test needs to be lengthened homogeneously to achieve a factor loading of a desired magnitude; and (3) the correlation between two tests, either or both of which have been altered in length, as a function of (a) the new factor loadings in the altered tests or (b) the original loadings in the unit-length tests. The appropriate use of the derived formulas depends upon the fulfillment of four assumptions enumerated.

In econometric literature a parameter in a theoretical model has been called identifiable if it can be uniquely determined in terms of the joint probability distribution of the observed variables. In this paper the identifiability of parameters in four different factor analysis models is considered. The last of these four models corresponds to Thurstone's factor analysis. In Sections 7 and 11, the possibility of a statistical testing of the models is discussed. Section 10 deals with the problem of actually determining the parameter r (the number of common factors) in terms of the probability distribution of the observed variables.

The theoretical basis for the Johnson-Neyman Technique is here presented for the first time in an American journal. In addition, a simplified working procedure is outlined, step-by-step, for an actual problem. The determination of significance is arrived at early in the analysis; and where no significant difference is found, the problem is complete at this point. The plotting of the region of significance where a significant difference does exist has also been simplified by using the procedure of rotation and translation of axes.

In a centroid factor analysis of the Multiphasic, Strong, Kuder, and Bell inventories using a population of 400 adult males, eight common factors dealing with aspects of personality as measured by these instruments were isolated. Seven of the factors were meaningful and one was a residual. This study indicates little overlap between the two personality and the two interest inventories. It would appear that factors found in these instruments measuring aspects of personality are dichotomous in nature and are not common to the two types of instruments included in this study. That is, two of the factors were common to the two personality inventories, and five of the factors were common to the two interest inventories.

A procedure is proposed for testing the significance of group differences in the standard error of measurement of a psychological test. Wilks' criterion is used to assure that the tests used in ascertaining reliability and hence variance of errors of measurement may be assumed parallel for each group. Votaw's criterion may be used to check whether the test scores of all the groups have the same mean, variance, and covariance. It is possible, however, for the variance and reliability of the test to differ widely from group to group, so that Votaw's criterion is not satisfied even though the variance of errors of measurement stays relatively constant. For this case a modification of Neyman and Pearson's criterion is developed to test agreement among standard errors of measurement despite group differences in mean, variance, and reliability of the test.

A mechanical device is described which materially reduces the time, labor, and probability of errors which are involved in the computation of certain moments about the mean. The use of the device is illustrated with the computation of the mean, standard deviation, g1 as a measure of skewness, and g2 as a measure of kurtosis of data from a given study.

A formula for estimating real scores on a multiple-choice test from a knowledge of raw scores is derived. This formula does not involve the assumption of a binomial distribution of real scores as does the Calandra formula. Other important formulas derived show: the variance of real scores in terms of the variance of raw scores and the correlation between real scores and raw scores. If the variance of real scores (or of raw scores also) is binomial, the regression of real scores on raw scores is linear; but, otherwise the regression is curvilinear. Yet the linear estimating formula is a close approximation to the curvilinear relationship. Factors affecting the regression of real scores on raw scores and the correlation coefficient are: (1) the number of choices per question; (2) the number of questions answered; (3) the ratio of the average group raw score to the variance of raw scores.

Scores from three mathematics tests of the College Entrance Examination Board were examined in order to determine the effect on the scores of (1) choice of test, (2) amount of training in mathematics, and (3) recency of training in mathematics. Groups of candidates were paired in a number of comparisons and matched by means of a regression technique which is described. On the average, students of similar ability made comparable scores on the mathematical section of the Scholastic Aptitude Test and on the Comprehensive Mathematics Test. The scores of candidates who took the Intermediate Mathematics Test averaged substantially higher than those of comparable students who took either of the other two tests. A greater amount of mathematical training and more recent training were both found to be positively related to scores on the mathematical section of the Scholastic Aptitude Test and on the Intermediate Mathematics Test, but the effect of recency appeared to be less than one might expect.

Some methods are presented for estimating the reliability of a partially speeded test without the use of a parallel form. The effect of these formulas on some test data is illustrated. Whenever an odd-even reliability is computed it is probably desirable to use one of the formulas noted in Section 2 of this paper in addition to the usual Spearman-Brown correction. Since the formulas given here involve the mean and the standard deviation of the “number unattempted score,” a method is given in Section 4 for computing this mean and standard deviation from item analysis data. If the item analysis data are available, this method will save considerable time as compared with rescoring answer sheets.

Nineteen different anthropometric measures were obtained on all members of each of two racial groups. A procedure was developed and applied to the data to give maximum differentiation between the groups. The method is applicable wherever we have a large number of independent variables and a dependent variable. In such cases, the conventional methods for determining multiple regression constants are very laborious. An iteration method is presented which is more rapid than any with which the writers are familiar. The method selects in sequence those variables which together yield the largest multiple correlation with the criterion. At each step in the procedure, rapid estimates of the regression weights and the multiple correlation at that point are available.

By using the concepts of antimetry and n-chain it is possible to define and to investigate some properties of connectivity in a sociometric group. It is shown that the number of elements in a group, the number of antimetries, and the degree of connectivity must satisfy certain inequalities. Using the ideas of connectivity, a generalized concept of clique, called an n-clique, is introduced. n-cliques are shown to have a very close relationship to the existence of cliques in an artificial structure defined on the same set of elements, thus permitting the determination of n-cliques by means of the same simple matrix procedures used to obtain the clique structures. The presence of two or more m-cliques, where m is the number of elements in the group, is proved to mean an almost complete splitting of the group.

Weights may be determined for combining tests so that the composite has maximum reliability, but the calculation of these weights by means of the original equations is cumbersome. It is shown that the desired weighted composite is the first principal axis of a matrix closely related to the intercorrelation matrix. Thus, simple and straightforward procedures are available for calculating the weights.